Staying Thermal with Hartree Ensemble Approximations
Salle, M; Vink, Jeroen C
2000-01-01
Using Hartree ensemble approximations to compute the real time dynamics of scalar fields in 1+1 dimension, we find that with suitable initial conditions, approximate thermalization is achieved much faster than found in our previous work. At large times, depending on the interaction strength and temperature, the particle distribution slowly changes: the Bose-Einstein distribution of the particle densities develops classical features. We also discuss variations of our method which are numerically more efficient.
Staying thermal with Hartree ensemble approximations
Salle, Mischa E-mail: msalle@science.uva.nl; Smit, Jan E-mail: jsmit@science.uva.nl; Vink, Jeroen C. E-mail: jcvink@science.uva.nl
2002-03-25
We study thermal behavior of a recently introduced Hartree ensemble approximation, which allows for non-perturbative inhomogeneous field configurations as well as for approximate thermalization, in the phi (cursive,open) Greek{sup 4} model in 1+1 dimensions. Using ensembles with a free field thermal distribution as out-of-equilibrium initial conditions we determine thermalization time scales. The time scale for which the system stays in approximate quantum thermal equilibrium is an indication of the time scales for which the approximation method stays reasonable. This time scale turns out to be two orders of magnitude larger than the time scale for thermalization, in the range of couplings and temperatures studied. We also discuss simplifications of our method which are numerically more efficient and make a comparison with classical dynamics.
Strong semiclassical approximation of Wigner functions for the Hartree dynamics
Athanassoulis, Agissilaos
2011-01-01
We consider the Wigner equation corresponding to a nonlinear Schrödinger evolution of the Hartree type in the semiclassical limit h → 0. Under appropriate assumptions on the initial data and the interaction potential, we show that the Wigner function is close in L 2 to its weak limit, the solution of the corresponding Vlasov equation. The strong approximation allows the construction of semiclassical operator-valued observables, approximating their quantum counterparts in Hilbert-Schmidt topology. The proof makes use of a pointwise-positivity manipulation, which seems necessary in working with the L 2 norm and the precise form of the nonlinearity. We employ the Husimi function as a pivot between the classical probability density and the Wigner function, which - as it is well known - is not pointwise positive in general.
The O(N) linear sigma model at finite temperature beyond the Hartree approximation
Baacke, J
2003-01-01
We study the O(N) linear sigma model with spontaneous symmetry breaking, using a Hartree-like ansatz with a classical field and variational masses. We go beyond the Hartree approximation by including the two-loop contribution, the sunset diagram, using the 2PPI expansion. We have computed numerically the effective potential at finite temperature. We find a phase transition of second order, while it is first order in the one-loop Hartree approximation. We also discuss some implications of the fact that in this order, the decay of the sigma into two pions affects the thermal diagrams.
O(N) linear sigma model beyond the Hartree approximation at finite temperature
Baacke, J; Michalski, Stefan
2003-01-01
We study the O(N) linear sigma model with spontaneous symmetry breaking at finite temperature in the framework of the two-particle point-irreducible (2PPI) effective action. We go beyond the Hartree approximation by including the two-loop contribution, i.e., the sunset diagram. A phase transition of second order is found, whereas it is of first order in the one-loop Hartree approximation. Furthermore, we show the temperature-dependence of the variational mass parameters and comment on their relation to the physical sigma and pion masses.
Bose-Einstein Condensation in Linear Sigma Model at Hartree Approximation
M. Agop; SHU Song; Camelia Popa; LI Jia-Rong; Anca Harabagiu
2008-01-01
The BEC of charged pions is investigated in the framework of O(4) linear sigma model. By using Cornwall Jackiw Tomboulis formalism, we have derived the gap equations for the effective masses of the mesons at finite tempera-ture and finite isospin density. The critical temperature and phase diagram of BEC are discussed in the non-chiral limit at Hartree approximation.
Hartree-Fock and Random Phase Approximation theories in a many-fermion solvable model
Co', Giampaolo
2016-01-01
We present an ideal system of interacting fermions where the solutions of the many-body Schroedinger equation can be obtained without making approximations. These exact solutions are used to test the validity of two many-body effective approaches, the Hartree-Fock and the Random Phase Approximation theories. The description of the ground state done by the effective theories improves with increasing number of particles.
Fractional Electron Loss in Approximate DFT and Hartree-Fock Theory.
Peach, Michael J G; Teale, Andrew M; Helgaker, Trygve; Tozer, David J
2015-11-10
Plots of electronic energy vs electron number, determined using approximate density functional theory (DFT) and Hartree-Fock theory, are typically piecewise convex and piecewise concave, respectively. The curves also commonly exhibit a minimum and maximum, respectively, in the neutral → anion segment, which lead to positive DFT anion HOMO energies and positive Hartree-Fock neutral LUMO energies. These minima/maxima are a consequence of using basis sets that are local to the system, preventing fractional electron loss. Ground-state curves are presented that illustrate the idealized behavior that would occur if the basis set were to be modified to enable fractional electron loss without changing the description in the vicinity of the system. The key feature is that the energy cannot increase when the electron number increases, so the slope cannot be anywhere positive, meaning frontier orbital energies cannot be positive. For the convex (DFT) case, the idealized curve is flat beyond a critical electron number such that any additional fraction of an electron added to the system is unbound. The anion HOMO energy is zero. For the concave (Hartree-Fock) case, the idealized curve is flat up to some critical electron number, beyond which it curves down to the anion energy. A minimum fraction of an electron is required before any binding occurs, but beyond that, the full fraction abruptly binds. The neutral LUMO energy is zero. Approximate DFT and Hartree-Fock results are presented for the F → F(-) segment, and results approaching the idealized behavior are recovered for highly diffuse basis sets. It is noted that if a DFT calculation using a highly diffuse basis set yields a negative LUMO energy then a fraction of an electron must bind and the electron affinity must be positive, irrespective of whether an electron binds experimentally. This is illustrated by calculations on Ne → Ne(-).
Complete equation of state for neutron stars using the relativistic Hartree-Fock approximation
Miyatsu, Tsuyoshi; Cheoun, Myung-Ki [Department of Physics, Soongsil University, Seoul 156-743 (Korea, Republic of); Yamamuro, Sachiko; Nakazato, Ken' ichiro [Department of Physics, Faculty of Science and Technology, Tokyo University of Science (TUS), Noda 278-8510 (Japan)
2014-05-02
We construct the equation of state in a wide-density range for neutron stars within relativistic Hartree-Fock approximation. The properties of uniform and nonuniform nuclear matter are studied consistently. The tensor couplings of vector mesons to baryons due to exchange contributions (Fock terms) are included, and the change of baryon internal structure in matter is also taken into account using the quark-meson coupling model. The Thomas-Fermi calculation is adopted to describe nonuniform matter, where the lattice of nuclei and the neutron drip out of nuclei are considered. Even if hyperons exist in the core of a neutron star, we obtain the maximum neutron-star mass of 1.95M{sub ⊙}, which is consistent with the recently observed massive pulsar, PSR J1614-2230. In addition, the strange vector (φ) meson also plays a important role in supporting a massive neutron star.
The trajectory-coherent approximation and the system of moments for the Hartree type equation
V. V. Belov
2002-01-01
Full Text Available The general construction of semiclassically concentrated solutions to the Hartree type equation, based on the complex WKB-Maslov method, is presented. The formal solutions of the Cauchy problem for this equation, asymptotic in small parameter ℏ (ℏ→0, are constructed with a power accuracy of O(ℏ N/2, where N is any natural number. In constructing the semiclassically concentrated solutions, a set of Hamilton-Ehrenfest equations (equations for centered moments is essentially used. The nonlinear superposition principle has been formulated for the class of semiclassically concentrated solutions of Hartree type equations. The results obtained are exemplified by a one-dimensional Hartree type equation with a Gaussian potential.
Mora, P J; Woodard, R P
2013-01-01
We use the Hartree approximation to the Einstein equation on de Sitter background to solve for the one loop correction to the graviton mode function. This should give a reasonable approximation to how the ensemble of inflationary gravitons affects a single external graviton. At late times we find that the one loop correction to the plane wave mode function $u(\\eta,k)$ goes like $G H^2 \\ln(a)/a^2$, where $a$ is the inflationary scale factor. One consequence is that the one loop corrections to the "electric" components of the linearized Weyl tensor grow compared to the tree order result.
The Dielectric Permittivity of Crystals in the reduced Hartree-Fock approximation
Cancès, Eric
2009-01-01
In a recent article (Canc\\`es, Deleurence and Lewin, Commun. Math. Phys., 281 (2008), pp. 129-177), we have rigorously derived, by means of bulk limit arguments, a new variational model to describe the electronic ground state of insulating or semiconducting crystals in the presence of local defects. In this so-called reduced Hartree-Fock model, the ground state electronic density matrix is decomposed as $\\gamma = \\gamma^0_{\\rm per} + Q_{\
Bernardos, P. [Universidad de Cantabria, Departamento de Matematica Aplicada y Ciencias de la Computacion, 39005, Santander (Spain); Fomenko, V.N. [St Petersburg University for Railway Engineering, Department of Mathematics, 190031, St Petersburg (Russian Federation); Marcos, S.; Niembro, R. [Universidad de Cantabria, Departamento de Fisica Moderna, 39005, Santander (Spain); Lopez-Quelle, M. [Universidad de Cantabria, Departamento de Fisica Aplicada, 39005, Santander (Spain); Savushkin, L.N. [St Petersburg University for Telecommunications, Department of Physics, 191186, St Petersburg (Russian Federation)
2001-02-01
An effective nuclear model describing {omega}-, {rho}- and axial-mesons as gauge fields is applied to nuclear matter in the relativistic Hartree-Fock approximation. The isoscalar two-pion exchange is simulated by a scalar field s similar to that used in the conventional relativistic mean-field approach. Two more scalar fields are essential ingredients of the present treatment: the {sigma}-field, the chiral partner of the pion, and the {sigma}-field, the Higgs field for the {omega}-meson. Two versions of the model are used depending on whether the {sigma}-field is considered as a dynamical variable or 'frozen', by taking its mass as infinite. The model contains four free parameters in the first case and three in the second one which are fitted to the nuclear matter saturation conditions. The nucleon and meson effective masses, compressibility modulus and symmetry energy are calculated. The results prove the reliability of the Dirac-Hartree-Fock approach within the linear realization of the chiral symmetry. (author)
Kambe, Takahide; Saito, Koichi
2016-01-01
As the interior density of a neutron star can become very high, it has been expected and discussed that quark matter may exist inside it. To describe the transition from hadron to quark phases (and vice versa), there are mainly two methods; one is the first-order phase transition, and the other is the crossover phenomenon. In the present study, using the flavor-SU (3) NJL model with the vector coupling interaction, we have calculated the equation of state for the quark phase at high density. Furthermore, for the hadron phase at low density, we have used two kinds of the equations of state; one is a relatively soft one by the QHD model, and the other is a stiff one calculated with relativistic Brueckner-Hartree-Fock approximation. Using those equations of state for the two phases, we have investigated the influence of various choices of parameters concerning the crossover region on the mass and radius of a neutron star.
KAUSHIK MAJI
2016-08-01
We propose a variant of the multiconfiguration time-dependent Hartree (MCTDH) method within the framework of Hermite-distributed approximating functional (HDAF) method. The discretized Hamiltonian is a highly banded Toeplitz matrix which significantly reduces computational cost in terms of both storage and number of operations. The method proposed is employed to carry out the study of tunnelling dynamics of two coupled double well oscillators. We have calculated the orthogonality time \\tau , which is a measure of the time interval for an initial state to evolve into its orthogonal state. It is observed that the coupling has a significant effect on \\tau .
Agrawal, B K
2004-01-01
We provide for the first time accurate assessments of the consequences of violations of self-consistency in the Hartree-Fock based random phase approximation (RPA) as commonly used to calculate the energy $E_c$ of the nuclear breathing mode. Using several Skyrme interactions we find that the self-consistency violated by ignoring the spin-orbit interaction in the RPA calculation causes a spurious enhancement of the breathing mode energy for spin unsaturated systems. Contrarily, neglecting the Coulomb interaction in the RPA or performing the RPA calculations in the TJ scheme underestimates the breathing mode energy. Surprisingly, our results for the $^{90}$Zr and $^{208}$Pb nuclei for several Skyrme type effective nucleon-nucleon interactions having a wide range of nuclear matter incompressibility ($K_{nm} \\sim 215 - 275$ MeV) and symmetry energy ($J \\sim 27 - 37$ MeV) indicate that the net uncertainty ($\\delta E_c \\sim 0.3$ MeV) is comparable to the experimental one.
Quiney, HM; Glushkov, VN; Wilson, S
2002-01-01
Using basis sets of distributed s-type Gaussian functions with positions and exponents optimized so as to support Hartree-Fock total energies with an accuracy approaching the sub-muHartree level, Dirac-Hartree-Fock-Coulomb calculations are reported for the ground states of the H-2, LiH, and BH molec
Hartree potential dependent exchange functional
Constantin, L A; Della Sala, F
2016-01-01
We introduce a novel non-local ingredient for the construction of exchange density functionals: the reduced Hartree parameter, which is invariant under the uniform scaling of the density and represents the exact exchange enhancement factor for one- and two-electron systems. The reduced Hartree parameter is used together with the conventional meta-generalized gradient approximation (meta-GGA) semilocal ingredients (i.e. the electron density, its gradient and the kinetic energy density) to construct a new generation exchange functional, termed u-meta-GGA. This u-meta-GGA functional is exact for {the exchange of} any one- and two-electron systems, is size-consistent and non-empirical, satisfies the uniform density scaling relation, and recovers the modified gradient expansion derived from the semiclassical atom theory. For atoms, ions, jellium spheres, and molecules, it shows a good accuracy, being often better than meta-GGA exchange functionals. Our construction validates the use of the reduced Hartree ingredie...
Koopmans' theorem in statistical Hartree-Fock theory
Pain, Jean-Christophe
2011-01-01
In this short paper, the validity of Koopmans' theorem in the Hartree-Fock theory at non-zero temperature (Hartree-Fock statistical theory) is investigated. It is shown that Koopmans' theorem does not apply in the grand-canonical ensemble, due to a missing contribution to the energy proportional to the interaction between two electrons belonging to the same orbital. Hartree-Fock statistical theory has also been applied in the canonical ensemble [Blenski et al., Phys. Rev. E 55, R4889 (1997)] for the purpose of photo-absorption calculations. In that case, the Hartree-Fock self-consistent-field equations are derived in the super-configuration approximation. It is shown that Koopmans' theorem does not hold in the canonical ensemble, but that a restricted version of the theorem can be obtained, by assuming that a particular quantity multiplying the interaction matrix element in the expression of the energy does not change during the removal of an electron.
Fully discrete Galerkin schemes for the nonlinear and nonlocal Hartree equation
Walter H. Aschbacher
2009-01-01
Full Text Available We study the time dependent Hartree equation in the continuum, the semidiscrete, and the fully discrete setting. We prove existence-uniqueness, regularity, and approximation properties for the respective schemes, and set the stage for a controlled numerical computation of delicate nonlinear and nonlocal features of the Hartree dynamics in various physical applications.
Particle unstable nuclei in the Hartree-Fock theory
Kruppa, A.T. [Magyar Tudomanyos Akademia, Debrecen (Hungary). Atommag Kutato Intezete; Heenen, P.H. [Brussels Univ. (Belgium). Service de Physique Nucleaire Theorique; Flocard, H. [Paris-11 Univ., 91 - Orsay (France). Inst. de Physique Nucleaire; Liotta, R.J. [Manne Siegbahn Inst. of Physics, Stockholm (Sweden)
1997-12-31
Ground state energies and decay widths of particle unstable nuclei are calculated within the Hartree-Fock approximation by performing a complex scaling of the many-body Hamiltonian. Through this transformation, the wave functions of the resonant state become square integrable. The method is implemented with Skyrme effective interactions. Several Skyrme parametrizations are tested on four unstable nuclei: {sup 10}He, {sup 12}O, {sup 26}O and {sup 28}O. (author). 24 refs.
Rezania, Hamed; Abdi, Ameneh
2017-04-01
We study the behaviors of both Hartree and correlation energies of undoped gapped armchair graphene nanoribbon using random phase approximation in the context of Hubbard model Hamiltonian. Specially, the effects of spin polarization and gap parameter on electron density dependence of Hartree and correlation energies of armchair graphene nanoribbon has been addressed. Our results show the variation of gap parameter leads to considerable effect on correlation and Hartree energy behavior of spin unpolarized gapped graphene in the middle electron density region. However local Hubbard interaction parameter affects the behaviors of Hartree and correlation energy on the whole range of electron density in zero magnetization case. We also show that a considerable reduction has been observed for density dependence of Hartree and correlation energies of spin polarized gapped graphene nanoribbon.
Nonuniqueness of solution of Hartree equations
Amus' ya, M.Ya.; Kuchiev, M.Yu. (AN SSSR, Leningrad. Fiziko-Tekhnicheskij Inst.)
1981-12-01
The problem of uniqueness of Hartree equations solution is studied. Two simple models for multielectron ''atoms'' are considered. The possibility of existence in nature of states corresponding to such solutions is discussed. It is shown that besides normal solution in both models considered at a certain interpartial interaction special solutions appear. The emergence of special solutions is related to nonlinearity of Hartree equations.
Ground state properties of graphene in Hartree-Fock theory
Hainzl, Christian; Sparber, Christof
2012-01-01
We study the Hartree-Fock approximation of graphene in infinite volume, with instantaneous Coulomb interactions. First we construct its translation-invariant ground state and we recover the well-known fact that, due to the exchange term, the effective Fermi velocity is logarithmically divergent at zero momentum. In a second step we prove the existence of a ground state in the presence of local defects and we discuss some properties of the linear response to an external electric field. All our results are non perturbative.
Restricted Closed Shell Hartree Fock Roothaan Matrix Method Applied to Helium Atom Using Mathematica
Acosta, César R.; Tapia, J. Alejandro; Cab, César
2014-01-01
Slater type orbitals were used to construct the overlap and the Hamiltonian core matrices; we also found the values of the bi-electron repulsion integrals. The Hartree Fock Roothaan approximation process starts with setting an initial guess value for the elements of the density matrix; with these matrices we constructed the initial Fock matrix.…
Goldstone modes in the random phase approximation
Neergård, Kai
2016-01-01
I show that the kernel of the random phase approximation (RPA) matrix based on a stable Hartree, Hartree-Fock, Hartree-Bogolyubov or Hartree-Fock-Bogolyubov mean field solution is decomposed into a subspace with a basis whose vectors are associated, in the equivalent formalism of a classical Hamiltonian linear in canonic coordinates, with conjugate momenta of cyclic coordinates (Goldstone modes) and a subspace with a basis whose vectors are associated with pairs of conjugate canonic coordinates that do not enter the Hamiltonian at all. In a subspace complementary to the one spanned by all these coordinates including the conjugate coordinates of the Goldstone momenta, the RPA matrix behaves as in the case of a zerodimensional kernel. This result was derived very recently by Nakada as a corollary to a general analysis of RPA matrices based on both stable and unstable mean field solutions. The present proof does not rest on Nakada's general results.
Correlated Electron Calculations with Hartree-Fock Scaling
Gebauer, Ralph; Car, Roberto
2013-01-01
We introduce an energy functional for ground-state electronic structure calculations with fundamental variables the natural spin orbitals and their joint occupation probabilities in an implied many-body trial wave function. We use a controlled approximation for the two-particle density matrix that greatly extends the accuracy compared to current functionals of the one-particle density matrix only. Algebraic scaling of computational cost with electron number is achieved in general, and Hartree-Fock scaling in the seniority-zero version of the theory. We present results obtained with the latter version for saturated small molecular systems for which highly accurate quantum chemical computations are available for comparison. The results are variational, capturing most of the correlation energy from equilibrium to dissociation.
Koopmans' theorem in the statistical Hartree-Fock theory
Pain, Jean-Christophe, E-mail: jean-christophe.pain@cea.fr [CEA, DAM, DIF, F-91297 Arpajon (France)
2011-07-28
In this short paper, the validity of Koopmans' theorem in the Hartree-Fock theory at non-zero temperature (Hartree-Fock statistical theory) is investigated. It is shown that Koopmans' theorem does not apply in the grand-canonical ensemble, due to a missing contribution to the energy proportional to the interaction between two electrons belonging to the same orbital. The Hartree-Fock statistical theory has also been applied in the canonical ensemble (Blenski et al 1997 Phys. Rev. E 55 R4889) for the purpose of photo-absorption calculations. In that case, the Hartree-Fock self-consistent field equations are derived in the super-configuration approximation. It is shown that Koopmans' theorem does not hold in the canonical ensemble, but a restricted version of the theorem can be obtained by assuming that a particular quantity multiplying the interaction matrix element in the expression of the energy does not change during the removal of an electron.
Thermal effects in gravitational Hartree systems
Aki, Gonca L. [Weierstrass-Institut fuer Angewandte Analysis und Stochastik (WIAS) im Forschungsverbund Berlin e.V. (Germany); Dolbeault, Jean [Paris-Dauphine Univ. (FR). Ceremade (UMR CNRS 7534); Sparber, Christof [Illinois Univ., Chicago, IL (United States). Dept. of Mathematics, Statistics, and Computer Science
2010-07-01
We consider the non-relativistic Hartree model in the gravitational case, i.e. with attractive Coulomb-Newton interaction. For a given mass M>0, we construct stationary states with non-zero temperature T by minimizing the corresponding free energy functional. It is proved that minimizers exist if and only if the temperature of the system is below a certain threshold T*>0 (possibly infinite), which itself depends on the specific choice of the entropy functional. We also investigate whether the corresponding minimizers are mixed or pure quantum states and characterize a critical temperature T{sub c} element of (0,T*) above which mixed states appear. (orig.)
Misfits in Skyrme-Hartree-Fock
Erler, J; Reinhard, P -G
2010-01-01
We address very briefly five critical points in the context of the Skyrme-Hartree-Fock (SHF) scheme: 1) the impossibility to consider it as an interaction, 2) a possible inconsistency of correlation corrections as, e.g., the center-of-mass correction, 3) problems to describe the giant dipole resonance (GDR) simultaneously in light and heavy nuclei, 4) deficiencies in the extrapolation of binding energies to super-heavy elements (SHE), and 5) a yet inappropriate trend in fission life-times when going to the heaviest SHE. While the first two points have more a formal bias, the other three points have practical implications and wait for solution.
Thermal Effects in Gravitational Hartree Systems
Aki, Gonca L.
2011-04-06
We consider the non-relativistic Hartree model in the gravitational case, i. e. with attractive Coulomb-Newton interaction. For a given mass M > 0, we construct stationary states with non-zero temperature T by minimizing the corresponding free energy functional. It is proved that minimizers exist if and only if the temperature of the system is below a certain threshold T* > 0 (possibly infinite), which itself depends on the specific choice of the entropy functional. We also investigate whether the corresponding minimizers are mixed or pure quantum states and characterize a critical temperature Tc ∈ (0,T*) above which mixed states appear. © 2011 Springer Basel AG.
Classical limit for semirelativistic Hartree systems
Aki, Gonca L.
2008-01-01
We consider the three-dimensional semirelativistic Hartree model for fast quantum mechanical particles moving in a self-consistent field. Under appropriate assumptions on the initial density matrix as a (fully) mixed quantum state we prove by using Wigner transformation techniques that its classical limit yields the well known relativistic Vlasov-Poisson system. The result holds for the case of attractive and repulsive mean-field interactions, with an additional size constraint in the attractive case. © 2008 American Institute of Physics.
Auxiliary Density Matrix Methods for Hartree-Fock Exchange Calculations.
Guidon, Manuel; Hutter, Jürg; VandeVondele, Joost
2010-08-10
The calculation of Hartree-Fock exchange (HFX) is computationally demanding for large systems described with high-quality basis sets. In this work, we show that excellent performance and good accuracy can nevertheless be obtained if an auxiliary density matrix is employed for the HFX calculation. Several schemes to derive an auxiliary density matrix from a high-quality density matrix are discussed. Key to the accuracy of the auxiliary density matrix methods (ADMM) is the use of a correction based on standard generalized gradient approximations for HFX. ADMM integrates seamlessly in existing HFX codes and, in particular, can be employed in linear scaling implementations. Demonstrating the performance of the method, the effect of HFX on the structure of liquid water is investigated in detail using Born-Oppenheimer molecular dynamics simulations (300 ps) of a system of 64 molecules. Representative for large systems are calculations on a solvated protein (Rubredoxin), for which ADMM outperforms the corresponding standard HFX implementation by approximately a factor 20.
A Hartree-Fock-Bogoliubov mass formula
Samyn, M; Heenen, P H; Pearson, J M; Tondeur, F
2002-01-01
In order to have more reliable predictions of nuclear masses at the neutron drip line, we here go beyond the recent mass formula HFBCS-1 and present a new mass formula, HFB-1, based on the Hartree-Fock-Bogoliubov method. As with the HFBCS-1 mass formula, we use a 10-parameter Skyrme force along with a 4-parameter delta-function pairing force and a 2-parameter phenomenological Wigner term. However, with the original HFBCS-1 Skyrme force (MSk7), the rms error becomes unacceptably large and a new force fit is required. With the isoscalar and isovector effective masses constrained to be equal, the remaining 15 degrees of freedom are fitted to the masses of all the 1754 measured nuclei with A>=16, |N-Z|>2 given in the 1995 Audi-Wapstra compilation. The rms error with respect to the masses of all the 1888 measured nuclei with Z,N>=8 is 0.764 MeV. A complete mass table, HFB-1 (available on the Web), has been constructed, giving all nuclei lying between the two drip lines over the range Z,N>=8 and Z<=120. A compar...
Many-body approximations for atomic binding energies
Schuster, Micah D; Staker, Joshua T
2011-01-01
We benchmark three approximations for the many-body problem -- the Hartree-Fock, projected Hartree-Fock, and random phase approximations -- against full numerical configuration-interaction calculations of the electronic structure of atoms, from Li through to Ne. Each method uses exactly the same input, i.e., the same single-particle basis and Coulomb matrix elements, so any differences are strictly due to the approximation itself. Although it consistently overestimates the ground state binding energy, the random phase approximation has the smallest overall errors; furthermore, we suggest it may be useful as a method for efficient optimization of single-particle basis functions.
Tensor numerical methods in quantum chemistry: from Hartree-Fock to excitation energies.
Khoromskaia, Venera; Khoromskij, Boris N
2015-12-21
We resume the recent successes of the grid-based tensor numerical methods and discuss their prospects in real-space electronic structure calculations. These methods, based on the low-rank representation of the multidimensional functions and integral operators, first appeared as an accurate tensor calculus for the 3D Hartree potential using 1D complexity operations, and have evolved to entirely grid-based tensor-structured 3D Hartree-Fock eigenvalue solver. It benefits from tensor calculation of the core Hamiltonian and two-electron integrals (TEI) in O(n log n) complexity using the rank-structured approximation of basis functions, electron densities and convolution integral operators all represented on 3D n × n × n Cartesian grids. The algorithm for calculating TEI tensor in a form of the Cholesky decomposition is based on multiple factorizations using algebraic 1D "density fitting" scheme, which yield an almost irreducible number of product basis functions involved in the 3D convolution integrals, depending on a threshold ε > 0. The basis functions are not restricted to separable Gaussians, since the analytical integration is substituted by high-precision tensor-structured numerical quadratures. The tensor approaches to post-Hartree-Fock calculations for the MP2 energy correction and for the Bethe-Salpeter excitation energies, based on using low-rank factorizations and the reduced basis method, were recently introduced. Another direction is towards the tensor-based Hartree-Fock numerical scheme for finite lattices, where one of the numerical challenges is the summation of electrostatic potentials of a large number of nuclei. The 3D grid-based tensor method for calculation of a potential sum on a L × L × L lattice manifests the linear in L computational work, O(L), instead of the usual O(L(3) log L) scaling by the Ewald-type approaches.
New Multithreaded Hybrid CPU/GPU Approach to Hartree-Fock.
Asadchev, Andrey; Gordon, Mark S
2012-11-13
In this article, a new multithreaded Hartree-Fock CPU/GPU method is presented which utilizes automatically generated code and modern C++ techniques to achieve a significant improvement in memory usage and computer time. In particular, the newly implemented Rys Quadrature and Fock Matrix algorithms, implemented as a stand-alone C++ library, with C and Fortran bindings, provides up to 40% improvement over the traditional Fortran Rys Quadrature. The C++ GPU HF code provides approximately a factor of 17.5 improvement over the corresponding C++ CPU code.
Superdeformed rotational bands in the mercury region. A cranked Skyrme-Hartree-Fock-Bogoliubov study
Gall, B. (Centre de Spectrometrie Nucleaire et de Spectrometrie de Masse, 91 Orsay (France)); Bonche, P. (Service de Physique Theorique, DSM, CE Saclay, 91 Gif-sur-Yvette (France)); Dobaczewski, J. (Inst. of Theoretical Physics, Warsaw Univ., Warsaw (Poland)); Flocard, H. (Div. de Physique Theorique, Inst. de Physique Nucleaire, 91 Orsay (France)); Heenen, P.H. (Physique Nucleaire Theorique, Univ. Libre de Bruxelles (Belgium))
1994-05-01
A study of rotational properties of the ground superdeformed bands in [sup 190]Hg, [sup 192]Hg, [sup 194]Hg, and [sup 194]Pb is presented. We use the cranked Hartree-Fock-Bogoliubov method with the SkM* parametrization of the Skyrme force in the particle-hole channel and a seniority interaction in the pairing channel. An approximate particle number projection is performed by means of the Lipkin-Nogami prescription. We analyze the proton and neutron quasiparticle routhians in connection with the present information on about thirty presently observed superdeformed bands in nuclei close neighbours of [sup 192]Hg (orig.)
Superdeformed rotational bands in the mercury region; a cranked Skyrme-Hartree-Fock-Bogoliubov study
Gall, B.; Bonche, P.; Dobaczewski, J.; Flocard, H.; Heenen, P. -H.
1994-01-01
URL: http://www-spht.cea.fr/articles/T94/011 http://fr.arxiv.org/abs/nucl-th/9312011; International audience; A study of rotational properties of the ground superdeformed bands in $ ^{190} {\\rm Hg,} $ $ ^{192} {\\rm Hg,} $ $ ^{194} {\\rm Hg,} $ and $ ^{194} {\\rm Pb} $ is presented. We use the cranked Hartree-Fock-Bogoliubov method with the SkM$ ^\\ast $ parametrization of the Skyrme force in the particle-hole channel and a seniority interaction in the pairing channel. An approximate particle num...
Superdeformed rotational bands in the mercury region. A cranked Skyrme-Hartree-Fock-Bogoliubov study
Gall, B.; Bonche, P.; Dobaczewski, J.; Flocard, H.; Heenen, P.-H.
1994-09-01
A study of rotational properties of the ground superdeformed bands in190Hg,192Hg,194Hg, and194Pb is presented. We use the cranked Hartree-Fock-Bogoliubov method with the SkM* parametrization of the Skyrme force in the particle-hole channel and a seniority interaction in the pairing channel. An approximate particle number projection is performed by means of the Lipkin-Nogami prescription. We analyze the proton and neutron quasiparticle routhians in connection with the present information on about thirty presently observed superdeformed bands in nuclei close neighbours of192Hg.
Superdeformed rotational bands in the Mercury region. A cranked Skyrme-Hartree-Fock-Bogoliubov study
Gall, B. [Paris-11 Univ., 91 - Orsay (France). Centre de Spectrometrie Nucleaire et de Spectrometrie de Masse; Bonche, P. [CEA Centre d`Etudes de Saclay, 91 - Gif-sur-Yvette (France). Service de Physique Theorique; Dobaczewski, J. [Warsaw Univ. (Poland). Inst. Fizyki Teoretycznej; Heenen, P.H. [Universite Libre de Bruxelles (Belgium). Physique Nucleaire Theorique; Flocard, H.
1993-12-17
A study of rotational properties of the ground superdeformed bands in {sup 190}Hg, {sup 192}Hg, {sup 194}Hg, and {sup 194}Pb is presented. The cranked Hartree-Fock-Bogolyubov method is used with the SkM* parametrization of the Skyrme force in the particle-hole channel and a seniority interaction in the pairing channel. An approximate particle number projection is performed by means of the Lipkin-Nogami prescription. The proton and neutron quasiparticle rhouthians are analyzed in connection with the present information on about thirty presently observed superdeformed bands in nuclei close neighbours of {sup 192}Hg. (authors). 53 refs., 14 figs.
Nuclear relativistic Hartree-Fock calculations including pions interacting with a scalar field
Marcos, S.; Lopez-Quelle, M.; Niembro, R.; Savushkin, L. N. [Departamento de Fisica Moderna, Universidad de Cantabria, Santander (Spain); Departamento de Fisica Aplicada, Universidad de Cantabria, Santander (Spain); Departamento de Fisica Moderna, Universidad de Cantabria, Santander (Spain); Department of Physics, St. Petersburg University for Telecommunications, St. Petersburg (Russian Federation)
2012-10-20
The effect of pions on the nuclear shell structure is analyzed in a relativistic Hartree-Fock approximation (RHFA). The Lagrangian includes, in particular, a mixture of {pi}N pseudoscalar (PS) and pseudovector (PV) couplings, self-interactions of the scalar field {sigma} and a {sigma} - {pi} interaction that dresses pions with an effective mass (m*{sub {pi}}). It is found that an increase of m*{sub {pi}} strongly reduces the unrealistic effect of pions, keeping roughly unchanged their contribution to the total binding energy.
Relativistic Brueckner-Hartree-Fock theory for finite nuclei
Shen, Shihang; Liang, Haozhao; Meng, Jie; Ring, Peter; Zhang, Shuangquan
2016-01-01
Starting with a bare nucleon-nucleon interaction, for the first time the full relativistic Brueckner-Hartree-Fock equations are solved for finite nuclei in a Dirac-Woods-Saxon basis. No free parameters are introduced to calculate the ground-state properties of finite nuclei. The nucleus $^{16}$O is investigated as an example. The resulting ground-state properties, such as binding energy and charge radius, are considerably improved as compared with the non-relativistic Brueckner-Hartree-Fock results and much closer to the experimental data. This opens the door for \\emph{ab initio} covariant investigations of heavy nuclei.
Pseudospin symmetry in finite nuclei within the relativistic Hartree-Fock framework
Lopez-Quelle, M [Departamento de Fisica Aplicada, Universidad de Cantabria, E-39005 Santander (Spain); Savushkin, L N [Department of Physics, St Petersburg University for Telecommunications, 191186 St Petersburg (Russian Federation); Marcos, S [Departamento de Fisica Moderna, Universidad de Cantabria, E-39005 Santander (Spain); Niembro, R [Departamento de Fisica Moderna, Universidad de Cantabria, E-39005 Santander (Spain)
2005-10-01
In the present work, we analyse the behaviour of the pseudospin symmetry (PSS) in heavy nuclei ({sup 208}Pb) in the framework of the relativistic Hartree-Fock approximation (RHFA). The quasidegeneracy of the pseudospin partners and the similarity of the small F components of their respective Dirac spinors have a somewhat lower degree of accuracy than in the relativistic mean field approximation (RMFA). Both properties improve when the number of nodes of the small component increases, as happens in the RMFA. The behaviour of the single-particle potentials appearing in the Dirac equation of the pseudospin partners is analysed. There is no dominance of the pseudocentrifugal barrier (PCB) compared to the pseudospin-orbit potential (PSOP). In the RHFA, the PSS is an approximately satisfied symmetry in nuclei and its dynamical character is reinforced with respect to the RMFA.
Olshanii, Maxim; Pricoupenko, Ludovic
2001-05-01
We introduce a novel one-parametric family of zero-range pseudopotentials hatV^Λ(r) = g_Λ δ(r) [ partialr + Λ ] (r \\cdot ) with g_Λ = fracg_01-Λ a and g0 = 4πhbar^2 a/m , whose scattering length a does not depend on the free parameter Λ. No exact (after the zero-range approximation has been made) many-body observable depends on it, although approximate treatments differ for different Λ (M. Olshanii and L. Pricoupenko, mat/0101275>). We incorporate these pseudopotentials in the Hartree-Fock-Bogoliubov variational formalism, whose conventional (Λ=0) version is known to exhibit UV-divergencies, inconsistencies with both Hugenholtz-Pines theorem and many-body T-matrix calculations, and inability to develop an energy minimum for the atomic condensate leading to a molecular condensate instead. Using Λ as a new variational parameter we resolve all inconsistencies of the Hartree-Fock-Bogoliubov formalism known so far, with no ad hoc modifications of the theory.
From the Hartree dynamics to the Vlasov equation
Benedikter, Niels Patriz; Porta, Marcello; Saffirio, Chiara;
2016-01-01
We consider the evolution of quasi-free states describing N fermions in the mean field limit, as governed by the nonlinear Hartree equation. In the limit of large N, we study the convergence towards the classical Vlasov equation. For a class of regular interaction potentials, we establish precise...... bounds on the 0rate of convergence....
Constrained Hartree-Fock and quasi-spin projection
Cambiaggio, M. C.; Plastino, A.; Szybisz, L.
1980-08-01
The constrained Hartree-Fock approach of Elliott and Evans is studied in detail with reference to two quasi-spin models, and their predictions compared with those arising from a projection method. It is found that the new approach works fairly well, although limitations to its applicability are encountered.
Karasiev, V.; López-Boada, R.
1998-09-01
The line-integral method developed by van Leeuwen and Baerends [Phys. Rev. A 51, 170 (1995)] is applied to the calculation of the differences of correlation energy functional values ΔEDFTc=EDFTc[ρHF]- EDFTc[ρexact], where ρHF is the Hartree-Fock density and ρexact is the near-exact one (DFT is density-functional theory). From the Kohn-Sham wave functions yielding Hartree-Fock and the near-exact densities, the corresponding noninteracting kinetic energies and the exchange energies are calculated. An approximate relation between EDFTc[ρHF] and the conventional quantum chemistry correlation energy is presented, accurate to <=4μ hartree for the isoelectronic series of He, and Li, and for the Be atom.
Constantin, Lucian A; Fabiano, Eduardo; Della Sala, Fabio
2017-09-12
Using the semiclassical neutral atom theory, we developed a modified fourth-order kinetic energy (KE) gradient expansion (GE4m) that keeps unchanged all the linear-response terms of the uniform electron gas and gives a significant improvement with respect to the known semilocal functionals for both large atoms and jellium surfaces. On the other hand, GE4m is not accurate for light atoms; thus, we modified the GE4m coefficients making them dependent on a novel ingredient, the reduced Hartree potential, recently introduced in the Journal of Chemical Physics 2016, 145, 084110, in the context of exchange functionals. The resulting KE gradient expansion functional, named uGE4m, belongs to the novel class of u-meta-generalized-gradient-approximations (uMGGA) whose members depend on the conventional ingredients (i.e., the reduced gradient and Laplacian of the density) as well as on the reduced Hartree potential. To test uGE4m, we defined an appropriate benchmark (including total KE and KE differences for atoms, molecules and jellium clusters) for gradient expansion functionals, that is, including only those systems which are mainly described by a slowly varying density regime. While most of the GGA and meta-GGA KE functionals (we tested 18 of them) are accurate for some properties and inaccurate for others, uGE4m shows a consistently good performance for all the properties considered. This represents a qualitative boost in the KE functional development and highlights the importance of the reduced Hartree potential for the construction of next-generation KE functionals.
A mathematical formulation of the random phase approximation for crystals
Cances, Eric
2011-01-01
This works extends the recent study on the dielectric permittivity of crystals within the Hartree model [E. Cances and M. Lewin, Arch. Rational Mech. Anal., 197 (2010) 139--177] to the time-dependent setting. In particular, we prove the existence and uniqueness of the nonlinear Hartree dynamics, also called the random phase approximation in the physics literature, in a suitable functional space allowing to describe a local defect embedded in a perfect crystal. We also give a rigorous mathematical definition of the microscopic frequency-dependent polarization matrix, and derive the macroscopic Maxwell-Gauss equation for insulating and semiconducting crystals, from a first order approximation of the nonlinear Hartree model, by means of homogenization arguments.
Time-Dependent Hartree-Fock Approach to Nuclear Pasta at Finite Temperature
Schuetrumpf, Bastian; Iida, Kei; Maruhn, Joachim; Mecke, Klaus; Reinhard, Paul-Gerhard
2013-01-01
We present simulations of neutron-rich matter at subnuclear densities, like supernova matter, with the time-dependent Hartree-Fock approximation at temperatures of several MeV. The initial state consists of $\\alpha$ particles randomly distributed in space that have a Maxwell-Boltzmann distribution in momentum space. Adding a neutron background initialized with Fermi distributed plane waves the calculations reflect a reasonable approximation of astrophysical matter. This matter evolves into spherical, rod-like, and slab-like shapes and mixtures thereof. The simulations employ a full Skyrme interaction in a periodic three-dimensional grid. By an improved morphological analysis based on Minkowski functionals, all eight pasta shapes can be uniquely identified by the sign of only two valuations, namely the Euler characteristic and the integral mean curvature.
Quasi-particle Continuum and Resonances in the Hartree-Fock-Bogoliubov Theory
Pei, J. C. [University of Tennessee, Knoxville (UTK) & Oak Ridge National Laboratory (ORNL); Kruppa, Andras Tibor [ORNL; Nazarewicz, Witold [ORNL
2011-01-01
The quasi-particle energy spectrum of the Hartree-Fock-Bogoliubov (HFB) equations contains discrete bound states, resonances, and non-resonant continuum states. We study the structure of the unbound quasi-particle spectrum of weakly bound nuclei within several methods that do not rely on imposing scattering or outgoing boundary conditions. Various approximations are examined to estimate resonance widths. It is shown that the stabilization method works well for all HFB resonances except for very narrow ones. The Thomas-Fermi approximation to the non-resonant continuum has been shown to be very effective, especially for coordinate-space HFB calculations in large boxes that involve huge amounts of discretized quasi-particle continuum states.
Pion tensor force and nuclear binding energy in the relativistic Hartree-Fock formalism
Marcos, S.; López-Quelle, M.; Niembro, R.; Savushkin, L. N.
2014-03-01
The binding energies of several isotopic families are studied within the relativistic Hartree-Fock approximation with the pseudovector coupling for the πN vertex, to find out a suitable strength for the effective pion tensor force (EPTF). An approximation for determining separately the contributions of the central and tensor forces generated by pion is considered. The results for heavy nuclei indicate that a realistic strength for the EPTF is smaller than a half of that appearing in the OPEP. This conclusion also applies to the results for the single-particle energies. Besides, it has been found that there is a genuine relativistic contribution of the EPTF in nuclear matter which is small but significant.
Quasiparticle continuum and resonances in the Hartree-Fock-Bogoliubov theory
Pei, Junchen [ORNL; Kruppa, A. T. [Joint Institute for Heavy Ion Research, Oak Ridge; Nazarewicz, W. [University of Tennessee, Knoxville (UTK) & Oak Ridge National Laboratory (ORNL)
2011-01-01
The quasi-particle energy spectrum of the Hartree-Fock-Bogoliubov (HFB) equations contains discrete bound states, resonances, and non-resonant continuum states. We study the structure of the unbound quasi-particle spectrum of weakly bound nuclei within several methods that do not rely on imposing scattering or outgoing boundary conditions. Various approximations are examined to estimate resonance widths. It is shown that the stabilization method works well for all HFB resonances except for very narrow ones. The Thomas-Fermi approximation to the non-resonant continuum has been shown to be very effective, especially for coordinate-space HFB calculations in large boxes that involve huge amounts of discretized quasi-particle continuum states.
Δ (1232 ) effects in density-dependent relativistic Hartree-Fock theory and neutron stars
Zhu, Zhen-Yu; Li, Ang; Hu, Jin-Niu; Sagawa, Hiroyuki
2016-10-01
The density-dependent relativistic Hartree-Fock (DDRHF) theory is extended to include Δ isobars for the study of dense nuclear matter and neutron stars. To this end, we solve the Rarita-Schwinger equation for spin-3/2 particle. Both the direct and exchange terms of the Δ isobars' self-energies are evaluated in detail. In comparison with the relativistic mean field theory (Hartree approximation), a weaker parameter dependence is found for DDRHF. An early appearance of Δ isobars is recognized at ρB˜0.28 fm-3, comparable with that of hyperons. Also, we find that the Δ isobars' softening of the equation of state is mainly due to the reduced Fock contributions from the coupling of the isoscalar mesons, while the pion contributions are negligibly small. We finally conclude that with typical parameter sets, neutron stars with Δ isobars in their interiors could be as heavy as the two massive pulsars whose masses are precisely measured, with slightly smaller radii than normal neutron stars.
$\\Delta$ (1232) effects in density-dependent relativistic Hartree-Fock theory and neutron stars
Zhu, Zhen-Yu; Hu, Jin-Niu; Sagawa, Hiroyuki
2016-01-01
The density-dependent relativistic Hartree-Fock (DDRHF) theory is extended to include $\\Delta$-isobars for the study of dense nuclear matter and neutron stars. To this end, we solve the Rarita-Schwinger equation for spin-3/2 particle. Both the direct and exchange terms of the $\\Delta$-isobars' self-energies are evaluated in details. In comparison with the relativistic mean field theory (Hartree approximation), a weaker parameter dependence is found for DDRHF. An early appearance of $\\Delta$-isobars is recognized at $\\rho_B\\sim0.27$fm$^{-3}$, comparable with that of hyperons. Also, we find that the $\\Delta$-isobars' softening of the equation of state is found to be mainly due to the reduced Fock contributions from the coupling of the isoscalar mesons, while the pion contributions are found negligibly small. We finally conclude that with typical parameter sets, neutron stars with $\\Delta$-isobars in their interiors could be as heavy as the two massive pulsars whose masses are precisely measured, with slightly s...
Accurate Hartree-Fock energy of extended systems using large Gaussian basis sets
Paier, Joachim; Diaconu, Cristian V.; Scuseria, Gustavo E.; Guidon, Manuel; Vandevondele, Joost; Hutter, Jürg
2009-11-01
Calculating highly accurate thermochemical properties of condensed matter via wave-function-based approaches (such as, e.g., Hartree-Fock or hybrid functionals) has recently attracted much interest. We here present two strategies providing accurate Hartree-Fock energies for solid LiH in a large Gaussian basis set and applying periodic boundary conditions. The total energies were obtained using two different approaches, namely, a supercell evaluation of Hartree-Fock exchange using a truncated Coulomb operator and an extrapolation toward the full-range Hartree-Fock limit of a Padé fit to a series of short-range screened Hartree-Fock calculations. These two techniques agreed to significant precision. We also present the Hartree-Fock cohesive energy of LiH (converged to within sub-millielectron volt) at the experimental equilibrium volume as well as the Hartree-Fock equilibrium lattice constant and bulk modulus.
Relativistic Hartree-Bogoliubov description of the halo nuclei
Meng, J.; Ring, P. [Universitaet Muenchen, Garching (Germany)
1996-12-31
Here the authors report the development of the relativistic Hartree-Bogoliubov theory in coordinate space. Pairing correlations are taken into account by both density dependent force of zero range and finite range Gogny force. As a primary application the relativistic HB theory is used to describe the chain of Lithium isotopes reaching from {sup 6}Li to {sup 11}Li. In contrast to earlier investigations within a relativistic mean field theory and a density dependent Hartree Fock theory, where the halo in {sup 11}Li could only be reproduced by an artificial shift of the 1p{sub 1/2} level close to the continuum limit, the halo is now reproduced in a self-consistent way without further modifications using the scattering of Cooper pairs to the 2s{sub 1/2} level in the continuum. Excellent agreement with recent experimental data is observed.
Qualitative breakdown of the unrestricted Hartree-Fock energy
Mori-Sánchez, Paula, E-mail: paula.mori@uam.es [Departamento de Química and Instituto de Física de la Materia Condensada (IFIMAC), Universidad Autónoma de Madrid, 28049 Madrid (Spain); Cohen, Aron J., E-mail: ajc54@cam.ac.uk [Department of Chemistry, Lensfield Road, University of Cambridge, Cambridge CB2 1EW (United Kingdom)
2014-10-28
The stretching of closed-shell molecules is a qualitative problem for restricted Hartree-Fock that is usually circumvented by the use of unrestricted Hartree-Fock (UHF). UHF is well known to break the spin symmetry at the Coulson-Fischer point, leading to a discontinuous derivative in the potential energy surface and incorrect spin density. However, this is generally not considered as a major drawback. In this work, we present a set of two electron molecules which magnify the problem of symmetry breaking and lead to drastically incorrect potential energy surfaces with UHF. These molecules also fail with unrestricted density-functional calculations where a functional such as B3LYP gives both symmetry breaking and an unphysically low energy due to the delocalization error. The implications for density functional theory are also discussed.
Hydrogen Beyond the Classic Approximation
Scivetti, I
2003-01-01
The classical nucleus approximation is the most frequently used approach for the resolution of problems in condensed matter physics.However, there are systems in nature where it is necessary to introduce the nuclear degrees of freedom to obtain a correct description of the properties.Examples of this, are the systems with containing hydrogen.In this work, we have studied the resolution of the quantum nuclear problem for the particular case of the water molecule.The Hartree approximation has been used, i.e. we have considered that the nuclei are distinguishable particles.In addition, we have proposed a model to solve the tunneling process, which involves the resolution of the nuclear problem for configurations of the system away from its equilibrium position
Twist-averaged boundary conditions for nuclear pasta Hartree-Fock calculations
Schuetrumpf, B
2015-01-01
Background: Nuclear pasta phases, present in the inner crust of neutron stars, are associated with nucleonic matter at sub-saturation densities arranged in regular shapes. Those complex phases, residing in a layer which is approximately 100 m thick, impact many features of neutron stars. Theoretical quantum-mechanical simulations of nuclear pasta are usually carried out in finite 3D boxes assuming periodic boundary conditions (PBC). The resulting solutions are affected by spurious finite-size effects. Purpose: In order to remove spurious finite-size effects, it is convenient to employ twist-averaged boundary conditions (TABC) used in condensed matter, nuclear matter, and lattice QCD applications. In this work, we study the effectiveness of TABC in the context of pasta phases simulations within nuclear density functional theory. Methods: We perform Skyrme-Hartree-Fock calculations in three dimensions by implementing Bloch boundary conditions. The TABC averages are obtained by means of Gauss-Legendre integratio...
Multidimensionally-constrained relativistic Hartree-Bogoliubov study of nuclear spontaneous fission
Zhao, Jie; Niksic, Tamara; Vretenar, Dario
2015-01-01
Recent microscopic studies, based on the theoretical framework of nuclear energy density functionals, have analyzed dynamic (least action) and static (minimum energy) fission paths, and it has been shown that in addition to the important role played by nonaxial and/or octupole collective degrees of freedom, fission paths crucially depend on the approximations adopted in calculating the collective inertia. The dynamics of spontaneous fission of $^{264}$Fm and $^{250}$Fm is explored. The fission paths, action integrals and the corresponding half-lives predicted by the functionals PC-PK1 and DD-PC1 are compared and, in the case of $^{264}$Fm, discussed in relation with recent results obtained using the HFB model based on the Skyrme functional SkM$^*$ and a density dependent mixed pairing interaction. Deformation energy surfaces, collective potentials, and perturbative and nonperturbative cranking collective inertia tensors are calculated using the multidimensionally-constrained relativistic Hartree-Bogoliubov (M...
Spiral magnetic phases on the Kondo Lattice Model: A Hartree-Fock approach
Costa, N. C.; Lima, J. P.; dos Santos, Raimundo R.
2017-02-01
We study the Kondo Lattice Model (KLM) on a square lattice through a Hartree-Fock approximation in which the local spins are treated semi-classically, in the sense that their average values are modulated by a magnetic wavevector Q while they couple with the conduction electrons through fermion operators. In this way, we obtain a ground state phase diagram in which spiral magnetic phases (in which the wavevector depends on the coupling constants and on the density) interpolate between the low-density ferromagnetic phase and the antiferromagnetic phase at half filling; within small regions of the phase diagram commensurate magnetic phases can coexist with Kondo screening. We have also obtained 'Doniach-like' diagrams, showing the effect of temperature on the ground state phases, and established that for some ranges of the model parameters (the exchange coupling and conduction electron density) the magnetic wavevector changes with temperature, either continuously or abruptly (e.g., from spiral to ferromagnetic).
Self-consistent Hartree-Fock RPA calculations in 208Pb
Taqi, Ali H.; Ali, Mohammed S.
2017-07-01
The nuclear structure of 208Pb is studied in the framework of the self-consistent random phase approximation (SCRPA). The Hartree-Fock mean field and single particle states are used to implement a completely SCRPA with Skyrme-type interactions. The Hamiltonian is diagonalised within a model space using five Skyrme parameter sets, namely LNS, SkI3, SkO, SkP and SLy4. In view of the huge number of the existing Skyrme-force parameterizations, the question remains which of them provide the best description of data. The approach attempts to accurately describe the structure of the spherical even-even nucleus 208Pb. To illustrate our approach, we compared the binding energy, charge density distribution, excitation energy levels scheme with the available experimental data. Moreover, we calculated isoscalar and isovector monopole, dipole, and quadrupole transition densities and strength functions.
Comparative study of metal cluster fission in Hartree-Fock and LDA
Lyalin, A; Greiner, W; Lyalin, Andrey; Solov'yov, Andrey; Greiner, Walter
2001-01-01
Fission of doubly charged metal clusters is studied using the open-shell two-center deformed jellium Hartree-Fock model and Local Density Approximation. Results of calculations of the electronic structure and fission barriers for the symmetric and asymmetric channels associated with the following processes Na_{10}^{2+} --> Na_{7}^{+} + Na_{3}^{+}, Na_{18}^{2+} --> Na_{15}^{+} + Na_{3}^{+} and Na_{18}^{2+} --> 2 Na_{9}^{+} are presented. The role of the exact exchange and many-body correlation effects in metal clusters fission is analysed. It is demonstrated that the influence of many-electron correlation effects on the height of the fission barrier is more profound if the barrier arises nearby or beyond the scission point. The importance of cluster deformation effects in the fission process is elucidated with the use of the overlapping-spheroids shape parametrization allowing one an independent variation of deformations in the parent and daughter clusters.
Static correlation beyond the random phase approximation
Olsen, Thomas; Thygesen, Kristian Sommer
2014-01-01
We investigate various approximations to the correlation energy of a H2 molecule in the dissociation limit, where the ground state is poorly described by a single Slater determinant. The correlation energies are derived from the density response function and it is shown that response functions...... derived from Hedin's equations (Random Phase Approximation (RPA), Time-dependent Hartree-Fock (TDHF), Bethe-Salpeter equation (BSE), and Time-Dependent GW) all reproduce the correct dissociation limit. We also show that the BSE improves the correlation energies obtained within RPA and TDHF significantly...
Kuwahara, Riichi; Tadokoro, Yoichi; Ohno, Kaoru
2014-08-28
In this paper, we calculate kinetic and potential energy contributions to the electronic ground-state total energy of several isolated atoms (He, Be, Ne, Mg, Ar, and Ca) by using the local density approximation (LDA) in density functional theory, the Hartree-Fock approximation (HFA), and the self-consistent GW approximation (GWA). To this end, we have implemented self-consistent HFA and GWA routines in our all-electron mixed basis code, TOMBO. We confirm that virial theorem is fairly well satisfied in all of these approximations, although the resulting eigenvalue of the highest occupied molecular orbital level, i.e., the negative of the ionization potential, is in excellent agreement only in the case of the GWA. We find that the wave function of the lowest unoccupied molecular orbital level of noble gas atoms is a resonating virtual bound state, and that of the GWA spreads wider than that of the LDA and thinner than that of the HFA.
Potential Energy Surface in Hartree-Fock Theory:Adiabatic or Configuration-Constrained?
GUO Lu; Sakata Fumihiko; ZHAO En-Guang
2004-01-01
Validity of adiabatic assumption is discussed within the constrained Hartree-Fock theory for self-conjugate nucleus 72Kr. It is shown that the adiabatic assumption does not provide a correct description for the nature of nucleus when a configuration change is involved. The excited Hartree-Fock states and the continuously-connected constrained Hartree-Fock states are given for the first time by applying the configuration dictated constrained Hartree-Fock theory with Gogny force. The importance of self-consistency between the mean-field and the single particle wave functions is emphasized even when a small number of nucleons are involved in the configuration change.
Relativistic Hartree-Fock-Bogoliubov model for deformed nuclei
Ebran, J -P; Arteaga, D Pena; Vretenar, D
2010-01-01
The Relativistic Hartree-Fock-Bogoliubov model for axially deformed nuclei (RHFBz) is introduced. The model is based on an effective Lagrangian with density-dependent meson-nucleon couplings in the particle-hole channel, and the pairing part of the Gogny force is used in the pairing channel. The RHFBz quasiparticle equations are solved by expansion in the basis of a deformed harmonic oscillator. Illustrative RHFBz calculations are performed for Carbon, Neon and Magnesium isotopes. The effect of the explicitly including the pion field is investigated for binding energies, deformation parameters, and charge radii.
Properties of the periodic Hartree-Fock minimizer
Ghimenti, Marco
2008-01-01
We study the periodic Hartree-Fock model used for the description of electrons in a crystal. The existence of a minimizer was previously shown by Catto, Le Bris and Lions (Ann. Inst. H. Poincare Anal. Non Lineaire} 18 (2001), no.6, 687--760). We prove in this paper that any minimizer is necessarily a projector and that it solves a certain nonlinear equation, similarly to the atomic case. In particular we show that the Fermi level is either empty or totally filled.
Basler, Mathias; Gindensperger, Etienne; Meyer, Hans-Dieter; Cederbaum, Lorenz S.
2008-05-01
We address the nonadiabatic quantum dynamics of (macro)systems involving a vast number of nuclear degrees of freedom (modes) in the presence of conical intersections. The macrosystem is first decomposed into a system part carrying a few, strongly coupled modes, and an environment, comprising the remaining modes. By successively transforming the modes of the environment, a hierarchy of effective Hamiltonians for the environment can be constructed. Each effective Hamiltonian depends on a reduced number of effective modes, which carry cumulative effects. The environment is described by a few effective modes augmented by a residual environment. In practice, the effective modes can be added to the system's modes and the quantum dynamics of the entire macrosystem can be accurately calculated on a limited time-interval. For longer times, however, the residual environment plays a role. We investigate the possibility to treat fully quantum mechanically the system plus a few effective environmental modes, augmented by the dynamics of the residual environment treated by the time-dependent Hartree (TDH) approximation. While the TDH approximation is known to fail to correctly reproduce the dynamics in the presence of conical intersections, it is shown that its use on top of the effective-mode formalism leads to much better results. Two numerical examples are presented and discussed; one of them is known to be a critical case for the TDH approximation.
Hellweg, Arnim
2016-01-01
Hartree--Fock theory is one of the most ancient methods of computational chemistry, but up to the present day quantum chemical calculations on Hartree--Fock level or with hybrid density functional theory can be excessively time consuming. We compare three currently available techniques to reduce the computational demands of such calculations in terms of timing and accuracy.
On Blowup for time-dependent generalized Hartree-Fock equations
Hainzl, Christian; Lewin, Mathieu; Schlein, Benjamin
2009-01-01
We prove finite-time blowup for spherically symmetric and negative energy solutions of Hartree-Fock and Hartree-Fock-Bogoliubov type equations, which describe the evolution of attractive fermionic systems (e. g. white dwarfs). Our main results are twofold: First, we extend the recent blowup result of [Hainzl and Schlein, Comm. Math. Phys. \\textbf{287} (2009), 705--714] to Hartree-Fock equations with infinite rank solutions and a general class of Newtonian type interactions. Second, we show the existence of finite-time blowup for spherically symmetric solutions of a Hartree-Fock-Bogoliubov model, where an angular momentum cutoff is introduced. We also explain the key difficulties encountered in the full Hartree-Fock-Bogoliubov theory.
Error estimates for the Skyrme-Hartree-Fock model
Erler, J
2014-01-01
There are many complementing strategies to estimate the extrapolation errors of a model which was calibrated in least-squares fits. We consider the Skyrme-Hartree-Fock model for nuclear structure and dynamics and exemplify the following five strategies: uncertainties from statistical analysis, covariances between observables, trends of residuals, variation of fit data, dedicated variation of model parameters. This gives useful insight into the impact of the key fit data as they are: binding energies, charge r.m.s. radii, and charge formfactor. Amongst others, we check in particular the predictive value for observables in the stable nucleus $^{208}$Pb, the super-heavy element $^{266}$Hs, $r$-process nuclei, and neutron stars.
Computational Nuclear Physics and Post Hartree-Fock Methods
Lietz, Justin; Jansen, Gustav R; Hagen, Gaute; Hjorth-Jensen, Morten
2016-01-01
We present a computational approach to infinite nuclear matter employing Hartree-Fock theory, many-body perturbation theory and coupled cluster theory. These lectures are closely linked with those of chapters 9, 10 and 11 and serve as input for the correlation functions employed in Monte Carlo calculations in chapter 9, the in-medium similarity renormalization group theory of dense fermionic systems of chapter 10 and the Green's function approach in chapter 11. We provide extensive code examples and benchmark calculations, allowing thereby an eventual reader to start writing her/his own codes. We start with an object-oriented serial code and end with discussions on strategies for porting the code to present and planned high-performance computing facilities.
Semiempirical Hartree-Fock calculations for $KNbO_{3}$
Eglitis, R I; Borstel, G
1996-01-01
In applying the semiempirical intermediate neglect of differential overlap (INDO) method based on the Hartree-Fock formalism to a cubic perovskite-based ferroelectric material KNbO3, it was demonstrated that the accuracy of the method is sufficient for adequately describing the small energy differences related to the ferroelectric instability. The choice of INDO parameters has been done for a system containing Nb. Based on the parametrization proposed, the electronic structure, equilibrium ground state structure of the orthorhombic and rhombohedral phases, and Gamma-TO phonon frequencies in cubic and rhombohedral phases of KNbO3 were calculated and found to be in good agreement with the experimental data and with the first-principles calculations available.
Using Hartree-Fock pseudopotentials in GW calculations
Hamann, D. R.; Vanderbilt, David
2010-03-01
The issue of including shallow ``semi-core'' states as valence has recently resurfaced in the context of self-consistent GW calculations.footnotetextF. Bruneval et al., Phys. Rev. Lett. 97, 267601 (2006). Supposing that semi-core-valence exchange is the dominant process necessitating the inclusion of semi-cores, we have investigated whether the use Hartree-Fock pseudopotentialsfootnotetextW. A. Al-Saidi, E. J. Walter, and A. M. Rappe, Phys. Rev. B 77, 075122 (2008). instead of density-functional psp's might obviate the need for semi-cores. The answers to this question appear to be ``yes'' for the case of CuCl (filled d shell), and ``semi-cores don't matter anyway'' for ScN (empty d shell). Opportunity permitting, additional examples will be discussed.
The Gogny-Hartree-Fock-Bogoliubov nuclear-mass model
Goriely, S. [Universite Libre de Bruxelles, Institut d' Astronomie et d' Astrophysique, CP-226, Brussels (Belgium); Hilaire, S.; Girod, M.; Peru, S. [CEA, DAM, DIF, Arpajon (France)
2016-07-15
We present the Gogny-Hartree-Fock-Bogoliubov model which reproduces nuclear masses with an accuracy comparable with the best mass formulas. In contrast to the Skyrme-HFB nuclear-mass models, an explicit and self-consistent account of all the quadrupole correlation energies is included within the 5D collective Hamiltonian approach. The final rms deviation with respect to the 2353 measured masses is 789 keV in the 2012 atomic mass evaluation. In addition, the D1M Gogny force is shown to predict nuclear and neutron matter properties in agreement with microscopic calculations based on realistic two- and three-body forces. The D1M properties and its predictions of various observables are compared with those of D1S and D1N. (orig.)
Exact exchange potential evaluated solely from occupied Kohn-Sham and Hartree-Fock solutions
Cinal, M
2011-01-01
The reported new algorithm determines the exact exchange potential v_x in a iterative way using energy and orbital shifts (ES, OS) obtained - with finite-difference formulas - from the solutions (occupied orbitals and their energies) of the Hartree-Fock-like equation and the Kohn-Sham-like equation, the former used for the initial approximation to v_x and the latter - for increments of ES and OS due to subsequent changes of v_x. Thus, solution of the differential equations for OS, used by Kummel and Perdew (KP) [Phys. Rev. Lett. 90, 043004 (2003)], is avoided. The iterated exchange potential, expressed in terms of ES and OS, is improved by modifying ES at odd iteration steps and OS at even steps. The modification formulas are related to the OEP equation (satisfied at convergence) written as the condition of vanishing density shift (DS) - they are obtained, respectively, by enforcing its satisfaction through corrections to approximate OS and by determining optimal ES that minimize the DS norm. The proposed met...
Angular-momentum projection for Hartree-Fock and RPA with realistic interactions
Erler, Bastian; Roth, Robert [Institut fuer Kernphysik, TU Darmstadt (Germany)
2012-07-01
Hartree-Fock (HF) with a Hamiltonian constructed from similarity transformed realistic NN potentials plus 3N contact interactions provides a good starting point for the description of closed shell nuclei. In conjunction with Many-Body-Perturbation-Theory, experimental ground-state energies and radii are well reproduced. To describe collective excitations, the Random-Phase-Approximation (RPA) is the method of choice. Beyond closed shells, e.g. in the sd-shell region, ground-states might exhibit intrinsic deformation, resulting in HF states where angular-momentum ceases to be a good quantum number. Lab-frame observables, like ground-state energies or rotational bands can be recovered from the intrinsic states via angular-momentum projection. We study axially deformed even-even sd-shell nuclei, namely {sup 20}Ne, {sup 28}Si and {sup 32}S. Starting from a HF ground state obtained by exact angular-momentum projection, we use the RPA to study collective excitations. The transition strengths obtained from the RPA are projected to good angular momentum in an exact formalism, without resorting to popular approximations. We investigate the effect of deformed intrinsic states on giant resonances.
Nuclear Pasta at Finite Temperature with the Time-Dependent Hartree-Fock Approach
Schuetrumpf, B.; Klatt, M. A.; Iida, K.; Maruhn, J. A.; Mecke, K.; Reinhard, P.-G.
2016-01-01
We present simulations of neutron-rich matter at sub-nuclear densities, like supernova matter. With the time-dependent Hartree-Fock approximation we can study the evolution of the system at temperatures of several MeV employing a full Skyrme interaction in a periodic three-dimensional grid [1]. The initial state consists of α particles randomly distributed in space that have a Maxwell-Boltzmann distribution in momentum space. Adding a neutron background initialized with Fermi distributed plane waves the calculations reflect a reasonable approximation of astrophysical matter. The matter evolves into spherical, rod-like, connected rod-like and slab-like shapes. Further we observe gyroid-like structures, discussed e.g. in [2], which are formed spontaneously choosing a certain value of the simulation box length. The ρ-T-map of pasta shapes is basically consistent with the phase diagrams obtained from QMD calculations [3]. By an improved topological analysis based on Minkowski functionals [4], all observed pasta shapes can be uniquely identified by only two valuations, namely the Euler characteristic and the integral mean curvature. In addition we propose the variance in the cell-density distribution as a measure to distinguish pasta matter from uniform matter.
Approximate Representations and Approximate Homomorphisms
Moore, Cristopher
2010-01-01
Approximate algebraic structures play a defining role in arithmetic combinatorics and have found remarkable applications to basic questions in number theory and pseudorandomness. Here we study approximate representations of finite groups: functions f:G -> U_d such that Pr[f(xy) = f(x) f(y)] is large, or more generally Exp_{x,y} ||f(xy) - f(x)f(y)||^2$ is small, where x and y are uniformly random elements of the group G and U_d denotes the unitary group of degree d. We bound these quantities in terms of the ratio d / d_min where d_min is the dimension of the smallest nontrivial representation of G. As an application, we bound the extent to which a function f : G -> H can be an approximate homomorphism where H is another finite group. We show that if H's representations are significantly smaller than G's, no such f can be much more homomorphic than a random function. We interpret these results as showing that if G is quasirandom, that is, if d_min is large, then G cannot be embedded in a small number of dimensi...
Application of a solvable model to test the accuracy of the time-dependent Hartree-Fock method
Bouayad, N. [Blida Univ. (Algeria). Inst. de Phys.; Zettili, N. [Blida Univ. (Algeria). Inst. de Phys.]|[Department of Physics, King Fahd University of Petroleum and Minerals, Dhahran 31261 (Saudi Arabia)
1996-11-11
This work deals with the application of a solvable model to study the accuracy of a nuclear many-body approximation method. Using a new exactly solvable model, we carry out here a quantitative test of the accuracy of the time-dependent Hartree-Fock (TDHF) method. The application of the TDHF method to the model reveals that the model is suitable for describing various forms of collective motion: harmonic and anharmonic oscillations as well as rotations. We then show that, by properly quantizing the TDHF results, the TDHF approximation method yields energy spectra that are in very good agreement with their exact counterparts. This work shows that the model offers a rich and comprehensive framework for studying the various aspects of the TDHF method and also for assessing quantitatively its accuracy. (orig.).
Application of a solvable model to test the accuracy of the time-dependent Hartree-Fock method
Bouayad, Nouredine; Zettili, Nouredine
1996-02-01
This work deals with the application of a solvable model to study the accuracy of a nuclear many-body approximation method. Using a new exactly solvable model, we carry out here a quantitative test of the accuracy of the time-dependent Hartree-Fock (TDHF) method. The application of the TDHF method to the model reveals that the model is suitable for describing various forms of collective motion: harmonic and anharmonic oscillations as well as rotations. We then show that, by properly quantizing the TDHF results, the TDHF approximation method yields energy spectra that are in very good agreement with their exact counterparts. This work shows that the model offers a rich and comprehensive framework for studying the various aspects of the TDHF method and also for assessing quantitatively its accuracy.
CERN. Geneva
2015-01-01
Most physics results at the LHC end in a likelihood ratio test. This includes discovery and exclusion for searches as well as mass, cross-section, and coupling measurements. The use of Machine Learning (multivariate) algorithms in HEP is mainly restricted to searches, which can be reduced to classification between two fixed distributions: signal vs. background. I will show how we can extend the use of ML classifiers to distributions parameterized by physical quantities like masses and couplings as well as nuisance parameters associated to systematic uncertainties. This allows for one to approximate the likelihood ratio while still using a high dimensional feature vector for the data. Both the MEM and ABC approaches mentioned above aim to provide inference on model parameters (like cross-sections, masses, couplings, etc.). ABC is fundamentally tied Bayesian inference and focuses on the “likelihood free” setting where only a simulator is available and one cannot directly compute the likelihood for the dat...
Cui, Yao; Bulik, Ireneusz W; Jiménez-Hoyos, Carlos A; Henderson, Thomas M; Scuseria, Gustavo E
2013-10-21
We study the spectra of the molecular orbital Hessian (stability matrix) and random-phase approximation (RPA) Hamiltonian of broken-symmetry Hartree-Fock solutions, focusing on zero eigenvalue modes. After all negative eigenvalues are removed from the Hessian by following their eigenvectors downhill, one is left with only positive and zero eigenvalues. Zero modes correspond to orbital rotations with no restoring force. These rotations determine states in the Goldstone manifold, which originates from a spontaneously broken continuous symmetry in the wave function. Zero modes can be classified as improper or proper according to their different mathematical and physical properties. Improper modes arise from symmetry breaking and their restoration always lowers the energy. Proper modes, on the other hand, correspond to degeneracies of the wave function, and their symmetry restoration does not necessarily lower the energy. We discuss how the RPA Hamiltonian distinguishes between proper and improper modes by doubling the number of zero eigenvalues associated with the latter. Proper modes in the Hessian always appear in pairs which do not double in RPA. We present several pedagogical cases exemplifying the above statements. The relevance of these results for projected Hartree-Fock methods is also addressed.
Approximate particle number projection for finite range density dependent forces
Valor, A; Robledo, L M
1996-01-01
The Lipkin-Nogami method is generalized to deal with finite range density dependent forces. New expressions are derived and realistic calculations with the Gogny force are performed for the nuclei ^{164}Er and ^{168}Er. The sharp phase transition predicted by the mean field approximation is washed out by the Lipkin-Nogami approach; a much better agreement with the experimental data is reached with the new approach than with the Hartree-Fock_Bogoliubov one, specially at high spins.
Hartree-Fock Many-Body Perturbation Theory for Nuclear Ground-States
Tichai, Alexander; Binder, Sven; Roth, Robert
2016-01-01
We investigate the order-by-order convergence behavior of many-body perturbation theory (MBPT) as a simple and efficient tool to approximate the ground-state energy of closed-shell nuclei. To address the convergence properties directly, we explore perturbative corrections up to 30th order and highlight the role of the partitioning for convergence. The use of a simple Hartree-Fock solution to construct the unperturbed basis leads to a convergent MBPT series for soft interactions, in contrast to, e.g., a harmonic oscillator basis. For larger model spaces and heavier nuclei, where a direct high-order MBPT calculation in not feasible, we perform third-order calculation and compare to advanced ab initio coupled-cluster calculations for the same interactions and model spaces. We demonstrate that third-order MBPT provides ground-state energies for nuclei up into tin isotopic chain that are in excellent agreement with the best available coupled-cluster results at a fraction of the computational cost.
Madsen, Lars Bojer; Jensen, Frank; Dnestryan, Andrey I.; Tolstikhin, Oleg I.
2017-07-01
In the leading-order approximation of the weak-field asymptotic theory (WFAT), the dependence of the tunneling ionization rate of a molecule in an electric field on its orientation with respect to the field is determined by the structure factor of the ionizing molecular orbital. The WFAT yields an expression for the structure factor in terms of a local property of the orbital in the asymptotic region. However, in general quantum chemistry approaches molecular orbitals are expanded in a Gaussian basis which does not reproduce their asymptotic behavior correctly. This hinders the application of the WFAT to polyatomic molecules, which are attracting increasing interest in strong-field physics. Recently, an integral-equation approach to the WFAT for tunneling ionization of one electron from an arbitrary potential has been developed. The structure factor is expressed in an integral form as a matrix element involving the ionizing orbital. The integral is not sensitive to the asymptotic behavior of the orbital, which resolves the difficulty mentioned above. Here, we extend the integral representation for the structure factor to many-electron systems treated within the Hartree-Fock method and show how it can be implemented on the basis of standard quantum chemistry software packages. We validate the methodology by considering noble-gas atoms and the CO molecule, for which accurate structure factors exist in the literature. We also present benchmark results for CO2 and for NH3 in the pyramidal and planar geometries.
Hassaneen, Khaled; Mansour, Hesham
2017-02-01
The single-particle potentials and other properties at absolute zero temperature in isospin asymmetric nuclear matter are investigated in the frame of an extended Brueckner theory. Also thermal quantities are calculated in asymmetric nuclear matter using CD-Bonn potential and the Urbana three-body forces (3BF). Also, the effects of the hole-hole contributions are investigated within the self-consistent Greens function approach. The inclusion of 3BF or the hole-hole contributions improves the predicted saturation property of symmetric nuclear matter within the Brueckner-Hartree-Fock approach and it leads to a significant stiffening of the density dependence of symmetry energy at high densities but the exact saturation point is not reproduced. This is of great importance in astrophysical calculation. A phenomenological term simulating the three-body interaction is introduced to assure the empirical saturation property. The hot properties of asymmetric nuclear matter such as the internal energy and the pressure are analyzed using T2-approximation method at low temperatures.
Parallel and Low-Order Scaling Implementation of Hartree-Fock Exchange Using Local Density Fitting.
Köppl, Christoph; Werner, Hans-Joachim
2016-07-12
Calculations using modern linear-scaling electron-correlation methods are often much faster than the necessary reference Hartree-Fock (HF) calculations. We report a newly implemented HF program that speeds up the most time-consuming step, namely, the evaluation of the exchange contributions to the Fock matrix. Using localized orbitals and their sparsity, local density fitting (LDF), and atomic orbital domains, we demonstrate that the calculation of the exchange matrix scales asymptotically linearly with molecular size. The remaining parts of the HF calculation scale cubically but become dominant only for very large molecular sizes or with many processing cores. The method is well parallelized, and the speedup scales well with up to about 100 CPU cores on multiple compute nodes. The effect of the local approximations on the accuracy of computed HF and local second-order Møller-Plesset perturbation theory energies is systematically investigated, and default values are established for the parameters that determine the domain sizes. Using these values, calculations for molecules with hundreds of atoms in combination with triple-ζ basis sets can be carried out in less than 1 h, with just a few compute nodes. The method can also be used to speed up density functional theory calculations with hybrid functionals that contain HF exchange.
Brueckner-Hartree-Fock calculations for finite nuclei with renormalized realistic forces
Hu, B. S.; Xu, F. R.; Wu, Q.; Ma, Y. Z.; Sun, Z. H.
2017-03-01
One can adopt two-step G -matrix approximations for the Brueckner-Hartree-Fock (BHF) calculations. The first G matrix is to soften the bare force, and the second one is to include the high-order correlations of the interaction in medium. The first G -matrix calculation for two-nucleon interaction should be done in the center-of-mass coordinate. As another alternative BHF approach, we have adopted the Vlow-k technique to soften the interaction and used the G matrix to include high-order correlations. The Vlow-k renormalization leads to high-momentum and low-momentum components of the interaction decoupled. With the Vlow-k potential, we have performed the BHF calculations for finite nuclei. The G -matrix elements with exact Pauli exclusions are calculated in the self-consistent BHF basis. To see effects from further possible correlations beyond BHF, we have simultaneously performed renormalized BHF (RBHF) calculations with the same potential. In RBHF, the mean field derived from realistic forces is modified by introducing the particle-occupation depletion resulting from many-body correlations. The ground-state energies and radii of the closed-shell nuclei, 4He, 16O, and 40Ca, have been investigated. The convergences of the BHF and RBHF calculations have been discussed and compared with other ab initio calculations with the same potential.
Thermal resonating Hartree-Bogoliubov theory based on the projection method
Nishiyama, Seiya; Ohnishi, Hiromasa
2013-01-01
We propose a rigorous thermal resonating mean-field theory (Res-MFT). A state is approximated by superposition of multiple MF wavefunctions (WFs) composed of non-orthogonal Hartree-Bogoliubov (HB) WFs. We adopt a Res-HB subspace spanned by Res-HB ground and excited states. A partition function (PF) in a SO(2N) coherent state representation |g> (N:Number of single-particle states) is expressed as Tr(e^{-\\beta H})=2^{N-1} \\int dg (\\beta=1/k_BT). Introducing a projection operator P to the Res-HB subspace, the PF in the Res-HB subspace is given as Tr(Pe^{-\\beta H}), which is calculated within the Res-HB subspace by using the Laplace transform of e^{-\\beta H} and the projection method. The variation of the Res-HB free energy is made, which leads to a thermal HB density matrix W_{Res}^{thermal} expressed in terms of a thermal Res-FB operator F_{Res}^{thermal} as W_{Res}^{thermal}={1_{2N}+exp(\\beta F_{Res}^{thermal})}^{-1}. A calculation of the PF by an infinite matrix continued fraction is cumbersome and a procedur...
On the problem of representability and the Bogolyubov-Hartree-Fock theory
Knoerr, Hans Konrad
2013-11-22
The general topic of this thesis is an approximation of the ground state energy for many-particle quantum systems. In particular the Bogolyubov-Hartree-Fock theory and the representability of one- and two-particle density matrices are studied. After an introductory chapter we specify some basic notation of many-body quantum mechanics in Chapter 2. In Chapter 3 we consider boson, as well as fermion systems. We first tackle the question of representability for bosons, i.e., the question which conditions a one- and a two-particle operator must satisfy to ensure that they are the one- and the two-particle density matrix of a state. For a particle number-conserving system, the representability conditions up to second order for bosons are well-known and called admissibility, P-, and G-conditions. Since, however, most physical systems consisting of bosons are not particle number-conserving, we give an alternative for such systems: Generalizing the two-particle density matrix, we observe that the representability conditions up to second order hold if and only if this generalized two-particle density matrix is positive semi-definite and the one- and the two-particle density matrices fulfill trace class and symmetry conditions. Moreover, we study the Bogolyubov-Hartree-Fock energy of boson and fermion systems. We generalize Lieb's variational principle which in its original formulation holds for purely repulsive particle interactions for fermions only. Our second main result is the following: for bosons, as well as for fermions the infimum of the energy for a variation over pure quasifree states coincides with the one for a variation over all quasifree states under the assumption that the Hamiltonian is bounded below. In the last section of Chapter 3 we specify the relation between centered quasifree states and their corresponding generalized one-particle density matrix, which finds an application in the variational process in the Bogolyubov-Hartree-Fock theory. It is
A collisional extension of time-dependent Hartree-Fock
Lacombe, L.; Reinhard, P.-G.; Dinh, P. M.; Suraud, E.
2016-12-01
We propose a collisional extension of time-dependent mean-field theories on the basis of a recently proposed stochastic extension of mean-field dynamics (stochastic time-dependent Hartree-Fock, STDHF). The latter theory is unfortunately too involved to envision practical applications in realistic systems in the near future and is thus bound to model systems. It is also hard to explore moderate to low energies with STDHF, because of vanishing transition probabilities that are impossible to sample properly. For such moderately excited situations covering small fluctuations, we compactify sampling by employing the same average mean field for all STDHF trajectories. The new approach, coined average STDHF (ASTDHF), ignores the fluctuations of the mean field but still accounts correctly for the collisional correlations responsible for dissipative features on top of mean-field dynamics. We detail the main features of the new approach in relation to existing equations, in particular quantum kinetic theories. The new theory is directly connected to STDHF, both formally and practically. We thus discuss in detail how the two approaches are related to each other. We apply the new scheme to illustrative examples taking as benchmark STDHF dynamics in 1D. ASTDHF provides results that are in remarkable agreement with the more elaborate STDHF. It makes it a promising approach to deal with dissipative dynamics in finite quantum systems, because of its moderate cost allowing applications in realistic systems and the possibility of exploring any excitation energy range where collisional correlations are expected to play a role.
High-frequency averaging in semi-classical Hartree-type equations
Giannoulis, Johannes; Sparber, Christof
2009-01-01
We investigate the asymptotic behavior of solutions to semi-classical Schroedinger equations with nonlinearities of Hartree type. For a weakly nonlinear scaling, we show the validity of an asymptotic superposition principle for slowly modulated highly oscillatory pulses. The result is based on a high-frequency averaging effect due to the nonlocal nature of the Hartree potential, which inhibits the creation of new resonant waves. In the proof we make use of the framework of Wiener algebras.
Momentum distribution of relativistic nuclei with Hartree-Fock mesonic correlations
Amaro, J.E. [Departamento de Fisica Moderna, Universidad de Granada, E-18071 Granada (Spain); Barbaro, M.B. [Dipartimento di Fisica Teorica, Universita di Torino and INFN, Sezione di Torino, Via P. Giuria 1, 10125 Torino (Italy); Departamento de Fisica Atomica, Molecular y Nuclear Universidad de Sevilla, Apdo. 1065, E-41080 Sevilla (Spain); Caballero, J.A. [Departamento de Fisica Atomica, Molecular y Nuclear Universidad de Sevilla, Apdo. 1065, E-41080 Sevilla (Spain); Donnelly, T.W. [Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics Massachusetts Institute of Technology, Cambridge, MA 02139 (United States); Molinari, A. [Dipartimento di Fisica Teorica, Universita di Torino and INFN, Sezione di Torino, Via P. Giuria 1, 10125 Torino (Italy)
2002-12-01
The impact of Hartree-Fock correlations on the nuclear momentum distribution is studied in a fully relativistic one-boson-exchange model. Hartree-Fock equations are exactly solved to first order in the coupling constants. The renormalization of the Dirac spinors in the medium is shown to affect the momentum distribution, as opposed to what happens in the non-relativistic case. The unitarity of the model is shown to be preserved by the present renormalization procedure. (orig.)
Lata, K. Ramani [State University of New York at Albany, Department of Physics (United States); Sahoo, N. [University of Texas M.D. Anderson Cancer Center, Department of Radiation Physics (United States); Dubey, Archana [University of Central Florida, Department of Physics (United States); Scheicher, R. H. [Uppsala University, Condensed Matter Theory Group, Department of Physics and Materials Science (Sweden); Badu, S. R.; Pink, R. H.; Mahato, Dip N. [State University of New York at Albany, Department of Physics (United States); Schulte, A. F.; Saha, H. P. [University of Central Florida, Department of Physics (United States); Maharjan, N. B. [State University of New York at Albany, Department of Physics (United States); Chow, Lee [University of Central Florida, Department of Physics (United States); Das, T. P., E-mail: tpd56@albany.edu [State University of New York at Albany, Department of Physics (United States)
2008-01-15
The electronic structure of the heme unit of deoxyhemoglobin including the proximal imidazole has been studied using the first-principles Hartree-Fock procedure. Our results for the {sup 57m}Fe isomer shift and asymmetry parameter are in very good agreement with the values obtained from Moessbauer spectroscopy measurements. The {sup 57m}Fe nuclear quadrupole coupling constant is smaller than the experimental result and possible ways to improve the agreement in the future are discussed. Improved analysis of the Moessbauer data, removing some approximations made for deriving the magnetic hyperfine tensor for the {sup 57m}Fe nucleus, is suggested to allow quantitative comparison with our results in the future.
Bierón, Jacek; Indelicato, Paul; Jönsson, Per; Pyykkö, Pekka
2009-01-01
The multiconfiguration Dirac-Hartree-Fock (MCDHF) model has been employed to calculate the expectation values for the hyperfine splittings of the 5d96s2 2D3/2 and 5d96s2 2D5/2 levels of atomic gold. One-, two-, and three-body electron correlation effects involving all 79 electrons have been included in a systematic manner. The approximation employed in this study is equivalent to a Complete Active Space (CAS) approach. Calculated electric field gradients, together with experimental values of the electric quadrupole hyperfine structure constants, allow us to extract a nuclear electric quadrupole moment Q(197Au)=521.5(5.0) mb.
Gould, Tim
2013-01-01
By considering the physics of non-interacting ensembles we better generalise the notion of `exact exchange' (EXX) to systems with fractional occupations in the frontier orbitals (called LEXX), in part by exploiting ambiguities in the definitions of `correlation', `exchange' and `Hartree' physics in ensemble systems. The LEXX is employed in an optimised effective potential (OEP) approach (OLEXX) to approximate groundstate energies, where it is bounded by the `ensemble EXX' (EEXX) energy and standard fractional OEXX energy via $E^{\\EEXX}\\leq E^{\\OLEXX} \\leq E^{\\OEXX}$. Analysis of the OLEXX explains the success of standard OEP methods for diatoms at large spacing, and why they can fail when both spins are allowed to be non-integer. The OLEXX is demonstrated on H, Li and Na with fractional electron number with improvements over OEXX for all cases.
Smeyers, Y.G.; Delgado-Barrio, G.
1976-05-01
The half-projected Hartree--Fock function for singlet states (HPHF) is analyzed in terms of natural electronic configurations. For this purpose the HPHF spinless density matrix and its natural orbitals are first deduced. It is found that the HPHF function does not contain any contribution from odd-times excited configurations. It is seen in addition, in the case of the singlet ground states, this function is approximately equivalent to two closed-shell configurations, although the nature of the excited one depends on the nuclear geometry. An example is given in the case of the LiH ground state. Finally, the application of this model for studying systems of more than two atoms is criticized.
Sil, T; Reinhard, P G; Shlomo, S; Sil, Tapas
2006-01-01
We provide accurate assessments of the consequences of violations of self-consistency in Hartree-Fock (HF) based random phase approximation (RPA) calculations of the centroid energy $E_{cen}$ of isoscalar and isovector giant resonances of multi-polarities $L=0-3$ in a wide range of nuclei. This is done by carrying out highly accurate HF-RPA calculations neglecting the particle-hole (ph) spin-orbit or Coulomb interaction in the RPA and comparing with the fully self-consistent HF-RPA results. We find that the shifts in the value of $E_{cen}$ due to self-consistency violation associated with the spin-orbit and Coulomb interactions are comparable or larger than the current experimental errors in $E_{cen}$.
The structure of approximate two electron wavefunctions in intense laser driven ionization dynamics
Sato, Takeshi
2014-01-01
The structure of approximate two electron wavefunction is deeply investigated, both theoretically and numerically, in the strong-field driven ionization dynamics. Theoretical analyses clarify that for two electron singlet systems, the previously proposed time-dependent extended Hartree-Fock (TD-EHF) method [Phys. Rev. A 51, 3999 (1995)] is equivalent to the multiconfiguration time-dependent Hartree-Fock method with two occupied orbitals. The latter wavefunction is further transformed into the natural expansion form, enabling the direct propagation of the natural orbitals (NOs). These methods, as well as the conventional time-dependent Hartree-Fock (TDHF) method, are numerically assessed for the description of ionization dynamics of one-dimensional helium atom model. This numerical analysis (i) explains the reason behind the well-known failure of TDHF method to describe tunneling ionization, (ii) demonstrates the interpretive power of the TD-EHF wavefunction both in the original nonorthogonal and the NO-based ...
Zhou, Fuyang; Li, Jiguang; Wang, Jianguo
2015-01-01
The multi-configuration Dirac-Hartree-Fock method was employed to calculate the total and excitation energies, oscillator strengths and hyperfine structure constants for low-lying levels of Sm I. In the first-order perturbation approximation, we systematically analyzed correlation effects from each electrons and electron pairs. It was found that the core correlations are of importance for physical quantities concerned. Based on the analysis, the important configuration state wave functions were selected to constitute atomic state wave functions. By using this computational model, our excitation energies, oscillator strengths, and hyperfine structure constants are in better agreement with experimental values than earlier theoretical works.
CASTRO EUSTÁQUIO V. R. DE
2001-01-01
Full Text Available The generator coordinate Hartree-Fock method is used to generate adapted Gaussian basis sets for the atoms from Li (Z=3 through Xe (Z=54. In this method the Griffin-Hill-Wheeler-Hartree-Fock equations are integrated through the integral discretization technique. The wave functions generated in this work are compared with the widely used Roothaan-Hartree-Fock wave functions of Clementi and Roetti (1974, and with other basis sets reported in the literature. For all atoms studied, the errors in our total energy values relatively to the numerical Hartree-Fock limits are always less than 7.426 mhartree.
On the problem of representability and the Bogolyubov-Hartree-Fock theory
Knoerr, Hans Konrad
2013-11-22
The general topic of this thesis is an approximation of the ground state energy for many-particle quantum systems. In particular the Bogolyubov-Hartree-Fock theory and the representability of one- and two-particle density matrices are studied. After an introductory chapter we specify some basic notation of many-body quantum mechanics in Chapter 2. In Chapter 3 we consider boson, as well as fermion systems. We first tackle the question of representability for bosons, i.e., the question which conditions a one- and a two-particle operator must satisfy to ensure that they are the one- and the two-particle density matrix of a state. For a particle number-conserving system, the representability conditions up to second order for bosons are well-known and called admissibility, P-, and G-conditions. Since, however, most physical systems consisting of bosons are not particle number-conserving, we give an alternative for such systems: Generalizing the two-particle density matrix, we observe that the representability conditions up to second order hold if and only if this generalized two-particle density matrix is positive semi-definite and the one- and the two-particle density matrices fulfill trace class and symmetry conditions. Moreover, we study the Bogolyubov-Hartree-Fock energy of boson and fermion systems. We generalize Lieb's variational principle which in its original formulation holds for purely repulsive particle interactions for fermions only. Our second main result is the following: for bosons, as well as for fermions the infimum of the energy for a variation over pure quasifree states coincides with the one for a variation over all quasifree states under the assumption that the Hamiltonian is bounded below. In the last section of Chapter 3 we specify the relation between centered quasifree states and their corresponding generalized one-particle density matrix, which finds an application in the variational process in the Bogolyubov-Hartree-Fock theory. It is
A systematic sequence of relativistic approximations.
Dyall, Kenneth G
2002-06-01
An approach to the development of a systematic sequence of relativistic approximations is reviewed. The approach depends on the atomically localized nature of relativistic effects, and is based on the normalized elimination of the small component in the matrix modified Dirac equation. Errors in the approximations are assessed relative to four-component Dirac-Hartree-Fock calculations or other reference points. Projection onto the positive energy states of the isolated atoms provides an approximation in which the energy-dependent parts of the matrices can be evaluated in separate atomic calculations and implemented in terms of two sets of contraction coefficients. The errors in this approximation are extremely small, of the order of 0.001 pm in bond lengths and tens of microhartrees in absolute energies. From this approximation it is possible to partition the atoms into relativistic and nonrelativistic groups and to treat the latter with the standard operators of nonrelativistic quantum mechanics. This partitioning is shared with the relativistic effective core potential approximation. For atoms in the second period, errors in the approximation are of the order of a few hundredths of a picometer in bond lengths and less than 1 kJ mol(-1) in dissociation energies; for atoms in the third period, errors are a few tenths of a picometer and a few kilojoule/mole, respectively. A third approximation for scalar relativistic effects replaces the relativistic two-electron integrals with the nonrelativistic integrals evaluated with the atomic Foldy-Wouthuysen coefficients as contraction coefficients. It is similar to the Douglas-Kroll-Hess approximation, and is accurate to about 0.1 pm and a few tenths of a kilojoule/mole. The integrals in all the approximations are no more complicated than the integrals in the full relativistic methods, and their derivatives are correspondingly easy to formulate and evaluate.
Projected Hartree Fock Theory as a Polynomial Similarity Transformation Theory of Single Excitations
Qiu, Yiheng; Scuseria, Gustavo E
2016-01-01
Spin-projected Hartree-Fock is introduced as a particle-hole excitation ansatz over a symmetry-adapted reference determinant. Remarkably, this expansion has an analytic expression that we were able to decipher. While the form of the polynomial expansion is universal, the excitation amplitudes need to be optimized. This is equivalent to the optimization of orbitals in the conventional projected Hartree-Fock framework of non-orthogonal determinants. Using the inverse of the particle-hole expansion, we similarity transform the Hamiltonian in a coupled-cluster style theory. The left eigenvector of the non-hermitian Hamiltonian is constructed in a similar particle-hole expansion fashion, and we show that to numerically reproduce variational projected Hartree-Fock results, one needs as many pair excitations in the bra as the number of strongly correlated entangled pairs in the system. This single-excitation polynomial similarity transformation theory is an alternative to our recently presented double excitation the...
A finite-temperature Hartree-Fock code for shell-model Hamiltonians
Bertsch, G. F.; Mehlhaff, J. M.
2016-10-01
The codes HFgradZ.py and HFgradT.py find axially symmetric minima of a Hartree-Fock energy functional for a Hamiltonian supplied in a shell model basis. The functional to be minimized is the Hartree-Fock energy for zero-temperature properties or the Hartree-Fock grand potential for finite-temperature properties (thermal energy, entropy). The minimization may be subjected to additional constraints besides axial symmetry and nucleon numbers. A single-particle operator can be used to constrain the minimization by adding it to the single-particle Hamiltonian with a Lagrange multiplier. One can also constrain its expectation value in the zero-temperature code. Also the orbital filling can be constrained in the zero-temperature code, fixing the number of nucleons having given Kπ quantum numbers. This is particularly useful to resolve near-degeneracies among distinct minima.
Application of Fourth Order Vibrational Perturbation Theory with Analytic Hartree-Fock Force Fields
Gong, Justin Z.; Matthews, Devin A.; Stanton, John F.
2014-06-01
Fourth-Order Rayleigh-Schrodinger Perturbation Theory (VPT4) is applied to a series of small molecules. The quality of results have been shown to be heavily dependent on the quality of the quintic and sextic force constants used and that numerical sextic force constants converge poorly and are unreliable for VPT4. Using analytic Hartree-Fock force constants, it is shown that these analytic higher-order force constants are comparable to corresponding force constants from numerical calculations at a higher level of theory. Calculations show that analytic Hartree-Fock sextic force constants are reliable and can provide good results with Fourth-Order Rayleigh-Schrodinger Perturbation Theory.
ON THE DECAY AND SCATTERING FOR THE KLEIN-GORDON-HARTREE EQUATION WITH RADIAL DATA
Wu Haigen; Zhang Junyong
2012-01-01
In this paper,we study the decay estimate and scattering theory for the Klein-Gordon-Hartree equation with radial data in space dimension d ≥ 3.By means of a compactness strategy and two Morawetz-type estimates which come from the linear and nonlinear parts of the equation,respectively,we obtain the corresponding theory for energy subcritical and critical cases.The exponent range of the decay estimates is extended to 0 ＜ γ ≤ 4 and γ＜ d with Hartree potential V(x) =|x|-γ.
Excess Charge for Pseudo-relativistic Atoms in Hartree-Fock Theory
Dall'Acqua, Anna; Solovej, Jan Philip
2010-01-01
We prove within the Hartree-Fock theory of pseudo-relativistic atoms that the maximal negative ionization charge and the ionization energy of an atom remain bounded independently of the nuclear charge $Z$ and the fine structure constant $\\alpha$ as long as $Z\\alpha$ is bounded.......We prove within the Hartree-Fock theory of pseudo-relativistic atoms that the maximal negative ionization charge and the ionization energy of an atom remain bounded independently of the nuclear charge $Z$ and the fine structure constant $\\alpha$ as long as $Z\\alpha$ is bounded....
Excess Charge for Pseudo-relativistic Atoms in Hartree-Fock Theory
Dall'Acqua, Anna; Solovej, Jan Philip
2010-01-01
We prove within the Hartree-Fock theory of pseudo-relativistic atoms that the maximal negative ionization charge and the ionization energy of an atom remain bounded independently of the nuclear charge $Z$ and the fine structure constant $\\alpha$ as long as $Z\\alpha$ is bounded.......We prove within the Hartree-Fock theory of pseudo-relativistic atoms that the maximal negative ionization charge and the ionization energy of an atom remain bounded independently of the nuclear charge $Z$ and the fine structure constant $\\alpha$ as long as $Z\\alpha$ is bounded....
Hartree-Fock Cluster Study of Interstitial Transition Metals in Silicon
Broer, R.; Aissing, G.; Nieuwpoort, W.C.; Feiner, L.F.
1986-01-01
Results are presented of a Hartree-Fock cluster study of interstitial Ti, V, Cr, and Mn impurities in silicon. A Si10 cluster models the nearest Si atoms around a tetrahedral interstitial site in crystalline Si. The dangling bonds of the Si atoms are saturated by hydrogens. The effect of the Si core
GEOMETRIC OPTICS FOR 3D-HARTREE-TYPE EQUATION WITH COULOMB POTENTIAL
无
2006-01-01
This article considers a family of 3D-Hartree-type equation with Coulomb potential |x|-1, whose initial data oscillates so that a caustic appears. In the linear geometric optics case, by using the Lagrangian integrals, a uniform description of the solution outside the caustic, and near the caustic are obtained.
Method of renormalization potential for one model of Hartree-Fock-Slater type
Zasorin, Y V
2002-01-01
A new method of the potential renormalization for the quasiclassical model of the Hartree-Fock-Slater real potential is proposed. The method makes it possible to easily construct the wave functions and contrary to the majority od similar methods it does not require the knowledge of the real-type potential
Robust Periodic Hartree-Fock Exchange for Large-Scale Simulations Using Gaussian Basis Sets.
Guidon, Manuel; Hutter, Jürg; VandeVondele, Joost
2009-11-10
Hartree-Fock exchange with a truncated Coulomb operator has recently been discussed in the context of periodic plane-waves calculations [Spencer, J.; Alavi, A. Phys. Rev. B: Solid State, 2008, 77, 193110]. In this work, this approach is extended to Gaussian basis sets, leading to a stable and accurate procedure for evaluating Hartree-Fock exchange at the Γ-point. Furthermore, it has been found that standard hybrid functionals can be transformed into short-range functionals without loss of accuracy. The well-defined short-range nature of the truncated exchange operator can naturally be exploited in integral screening procedures and makes this approach interesting for both condensed phase and gas phase systems. The presented Hartree-Fock implementation is massively parallel and scales up to ten thousands of cores. This makes it feasible to perform highly accurate calculations on systems containing thousands of atoms or ten thousands of basis functions. The applicability of this scheme is demonstrated by calculating the cohesive energy of a LiH crystal close to the Hartree-Fock basis set limit and by performing an electronic structure calculation of a complete protein (rubredoxin) in solution with a large and flexible basis set.
Qiu, Yiheng; Henderson, Thomas M.; Scuseria, Gustavo E.
2016-09-01
Spin-projected Hartree-Fock is written as a particle-hole excitation ansatz over a symmetry-adapted reference determinant. Remarkably, this expansion has an analytic expression that we were able to decipher. While the form of the polynomial expansion is universal, the excitation amplitudes need to be optimized. This is equivalent to the optimization of orbitals in the conventional projected Hartree-Fock framework of non-orthogonal determinants. Using the inverse of the particle-hole expansion, we similarity transform the Hamiltonian in a coupled-cluster style theory. The left eigenvector of the non-Hermitian Hamiltonian is constructed in a similar particle-hole expansion fashion, and we show that to numerically reproduce variational projected Hartree-Fock results, one needs as many pair excitations in the bra as the number of strongly correlated entangled pairs in the system. This single-excitation polynomial similarity transformation theory is an alternative to our recently presented double excitation theory, but supports projected Hartree-Fock and coupled cluster simultaneously rather than interpolating between them.
Hartree-Fock Cluster Study of Interstitial Transition Metals in Silicon
Broer, R.; Aissing, G.; Nieuwpoort, W.C.; Feiner, L.F.
Results are presented of a Hartree-Fock cluster study of interstitial Ti, V, Cr, and Mn impurities in silicon. A Si10 cluster models the nearest Si atoms around a tetrahedral interstitial site in crystalline Si. The dangling bonds of the Si atoms are saturated by hydrogens. The effect of the Si core
Perturbative calculation of the Sternheimer anti-shielding factor with Hartree-Fock atomic orbitals
2012-01-01
We report a calculation of the Sternheimer anti-shielding factor, \\gamma, by means of first order perturbation theory. In quality of basis functions, we use Hartree-Fock electronic orbitals, expanded on hydrogenic atomic states. The computed \\gamma(r) for Fe^{3+} and Cu^{1+} inner electronic cores are reported and compared with literature values, obtained from alternative methodologies.
GUSEINOV,Israfil; ERT(U)RK,Murat; SAHIN,Ercan; AKSU,Hüseyin
2008-01-01
Using integer and noninteger n-Slater type orbitals in single- and double-zeta approximations, the Hartree-Fock-Roothaan calculations were performed for the ground states of first ten cationic members of the isoelectronic series of He atom. All the noninteger parameters and orbital exponents were fully optimized. In the case of noninteger n-Slater type orbitals in double zeta basis sets, the results of calculations obtained are more close to the numerical Hatree-Fock values and the average deviations of our ground state energies do not exceed 2×10-6 hartrees of their numerical results.
Relativistic quasiparticle random phase approximation in deformed nuclei
Pena Arteaga, D.
2007-06-25
Covariant density functional theory is used to study the influence of electromagnetic radiation on deformed superfluid nuclei. The relativistic Hartree-Bogolyubov equations and the resulting diagonalization problem of the quasiparticle random phase approximation are solved for axially symmetric systems in a fully self-consistent way by a newly developed parallel code. Three different kinds of high precision energy functionals are investigated and special care is taken for the decoupling of the Goldstone modes. This allows the microscopic investigation of Pygmy and scissor resonances in electric and magnetic dipole fields. Excellent agreement with recent experiments is found and new types of modes are predicted for deformed systems with large neutron excess. (orig.)
Expectation values of single-particle operators in the random phase approximation ground state
Kosov, Daniel S
2016-01-01
We developed a method for computing matrix elements of single-particle operators in the correlated random phase approximation ground state. Working with the explicit random phase approximation ground state wavefunction, we derived practically useful and simple expression for a molecular property in terms of random phase approximation amplitudes. The theory is illustrated by the calculation of molecular dipole moments. It is shown that Hartree-Fock based random phase approximation provides a systematic improvement of molecular dipole moment values in comparison to M{\\o}ller-Plesset second order perturbation theory and coupled cluster method for a considered set of molecules.
Nishiyama, Seiya; da Providência, João
2015-02-01
In this paper we present the induced representation of SO(2N) canonical transformation group and introduce (SO(2N))/(U(N)) coset variables. We give a derivation of the time-dependent Hartree-Bogoliubov (TDHB) equation on the Kähler coset space (G)/(H) = (SO(2N))/(U(N)) from the Euler-Lagrange equation of motion for the coset variables. The TDHB wave function represents the TD behavior of Bose condensate of fermion pairs. It is a good approximation for the ground state of the fermion system with a pairing interaction, producing the spontaneous Bose condensation. To describe the classical motion on the coset manifold, we start from the local equation of motion. This equation becomes a Riccati-type equation. After giving a simple two-level model and a solution for a coset variable, we can get successfully a general solution of time-dependent Riccati-Hartree-Bogoliubov equation for the coset variables. We obtain the Harish-Chandra decomposition for the SO(2N) matrix based on the nonlinear Möbius transformation together with the geodesic flow on the manifold.
Fission dynamics within time-dependent Hartree-Fock: boost-induced fission
Goddard, P M; Rios, A
2015-01-01
Background: Nuclear fission is a complex large-amplitude collective decay mode in heavy nuclei. Microscopic density functional studies of fission have previously concentrated on adiabatic approaches based on constrained static calculations ignoring dynamical excitations of the fissioning nucleus, and the daughter products. Purpose: To explore the ability of dynamic mean-field methods to describe induced fission processes, using quadrupole boosts in the nuclide $^{240}$Pu as an example. Methods: Quadrupole constrained Hartree-Fock calculations are used to create a potential energy surface. An isomeric state and a state beyond the second barrier peak are excited by means of instantaneous as well as temporally extended gauge boosts with quadrupole shapes. The subsequent deexcitation is studied in a time-dependent Hartree-Fock simulation, with emphasis on fissioned final states. The corresponding fission fragment mass numbers are studied. Results: In general, the energy deposited by the quadrupole boost is quickl...
Multiconfiguration hartree-fock theory for pseudorelativistic systems: The time-dependent case
Hajaiej, Hichem
2014-03-01
In [Setting and analysis of the multi-configuration time-dependent Hartree-Fock equations, Arch. Ration. Mech. Anal. 198 (2010) 273-330] the third author has studied in collaboration with Bardos, Catto and Mauser the nonrelativistic multiconfiguration time-dependent Hartree-Fock system of equations arising in the modeling of molecular dynamics. In this paper, we extend the previous work to the case of pseudorelativistic atoms. We show the existence and the uniqueness of global-in-time solution to the underlying system under technical assumptions on the energy of the initial data and the charge of the nucleus. Moreover, we prove that the result can be extended to the case of neutron stars when the number of electrons is less than a critical number N cr. © 2014 World Scientific Publishing Company.
Brueckner-Hartree-Fock and its renormalized calculations for finite nuclei
Hu, B S; Ma, Y Z; Wu, Q; Sun, Z H
2016-01-01
We have performed self-consistent Brueckner-Hartree-Fock (BHF) and its renormalized theory to the structure calculations of finite nuclei. The $G$-matrix is calculated within the BHF basis, and the exact Pauli exclusion operator is determined by the BHF spectrum. Self-consistent occupation probabilities are included in the renormalized Brueckner-Hartree-Fock (RBHF). Various systematics and convergences are studies. Good results are obtained for the ground-state energy and radius. RBHF can give a more reasonable single-particle spectrum and radius. We present a first benchmark calculation with other {\\it ab initio} methods using the same effective Hamiltonian. We find that the BHF and RBHF results are in good agreement with other $\\it{ab}$ $\\it{initio}$ methods.
RONG; Jian; MA; Zhongyu
2004-01-01
The relativistic microscopic optical potential in the asymmetric nuclear matter is studied in the framework of the Dirac Brueckner-Hartree-Fock method. A new decomposition of the Dirac structure of the nuclear self-energy in nuclear matter is adopted. The self-energy of a nucleon with E＞ 0 in nuclear matter is calculated with the G matrix in the Hartree-Fock approach. The optical potential of a nucleon in the nuclear medium is identified with the nucleon self-energy. The energy and asymmetric parameter dependence of the relativistic optical potentials for proton and neutron are discussed. The resulting Schroedinger equivalent potentials have reasonable behaviors of the energy dependence. The asymmetric parameter dependence of relativistic optical potentials and Schroedinger potentials are emphasized.
Fission dynamics within time-dependent Hartree-Fock: deformation-induced fission
Goddard, P M; Rios, A
2015-01-01
Background: Nuclear fission is a complex large-amplitude collective decay mode in heavy nuclei. Microscopic density functional studies of fission have previously concentrated on adiabatic approaches based on constrained static calculations ignoring dynamical excitations of the fissioning nucleus, and the daughter products. Purpose: To explore the ability of dynamic mean-field methods to describe fast fission processes beyond the fission barrier, using the nuclide $^{240}$Pu as an example. Methods: Time-dependent Hartree-Fock calculations based on the Skyrme interaction are used to calculate non-adiabatic fission paths, beginning from static constrained Hartree-Fock calculations. The properties of the dynamic states are interpreted in terms of the nature of their collective motion. Fission product properties are compared to data. Results: Parent nuclei constrained to begin dynamic evolution with a deformation less than the fission barrier exhibit giant-resonance-type behaviour. Those beginning just beyond the ...
Pototzky, K J; Reinhard, P -G; Nesterenko, V O
2010-01-01
We present a systematic analysis of the description of odd nuclei by the Skyrme-Hartree-Fock approach augmented with pairing in BCS approximation and blocking of the odd nucleon. Current and spin densities in the Skyrme functional produce time-odd mean fields (TOMF) for odd nuclei. Their effect on basic properties (binding energies, odd-even staggering, separation energies and spectra) is investigated for the three Skyrme parameterizations SkI3, SLy6, and SV-bas. About 1300 spherical and axially-deformed odd nuclei with 16 < Z < 92 are considered. The calculations demonstrate that the TOMF effect is generally small, although not fully negligible. The influence of the Skyrme parameterization and the consistency of the calculations are much more important. With a proper choice of the parameterization, a good description of binding energies and their differences is obtained, comparable to that for even nuclei. The description of low-energy excitation spectra of odd nuclei is of varying quality depending on...
Xu, Ruirui; Ma, Zhongyu; Zhang, Yue; Tian, Yuan; van Dalen, E. N. E.; Müther, H.
2016-09-01
Background: For the study of exotic nuclei it is important to have an optical model potential that is reliable not only for stable nuclei but can also be extrapolated to nuclear systems with exotic numbers of protons and neutrons. An efficient way to obtain such a potential is to develop a microscopic optical potential (MOP) based on a fundamental theory with a minimal number of free parameters, which are adjusted to describe stable nuclei all over the nuclide chart. Purpose: The choice adopted in the present work is to develop the MOP within a relativistic scheme which provides a natural and consistent relation between the spin-orbit part and the central part of the potential. The Dirac-Brueckner-Hartree-Fock (DBHF) approach provides such a microscopic relativistic scheme, which is based on a realistic nucleon-nucleon interaction and reproduces the saturation properties of symmetric nuclear matter without any adjustable parameter. Its solution using the projection technique within the subtracted T -matrix representation provides a reliable extension to asymmetric nuclear matter, which is important to describe the features of isospin asymmetric nuclei. The present work performs a global analysis of the isospin dependent nucleon-nucleus MOP based on the DBHF calculation in symmetric and asymmetric nuclear matter. Methods: The DBHF approach is used to evaluate the relativistic structure of the nucleon self-energies in nuclear matter at various densities and asymmetries. The Schrödinger equivalent potentials of finite nuclei are derived from these Dirac components by a local density approximation (LDA). The density distributions of finite nuclei are taken from the Hartree-Fock-Bogoliubov approach with Gogny D1S force. An improved LDA approach (ILDA) is employed to get a better prediction of the scattering observables. A χ2 assessment system based on the global simulated annealing algorithm is developed to optimize the very few free components in this study. Results
GUO Lu; ZHAO En-Guang; SAKATA Fumihiko
2003-01-01
Ground-state.properties of C, O, and Ne isotopes are described in the framework of Hartree-FockBogoliubov theory with density-dependent finite-range Gogny interaction D1S. We include all the contributions to the Hartree-Fock and pairing field arising from Gogny and Coulomb interaction as well as the center of mass correction in the numerical calculations. These ground-state properties of C, O, and Ne isotopes are compared with available experimental results, Hartree-Fock plus BCS, shell model and relativistic Hartree-Bogoliubov calculations. The agreement between experiments and our theoretical results is pretty well. The predicted drip-line is dependent strongly on the model and effective interaction due to their sensitivity to various theoretical details. The calculations predict no evidence for halo structure predicted for C, O, and Ne isotopes in a previous RHB study.
GUOLu; ZHAOEn-Guang; SAKATAFumihiko
2003-01-01
Ground-state properties of C, O, and Ne isotopes are described in the framework of Hartree-Fock-Bogoliubov theory with density-dependent finite-range Gogny interaction D1S. We include all the contributions to the Hartree-Fock and pairing feld arising from Gogny and Coulomb interaction as well as the center of mass correction in the numerical calcu/ations. These ground-state properties of C, O, and Ne isotopes are compared with available experimental results, Hartree-Fock plus BCS, shell model and relativistic Hartree--Bogoliubov calculations. The agreement between experiments and our theoretical results is pretty well. The predicted drip-line is dependent strongly on the model and effective interaction due to their sensitivity to various theoretical details. The calculations predict no evidence for halo structure predicted for C,O, and Ne isotopes in a previous RHB study.
Mazurek, A. P.; Sadlej-Sosnowska, N.
2000-11-01
A comparison of the ab initio quantum chemical methods: Hartree-Fock (HF) and hybrid density functional theory (DFT)/B3LYP for the treatment of tautomeric equilibria both in the gas phase and in the solution is made. The solvent effects were investigated in terms of the self-consistent reaction field (SCRF). Ionization potentials (IP), calculated by DFT/B3LYP, are also compared with those calculated previously within the HF frame.
Generalization of Cramer's rule and its application to the projection of Hartree-Fock wave function
Hage-Hassan, Mehdi
2009-01-01
We generalize the Cramer's rule of linear algebra. We apply it to calculate the spectra of nucleus by applying Hill-Wheeler projection operator to Hartree-Fock wave function, and to derive L\\"owdin formula and Thouless theorem. We derive by an elementary method the infinitesimal or L\\"owdin projection operators and its integral representation to be useful for the projection of Slater determinant.
On the solution of the Hartree-Fock-Bogoliubov equations by the conjugate gradient method
Egido, J.L. [Universidad Autonoma de Madrid (Spain). Dept. de Fisica Teorica; Lessing, J. [Universidad Autonoma de Madrid (Spain). Dept. de Fisica Teorica; Martin, V. [Analisis Numerico, Facultad de Informatica, Universidad Politecnica de Madrid, E-28660 Boadilla del Monte, Madrid (Spain); Robledo, L.M. [Universidad Autonoma de Madrid (Spain). Dept. de Fisica Teorica
1995-11-06
The conjugate gradient method is formulated in the Hilbert space for density and non-density dependent Hamiltonians. We apply it to the solution of the Hartree-Fock-Bogoliubov equations with constraints. As a numerical application we show calculations with the finite range density dependent Gogny force. The number of iterations required to reach convergence is reduced by a factor of three to four as compared with the standard gradient method. (orig.).
Can X-ray constrained Hartree-Fock wavefunctions retrieve electron correlation?
Genoni, Alessandro; Dos Santos, Leonardo H R; Meyer, Benjamin; Macchi, Piero
2017-03-01
The X-ray constrained wavefunction (XC-WF) method proposed by Jayatilaka [Jayatilaka & Grimwood (2001) ▸, Acta Cryst. A57, 76-86] has attracted much attention because it represents a possible third way of theoretically studying the electronic structure of atoms and molecules, combining features of the more popular wavefunction- and DFT-based approaches. In its original formulation, the XC-WF technique extracts statistically plausible wavefunctions from experimental X-ray diffraction data of molecular crystals. A weight is used to constrain the pure Hartree-Fock solution to the observed X-ray structure factors. Despite the wavefunction being a single Slater determinant, it is generally assumed that its flexibility could guarantee the capture, better than any other experimental model, of electron correlation effects, absent in the Hartree-Fock Hamiltonian but present in the structure factors measured experimentally. However, although the approach has been known for long time, careful testing of this fundamental hypothesis is still missing. Since a formal demonstration is impossible, the validation can only be done heuristically and, to accomplish this task, X-ray constrained Hartree-Fock calculations have been performed using structure factor amplitudes computed at a very high correlation level (coupled cluster) for selected molecules in isolation, in order to avoid the perturbations due to intermolecular interactions. The results show that a single-determinant XC-WF is able to capture the electron correlation effects only partially. The largest amount of electron correlation is extracted when: (i) a large external weight is used (much larger than what has normally been used in XC-WF calculations using experimental data); and (ii) the high-order reflections, which carry less information on the electron correlation, are down-weighted (or even excluded), otherwise they would bias the fitting towards the unconstrained Hartree-Fock wavefunction.
On the NP-completeness of the Hartree-Fock method for translationally invariant systems
Whitfield, James D
2014-01-01
The self-consistent field method utilized for solving the Hartree-Fock (HF) problem and the closely related Kohn-Sham problem, is typically thought of as one of the cheapest methods available to quantum chemists. This intuition has been developed from the numerous applications of the self-consistent field method to a large variety of molecular systems. However, in terms of its worst-case computational complexity, the HF problem is NP-complete. In this work, we investigate how far one can push the boundaries of the NP-completeness by investigating restricted instances of HF. We have constructed two new NP-complete variants of the problem. The first is a set of Hamiltonians whose translationally invariant Hartree-Fock solutions are trivial, but whose broken symmetry solutions are NP-complete. Second, we demonstrate how to embed instances of spin glasses into translationally invariant Hartree-Fock instances and provide a numerical example. These findings are the first steps towards understanding in which cases t...
Diophantine approximation and badly approximable sets
Kristensen, S.; Thorn, R.; Velani, S.
2006-01-01
Let (X,d) be a metric space and (Omega, d) a compact subspace of X which supports a non-atomic finite measure m. We consider `natural' classes of badly approximable subsets of Omega. Loosely speaking, these consist of points in Omega which `stay clear' of some given set of points in X. The clas......Let (X,d) be a metric space and (Omega, d) a compact subspace of X which supports a non-atomic finite measure m. We consider `natural' classes of badly approximable subsets of Omega. Loosely speaking, these consist of points in Omega which `stay clear' of some given set of points in X....... The classical set Bad of `badly approximable' numbers in the theory of Diophantine approximation falls within our framework as do the sets Bad(i,j) of simultaneously badly approximable numbers. Under various natural conditions we prove that the badly approximable subsets of Omega have full Hausdorff dimension...
Leike, Reimar H
2016-01-01
In Bayesian statistics probability distributions express beliefs. However, for many problems the beliefs cannot be computed analytically and approximations of beliefs are needed. We seek a ranking function that quantifies how "embarrassing" it is to communicate a given approximation. We show that there is only one ranking under the requirements that (1) the best ranked approximation is the non-approximated belief and (2) that the ranking judges approximations only by their predictions for actual outcomes. We find that this ranking is equivalent to the Kullback-Leibler divergence that is frequently used in the literature. However, there seems to be confusion about the correct order in which its functional arguments, the approximated and non-approximated beliefs, should be used. We hope that our elementary derivation settles the apparent confusion. We show for example that when approximating beliefs with Gaussian distributions the optimal approximation is given by moment matching. This is in contrast to many su...
Bouguettoucha, A.
1996-06-14
A possible effects of the C{sub 4}-symmetry in the superdeformed nuclei of the A {approx}150 mass range has been studied microscopically using cranking Strutinsky method with the deformed Woods-Saxon potential and the Hartree-Fock approach with the Skyrme interaction. If the existence of such a symmetry is judged by the moments Q{sub 44}, the results of the calculation indicate a very weak effect of this kind. Four new superdeformed bands in the {sup 148}Gd nucleus have been studied in reaction to the recent experimental observations (Eurogam Phase 2): a backbending has been tentatively observed at very high rotational frequency in the third excited band. One of the other bands exhibits a J{sup (2)} moment very similar to that of the yrast band in {sup 152}Dy, and this is the first example of identical bands which differ by four mass units. Calculations with the methods mentioned above have been used to analyse the band structure in terms of the nucleonic configurations. Calculation have been performed for some nuclear configurations predicted to involve the exotic octupole deformations (Y{sub 30-}`pear shapes`; Y{sub 31-}`banana mode`; Y{sub 32-}`T{sub d}-symmetry` and Y{sub 33-}`C{sub 3}-symmetry`). While the previous calculations based on the Strutinsky method could not treat the coupling between those modes, the Hartree-Fock approach allows to see for the first time in which propositions the various modes couple. (author). 116 refs.
Schimeczek, C.; Engel, D.; Wunner, G.
2014-05-01
Our previously published code for calculating energies and bound-bound transitions of medium-Z elements at neutron star magnetic field strengths [D. Engel, M. Klews, G. Wunner, Comp. Phys. Comm. 180, 3-2-311 (2009)] was based on the adiabatic approximation. It assumes a complete decoupling of the (fast) gyration of the electrons under the action of the magnetic field and the (slow) bound motion along the field under the action of the Coulomb forces. For the single-particle orbitals this implied that each is a product of a Landau state and an (unknown) longitudinal wave function whose B-spline coefficients were determined self-consistently by solving the Hartree-Fock equations for the many-electron problem on a finite-element grid. In the present code we go beyond the adiabatic approximation, by allowing the transverse part of each orbital to be a superposition of Landau states, while assuming that the longitudinal part can be approximated by the same wave function in each Landau level. Inserting this ansatz into the energy variational principle leads to a system of coupled equations in which the B-spline coefficients depend on the weights of the individual Landau states, and vice versa, and which therefore has to be solved in a doubly self-consistent manner. The extended ansatz takes into account the back-reaction of the Coulomb motion of the electrons along the field direction on their motion in the plane perpendicular to the field, an effect which cannot be captured by the adiabatic approximation. The new code allows for the inclusion of up to 8 Landau levels. This reduces the relative error of energy values as compared to the adiabatic approximation results by typically a factor of three (1/3 of the original error) and yields accurate results also in regions of lower neutron star magnetic field strengths where the adiabatic approximation fails. Further improvements in the code are a more sophisticated choice of the initial wave functions, which takes into
Rašin, Andrija
1994-01-01
We discuss the idea of approximate flavor symmetries. Relations between approximate flavor symmetries and natural flavor conservation and democracy models is explored. Implications for neutrino physics are also discussed.
On Element SDD Approximability
Avron, Haim; Toledo, Sivan
2009-01-01
This short communication shows that in some cases scalar elliptic finite element matrices cannot be approximated well by an SDD matrix. We also give a theoretical analysis of a simple heuristic method for approximating an element by an SDD matrix.
Approximate iterative algorithms
Almudevar, Anthony Louis
2014-01-01
Iterative algorithms often rely on approximate evaluation techniques, which may include statistical estimation, computer simulation or functional approximation. This volume presents methods for the study of approximate iterative algorithms, providing tools for the derivation of error bounds and convergence rates, and for the optimal design of such algorithms. Techniques of functional analysis are used to derive analytical relationships between approximation methods and convergence properties for general classes of algorithms. This work provides the necessary background in functional analysis a
Collapse of the random phase approximation: examples and counter-examples from the shell model
Johnson, Calvin W
2009-01-01
The Hartree-Fock approximation to the many-fermion problem can break exact symmetries, and in some cases by changing a parameter in the interaction one can drive the Hartree-Fock minimum from a symmetry-breaking state to a symmetry-conserving state (also referred to as a ``phase transition'' in the literature). The order of the transition is important when one applies the random phase approximation (RPA) to the of the Hartree-Fock wavefunction: if first order, RPA is stable through the transition, but if second-order, then the RPA amplitudes become large and lead to unphysical results. The latter is known as ``collapse'' of the RPA. While the difference between first- and second-order transitions in the RPA was first pointed out by Thouless, we present for the first time non-trivial examples of both first- and second-order transitions in a uniform model, the interacting shell-model, where we can compare to exact numerical results.
Big Data Meets Quantum Chemistry Approximations: The Δ-Machine Learning Approach.
Ramakrishnan, Raghunathan; Dral, Pavlo O; Rupp, Matthias; von Lilienfeld, O Anatole
2015-05-12
Chemically accurate and comprehensive studies of the virtual space of all possible molecules are severely limited by the computational cost of quantum chemistry. We introduce a composite strategy that adds machine learning corrections to computationally inexpensive approximate legacy quantum methods. After training, highly accurate predictions of enthalpies, free energies, entropies, and electron correlation energies are possible, for significantly larger molecular sets than used for training. For thermochemical properties of up to 16k isomers of C7H10O2 we present numerical evidence that chemical accuracy can be reached. We also predict electron correlation energy in post Hartree-Fock methods, at the computational cost of Hartree-Fock, and we establish a qualitative relationship between molecular entropy and electron correlation. The transferability of our approach is demonstrated, using semiempirical quantum chemistry and machine learning models trained on 1 and 10% of 134k organic molecules, to reproduce enthalpies of all remaining molecules at density functional theory level of accuracy.
Big Data meets Quantum Chemistry Approximations: The $\\Delta$-Machine Learning Approach
Ramakrishnan, Raghunathan; Rupp, Matthias; von Lilienfeld, O Anatole
2015-01-01
Chemically accurate and comprehensive studies of the virtual space of all possible molecules are severely limited by the computational cost of quantum chemistry. We introduce a composite strategy that adds machine learning corrections to computationally inexpensive approximate legacy quantum methods. After training, highly accurate predictions of enthalpies, free energies, entropies, and electron correlation energies are possible, for significantly larger molecular sets than used for training. For thermochemical properties of up to 16k constitutional isomers of C$_7$H$_{10}$O$_2$ we present numerical evidence that chemical accuracy can be reached. We also predict electron correlation energy in post Hartree-Fock methods, at the computational cost of Hartree-Fock, and we establish a qualitative relationship between molecular entropy and electron correlation. The transferability of our approach is demonstrated, using semi-empirical quantum chemistry and machine learning models trained on 1 and 10\\% of 134k organ...
Benchmarking mean-field approximations to level densities
Alhassid, Y; Gilbreth, C N; Nakada, H
2015-01-01
We assess the accuracy of finite-temperature mean-field theory using as a standard the Hamiltonian and model space of the shell model Monte Carlo calculations. Two examples are considered: the nucleus $^{162}$Dy, representing a heavy deformed nucleus, and $^{148}$Sm, representing a nearby heavy spherical nucleus with strong pairing correlations. The errors inherent in the finite-temperature Hartree-Fock and Hartree-Fock-Bogoliubov approximations are analyzed by comparing the entropies of the grand canonical and canonical ensembles, as well as the level density at the neutron resonance threshold, with shell model Monte Carlo (SMMC) calculations, which are accurate up to well-controlled statistical errors. The main weak points in the mean-field treatments are seen to be: (i) the extraction of number-projected densities from the grand canonical ensembles, and (ii) the symmetry breaking by deformation or by the pairing condensate. In the absence of a pairing condensate, we confirm that the usual saddle-point appr...
Approximation of distributed delays
Lu, Hao; Eberard, Damien; Simon, Jean-Pierre
2010-01-01
We address in this paper the approximation problem of distributed delays. Such elements are convolution operators with kernel having bounded support, and appear in the control of time-delay systems. From the rich literature on this topic, we propose a general methodology to achieve such an approximation. For this, we enclose the approximation problem in the graph topology, and work with the norm defined over the convolution Banach algebra. The class of rational approximates is described, and a constructive approximation is proposed. Analysis in time and frequency domains is provided. This methodology is illustrated on the stabilization control problem, for which simulations results show the effectiveness of the proposed methodology.
Diophantine approximation and badly approximable sets
Kristensen, S.; Thorn, R.; Velani, S.
2006-01-01
Let (X,d) be a metric space and (Omega, d) a compact subspace of X which supports a non-atomic finite measure m. We consider `natural' classes of badly approximable subsets of Omega. Loosely speaking, these consist of points in Omega which `stay clear' of some given set of points in X. The clas......Let (X,d) be a metric space and (Omega, d) a compact subspace of X which supports a non-atomic finite measure m. We consider `natural' classes of badly approximable subsets of Omega. Loosely speaking, these consist of points in Omega which `stay clear' of some given set of points in X...
Veeraraghavan, Srikant; Mazziotti, David A
2014-03-28
We present a density matrix approach for computing global solutions of restricted open-shell Hartree-Fock theory, based on semidefinite programming (SDP), that gives upper and lower bounds on the Hartree-Fock energy of quantum systems. While wave function approaches to Hartree-Fock theory yield an upper bound to the Hartree-Fock energy, we derive a semidefinite relaxation of Hartree-Fock theory that yields a rigorous lower bound on the Hartree-Fock energy. We also develop an upper-bound algorithm in which Hartree-Fock theory is cast as a SDP with a nonconvex constraint on the rank of the matrix variable. Equality of the upper- and lower-bound energies guarantees that the computed solution is the globally optimal solution of Hartree-Fock theory. The work extends a previously presented method for closed-shell systems [S. Veeraraghavan and D. A. Mazziotti, Phys. Rev. A 89, 010502-R (2014)]. For strongly correlated systems the SDP approach provides an alternative to the locally optimized Hartree-Fock energies and densities with a certificate of global optimality. Applications are made to the potential energy curves of C2, CN, Cr2, and NO2.
Veeraraghavan, Srikant; Mazziotti, David A., E-mail: damazz@uchicago.edu [Department of Chemistry and The James Franck Institute, The University of Chicago, Chicago, Illinois 60637 (United States)
2014-03-28
We present a density matrix approach for computing global solutions of restricted open-shell Hartree-Fock theory, based on semidefinite programming (SDP), that gives upper and lower bounds on the Hartree-Fock energy of quantum systems. While wave function approaches to Hartree-Fock theory yield an upper bound to the Hartree-Fock energy, we derive a semidefinite relaxation of Hartree-Fock theory that yields a rigorous lower bound on the Hartree-Fock energy. We also develop an upper-bound algorithm in which Hartree-Fock theory is cast as a SDP with a nonconvex constraint on the rank of the matrix variable. Equality of the upper- and lower-bound energies guarantees that the computed solution is the globally optimal solution of Hartree-Fock theory. The work extends a previously presented method for closed-shell systems [S. Veeraraghavan and D. A. Mazziotti, Phys. Rev. A 89, 010502–R (2014)]. For strongly correlated systems the SDP approach provides an alternative to the locally optimized Hartree-Fock energies and densities with a certificate of global optimality. Applications are made to the potential energy curves of C{sub 2}, CN, Cr {sub 2}, and NO {sub 2}.
A importância do método de Hartree no ensino de química quântica
Silmar A. do Monte
2011-01-01
Full Text Available Hartree's original ideas are described. Its connection with electrostatics can be explored in order to decrease the gap between teaching of Physics and Chemistry. As a consequence of its simplicity and connection with electrostatics, it is suggested that Hartree's method should be presented before the Hartree-Fock method. Besides, since the fundamental concepts of indistinguishibility of electrons along with the antissimetry of the wave function are missing in the Hartree's product, the method itself can be used to introduce these concepts. Despite the fact that these features are not included in the trial wavefunction, important qualitatively correct results can be obtained.
Sparse approximation with bases
2015-01-01
This book systematically presents recent fundamental results on greedy approximation with respect to bases. Motivated by numerous applications, the last decade has seen great successes in studying nonlinear sparse approximation. Recent findings have established that greedy-type algorithms are suitable methods of nonlinear approximation in both sparse approximation with respect to bases and sparse approximation with respect to redundant systems. These insights, combined with some previous fundamental results, form the basis for constructing the theory of greedy approximation. Taking into account the theoretical and practical demand for this kind of theory, the book systematically elaborates a theoretical framework for greedy approximation and its applications. The book addresses the needs of researchers working in numerical mathematics, harmonic analysis, and functional analysis. It quickly takes the reader from classical results to the latest frontier, but is written at the level of a graduate course and do...
Time-dependent Hartree-Fock studies of the dynamical fusion threshold
Nakatsukasa Takashi
2012-12-01
Full Text Available A microscopic description of dynamical fusion threshold in heavy ion collisions is performed in the framework of time-dependent Hartree-Fock (TDHF theory using Skyrme energy density functional (EDF. TDHF fusion threshold is in a better agreement with experimental fusion barrier. We find that the onset of extra push lies at the effective fissility 33, which is consistent with the prediction of Swiateckis macroscopic model. The extra push energy in our TDHF simulation is systematically smaller than the prediction in macroscopic model. The important dynamical effects and the way to fit the parameter might be responsible for the different results.
Wang, Haobin; Thoss, Michael
2016-10-01
The multilayer multiconfiguration time-dependent Hartree (ML-MCTDH) method is implemented in the interaction picture to allow a more effective description of correlation effects. It is shown that the artificial correlation present in the original Schrödinger picture can be removed with an appropriate choice of the zeroth-order Hamiltonian. Thereby, operators in the interaction picture are obtained through time-dependent unitary transformations, which have negligible computational cost compared with other parts of the ML-MCTDH algorithm. The efficiency of the method is demonstrated by application to a model of vibrationally coupled charge transport in molecular junctions.
Extreme Exotic Calcium Lambda Hypernuclei in the Relativistic Continuum Hartree-Bogoliubov Theory
L(U) Hong-Feng
2008-01-01
Exotic calcium lambda hypernuclei properties with the neutron number of 20-400 by a step of 20 are discussed by employing the relativistic continuum Hartree-Bogoliubov theory with a zero range pairing interaction. The Bethe-Weizs(a)cker mass formula of a multi-strange system and the Woods-Saxon-type potential of larnbda need to be modified for exotic calcium hypernuclei with unusual number of neutrons and lambdas. The possible neutron and lambda limits of exotic Ca larnbda hypernuclei are also investigated.
Evolution of $N = 28$ shell closure in relativistic continuum Hartree-Bogoliubov theory
Xia, Xuewei
2015-01-01
The $N = 28$ shell gap in sulfur, argon, calcium and titanium isotopes is investigated in the framework of relativistic continuum Hartree-Bogoliubov (RCHB) theory. The evolutions of neutron shell gap, separation energy, single particle energy and pairing energy are analyzed, and it is found that $N = 28$ shell gap is quenched in sulfur isotopes but persists in argon, calcium and titanium isotopes. The evolution of $N = 28$ shell gap in $N = 28$ isotonic chain is discussed, and the erosion of $N = 28$ shell gap is understood with the evolution of potential with proton number.
Projected gradient algorithms for Hartree-Fock and density matrix functional theory calculations
Cancès, Eric; Pernal, Katarzyna
2008-04-01
We present projected gradient algorithms designed for optimizing various functionals defined on the set of N-representable one-electron reduced density matrices. We show that projected gradient algorithms are efficient in minimizing the Hartree-Fock or the Müller-Buijse-Baerends functional. On the other hand, they converge very slowly when applied to the recently proposed BBk (k =1,2,3) functionals [O. Gritsenko et al., J. Chem. Phys. 122, 204102 (2005)]. This is due to the fact that the BBk functionals are not proper functionals of the density matrix.
Neese, Frank
2007-10-28
The zero-field splitting (ZFS) (expressed in terms of the D tensor) is the leading spin-Hamiltonian parameter for systems with a ground state spin S>12. To first order in perturbation theory, the ZFS arises from the direct spin-spin dipole-dipole interaction. To second order, contributions arise from spin-orbit coupling (SOC). The latter contributions are difficult to treat since the SOC mixes states of different multiplicities. This is an aspect of dominant importance for the correct prediction of the D tensor. In this work, the theory of the D tensor is discussed from the point of view of analytic derivative theory. Starting from a general earlier perturbation treatment [F. Neese and E. I. Soloman, Inorg. Chem. 37, 6568 (1998)], straightforward response equations are derived that are readily transferred to the self-consistent field (SCF) Hartree-Fock (HF) or density functional theory (DFT) framework. The main additional effort in such calculations arises from the solution of nine sets of nonstandard coupled-perturbed SCF equations. These equations have been implemented together with the spin-orbit mean-field representation of the SOC operator and a mean-field treatment of the direct spin-spin interaction into the ORCA electronic structure program. A series of test calculations on diatomic molecules with accurately known zero-field splittings shows that the new approach corrects most of the shortcomings of previous DFT based methods and, on average, leads to predictions within 10% of the experimental values. The slope of the correlation line is essentially unity for the B3LYP and BLYP functionals compared to approximately 0.5 in previous treatments.
Yoshida, Tatsusada; Hayashi, Takahisa; Mashima, Akira; Chuman, Hiroshi
2015-10-01
One of the most challenging problems in computer-aided drug discovery is the accurate prediction of the binding energy between a ligand and a protein. For accurate estimation of net binding energy ΔEbind in the framework of the Hartree-Fock (HF) theory, it is necessary to estimate two additional energy terms; the dispersion interaction energy (Edisp) and the basis set superposition error (BSSE). We previously reported a simple and efficient dispersion correction, Edisp, to the Hartree-Fock theory (HF-Dtq). In the present study, an approximation procedure for estimating BSSE proposed by Kruse and Grimme, a geometrical counterpoise correction (gCP), was incorporated into HF-Dtq (HF-Dtq-gCP). The relative weights of the Edisp (Dtq) and BSSE (gCP) terms were determined to reproduce ΔEbind calculated with CCSD(T)/CBS or /aug-cc-pVTZ (HF-Dtq-gCP (scaled)). The performance of HF-Dtq-gCP (scaled) was compared with that of B3LYP-D3(BJ)-bCP (dispersion corrected B3LYP with the Boys and Bernadi counterpoise correction (bCP)), by taking ΔEbind (CCSD(T)-bCP) of small non-covalent complexes as 'a golden standard'. As a critical test, HF-Dtq-gCP (scaled)/6-31G(d) and B3LYP-D3(BJ)-bCP/6-31G(d) were applied to the complex model for HIV-1 protease and its potent inhibitor, KNI-10033. The present results demonstrate that HF-Dtq-gCP (scaled) is a useful and powerful remedy for accurately and promptly predicting ΔEbind between a ligand and a protein, albeit it is a simple correction procedure.
Approximation techniques for engineers
Komzsik, Louis
2006-01-01
Presenting numerous examples, algorithms, and industrial applications, Approximation Techniques for Engineers is your complete guide to the major techniques used in modern engineering practice. Whether you need approximations for discrete data of continuous functions, or you''re looking for approximate solutions to engineering problems, everything you need is nestled between the covers of this book. Now you can benefit from Louis Komzsik''s years of industrial experience to gain a working knowledge of a vast array of approximation techniques through this complete and self-contained resource.
Achieser, N I
2004-01-01
A pioneer of many modern developments in approximation theory, N. I. Achieser designed this graduate-level text from the standpoint of functional analysis. The first two chapters address approximation problems in linear normalized spaces and the ideas of P. L. Tchebysheff. Chapter III examines the elements of harmonic analysis, and Chapter IV, integral transcendental functions of the exponential type. The final two chapters explore the best harmonic approximation of functions and Wiener's theorem on approximation. Professor Achieser concludes this exemplary text with an extensive section of pr
Hu, Jinniu; Shen, Hong
2016-01-01
We study the properties of nuclear matter with lattice nucleon-nucleon ($NN$) potential in the relativistic Brueckner-Hartree-Fock (RBHF) theory. To use this potential in such a microscopic many-body theory, we firstly have to construct a one-boson-exchange potential (OBEP) based on the latest lattice $NN$ potential. Three mesons, pion, $\\sigma$ meson, and $\\omega$ meson, are considered. Their coupling constants and cut-off momenta are determined by fitting the on-shell behaviors and phase shifts of the lattice force, respectively. Therefore, we obtain two parameter sets of the OBEP potential (named as LOBEP1 and LOBEP2) with these two fitting ways. We calculate the properties of symmetric and pure neutron matter with LOBEP1 and LOBEP2. In non-relativistic Brueckner-Hartree-Fock case, the binding energies of symmetric nuclear matter are around $-3$ and $-5$ MeV at saturation densities, while it becomes $-8$ and $-12$ MeV in relativistic framework with $^1S_0,~^3S_1,$ and $^3D_1$ channels using our two paramet...
A divide and conquer real space finite-element Hartree-Fock method
Alizadegan, R.; Hsia, K. J.; Martinez, T. J.
2010-01-01
Since the seminal contribution of Roothaan, quantum chemistry methods are traditionally expressed using finite basis sets comprised of smooth and continuous functions (atom-centered Gaussians) to describe the electronic degrees of freedom. Although this approach proved quite powerful, it is not well suited for large basis sets because of linear dependence problems and ill conditioning of the required matrices. The finite element method (FEM), on the other hand, is a powerful numerical method whose convergence is also guaranteed by variational principles and can be achieved systematically by increasing the number of degrees of freedom and/or the polynomial order of the shape functions. Here we apply the real-space FEM to Hartree-Fock calculations in three dimensions. The method produces sparse, banded Hermitian matrices while allowing for variable spatial resolution. This local-basis approach to electronic structure theory allows for systematic convergence and promises to provide an accurate and efficient way toward the full ab initio analysis of materials at larger scales. We introduce a new acceleration technique for evaluating the exchange contribution within FEM and explore the accuracy and robustness of the method for some selected test atoms and molecules. Furthermore, we applied a divide-and-conquer (DC) method to the finite-element Hartree-Fock ab initio electronic-structure calculations in three dimensions. This DC approach leads to facile parallelization and should enable reduced scaling for large systems.
Expectation Consistent Approximate Inference
Opper, Manfred; Winther, Ole
2005-01-01
We propose a novel framework for approximations to intractable probabilistic models which is based on a free energy formulation. The approximation can be understood from replacing an average over the original intractable distribution with a tractable one. It requires two tractable probability dis...
Ordered cones and approximation
Keimel, Klaus
1992-01-01
This book presents a unified approach to Korovkin-type approximation theorems. It includes classical material on the approximation of real-valuedfunctions as well as recent and new results on set-valued functions and stochastic processes, and on weighted approximation. The results are notonly of qualitative nature, but include quantitative bounds on the order of approximation. The book is addressed to researchers in functional analysis and approximation theory as well as to those that want to applythese methods in other fields. It is largely self- contained, but the readershould have a solid background in abstract functional analysis. The unified approach is based on a new notion of locally convex ordered cones that are not embeddable in vector spaces but allow Hahn-Banach type separation and extension theorems. This concept seems to be of independent interest.
Approximate Modified Policy Iteration
Scherrer, Bruno; Ghavamzadeh, Mohammad; Geist, Matthieu
2012-01-01
Modified policy iteration (MPI) is a dynamic programming (DP) algorithm that contains the two celebrated policy and value iteration methods. Despite its generality, MPI has not been thoroughly studied, especially its approximation form which is used when the state and/or action spaces are large or infinite. In this paper, we propose three approximate MPI (AMPI) algorithms that are extensions of the well-known approximate DP algorithms: fitted-value iteration, fitted-Q iteration, and classification-based policy iteration. We provide an error propagation analysis for AMPI that unifies those for approximate policy and value iteration. We also provide a finite-sample analysis for the classification-based implementation of AMPI (CBMPI), which is more general (and somehow contains) than the analysis of the other presented AMPI algorithms. An interesting observation is that the MPI's parameter allows us to control the balance of errors (in value function approximation and in estimating the greedy policy) in the fina...
Approximate calculation of integrals
Krylov, V I
2006-01-01
A systematic introduction to the principal ideas and results of the contemporary theory of approximate integration, this volume approaches its subject from the viewpoint of functional analysis. In addition, it offers a useful reference for practical computations. Its primary focus lies in the problem of approximate integration of functions of a single variable, rather than the more difficult problem of approximate integration of functions of more than one variable.The three-part treatment begins with concepts and theorems encountered in the theory of quadrature. The second part is devoted to t
Approximate and renormgroup symmetries
Ibragimov, Nail H. [Blekinge Institute of Technology, Karlskrona (Sweden). Dept. of Mathematics Science; Kovalev, Vladimir F. [Russian Academy of Sciences, Moscow (Russian Federation). Inst. of Mathematical Modeling
2009-07-01
''Approximate and Renormgroup Symmetries'' deals with approximate transformation groups, symmetries of integro-differential equations and renormgroup symmetries. It includes a concise and self-contained introduction to basic concepts and methods of Lie group analysis, and provides an easy-to-follow introduction to the theory of approximate transformation groups and symmetries of integro-differential equations. The book is designed for specialists in nonlinear physics - mathematicians and non-mathematicians - interested in methods of applied group analysis for investigating nonlinear problems in physical science and engineering. (orig.)
Approximating Stationary Statistical Properties
Xiaoming WANG
2009-01-01
It is well-known that physical laws for large chaotic dynamical systems are revealed statistically. Many times these statistical properties of the system must be approximated numerically. The main contribution of this manuscript is to provide simple and natural criterions on numerical methods (temporal and spatial discretization) that are able to capture the stationary statistical properties of the underlying dissipative chaotic dynamical systems asymptotically. The result on temporal approximation is a recent finding of the author, and the result on spatial approximation is a new one. Applications to the infinite Prandtl number model for convection and the barotropic quasi-geostrophic model are also discussed.
Coulomb and spin-orbit interactions in random phase approximation calculations
De Donno, V; Anguiano, M; Lallena, A M
2013-01-01
We present a fully self-consistent computational framework composed by Hartree-Fock plus ran- dom phase approximation where the spin-orbit and Coulomb terms of the interaction are included in both steps of the calculations. We study the effects of these terms of the interaction on the random phase approximation calculations, where they are usually neglected. We carry out our investigation of excited states in spherical nuclei of oxygen, calcium, nickel, zirconium, tin and lead isotope chains. We use finite-range effective nucleon-nucleon interactions of Gogny type. The size of the effects we find is, usually, of few hundreds of keV. There are not simple approximations which can be used to simulate these effects since they strongly depend on all the variables related to the excited states, angular momentum, parity, excitation energy, isoscalar and isovector characters. Even the Slater approximation developed to account for the Coulomb exchange terms in Hartree-Fock is not valid in random phase approximation ca...
Linear $\\Sigma$ Model in the Gaussian Functional Approximation
Nakamura, I
2001-01-01
We apply a self-consistent relativistic mean-field variational ``Gaussian functional'' (or Hartree) approximation to the linear $\\sigma$ model with spontaneously and explicitly broken chiral O(4) symmetry. We set up the self-consistency, or ``gap'' and the Bethe-Salpeter equations. We check and confirm the chiral Ward-Takahashi identities, among them the Nambu-Goldstone theorem and the (partial) axial current conservation [CAC], both in and away from the chiral limit. With explicit chiral symmetry breaking we confirm the Dashen relation for the pion mass and partial CAC. We solve numerically the gap and Bethe-Salpeter equations, discuss the solutions' properties and the particle content of the theory.
Malvina Baica
1985-01-01
Full Text Available The author uses a new modification of Jacobi-Perron Algorithm which holds for complex fields of any degree (abbr. ACF, and defines it as Generalized Euclidean Algorithm (abbr. GEA to approximate irrationals.
Approximations in Inspection Planning
Engelund, S.; Sørensen, John Dalsgaard; Faber, M. H.
2000-01-01
Planning of inspections of civil engineering structures may be performed within the framework of Bayesian decision analysis. The effort involved in a full Bayesian decision analysis is relatively large. Therefore, the actual inspection planning is usually performed using a number of approximations....... One of the more important of these approximations is the assumption that all inspections will reveal no defects. Using this approximation the optimal inspection plan may be determined on the basis of conditional probabilities, i.e. the probability of failure given no defects have been found...... by the inspection. In this paper the quality of this approximation is investigated. The inspection planning is formulated both as a full Bayesian decision problem and on the basis of the assumption that the inspection will reveal no defects....
The Karlqvist approximation revisited
Tannous, C
2015-01-01
The Karlqvist approximation signaling the historical beginning of magnetic recording head theory is reviewed and compared to various approaches progressing from Green, Fourier, Conformal mapping that obeys the Sommerfeld edge condition at angular points and leads to exact results.
Approximations in Inspection Planning
Engelund, S.; Sørensen, John Dalsgaard; Faber, M. H.
2000-01-01
Planning of inspections of civil engineering structures may be performed within the framework of Bayesian decision analysis. The effort involved in a full Bayesian decision analysis is relatively large. Therefore, the actual inspection planning is usually performed using a number of approximations....... One of the more important of these approximations is the assumption that all inspections will reveal no defects. Using this approximation the optimal inspection plan may be determined on the basis of conditional probabilities, i.e. the probability of failure given no defects have been found...... by the inspection. In this paper the quality of this approximation is investigated. The inspection planning is formulated both as a full Bayesian decision problem and on the basis of the assumption that the inspection will reveal no defects....
Gautschi, Walter; Rassias, Themistocles M
2011-01-01
Approximation theory and numerical analysis are central to the creation of accurate computer simulations and mathematical models. Research in these areas can influence the computational techniques used in a variety of mathematical and computational sciences. This collection of contributed chapters, dedicated to renowned mathematician Gradimir V. Milovanovia, represent the recent work of experts in the fields of approximation theory and numerical analysis. These invited contributions describe new trends in these important areas of research including theoretic developments, new computational alg
Approximation Behooves Calibration
da Silva Ribeiro, André Manuel; Poulsen, Rolf
2013-01-01
Calibration based on an expansion approximation for option prices in the Heston stochastic volatility model gives stable, accurate, and fast results for S&P500-index option data over the period 2005–2009.......Calibration based on an expansion approximation for option prices in the Heston stochastic volatility model gives stable, accurate, and fast results for S&P500-index option data over the period 2005–2009....
Symbolic computation of the Hartree-Fock energy from a chiral EFT three-nucleon interaction at N 2LO
Gebremariam, B.; Bogner, S. K.; Duguet, T.
2010-06-01
We present the first of a two-part Mathematica notebook collection that implements a symbolic approach for the application of the density matrix expansion (DME) to the Hartree-Fock (HF) energy from a chiral effective field theory (EFT) three-nucleon interaction at N 2LO. The final output from the notebooks is a Skyrme-like energy density functional that provides a quasi-local approximation to the non-local HF energy. In this paper, we discuss the derivation of the HF energy and its simplification in terms of the scalar/vector-isoscalar/isovector parts of the one-body density matrix. Furthermore, a set of steps is described and illustrated on how to extend the approach to other three-nucleon interactions. Program summaryProgram title: SymbHFNNN Catalogue identifier: AEGC_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEGC_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 96 666 No. of bytes in distributed program, including test data, etc.: 378 083 Distribution format: tar.gz Programming language: Mathematica 7.1 Computer: Any computer running Mathematica 6.0 and later versions Operating system: Windows Xp, Linux/Unix RAM: 256 Mb Classification: 5, 17.16, 17.22 Nature of problem: The calculation of the HF energy from the chiral EFT three-nucleon interaction at N 2LO involves tremendous spin-isospin algebra. The problem is compounded by the need to eventually obtain a quasi-local approximation to the HF energy, which requires the HF energy to be expressed in terms of scalar/vector-isoscalar/isovector parts of the one-body density matrix. The Mathematica notebooks discussed in this paper solve the latter issue. Solution method: The HF energy from the chiral EFT three-nucleon interaction at N 2LO is cast into a form suitable for an automatic simplification of
Approximate kernel competitive learning.
Wu, Jian-Sheng; Zheng, Wei-Shi; Lai, Jian-Huang
2015-03-01
Kernel competitive learning has been successfully used to achieve robust clustering. However, kernel competitive learning (KCL) is not scalable for large scale data processing, because (1) it has to calculate and store the full kernel matrix that is too large to be calculated and kept in the memory and (2) it cannot be computed in parallel. In this paper we develop a framework of approximate kernel competitive learning for processing large scale dataset. The proposed framework consists of two parts. First, it derives an approximate kernel competitive learning (AKCL), which learns kernel competitive learning in a subspace via sampling. We provide solid theoretical analysis on why the proposed approximation modelling would work for kernel competitive learning, and furthermore, we show that the computational complexity of AKCL is largely reduced. Second, we propose a pseudo-parallelled approximate kernel competitive learning (PAKCL) based on a set-based kernel competitive learning strategy, which overcomes the obstacle of using parallel programming in kernel competitive learning and significantly accelerates the approximate kernel competitive learning for large scale clustering. The empirical evaluation on publicly available datasets shows that the proposed AKCL and PAKCL can perform comparably as KCL, with a large reduction on computational cost. Also, the proposed methods achieve more effective clustering performance in terms of clustering precision against related approximate clustering approaches.
Maksim Duškin
2015-11-01
Full Text Available Approximation and supposition This article compares exponents of approximation (expressions like Russian около, примерно, приблизительно, более, свыше and the words expressing supposition (for example Russian скорее всего, наверное, возможно. These words are often confused in research, in particular researchers often mention exponents of supposition in case of exponents of approximation. Such approach arouses some objections. The author intends to demonstrate in this article a notional difference between approximation and supposition, therefore the difference between exponents of these two notions. This difference could be described by specifying different attitude of approximation and supposition to the notion of knowledge. Supposition implies speaker’s ignorance of the exact number, while approximation does not mean such ignorance. The article offers examples proving this point of view.
Boblest, S.; Meyer, D.; Wunner, G.
2014-11-01
-Fock calculations. The guiding functions are created from single-electron orbitals ψi which are either products of a wave function in the z-direction (the direction of the magnetic field) and an expansion of the wave function perpendicular to the direction of the magnetic field in terms of Landau states, ψi(ρ,φ,z)=Pi(z)∑n=0NLtinϕni(ρ,φ), or a full two-dimensional expansion using separate z-wave functions for each Landau level, i.e. ψi(ρ,φ,z)=∑n=0NLPni(z)ϕni(ρ,φ). In the first form, the tin are expansion coefficients, and the expansion is cut off at some maximum Landau level quantum number NL. In the second form, the expansion coefficients are contained in the respective Pni. Restrictions: The method itself is very flexible and not restricted to a certain interval of magnetic field strengths. However, it is only variational for the lowest-lying state in each symmetry subspace and the accompanying Hartree-Fock method can only obtain guiding functions in the regime of strong magnetic fields. Unusual features: The program needs approximate wave functions computed with another method as input. Running time: 1 min-several days. The example provided takes approximately 50 min to run on 1 processor.
Barbosa, Rugles Cesar
2002-07-01
The present thesis is divided into two parts. The first part describes the many kind of the formalisms of the Generator Coordinate Hartree-Fock method (GCHFM) and second part describes the computational aspect applied to the GCHFM formalism in its discreet form. The major aim of this work is the development of an alternative method to non-linear parameters optimization (basis set) and later uses these optimized parameters to adjust the weight function into GCHFM method. The study of the weight function when N {yields} {infinity} (or for large N), where N represents the number of mesh, is important since the GCHFM theory in its continuous form depend on understanding of such behavior. In this thesis, a detailed study is carried out about the methodologies of the self-consistent solution of the GCHFM and some methodology aspects of non-linear parameters optimization. This work shows that the Generator Coordinate Hartree-Fock method is general and it has as particular case the Hartree-Fock Roothaan method. Some possible variations or combinations around of the characteristics of the GCHFM and a comparison with conventional SCF procedure are reported in this thesis. The piecewise weight function method developed in this work shows to be very good for collective parameter optimizations of the Generator Coordinate (GC). The GCHFM calculations are necessary restrict (GCM-RHF), especially when the calculated value energies approach of its numerical values or Hartree-Fock limit. In the optimization methods of state functions for atomic electronic systems is very common the application of the gradient method and its efficacy is not contested. However, the method describes above allow us to obtain results as good as the gradient method. The basis set generated using the piecewise weight function method for Gaussian type function were used in the Restrict Hartree-Fock (RHF) calculations to obtain the total energies for some atomic electronic systems, such as neutron atoms and
Amour, Laurent; Nourrigat, Jean
2011-01-01
We study the Wick symbol of a solution of the time dependent Hartree Fock equation, under weaker hypotheses than those needed for the Weyl symbol in the first paper with thesame title. With similar, we prove some kind of Ehrenfest theorem for observables that are not pseudo-differential operators.
Jiménez-Hoyos, Carlos A; Scuseria, Gustavo E
2013-01-01
Recent work from our research group has demonstrated that symmetry-projected Hartree--Fock (HF) methods provide a compact representation of molecular ground state wavefunctions based on a superposition of non-orthogonal Slater determinants. The symmetry-projected ansatz can account for static correlations in a computationally efficient way. Here we present a variational extension of this methodology applicable to excited states of the same symmetry as the ground state. Benchmark calculations on the C$_2$ dimer with a modest basis set, which allows comparison with full configuration interaction results, indicate that this extension provides a high quality description of the low-lying spectrum for the entire dissociation profile. We apply the same methodology to obtain the full low-lying vertical excitation spectrum of formaldehyde, in good agreement with available theoretical and experimental data, as well as to a challenging model $C_{2v}$ insertion pathway for BeH$_2$. The variational excited state methodolo...
Constrained Hartree-Fock Theory and Study of Deformed Structures of Closed Shell Nuclei
Praharaj, Choudhury
2016-03-01
We have studied some N or Z = 50 nuclei in a microscopic model with effective interaction in a reasonably large shell model space. Excitation of particles across 50 shell closure leads to well-deformed excited prolate configurations. The potential energy surfaces of nuclei are studied using Hartree-Fock theory with quadrupole constraint to explore the various deformed configurations of N = 50 nuclei 82Ge , 84Se and 86Kr . Energy spectra are calculated from various intrinsic states using Peierls-Yoccoz angular momentum projection technique. Results of spectra and electromagnetic moments and transitions will be presented for N = 50 nuclei and for Z = 50 114Sn nucleus. Supported by Grant No SB/S2/HEP-06/2013 of DST.
A New Decomposition Approach of Dirac Brueckner Hartree-Fock G Matrix for Asymmetric Nuclear Matter
刘玲; 马中玉
2002-01-01
Asymmetric nuclear matter is investigated by the Dirac Brueckner Hartree-Fock (DBHF) approach with a new decomposition of the Dirac structure of nucleon self-energy from the G matrix. It is found that the isospin dependence of the scalar and vector potentials is relatively weak, although both potentials for neutron (proton)become deep (shallow) in the neutron-rich nuclear matter. The results in asymmetric nuclear matter are rather different from those obtained by a simple method, where the nucleon self-energy is deduced from the single-particle energy. The nuclear binding energy as a function of the asymmetry parameter fulfils the empirical parabolic law up to very extreme isospin asymmetric nuclear matter in the DBHF approach. The behaviour of the density dependence of the asymmetry energy is different from that obtained by non-relativistic approaches, although both give similar asymmetry energy at the nuclear saturation density.
The neutron halo structure of 17B studied with the relativistic Hartree-Bogoliubov theory
JI Juan-Xia; LI Jia-Xing; HAN Rui; WANG Jian-Song; HU Qiang
2012-01-01
The properties of neutron-rich boron isotopes are studied in the relativistic continuum HartreeBogoliubov theory in coordinate space with NL-SH,PK1 and TM2 effective interactions.Pairing corrections are taken into account by a density dependent force of zero range.The binding energies calculated for these nuclei agree with the experimental data quite well.The neutron-rich nucleus 17B has been predicted to have a two-neutron halo structure in its ground state.The halo structure of 17B is reproduced in a self-consistent way,and this halo is shown to be formed by the valence neutron level 2s1/2.
A relativistic continuum Hartree-Bogoliubov theory description of N=3 isotones
HAN Rui; JI Juan-Xia; LI Jia-Xing
2011-01-01
The ground-state properties of N=3 isotones and mirror nuclei have been investigated in the Rrelativistic Continuum Hartree-Bogoliubov theory with the NLSH effective interaction.Pairing correlations are taken into account by a density-dependent δ-force.The calculations show that the proton density distributions of 8B and 9C have a long tail,the core has an increasing tendency of 9C and the paired off valence protons make the halo distribution shrink.The cross sections for the 8B(9C)+12C reaction which are consistent with the experimental data are calculated using the Glauber model.On the whole,we think that 8B is a one-proton halo nucleus and 9C is a two-proton halo nucleus.
Pairing phase transition: A Finite-Temperature Relativistic Hartree-Fock-Bogoliubov study
Li, Jia Jie; Long, Wen Hui; Van Giai, Nguyen
2015-01-01
Background: The relativistic Hartree-Fock-Bogoliubov (RHFB) theory has recently been developed and it provides a unified and highly predictive description of both nuclear mean field and pairing correlations. Ground state properties of finite nuclei can accurately be reproduced without neglecting exchange (Fock) contributions. Purpose: Finite-temperature RHFB (FT-RHFB) theory has not yet been developed, leaving yet unknown its predictions for phase transitions and thermal excitations in both stable and weakly bound nuclei. Method: FT-RHFB equations are solved in a Dirac Woods-Saxon (DWS) basis considering two kinds of pairing interactions: finite or zero range. Such a model is appropriate for describing stable as well as loosely bound nuclei since the basis states have correct asymptotic behaviour for large spatial distributions. Results: Systematic FT-RH(F)B calculations are performed for several semi-magic isotopic/isotonic chains comparing the predictions of a large number of Lagrangians, among which are PK...
Hartree-Fock mean-field theory for trapped dirty bosons
Khellil, Tama; Pelster, Axel
2016-06-01
Here we work out in detail a non-perturbative approach to the dirty boson problem, which relies on the Hartree-Fock theory and the replica method. For a weakly interacting Bose gas within a trapped confinement and a delta-correlated disorder potential at finite temperature, we determine the underlying free energy. From it we determine via extremization self-consistency equations for the three components of the particle density, namely the condensate density, the thermal density, and the density of fragmented local Bose-Einstein condensates within the respective minima of the random potential landscape. Solving these self-consistency equations in one and three dimensions in two other publications has revealed how these three densities change for increasing disorder strength.
Deformed Relativistic Hartree Theory in Coordinate Space and in Harmonic Oscillator Basis
ZHOU Shan-Gui; MENG Jie; Shuhei YAMAJI; YANG Si-Chun
2000-01-01
The deformed relativistic Hartree theory (DRH) is solved both in coordinate space (DRH-c) and in harmonic oscillator basis (DRH-o). Results obtained from these two methods are compared in details. The DRH-c and DRH-o calculations give similar total binding energies, deformation, level structures and radii for nitrogen iso topes, while their descriptions on the density distributions for drip-line nuclei are very different. The large spatiai istributions of nucleon densities, which is crucial to understand a weakly bound system, can only be obtained by DRH-c calculations. This implies that the DRH theory should be solved in coordinate space in order to describe uclei close to the drip line.
Phase structure of the massive chiral Gross-Neveu model from Hartree-Fock
Boehmer, Christian; Kraus, Sebastian; Thies, Michael
2008-01-01
The phase diagram of the massive chiral Gross-Neveu model (the massive Nambu-Jona-Lasinio model in 1+1 dimensions) is constructed. In the large N limit, the Hartree-Fock approach can be used. We find numerically a chiral crystal phase separated from a massive Fermi gas phase by a 1st order transition. Using perturbation theory, we also construct the critical sheet where the homogeneous phase becomes unstable in a 2nd order transition. A tricritical curve is located. The phase diagram is mapped out as a function of fermion mass, chemical potential and temperature and compared with the one of the discrete chiral Gross-Neveu model. As a by-product, we illustrate the crystal structure of matter at zero temperature for various densities and fermion masses.
Ab-initio Hartree-Fock study of tritium desorption from Li{sub 2}O
Taniguchi, Masaki; Tanaka, Satoru [Tokyo Univ. (Japan). Faculty of Engineering
1998-03-01
Dissociative adsorption of hydrogen on Li{sub 2}O (110) surface has been investigated with ab-initio Hartree-Fock quantum chemical calculation technique. Heat of adsorption and potential energy surface for H{sub 2} dissociative adsorption was evaluated by calculating the total energy of the system. Calculation results on adsorption heat indicated that H{sub 2} adsorption is endothermic. However, when oxygen vacancy exists adjacent to the adsorption sites, heat of adsorption energy became less endothermic and the activation energy required to dissociate the H-H bonding was smaller than that for the terrace site. This is considered to be caused by the excess charge localized near the defect. (author)
Covariant approximation averaging
Shintani, Eigo; Blum, Thomas; Izubuchi, Taku; Jung, Chulwoo; Lehner, Christoph
2014-01-01
We present a new class of statistical error reduction techniques for Monte-Carlo simulations. Using covariant symmetries, we show that correlation functions can be constructed from inexpensive approximations without introducing any systematic bias in the final result. We introduce a new class of covariant approximation averaging techniques, known as all-mode averaging (AMA), in which the approximation takes account of contributions of all eigenmodes through the inverse of the Dirac operator computed from the conjugate gradient method with a relaxed stopping condition. In this paper we compare the performance and computational cost of our new method with traditional methods using correlation functions and masses of the pion, nucleon, and vector meson in $N_f=2+1$ lattice QCD using domain-wall fermions. This comparison indicates that AMA significantly reduces statistical errors in Monte-Carlo calculations over conventional methods for the same cost.
Diophantine approximations on fractals
Einsiedler, Manfred; Shapira, Uri
2009-01-01
We exploit dynamical properties of diagonal actions to derive results in Diophantine approximations. In particular, we prove that the continued fraction expansion of almost any point on the middle third Cantor set (with respect to the natural measure) contains all finite patterns (hence is well approximable). Similarly, we show that for a variety of fractals in [0,1]^2, possessing some symmetry, almost any point is not Dirichlet improvable (hence is well approximable) and has property C (after Cassels). We then settle by similar methods a conjecture of M. Boshernitzan saying that there are no irrational numbers x in the unit interval such that the continued fraction expansions of {nx mod1 : n is a natural number} are uniformly eventually bounded.
Monotone Boolean approximation
Hulme, B.L.
1982-12-01
This report presents a theory of approximation of arbitrary Boolean functions by simpler, monotone functions. Monotone increasing functions can be expressed without the use of complements. Nonconstant monotone increasing functions are important in their own right since they model a special class of systems known as coherent systems. It is shown here that when Boolean expressions for noncoherent systems become too large to treat exactly, then monotone approximations are easily defined. The algorithms proposed here not only provide simpler formulas but also produce best possible upper and lower monotone bounds for any Boolean function. This theory has practical application for the analysis of noncoherent fault trees and event tree sequences.
Particle-number projection in the finite-temperature mean-field approximation
Fanto, P; Bertsch, G F
2016-01-01
Calculation of statistical properties of nuclei in a finite-temperature mean-field theory requires projection onto good particle number, since the theory is formulated in the grand canonical ensemble. This projection is usually carried out in a saddle-point approximation. Here we derive formulas for an exact particle-number projection of the finite-temperature mean-field solution. We consider both deformed nuclei, in which the pairing condensate is weak and the Hartree-Fock (HF) approximation is the appropriate mean-field theory, and nuclei with strong pairing condensates, in which the appropriate theory is the Hartree-Fock-Bogoliubov (HFB) approximation, a method that explicitly violates particle-number conservation. For the HFB approximation, we present a general projection formula for a condensate that is time-reversal invariant and a simpler formula for the Bardeen-Cooper-Schrieffer (BCS) limit, which is realized in nuclei with spherical condensates. We apply the method to three heavy nuclei: a typical de...
Prestack wavefield approximations
Alkhalifah, Tariq
2013-09-01
The double-square-root (DSR) relation offers a platform to perform prestack imaging using an extended single wavefield that honors the geometrical configuration between sources, receivers, and the image point, or in other words, prestack wavefields. Extrapolating such wavefields, nevertheless, suffers from limitations. Chief among them is the singularity associated with horizontally propagating waves. I have devised highly accurate approximations free of such singularities which are highly accurate. Specifically, I use Padé expansions with denominators given by a power series that is an order lower than that of the numerator, and thus, introduce a free variable to balance the series order and normalize the singularity. For the higher-order Padé approximation, the errors are negligible. Additional simplifications, like recasting the DSR formula as a function of scattering angle, allow for a singularity free form that is useful for constant-angle-gather imaging. A dynamic form of this DSR formula can be supported by kinematic evaluations of the scattering angle to provide efficient prestack wavefield construction. Applying a similar approximation to the dip angle yields an efficient 1D wave equation with the scattering and dip angles extracted from, for example, DSR ray tracing. Application to the complex Marmousi data set demonstrates that these approximations, although they may provide less than optimal results, allow for efficient and flexible implementations. © 2013 Society of Exploration Geophysicists.
On Convex Quadratic Approximation
den Hertog, D.; de Klerk, E.; Roos, J.
2000-01-01
In this paper we prove the counterintuitive result that the quadratic least squares approximation of a multivariate convex function in a finite set of points is not necessarily convex, even though it is convex for a univariate convex function. This result has many consequences both for the field of
Norton, Andrew H.
1991-01-01
Local spline approximants offer a means for constructing finite difference formulae for numerical solution of PDEs. These formulae seem particularly well suited to situations in which the use of conventional formulae leads to non-linear computational instability of the time integration. This is explained in terms of frequency responses of the FDF.
On Convex Quadratic Approximation
den Hertog, D.; de Klerk, E.; Roos, J.
2000-01-01
In this paper we prove the counterintuitive result that the quadratic least squares approximation of a multivariate convex function in a finite set of points is not necessarily convex, even though it is convex for a univariate convex function. This result has many consequences both for the field of
Approximation by Cylinder Surfaces
Randrup, Thomas
1997-01-01
We present a new method for approximation of a given surface by a cylinder surface. It is a constructive geometric method, leading to a monorail representation of the cylinder surface. By use of a weighted Gaussian image of the given surface, we determine a projection plane. In the orthogonal...
Lacivita, Valentina; Rérat, Michel; Kirtman, Bernard; Ferrero, Mauro; Orlando, Roberto; Dovesi, Roberto
2009-11-01
The high-frequency dielectric ɛ and the first nonlinear electric susceptibility χ(2) tensors of crystalline potassium dihydrogen phosphate (KH2PO4) are calculated by using the coupled perturbed Hartree-Fock and Kohn-Sham methods as implemented in the CRYSTAL code. The effect of basis sets of increasing size on ɛ and χ(2) is explored. Five different levels of theory, namely, local-density approximation, generalized gradient approximation (PBE), hybrids (B3LYP and PBE0), and HF are compared using the experimental and theoretical structures corresponding not only to the tetragonal geometry I4d2 at room temperature but also to the orthorhombic phase Fdd2 at low temperature. Comparison between the two phases and their optical behavior is made. The calculated results for the tetragonal phase are in good agreement with the experimental data.
Electron correlation within the relativistic no-pair approximation
Almoukhalalati, Adel; Knecht, Stefan; Jensen, Hans Jørgen Aa.; Dyall, Kenneth G.; Saue, Trond
2016-08-01
This paper addresses the definition of correlation energy within 4-component relativistic atomic and molecular calculations. In the nonrelativistic domain the correlation energy is defined as the difference between the exact eigenvalue of the electronic Hamiltonian and the Hartree-Fock energy. In practice, what is reported is the basis set correlation energy, where the "exact" value is provided by a full Configuration Interaction (CI) calculation with some specified one-particle basis. The extension of this definition to the relativistic domain is not straightforward since the corresponding electronic Hamiltonian, the Dirac-Coulomb Hamiltonian, has no bound solutions. Present-day relativistic calculations are carried out within the no-pair approximation, where the Dirac-Coulomb Hamiltonian is embedded by projectors eliminating the troublesome negative-energy solutions. Hartree-Fock calculations are carried out with the implicit use of such projectors and only positive-energy orbitals are retained at the correlated level, meaning that the Hartree-Fock projectors are frozen at the correlated level. We argue that the projection operators should be optimized also at the correlated level and that this is possible by full Multiconfigurational Self-Consistent Field (MCSCF) calculations, that is, MCSCF calculations using a no-pair full CI expansion, but including orbital relaxation from the negative-energy orbitals. We show by variational perturbation theory that the MCSCF correlation energy is a pure MP2-like correlation expression, whereas the corresponding CI correlation energy contains an additional relaxation term. We explore numerically our theoretical analysis by carrying out variational and perturbative calculations on the two-electron rare gas atoms with specially tailored basis sets. In particular, we show that the correlation energy obtained by the suggested MCSCF procedure is smaller than the no-pair full CI correlation energy, in accordance with the underlying
Matsuo, Masayuki
2014-01-01
We formulate a many-body theory to calculate the cross section of direct radiative neutron capture reaction by means of the Hartree-Fock-Bogoliubov mean-field model and the continuum quasiparticle random phase approximation (QRPA). A focus is put on very neutron-rich nuclei and low-energy neutron kinetic energy in the range of O(1 keV) - O(1 MeV), relevant for the rapid neutron-capture process of nucleosynthesis. We begin with the photo-absorption cross section and the E1 strength function, t...
Guseinov I. Israfil; Erturk Murat
2008-01-01
Using complete orthonormal sets of Ψα -exponential type orbitals in single exponent approximation the new approach has been suggested for construction of different kinds of functions which can be useful in the theory of linear combination of atomic orbitals. These functions can be chosen properly according to the nature of the problems under consideration. This is rather important because the choice of the basis set may be play a crucial role in applications to atomic and molecular problems. As an example of application, different atomic orbitals for the ground states of the neutral and the first ten cationic members of the isoelectronic series of He atom are constructed by the solution of Hartree-Fock Roothaan equations using Ψ1, Ψ0 and Ψ-1 basis sets. The calculated results are close to the numerical Hartree-Fock values. The total energy, expansion coefficients, orbital exponents and virial ratio for each atom are presented.
Topology, calculus and approximation
Komornik, Vilmos
2017-01-01
Presenting basic results of topology, calculus of several variables, and approximation theory which are rarely treated in a single volume, this textbook includes several beautiful, but almost forgotten, classical theorems of Descartes, Erdős, Fejér, Stieltjes, and Turán. The exposition style of Topology, Calculus and Approximation follows the Hungarian mathematical tradition of Paul Erdős and others. In the first part, the classical results of Alexandroff, Cantor, Hausdorff, Helly, Peano, Radon, Tietze and Urysohn illustrate the theories of metric, topological and normed spaces. Following this, the general framework of normed spaces and Carathéodory's definition of the derivative are shown to simplify the statement and proof of various theorems in calculus and ordinary differential equations. The third and final part is devoted to interpolation, orthogonal polynomials, numerical integration, asymptotic expansions and the numerical solution of algebraic and differential equations. Students of both pure an...
Prestack traveltime approximations
Alkhalifah, Tariq Ali
2011-01-01
Most prestack traveltime relations we tend work with are based on homogeneous (or semi-homogenous, possibly effective) media approximations. This includes the multi-focusing or double square-root (DSR) and the common reflection stack (CRS) equations. Using the DSR equation, I analyze the associated eikonal form in the general source-receiver domain. Like its wave-equation counterpart, it suffers from a critical singularity for horizontally traveling waves. As a result, I derive expansion based solutions of this eikonal based on polynomial expansions in terms of the reflection and dip angles in a generally inhomogenous background medium. These approximate solutions are free of singularities and can be used to estimate travetimes for small to moderate offsets (or reflection angles) in a generally inhomogeneous medium. A Marmousi example demonstrates the usefulness of the approach. © 2011 Society of Exploration Geophysicists.
Optimization and approximation
Pedregal, Pablo
2017-01-01
This book provides a basic, initial resource, introducing science and engineering students to the field of optimization. It covers three main areas: mathematical programming, calculus of variations and optimal control, highlighting the ideas and concepts and offering insights into the importance of optimality conditions in each area. It also systematically presents affordable approximation methods. Exercises at various levels have been included to support the learning process.
Topics in Metric Approximation
Leeb, William Edward
This thesis develops effective approximations of certain metrics that occur frequently in pure and applied mathematics. We show that distances that often arise in applications, such as the Earth Mover's Distance between two probability measures, can be approximated by easily computed formulas for a wide variety of ground distances. We develop simple and easily computed characterizations both of norms measuring a function's regularity -- such as the Lipschitz norm -- and of their duals. We are particularly concerned with the tensor product of metric spaces, where the natural notion of regularity is not the Lipschitz condition but the mixed Lipschitz condition. A theme that runs throughout this thesis is that snowflake metrics (metrics raised to a power less than 1) are often better-behaved than ordinary metrics. For example, we show that snowflake metrics on finite spaces can be approximated by the average of tree metrics with a distortion bounded by intrinsic geometric characteristics of the space and not the number of points. Many of the metrics for which we characterize the Lipschitz space and its dual are snowflake metrics. We also present applications of the characterization of certain regularity norms to the problem of recovering a matrix that has been corrupted by noise. We are able to achieve an optimal rate of recovery for certain families of matrices by exploiting the relationship between mixed-variable regularity conditions and the decay of a function's coefficients in a certain orthonormal basis.
Schunck, N; McDonnell, J; Satula, W; Sheikh, J A; Staszczak, A; Stoitsov, M; Toivanen, P
2011-01-01
We describe the new version (v2.49s) of the code HFODD which solves the nuclear Skyrme Hartree-Fock (HF) or Skyrme Hartree-Fock-Bogolyubov (HFB) problem by using the Cartesian deformed harmonic-oscillator basis. In the new version, we have implemented the following physics features: (i) the isospin mixing and projection, (ii) the finite temperature formalism for the HFB and HF+BCS methods, (iii) the Lipkin translational energy correction method, (iv) the calculation of the shell correction. A number of specific numerical methods have also been implemented in order to deal with large-scale multi-constraint calculations and hardware limitations: (i) the two-basis method for the HFB method, (ii) the Augmented Lagrangian Method (ALM) for multi-constraint calculations, (iii) the linear constraint method based on the approximation of the RPA matrix for multi-constraint calculations, (iv) an interface with the axial and parity-conserving Skyrme-HFB code HFBTHO, (v) the mixing of the HF or HFB matrix elements instead...
Hamel, Sébastien; Casida, Mark E.; Salahub, Dennis R.
2001-05-01
The Roothaan-Hartree-Fock (HF) method has been implemented in deMon-DynaRho within the resolution-of-the-identity (RI) auxiliary-function approximation. While previous studies have focused primarily upon the effect of the RI approximation on total energies, very little information has been available regarding the effect of the RI approximation on orbital energies, even though orbital energies play a central role in many theories of ionization and excitation. We fill this gap by testing the accuracy of the RI approximation against non-RI-HF calculations using the same basis sets, for the occupied orbital energies and an equal number of unoccupied orbital energies of five small molecules, namely CO, N2, CH2O, C2H4, and pyridine (in total 102 orbitals). These molecules have well-characterized excited states and so are commonly used to test and validate molecular excitation spectra computations. Of the deMon auxiliary basis sets tested, the best results are obtained with the (44) auxiliary basis sets, yielding orbital energies to within 0.05 eV, which is adequate for analyzing typical low resolution polyatomic molecule ionization and excitation spectra. Interestingly, we find that the error in orbital energies due to the RI approximation does not seem to increase with the number of electrons. The absolute RI error in the orbital energies is also roughly related to their absolute magnitude, being larger for the core orbitals where the magnitude of orbital energy is large and smallest where the molecular orbital energy is smallest. Two further approximations were also considered, namely uniterated ("zero-order") and single-iteration ("first-order") calculations of orbital energies beginning with a local density approximation initial guess. We find that zero- and first-order orbital energies are very similar for occupied but not for unoccupied orbitals, and that the first-order orbital energies are fairly close to the corresponding fully converged values. Typical root
Adaptive multi-resolution 3D Hartree-Fock-Bogoliubov solver for nuclear structure
Pei, J. C.; Fann, G. I.; Harrison, R. J.; Nazarewicz, W.; Shi, Yue; Thornton, S.
2014-08-01
Background: Complex many-body systems, such as triaxial and reflection-asymmetric nuclei, weakly bound halo states, cluster configurations, nuclear fragments produced in heavy-ion fusion reactions, cold Fermi gases, and pasta phases in neutron star crust, are all characterized by large sizes and complex topologies in which many geometrical symmetries characteristic of ground-state configurations are broken. A tool of choice to study such complex forms of matter is an adaptive multi-resolution wavelet analysis. This method has generated much excitement since it provides a common framework linking many diversified methodologies across different fields, including signal processing, data compression, harmonic analysis and operator theory, fractals, and quantum field theory. Purpose: To describe complex superfluid many-fermion systems, we introduce an adaptive pseudospectral method for solving self-consistent equations of nuclear density functional theory in three dimensions, without symmetry restrictions. Methods: The numerical method is based on the multi-resolution and computational harmonic analysis techniques with a multi-wavelet basis. The application of state-of-the-art parallel programming techniques include sophisticated object-oriented templates which parse the high-level code into distributed parallel tasks with a multi-thread task queue scheduler for each multi-core node. The internode communications are asynchronous. The algorithm is variational and is capable of solving coupled complex-geometric systems of equations adaptively, with functional and boundary constraints, in a finite spatial domain of very large size, limited by existing parallel computer memory. For smooth functions, user-defined finite precision is guaranteed. Results: The new adaptive multi-resolution Hartree-Fock-Bogoliubov (HFB) solver madness-hfb is benchmarked against a two-dimensional coordinate-space solver hfb-ax that is based on the B-spline technique and a three-dimensional solver
Chalasani, P.; Saias, I. [Los Alamos National Lab., NM (United States); Jha, S. [Carnegie Mellon Univ., Pittsburgh, PA (United States)
1996-04-08
As increasingly large volumes of sophisticated options (called derivative securities) are traded in world financial markets, determining a fair price for these options has become an important and difficult computational problem. Many valuation codes use the binomial pricing model, in which the stock price is driven by a random walk. In this model, the value of an n-period option on a stock is the expected time-discounted value of the future cash flow on an n-period stock price path. Path-dependent options are particularly difficult to value since the future cash flow depends on the entire stock price path rather than on just the final stock price. Currently such options are approximately priced by Monte carlo methods with error bounds that hold only with high probability and which are reduced by increasing the number of simulation runs. In this paper the authors show that pricing an arbitrary path-dependent option is {number_sign}-P hard. They show that certain types f path-dependent options can be valued exactly in polynomial time. Asian options are path-dependent options that are particularly hard to price, and for these they design deterministic polynomial-time approximate algorithms. They show that the value of a perpetual American put option (which can be computed in constant time) is in many cases a good approximation to the value of an otherwise identical n-period American put option. In contrast to Monte Carlo methods, the algorithms have guaranteed error bounds that are polynormally small (and in some cases exponentially small) in the maturity n. For the error analysis they derive large-deviation results for random walks that may be of independent interest.
Finite elements and approximation
Zienkiewicz, O C
2006-01-01
A powerful tool for the approximate solution of differential equations, the finite element is extensively used in industry and research. This book offers students of engineering and physics a comprehensive view of the principles involved, with numerous illustrative examples and exercises.Starting with continuum boundary value problems and the need for numerical discretization, the text examines finite difference methods, weighted residual methods in the context of continuous trial functions, and piecewise defined trial functions and the finite element method. Additional topics include higher o
Approximate Bayesian computation.
Mikael Sunnåker
Full Text Available Approximate Bayesian computation (ABC constitutes a class of computational methods rooted in Bayesian statistics. In all model-based statistical inference, the likelihood function is of central importance, since it expresses the probability of the observed data under a particular statistical model, and thus quantifies the support data lend to particular values of parameters and to choices among different models. For simple models, an analytical formula for the likelihood function can typically be derived. However, for more complex models, an analytical formula might be elusive or the likelihood function might be computationally very costly to evaluate. ABC methods bypass the evaluation of the likelihood function. In this way, ABC methods widen the realm of models for which statistical inference can be considered. ABC methods are mathematically well-founded, but they inevitably make assumptions and approximations whose impact needs to be carefully assessed. Furthermore, the wider application domain of ABC exacerbates the challenges of parameter estimation and model selection. ABC has rapidly gained popularity over the last years and in particular for the analysis of complex problems arising in biological sciences (e.g., in population genetics, ecology, epidemiology, and systems biology.
Gharabaghi, Masumeh
2016-01-01
In this letter the conceptual and computational implications of the Hartree product type nuclear wavefunction introduced recently within context of the ab initio non-Born-Oppenheimer Nuclear-electronic orbital (NEO) methodology are considered. It is demonstrated that this wavefunction may imply a pseudo-adiabatic separation of the nuclei and electrons and each nucleus is conceived as a quantum oscillator while a non-Coulombic effective Hamiltonian is deduced for electrons. Using variational principle this Hamiltonian is used to derive a modified set of single-component Hartree-Fock equations which are equivalent to the multi-component version derived previously within context of the NEO and, easy to be implemented computationally.
Gharabaghi, Masumeh; Shahbazian, Shant
2016-12-01
In this letter the conceptual and computational implications of the Hartree product type nuclear wavefunction introduced recently within the context of the ab initio non-Born-Oppenheimer Nuclear-electronic orbital (NEO) methodology are considered. It is demonstrated that this wavefunction may imply a pseudo-adiabatic separation of the nuclei and electrons and each nucleus is conceived as a quantum oscillator while a non-Coulombic effective Hamiltonian is deduced for electrons. Using the variational principle this Hamiltonian is employed to derive a modified set of single-component Hartree-Fock equations which are equivalent to the multi-component version derived previously within the context of the NEO and, easy to be implemented computationally.
Away from generalized gradient approximation: orbital-dependent exchange-correlation functionals.
Baerends, E J; Gritsenko, O V
2005-08-08
The local-density approximation of density functional theory (DFT) is remarkably accurate, for instance, for geometries and frequencies, and the generalized gradient approximations have also made bond energies quite reliable. Sometimes, however, one meets with failure in individual cases. One of the possible routes towards better functionals would be the incorporation of orbital dependence (which is an implicit density dependency) in the functionals. We discuss this approach both for energies and for response properties. One possibility is the use of the Hartree-Fock-type exchange energy expression as orbital-dependent functional. We will argue that in spite of the increasing popularity of this approach, it does not offer any advantage over Hartree-Fock for energies. We will advocate not to apply the separation of exchange and correlation, which is so ingrained in quantum chemistry, but to model both simultaneously. For response properties the energies and shapes of the virtual orbitals are crucial. We will discuss the benefits that Kohn-Sham potentials can offer which are derived from either an orbital-dependent energy functional, including the exact-exchange functional, or which can be obtained directly as orbital-dependent functional. We highlight the similarity of the Hartree-Fock and Kohn-Sham occupied orbitals and orbital energies, and the essentially different meanings the virtual orbitals and orbital energies have in these two models. We will show that these differences are beneficial for DFT in the case of localized excitations (in a small molecule or in a fragment), but are detrimental for charge-transfer excitations. Again, orbital dependency, in this case in the exchange-correlation kernel, offers a solution.
Nishiyama, Seiya
2014-01-01
In this paper we present the induced representation of SO(2N) canonical transformation group and introduce SO(2N)/U(N) coset variables. We give a derivation of the time dependent Hartree-Bogoliubov (TDHB) equation on the Kaehler coset space G/H=SO(2N)/U(N) from the Euler-Lagrange equation of motion for the coset variables. The TDHB wave function represents the TD behavior of Bose condensate of fermion pairs. It is a good approximation for the ground state of the fermion system with a pairing interaction, producing the spontaneous Bose condensation. To describe the classical motion on the coset manifold, we start from the local equation of motion. This equation becomes a Riccati-type equation. After giving a simple two-level model and a solution for a coset variable, we can get successfully a general solution of TDRHB equation for the coset variables. We obtain the Harish-Chandra decomposition for the SO(2N) matrix based on the nonlinear Moebius transformation together with the geodesic flow on the manifold.
Langhoff, S. R.; Scott, W. R.; Suzuki, N.; Chong, D. P.
1979-01-01
Ordinary Rayleigh-Schroudinger perturbation theory with Moller-Plesset (RSMP) partitioning is used to calculate second- and third-order correlation corrections to the CHF polarizability and dipole moment of the water molecule by a finite-field procedure. Pade approximants are found to be useful in accelerating the convergence of the property perturbation expansions. Field-induced polarization functions suitable for polarizability calculations are determined. The average polarizability calculated, neglecting vibrational averaging, with Dunning's (9s5p/4s-4s2p/2s) contracted GTO basis set augmented by field-induced lslp2d/lp polarization functions is within 3 per cent of the experimental result. Correlation corrections to the dipole moment and polarizability of the water molecule calculated by the finite-field RSMP and single + double excitation CI(SDCI) methods for the same basis set are found to be in close agreement. The RSMP approach has the advantages of being size-consistent and of being capable of greater efficiency than the SCDI method. Comparative calculations carried out using Epstein-Nesbet partitioning show that through third order RSEN correlation perturbation expansions for the dipole moment and polarizability are less rapidly convergent than RSMP expansions. However, reasonable accord with RSMP results can be achieved by using Pade approximants to accelerate the convergence of RSEN energy perturbation expansions. The convergence of RSMP property correlation expansions based on the zeroth-order uncoupled-Hartree-Fock (UCHF) and coupled-Hartree-Fock (CHF) approximations are compared through third order. Whereas the CHF + RSMP expansions are for practical purposes fully converged, the UCHF + RSMP expansions are not adequately converged.
Cho, Yonggeun
2016-05-04
This paper is devoted to the mathematical analysis of a class of nonlinear fractional Schrödinger equations with a general Hartree-type integrand. We show the well-posedness of the associated Cauchy problem and prove the existence and stability of standing waves under suitable assumptions on the nonlinearity. Our proofs rely on a contraction argument in mixed functional spaces and the concentration-compactness method. © 2015 World Scientific Publishing Company
Ebran, J-P [CEA/DAM/DIF, F-91297 Arpajon (France); Khan, E; Arteaga, D Pena [Institut de Physique Nucleaire, University Paris-Sud, IN2P3-CNRS, F-91406 Orsay Cedex (France); Vretenar, D, E-mail: jean-paul.ebran@cea.fr [Physics Department, Faculty of Science, University of Zagreb, 10000 Zagreb (Croatia)
2011-09-16
The Relativistic Hartree-Fock-Bogoliubov model for axially deformed nuclei (RHFBz) is presented. The model involves a phenomenological Lagrangian with density-dependent meson-nucleon couplings in the particle-hole channel and the central part of the Gogny force in the particle-particle channel. The RHFBz equations are solved by expansion in the basis of a deformed harmonic oscillator. Illustrative RHFBz calculations are performed for Neon isotopes.
The classical limit of the time dependent Hartree-Fock equation. I. The Weyl symbol of the solution
Amour, Laurent; Nourrigat, Jean
2011-01-01
We study the time evolution of the Weyl symbol of a solution of the time dependent Hartree Fock equation, assuming that for t=0, it has a Weyl symbol which is integrable in the phase space, such as all its derivatives. We prove that the solution has the same property for all t, and we give an asymptotic expansion, in L1 sense, of this Weyl symbol.
Systematic study of even-even nuclei with Hartree-Fock+BCS method using Skyrme SIII force
Tajima, Naoki; Takahara, Satoshi; Onishi, Naoki [Tokyo Univ. (Japan). Coll. of Arts and Sciences
1997-03-01
We have applied the Hartree-Fock+BCS method with Skyrme SIII force formulated in a three-dimensional Cartesian-mesh representation to even-even nuclei with 2 {<=} Z {<=} 114. We discuss the results concerning the atomic masses, the quadrupole (m=0, 2) and hexadecapole (m=0, 2, 4) deformations, the skin thicknesses, and the halo radii. We also discuss the energy difference between oblate and prolate solutions and the shape difference between protons and neutrons. (author)
Fischer, C F
1977-11-01
This technical report contains a listing of a general program for multi-configuration Hartree--Fock (MCHF) calculations, including its documentation. Several examples are given showing how the program may be used. Typical output for several cases is also presented. This program has been tested over an extended period of time for a large variety of cases. This program is written for the IBM 360 or 370 in double-precision arithmetic.
General technique for analytical derivatives of post-projected Hartree-Fock
Tsuchimochi, Takashi; Ten-no, Seiichiro
2017-02-01
In electronic structure theory, the availability of an analytical derivative is one of the desired features for a method to be useful in practical applications, as it allows for geometry optimization as well as computation of molecular properties. With the recent advances in the development of symmetry-projected Hartree-Fock (PHF) methods, we here aim at further extensions by devising the analytic gradients of post-PHF approaches with a special focus on spin-extended (spin-projected) configuration interaction with single and double substitutions (ECISD). Just like standard single-reference methods, the mean-field PHF part does not require the corresponding coupled-perturbed equation to be solved, while the correlation energy term needs the orbital relaxation effect to be accounted for, unless the underlying molecular orbitals are variationally optimized in the presence of the correlation energy. We present a general strategy for post-PHF analytical gradients, which closely parallels that for single-reference methods, yet addressing the major difference between them. The similarity between ECISD and multi-reference CI not only in the energy but also in the optimized geometry is clearly demonstrated by the numerical examples of ozone and cyclobutadiene.
Exploration of (super-)heavy elements using the Skyrme-Hartree-Fock model
Erler, Jochen
2011-01-31
Motivated by the steadily increasing number of known nuclei and nuclear properties, theories of nuclear structure are presently a field of intense research. This work concentrates on the self-consistent description of nuclei in terms of the Skyrme-Hartree-Fock (SHF) approach. The extrapolation of nuclear shell structure to the region of super-heavy elements (SHE) using the SHF model, the dependence on different parameterization and the influence of collective correlation will be studied. The general scope of this work are large scale calculation for a global survey of properties of SHE like binding energies, separation energies and decay characteristics and lifetimes. These calculations were done in a collaboration with the theory group of the GSI in Darmstadt and have the aim to develop a database of lifetimes and reaction rates for {alpha}, {beta}-decay and spontaneous fission in a very wide range with proton numbers 86 {<=} Z {<=} 120 and neutron numbers up to N {approx} 260 relevant for the astrophysical r-process. The results of this study for example predictions of a possible islands of very stable nuclei and information of favored decay mode for each nuclei are also applicable in the recent experimental synthesis of exotic SHE. For these calculation a framework to calculate {beta}-decay half-lives within the SHF model has been developed and the existing axial SHF code has been extended to compute {beta}-transition matrix elements and so to provide an estimation of half-lives. (orig.)
The Hartree-Fock exchange effect on the CO adsorption by the boron nitride nanocage
Vessally, E.; Soleimani-Amiri, S.; Hosseinian, A.; Edjlali, L.; Bekhradnia, A.
2017-03-01
We studied the effect of Hartree-Fock (HF) exchange percentage of a density functional on the adsorption properties and electronic sensitivity of the B12N12 nanocluster to CO molecule. It was found that by an increase in the %HF, the LUMO level is nearly constant while the HOMO level is strongly stabilized, expanding the HOMO-LUMO gap (Eg). Also, the volume of the all structures decreased and the sensitivity of the B12N12 is slightly increased to CO molecule. For the pristine B12N12 cluster, the B66 and B64 bonds are about 1.43 and 1.49 Å at 10% HF, and 1.23 and 1.26 Å at 100% HF, respectively. The HF exchange between 10-20% may predict an accurate Eg for the B12N12 system. We concluded that functionals with a large %HF such as M06-HF, and M06-2X may significantly overestimate the Eg, and bond strength. We obtained a parabolic relationship between the %HF and the adsorption energy of CO molecule on the B12N12 cluster. Also, an increase in the %HF predicts a larger charge transfer from the CO molecule to the cage.
Generalized Hartree-Fock-Bogoliubov description of the Fröhlich polaron
Kain, Ben; Ling, Hong Y.
2016-07-01
We adapt the generalized Hartree-Fock-Bogoliubov (HFB) method to an interacting many-phonon system free of impurities. The many-phonon system is obtained from applying the Lee-Low-Pine (LLP) transformation to the Fröhlich model which describes a mobile impurity coupled to noninteracting phonons. We specialize our general HFB description of the Fröhlich polaron to Bose polarons in quasi-one-dimensional cold-atom mixtures. The LLP-transformed many-phonon system distinguishes itself with an artificial phonon-phonon interaction which is very different from the usual two-body interaction. We use the quasi-one-dimensional model, which is free of an ultraviolet divergence that exists in higher dimensions, to better understand how this unique interaction affects polaron states and how the density and pair correlations inherent to the HFB method conspire to create a polaron ground state with an energy in good agreement with and far closer to the prediction from Feynman's variational path integral approach than mean-field theory where HFB correlations are absent.
Aryal, M. M.; Maharjan, N. B.; Paudyal, D. D.; Mishra, D. R.; Byahut, S. R.; Scheicher, R. H.; Badu, S. R.; Jeong, J.; Chow, Lee; Das, T. P.
2008-03-01
Using the first-principles Hartree-Fock Cluster Procedure, we have studied the electronic structures of pure chain like Selenium and Tellurium, pure ring structured Selenium, Tellurium impurity in chain and ring-structured Selenium and Selenium impurity in chain-structured Tellurium chain. For our investigations in all the systems we have carried out convergence studies with respect to variational basis set sizes,sizes of clusters and electron correlation effects using many-body perturbation theory. Using our calculated electronic field-gradient parameters q in the pure chain systems and employing the experimental quadrupole coupling constants (e^2qQ), the values Q(^77Se)=(0.50±0.04) 10-28 m^2 and Q(^125Te)=-(0.2±0.02) 10-28m^2. Results will also be presented for the asymmetry parameters η for the pure chain systems and the e^2qQ and η for ^77Se in selenium ring. Our calculated values for e^2qQ and η for the impurity systems will also be presented and compared with available experimental data and earlier theoretical results.
Coordinate-Space Hartree-Fock-Bogoliubov Solvers for Superfluid Fermi Systems in Large Boxes
Pei, J. C. [University of Tennessee (UTK) and Oak Ridge National Laboratory (ORNL); Fann, George I [ORNL; Harrison, Robert J [ORNL; Nazarewicz, W. [University of Tennessee (UTK) and Oak Ridge National Laboratory (ORNL); Hill, Judith C [ORNL; Galindo, Diego A [ORNL; Jia, Jun [ORNL
2012-01-01
The self-consistent Hartree-Fock-Bogoliubov problem in large boxes can be solved accurately in the coordinate space with the recently developed solvers HFB-AX (2D) and MADNESS-HFB (3D). This is essential for the description of superfluid Fermi systems with complicated topologies and significant spatial extend, such as fissioning nuclei, weakly-bound nuclei, nuclear matter in the neutron star rust, and ultracold Fermi atoms in elongated traps. The HFB-AX solver based on B-spline techniques uses a hybrid MPI and OpenMP programming model for parallel computation for distributed parallel computation, within a node multi-threaded LAPACK and BLAS libraries are used to further enable parallel calculations of large eigensystems. The MADNESS-HFB solver uses a novel multi-resolution analysis based adaptive pseudo-spectral techniques to enable fully parallel 3D calculations of very large systems. In this work we present benchmark results for HFB-AX and MADNESS-HFB on ultracold trapped fermions.
A simple and efficient dispersion correction to the Hartree-Fock theory.
Yoshida, Tatsusada; Mashima, Akira; Sasahara, Katsunori; Chuman, Hiroshi
2014-02-15
One of the most challenging problems in computational chemistry and in drug discovery is the accurate prediction of the binding energy between a ligand and a protein receptor. It is well known that the binding energy calculated with the Hartree-Fock molecular orbital theory (HF) lacks the dispersion interaction energy that significantly affects the accuracy of the total binding energy of a large molecular system. We propose a simple and efficient dispersion energy correction to the HF theory (HF-Dtq). The performance of HF-Dtq was compared with those of several recently proposed dispersion corrected density functional theory methods (DFT-Ds) as to the binding energies of 68 small non-covalent complexes. The overall performance of HF-Dtq was found to be nearly equivalent to that of more sophisticated B3LYP-D3. HF-Dtq will thus be a useful and powerful method for accurately predicting the binding energy between a ligand and a protein, albeit it is a simple correction procedure based on HF.
Cho, Daeheum; Ko, Kyoung Chul; Lee, Jin Yong, E-mail: jinylee@skku.edu [Department of Chemistry, Sungkyunkwan University, Suwon 440-746 (Korea, Republic of); Ikabata, Yasuhiro; Wakayama, Kazufumi; Yoshikawa, Takeshi [Department of Chemistry and Biochemistry, School of Advanced Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555 (Japan); Nakai, Hiromi, E-mail: nakai@waseda.jp [Department of Chemistry and Biochemistry, School of Advanced Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555 (Japan); Research Institute for Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555 (Japan); CREST, Japan Science and Technology Agency, Tokyo 102-0075 (Japan); Elements Strategy Initiative for Catalysts and Batteries (ESICB), Kyoto University, Katsura, Kyoto 615-8520 (Japan)
2015-01-14
The intramolecular magnetic coupling constant (J) of diradical systems linked with five- or six-membered aromatic rings was calculated to obtain the scaling factor (experimental J/calculated J ratio) for various density functional theory (DFT) functionals. Scaling factors of group A (PBE, TPSSh, B3LYP, B97-1, X3LYP, PBE0, and BH and HLYP) and B (M06-L, M06, M06-2X, and M06-HF) were shown to decrease as the amount of Hartree-Fock exact exchange (HFx) increases, in other words, overestimation of calculated J becomes more severe as the HFx increases. We further investigated the effect of HFx fraction of DFT functional on J value, spin contamination, and spin density distributions by comparing the B3LYP analogues containing different amount of HFx. It was revealed that spin contamination and spin densities at each atom increases as the HFx increases. Above all, newly developed BLYP-5 functional, which has 5% of HFx, was found to have the scaling factor of 1.029, indicating that calculated J values are very close to that of experimental values without scaling. BLYP-5 has potential to be utilized for accurate evaluation of intramolecular magnetic coupling constant (J) of diradicals linked by five- or six-membered aromatic ring couplers.
Basis set effects on the Hartree-Fock description of confined many-electron atoms
Garza, Jorge; Hernández-Pérez, Julio M.; Ramírez, José-Zeferino; Vargas, Rubicelia
2012-01-01
In this work, the basis sets designed by Clementi, Bunge and Thakkar, for atomic systems, have been used to obtain the electronic structure of confined many-electron atoms by using Roothaan's approach in the Hartree-Fock context with a new code written in C, which uses the message-passing interface library. The confinement was imposed as Ludeña suggested to simulate walls with infinity potential. For closed-shell atoms, the Thakkar basis set functions give the best total energies (TE) as a function of the confinement radius, obtaining the following ordering: TE(Thakkar) Clementi). However, for few open-shell atoms this ordering is not preserved and a trend, for the basis sets, is not observed. Although there are differences between the TE predicted by these basis set functions, the corresponding pressures are similar to each other; it means that changes in the total energy are described almost in the same way by using any of these basis sets. By analysing the total energy as a function of the inverse of the volume we propose an equation of state; for regions of small volumes, this equation predicts that the pressure is inversely proportional to the square of the volume.
Aerts, Patrick Johan Coenraad
1986-01-01
Computational Theoretical Chemnistry is a research area which, as far as electronic structure problems are concerned, encompasses essentially the development of theoretically sound, yet computionally feasable quantum mechanical models for atoms melecules and the solid state. ... Zie: Introduction
Bassem, Y El
2016-01-01
In a previous work [Int. J. Mod. Phys. E 24, 1550073 (2015)], hereafter referred as paper I, we have investigated the ground-state properties of Nd, Ce and Sm isotopes within Hartree-Fock-Bogoliubov method with SLy5 skyrme force in which the pairing strength has been generalized with a new proposed formula. However, that formula is more appropriate for the region of Nd. In this work, we have studied the ground-state properties of both even-even and odd Mo and Ru isotopes. For this, we have used Hartree- Fock-Bogoliubov method with SLy4 skyrme force, and a new formula of the pairing strength which is more accurate for this region of nuclei. The results have been compared with available experimental data, the results of Hartree-Fock-Bogoliubov calculations based on the D1S Gogny effective nucleon-nucleon interaction and predictions of some nuclear models such as Finite Range Droplet Model (FRDM) and Relativistic Mean Field (RMF) theory.
Ragot, Sébastien
2008-04-01
The ground-state Hartree-Fock (HF) wavefunction of Hooke's atom is not known in closed form, contrary to the exact solution. The single HF orbital involved has thus far been studied using expansion techniques only, leading to slightly disparate energies. Therefore, the present letter aims at proposing alternative definitions of the HF wavefunction. First, the HF limit is ascertained using a simple expansion, which makes it possible to formulate explicit expressions of HF properties. The resulting energy, 2.038 438 871 8 Eh, is found stable at the tenth digit. Second and more instructive, an analysis of the Hartree equation makes it possible to infer a remarkably simple and accurate HF orbital, i.e., φHF(r)=nHFe-αr2√r2+β2, leading to an energy exceeding by 5.76×10-7 Eh only the above HF limit. This orbital makes it possible to obtain (near) Hartree-Fock properties in closed form, which in turn enables handy comparisons with exact quantities.
Rayka, Milad; Shahbazian, Shant
2016-01-01
In this communication, an effective set of the Hartree-Fock equations are derived only for electrons of the muonic systems, i.e., molecules containing a positively charged muon, conceiving the muon as a quantum oscillator. In these equations, a non-Coulombic potential is added to the orthodox coulomb and exchange potential energy terms, which describes the interaction of the muon and electrons effectively. The explicit form of the effective potential depends on the nature of muon vibrations and is derived for a combination of Cartesian Gaussian functions that are used to expand the muonic spatial orbital. The resulting effective Hartree-Fock equations are implemented computationally and used successfully, as a proof of concept, in the case of MuCN molecule, which results from replacing the proton of HCN molecule with a muon. The developed effective Hartree-Fock theory is quite general and in principle can be used for any muonic system while it is the starting point for a general effective electronic structure...
Roy, Swapnoneel; Thakur, Ashok Kumar
2008-01-01
Genome rearrangements have been modelled by a variety of primitives such as reversals, transpositions, block moves and block interchanges. We consider such a genome rearrangement primitive Strip Exchanges. Given a permutation, the challenge is to sort it by using minimum number of strip exchanges. A strip exchanging move interchanges the positions of two chosen strips so that they merge with other strips. The strip exchange problem is to sort a permutation using minimum number of strip exchanges. We present here the first non-trivial 2-approximation algorithm to this problem. We also observe that sorting by strip-exchanges is fixed-parameter-tractable. Lastly we discuss the application of strip exchanges in a different area Optical Character Recognition (OCR) with an example.
Approximation by Cylinder Surfaces
Randrup, Thomas
1997-01-01
We present a new method for approximation of a given surface by a cylinder surface. It is a constructive geometric method, leading to a monorail representation of the cylinder surface. By use of a weighted Gaussian image of the given surface, we determine a projection plane. In the orthogonal...... projection of the surface onto this plane, a reference curve is determined by use of methods for thinning of binary images. Finally, the cylinder surface is constructed as follows: the directrix of the cylinder surface is determined by a least squares method minimizing the distance to the points...... in the projection within a tolerance given by the reference curve, and the rulings are lines perpendicular to the projection plane. Application of the method in ship design is given....
S-Approximation: A New Approach to Algebraic Approximation
M. R. Hooshmandasl
2014-01-01
Full Text Available We intend to study a new class of algebraic approximations, called S-approximations, and their properties. We have shown that S-approximations can be used for applied problems which cannot be modeled by inclusion based approximations. Also, in this work, we studied a subclass of S-approximations, called Sℳ-approximations, and showed that this subclass preserves most of the properties of inclusion based approximations but is not necessarily inclusionbased. The paper concludes by studying some basic operations on S-approximations and counting the number of S-min functions.
On the construction of a new solvable model and validity of many-body approximation methods
Zettili, Nouredine; Villars, Felix M. H.
1987-07-01
This work deals both with the construction of a new analytically solvable model and with the quantitative test of the time-dependent Hartree-Fock (TDHF) method. First, we construct a new analytically solvable model, which serves as a testing ground for the various many-body approximation methods. The construction is based on two vector operators that are the generators of a Lie algebra. The model consists of a one-dimensional system of two distinguishable sets of fermions interacting via a schematic two-body force. The model has a simple analytic energy spectrum. Second, we use this model to test the validity of the TDHF approximation. Exact eigenvalues are compared with the corresponding solutions of the TDHF method. The TDHF approximation is shown to be reasonably accurate in the description of the system's eigenstates.
On the construction of a new solvable model and validity of many-body approximation methods
Zettili, N.; Villars, F.M.H.
1987-07-20
This work deals both with the construction of a new analytically solvable model and with the quantitative test of the time-dependent Hartree-Fock (TDHF) method. First, we construct a new analytically solvable model, which serves as a testing ground for the various many-body approximation methods. The construction is based on two vector operators that are the generators of a Lie algebra. The model consists of a one-dimensional system of two distinguishable sets of fermions interacting via a schematic two-body force. The model has a simple analytic energy spectrum. Second, we use this model to test the validity of the TDHF approximation. Exact eigenvalues are compared with the corresponding solutions of the TDHF method. The TDHF approximation is shown to be reasonably accurate in the description of the system's eigenstates.
Toulouse, Julien; Angyan, Janos G; Savin, Andreas
2010-01-01
Using Green-function many-body theory, we present the details of a formally exact adiabatic-connection fluctuation-dissipation density-functional theory based on range separation, which was sketched in Toulouse, Gerber, Jansen, Savin and Angyan, Phys. Rev. Lett. 102, 096404 (2009). Range-separated density-functional theory approaches combining short-range density functional approximations with long-range random phase approximations (RPA) are then obtained as well-identified approximations on the long-range Green-function self-energy. Range-separated RPA-type schemes with or without long-range Hartree-Fock exchange response kernel are assessed on rare-gas and alkaline-earth dimers, and compared to range-separated second-order perturbation theory and range-separated coupled-cluster theory.
Wang, Haobin; Thoss, Michael
2016-12-01
The accuracy of the noninteracting electron approximation is examined for a model of vibrationally coupled electron transport in single molecule junction. In the absence of electronic-vibrational coupling, steady state transport in this model is described exactly by Landauer theory. Including coupling, both electronic-vibrational and vibrationally induced electron-electron correlation effects may contribute to the real time quantum dynamics. Using the multilayer multiconfiguration time-dependent Hartree (ML-MCTDH) theory to describe nuclear dynamics exactly while maintaining the noninteracting electron approximation for the electronic dynamics, the correlation effects are analyzed in different physical regimes. It is shown that although the noninteracting electron approximation may be reasonable for describing short time dynamics, it does not give the correct long time limit for certain initial conditions.
Prestack traveltime approximations
Alkhalifah, Tariq Ali
2012-05-01
Many of the explicit prestack traveltime relations used in practice are based on homogeneous (or semi-homogenous, possibly effective) media approximations. This includes the multifocusing, based on the double square-root (DSR) equation, and the common reflection stack (CRS) approaches. Using the DSR equation, I constructed the associated eikonal form in the general source-receiver domain. Like its wave-equation counterpart, it suffers from a critical singularity for horizontally traveling waves. As a result, I recasted the eikonal in terms of the reflection angle, and thus, derived expansion based solutions of this eikonal in terms of the difference between the source and receiver velocities in a generally inhomogenous background medium. The zero-order term solution, corresponding to ignoring the lateral velocity variation in estimating the prestack part, is free of singularities and can be used to estimate traveltimes for small to moderate offsets (or reflection angles) in a generally inhomogeneous medium. The higher-order terms include limitations for horizontally traveling waves, however, we can readily enforce stability constraints to avoid such singularities. In fact, another expansion over reflection angle can help us avoid these singularities by requiring the source and receiver velocities to be different. On the other hand, expansions in terms of reflection angles result in singularity free equations. For a homogenous background medium, as a test, the solutions are reasonably accurate to large reflection and dip angles. A Marmousi example demonstrated the usefulness and versatility of the formulation. © 2012 Society of Exploration Geophysicists.
Calculation of electric dipole hypershieldings at the nuclei in the Hellmann-Feynman approximation.
Soncini, Alessandro; Lazzeretti, Paolo; Bakken, Vebjørn; Helgaker, Trygve
2004-02-15
The third-rank electric hypershieldings at the nuclei of four small molecules have been evaluated at the Hartree-Fock level of theory in the Hellmann-Feynman approximation. The nuclear electric hypershieldings are closely related to molecular vibrational absorption intensities and a generalization of the atomic polar tensors (expanded in powers of the electric field strength) is proposed to rationalize these intensities. It is shown that the sum rules for rototranslational invariance and the constraints imposed by the virial theorem provide useful criteria for basis-set completeness and for near Hartree-Fock quality of nuclear shieldings and hypershieldings evaluated in the Hellmann-Feynman approximation. Twelve basis sets of different size and quality have been employed for the water molecule in an extended numerical test on the practicality of the proposed scheme. The best results are obtained with the R12 and R12+ basis sets, designed for the calculation of electronic energies by the explicitly correlated R12 method. The R12 basis set is subsequently used to investigate three other molecules, CO, N2, and NH3, verifying that the R12 basis consistently performs very well.
Study of superdeformation at zero spin with Skyrme-Hartree-Fock method
Takahara, S.; Tajima, N.; Onishi, N. [Tokyo Univ. (Japan)
1998-03-01
Superdeformed (SD) bands have been studied extensively both experimentally and theoretically in the last decade. Since the first observation in {sup 152}Dy in 1986, SD bands have been found in four mass regions, i.e., A {approx} 80, 130, 150 and 190. While these SD bands have been observed only at high spins so far, they may also be present at zero spin like fission isomers in actinide nuclei: The familiar generic argument on the strong shell effect at axis ratio 2:1 does not assume rotations. If non-fissile SD isomers exist at zero spin, they may be utilized to develop new experimental methods to study exotic states, in a similar manner as short-lived high-spin isomers are planned to be utilized as projectiles of fusion reactions in order to populate very high-spin near-yrast states. They will also be useful to test theoretical models whether the models can describe correctly the large deformations of rare-earth nuclei without further complications due to rotations. In this report, we employ the Skyrme-Hartree-Fock method to study the SD states at zero spin. First, we compare various Skyrme force parameter sets to test whether they can reproduce the extrapolated excitation energy of the SD band head of {sup 194}Hg. Second, we systematically search large-deformation solutions with the SkM{sup *} force. The feature of our calculations is that the single-particle wavefunctions are expressed in a three-dimensional-Cartesian-mesh representation. This representation enables one to obtain solutions of various shapes (including SD) without preparing a basis specific to each shape. Solving the mean-field equations in this representation requires, however, a large amount of computation which can be accomplished only with present supercomputers. (author)
Operators of Approximations and Approximate Power Set Spaces
ZHANG Xian-yong; MO Zhi-wen; SHU Lan
2004-01-01
Boundary inner and outer operators are introduced; and union, intersection, complement operators of approximations are redefined. The approximation operators have a good property of maintaining union, intersection, complement operators, so the rough set theory has been enriched from the operator-oriented and set-oriented views. Approximate power set spaces are defined, and it is proved that the approximation operators are epimorphisms from power set space to approximate power set spaces. Some basic properties of approximate power set space are got by epimorphisms in contrast to power set space.
International Conference Approximation Theory XV
Schumaker, Larry
2017-01-01
These proceedings are based on papers presented at the international conference Approximation Theory XV, which was held May 22–25, 2016 in San Antonio, Texas. The conference was the fifteenth in a series of meetings in Approximation Theory held at various locations in the United States, and was attended by 146 participants. The book contains longer survey papers by some of the invited speakers covering topics such as compressive sensing, isogeometric analysis, and scaling limits of polynomials and entire functions of exponential type. The book also includes papers on a variety of current topics in Approximation Theory drawn from areas such as advances in kernel approximation with applications, approximation theory and algebraic geometry, multivariate splines for applications, practical function approximation, approximation of PDEs, wavelets and framelets with applications, approximation theory in signal processing, compressive sensing, rational interpolation, spline approximation in isogeometric analysis, a...
Dobaczewski, J.; Olbratowski, P.
2005-05-01
We describe the new version (v2.08k) of the code HFODD which solves the nuclear Skyrme-Hartree-Fock or Skyrme-Hartree-Fock-Bogolyubov problem by using the Cartesian deformed harmonic-oscillator basis. Similarly as in the previous version (v2.08i), all symmetries can be broken, which allows for calculations with angular frequency and angular momentum tilted with respect to the mass distribution. In the new version, three minor errors have been corrected. New Version Program SummaryTitle of program: HFODD; version: 2.08k Catalogue number: ADVA Catalogue number of previous version: ADTO (Comput. Phys. Comm. 158 (2004) 158) Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADVA Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland Does the new version supersede the previous one: yes Computers on which this or another recent version has been tested: SG Power Challenge L, Pentium-II, Pentium-III, AMD-Athlon Operating systems under which the program has been tested: UNIX, LINUX, Windows-2000 Programming language used: Fortran Memory required to execute with typical data: 10M words No. of bits in a word: 64 No. of lines in distributed program, including test data, etc.: 52 631 No. of bytes in distributed program, including test data, etc.: 266 885 Distribution format:tar.gz Nature of physical problem: The nuclear mean-field and an analysis of its symmetries in realistic cases are the main ingredients of a description of nuclear states. Within the Local Density Approximation, or for a zero-range velocity-dependent Skyrme interaction, the nuclear mean-field is local and velocity dependent. The locality allows for an effective and fast solution of the self-consistent Hartree-Fock equations, even for heavy nuclei, and for various nucleonic ( n-particle n-hole) configurations, deformations, excitation energies, or angular momenta. Similar Local Density Approximation in the particle-particle channel, which is equivalent to using a zero
Two-component hybrid time-dependent density functional theory within the Tamm-Dancoff approximation
Kühn, Michael [Institut für Physikalische Chemie, Karlsruher Institut für Technologie, Kaiserstraße 12, 76131 Karlsruhe (Germany); Weigend, Florian, E-mail: florian.weigend@kit.edu [Institut für Physikalische Chemie, Karlsruher Institut für Technologie, Kaiserstraße 12, 76131 Karlsruhe (Germany); Institut für Nanotechnologie, Karlsruher Institut für Technologie, Postfach 3640, 76021 Karlsruhe (Germany)
2015-01-21
We report the implementation of a two-component variant of time-dependent density functional theory (TDDFT) for hybrid functionals that accounts for spin-orbit effects within the Tamm-Dancoff approximation (TDA) for closed-shell systems. The influence of the admixture of Hartree-Fock exchange on excitation energies is investigated for several atoms and diatomic molecules by comparison to numbers for pure density functionals obtained previously [M. Kühn and F. Weigend, J. Chem. Theory Comput. 9, 5341 (2013)]. It is further related to changes upon switching to the local density approximation or using the full TDDFT formalism instead of TDA. Efficiency is demonstrated for a comparably large system, Ir(ppy){sub 3} (61 atoms, 1501 basis functions, lowest 10 excited states), which is a prototype molecule for organic light-emitting diodes, due to its “spin-forbidden” triplet-singlet transition.
Staker, Joshua T
2013-01-01
We make numerical comparison of spectra from angular-momentum projection on Hartree-Fock states with spectra from configuration-interaction nuclear shell-model calculations, all carried out in the same model spaces (in this case the sd, lower pf, and p-sd_5/2 shells) and using the same input Hamiltonians. We find, unsurprisingly, that the low-lying excitation spectra for rotational nuclides are well reproduced, but the spectra for vibrational nuclides, and more generally the complex specta for odd-A and odd-odd nuclides are less well reproduced in detail.
Miyasita, Mitiyasu, E-mail: miyasita.mitiyasu@gmail.com [Graduate School of Science and Engineering, Shinshu University, Ueda 386-8567 (Japan); Higuchi, Katsuhiko [Graduate School of Advanced Science of Matter, Hiroshima University, Higashi-Hiroshima 739-8527 (Japan); Higuchi, Masahiko [Department of Physics, Faculty of Science, Shinshu University, Matsumoto 390-8621 (Japan)
2012-07-15
We present an alternative scheme for calculating the unrestricted Hartree-Fock (HF) equation. The scheme is based on the variational method utilizing the sophisticated basis functions that include no adjustable parameters. The validity of the present scheme is confirmed by actual calculations of the boron and neon atoms. The total energy of the present scheme is lower than that of the conventional restrictive HF equation, but higher than that of the CI method. Also, the resultant wave function satisfies the electron-nucleus cusp condition.
Hierarchical low-rank approximation for high dimensional approximation
Nouy, Anthony
2016-01-07
Tensor methods are among the most prominent tools for the numerical solution of high-dimensional problems where functions of multiple variables have to be approximated. Such high-dimensional approximation problems naturally arise in stochastic analysis and uncertainty quantification. In many practical situations, the approximation of high-dimensional functions is made computationally tractable by using rank-structured approximations. In this talk, we present algorithms for the approximation in hierarchical tensor format using statistical methods. Sparse representations in a given tensor format are obtained with adaptive or convex relaxation methods, with a selection of parameters using crossvalidation methods.
Mackrodt, W. C.
1999-02-01
First principles periodic Hartree-Fock calculations are reported for the P4 2/ mnm(rutile), I4 1/ amd(anatase), Pbca(brookite), Pnma(ramsdellite), Pcbn(colombite), Fdoverline3m(spinel), and Imma(orthorhombic) polymorphs of TiO 2, from which the predicted order of stability is The calculated difference in energy between the rutile and anatase structures is 0.02-0.06 eV, in good agreement with a recent local density approximation (LDA) estimate of 0.033 eV and an experiment enthalpy difference of 0.05 eV. The corresponding Hartree-Fock and LDA differences for the brookite structure are 0.06 and 0.058 eV, respectively. The calculated volumes, which are based on isotropic volume-optimized Hartree-Fock energies, are also in good agreement with recent LDA calculations and with experiment. Spin-unrestricted calculations are reported for the Fmoverline3m, Imma, Pnma, and P4 2/ mmmof LiTiO 2, where the stability is in the order The only reported phase for LiTiO 2is Fmoverline3m, for which the calculated volume is in good agreement with experiment. From the relative stabilities of TiO 2and LiTiO 2, the relative lithium insertion potentials corresponding to TiO 2 → LiTiO 2are deduced, with a maximum variation of 1.6 eV for the different polymorphic routes. The maximum voltage predicted is that for the Immaroute which is ˜1 eV larger than that for Pnma. Direct comparisons with the calculated energy for C2/ mLi 0.5MnO 2 → LiMnO 2lead to an estimate of the voltage for ImmaTiO 2 → LiTiO 2of ˜1.3 eV, which is ˜2.5 eV anodicto the Mn system. The corresponding values for the Pnmapolymorphic route are ˜3 and ˜3.5 eV, respectively. Mulliken population analyses indicate that lithium is completely ionized in LiTiO 2and that the charge transfer is predominantly to the oxygen sublattice. There is a rehybridization of the titanium valence orbitals leading to a slight increase in the 3 dpopulation and strong localization of spin density at the titanium sites with local moments of
Nonlinear Approximation Using Gaussian Kernels
Hangelbroek, Thomas
2009-01-01
It is well-known that non-linear approximation has an advantage over linear schemes in the sense that it provides comparable approximation rates to those of the linear schemes, but to a larger class of approximands. This was established for spline approximations and for wavelet approximations, and more recently for homogeneous radial basis function (surface spline) approximations. However, no such results are known for the Gaussian function. The crux of the difficulty lies in the necessity to vary the tension parameter in the Gaussian function spatially according to local information about the approximand: error analysis of Gaussian approximation schemes with varying tension are, by and large, an elusive target for approximators. We introduce and analyze in this paper a new algorithm for approximating functions using translates of Gaussian functions with varying tension parameters. Our scheme is sophisticated to a degree that it employs even locally Gaussians with varying tensions, and that it resolves local ...
Forms of Approximate Radiation Transport
Brunner, G
2002-01-01
Photon radiation transport is described by the Boltzmann equation. Because this equation is difficult to solve, many different approximate forms have been implemented in computer codes. Several of the most common approximations are reviewed, and test problems illustrate the characteristics of each of the approximations. This document is designed as a tutorial so that code users can make an educated choice about which form of approximate radiation transport to use for their particular simulation.
Approximation by Multivariate Singular Integrals
Anastassiou, George A
2011-01-01
Approximation by Multivariate Singular Integrals is the first monograph to illustrate the approximation of multivariate singular integrals to the identity-unit operator. The basic approximation properties of the general multivariate singular integral operators is presented quantitatively, particularly special cases such as the multivariate Picard, Gauss-Weierstrass, Poisson-Cauchy and trigonometric singular integral operators are examined thoroughly. This book studies the rate of convergence of these operators to the unit operator as well as the related simultaneous approximation. The last cha
Approximations of fractional Brownian motion
Li, Yuqiang; 10.3150/10-BEJ319
2012-01-01
Approximations of fractional Brownian motion using Poisson processes whose parameter sets have the same dimensions as the approximated processes have been studied in the literature. In this paper, a special approximation to the one-parameter fractional Brownian motion is constructed using a two-parameter Poisson process. The proof involves the tightness and identification of finite-dimensional distributions.
Approximation by planar elastic curves
Brander, David; Gravesen, Jens; Nørbjerg, Toke Bjerge
2016-01-01
We give an algorithm for approximating a given plane curve segment by a planar elastic curve. The method depends on an analytic representation of the space of elastic curve segments, together with a geometric method for obtaining a good initial guess for the approximating curve. A gradient......-driven optimization is then used to find the approximating elastic curve....
Relativistic Quasiparticle Random Phase Approximation with a Separable Pairing Force
TIAN Yuan; MA Zhong-Yu; Ring Peter
2009-01-01
In our previous work [Phys. Lett. (to be published), Chin. Phys. Lett. 23 (2006) 3226], we introduced a separable pairing force for relativistic Hartree-Bogoliubov calculations. This force was adjusted to reproduce the pairing properties of the Gogny force in nuclear matter. By using the well known techniques of Talmi and Moshinsky it can be expanded in a series of separable terms and converges quickly after a few terms. It was found that the pairing properties can be depicted on almost the same footing as the original pairing interaction, not only in nuclear matter, but also in finite nuclei. In this study, we construct a relativistic quasiparticle random phase approximation (RQRPA ) with this separable pairing interaction and calculate the excitation energies of the first excited 2+ .states and reduced B(E2; 0+ → 2+) transition rates for a chain of Sn isotopes in RQRPA. Compared with the results of the full Gogny force, we find that this simple separable pairing interaction can describe the pairing properties of the excited vibrational states as well as the original pairing interaction.
International Conference Approximation Theory XIV
Schumaker, Larry
2014-01-01
This volume developed from papers presented at the international conference Approximation Theory XIV, held April 7–10, 2013 in San Antonio, Texas. The proceedings contains surveys by invited speakers, covering topics such as splines on non-tensor-product meshes, Wachspress and mean value coordinates, curvelets and shearlets, barycentric interpolation, and polynomial approximation on spheres and balls. Other contributed papers address a variety of current topics in approximation theory, including eigenvalue sequences of positive integral operators, image registration, and support vector machines. This book will be of interest to mathematicians, engineers, and computer scientists working in approximation theory, computer-aided geometric design, numerical analysis, and related approximation areas.
Exact constants in approximation theory
Korneichuk, N
1991-01-01
This book is intended as a self-contained introduction for non-specialists, or as a reference work for experts, to the particular area of approximation theory that is concerned with exact constants. The results apply mainly to extremal problems in approximation theory, which in turn are closely related to numerical analysis and optimization. The book encompasses a wide range of questions and problems: best approximation by polynomials and splines; linear approximation methods, such as spline-approximation; optimal reconstruction of functions and linear functionals. Many of the results are base
Small, David W; Sundstrom, Eric J; Head-Gordon, Martin
2015-01-14
Restricted Hartree Fock using complex-valued orbitals (cRHF) is studied. We introduce an orbital pairing theorem, with which we obtain a concise connection between cRHF and real-valued RHF, and use it to uncover the close relationship between cRHF, unrestricted Hartree Fock, and generalized valence bond perfect pairing. This enables an intuition for cRHF, contrasting with the generally unintuitive nature of complex orbitals. We also describe an efficient computer implementation of cRHF and its corresponding stability analysis. By applying cRHF to the Be + H2 insertion reaction, a Woodward-Hoffmann violating reaction, and a symmetry-driven conical intersection, we demonstrate in genuine molecular systems that cRHF is capable of removing certain potential energy surface singularities that plague real-valued RHF and related methods. This complements earlier work that showed this capability in a model system. We also describe how cRHF is the preferred RHF method for certain radicaloid systems like singlet oxygen and antiaromatic molecules. For singlet O2, we show that standard methods fail even at the equilibrium geometry. An implication of this work is that, regardless of their individual efficacies, cRHF solutions to the HF equations are fairly commonplace.
Kupka, T.; Ruscic, B.; Botto, R. E.; Chemistry
2003-05-01
The nuclear shielding anisotropy and shielding tensor components calculated using the hybrid density functional B3PW91 are reported for a model set of compounds comprised of N{sub 2}, NH{sub 3}, CH{sub 4}, C{sub 2}H{sub 4}, HCN and CH{sub 3}CN. An estimation of density functional theory (DFT) and Hartree-Fock complete basis-set limit (CBS) parameters from a 2 (3) point exact fit vs. least-squares fit was obtained with the cc-pVxZ and aug-cc-pVxZ basis sets (x=D, T, Q, 5, 6). Both Hartree-Fock- and DFT-predicted CBS shielding anisotropies and shielding tensor components of the model molecules were in reasonable agreement with available experimental data. The utility of using a limited CBS approach for calculating accurate anisotropic shielding parameters of larger molecules as complementary methods to solid-state NMR is proposed.
Weimer, Martin; Hieringer, Wolfgang; Sala, Fabio Della; Goerling, Andreas
2005-02-21
The electronic and optical properties of extended functionalized carbyne chains, polyynes and cumulenes, are investigated with the localized Hartree-Fock method, with conventional Kohn-Sham methods, and with the Hartree-Fock method. It is found that even for very long polyynes the carbon-carbon bond lengths within a polyyne alternate while for long cumulenes no carbon-carbon bond length alternation occurs. Polyynes exhibit a finite HOMO-LUMO gap even if they become very long while cumulenes are found to become metallic in the limit of long chain lengths. The geometry and the electro-optical properties of polyynes cannot be influenced significantly by simple sp-{sigma}-bonded end groups. The optically active {sup 1}{sigma}{sub u}{sup +} <- X{sup 1}{sigma}{sub g}{sup +} electronic transition in polyynes is investigated by time-dependent density-functional theory (TDDFT). The known systematic underestimation of excitation energies in large chain-like systems by TDDFT methods is also found for the systems considered here. Deficiencies in the commonly used exchange-correlation kernels are identified as the main source of this shortcoming of TDDFT methods. Unphysical Coulomb self-interactions present in conventional Kohn-Sham potentials seem to not contribute significantly to the problem.
Gould, Tim; Dobson, John F.
2013-01-01
By exploiting freedoms in the definitions of "correlation," "exchange," and "Hartree" physics in ensemble systems, we better generalise the notion of "exact exchange" (EXX) to systems with fractional occupations of the frontier orbitals, arising in the dissociation limit of some molecules. We introduce the linear EXX ("LEXX") theory whose pair distribution and energy are explicitly piecewise linear in the occupations f^{σ }i. We provide explicit expressions for these functions for frontier s and p shells. Used in an optimised effective potential (OEP) approach the LEXX yields energies bounded by the piecewise linear "ensemble EXX" (EEXX) energy and standard fractional optimised EXX energy: EEEXX ⩽ ELEXX ⩽ EEXX. Analysis of the LEXX explains the success of standard OEP methods for diatoms at large spacing, and why they can fail when both spins are allowed to be non-integer so that "ghost" Hartree interactions appear between opposite spin electrons in the usual formula. The energy ELEXX contains a cancellation term for the spin ghost case. It is evaluated for H, Li, and Na fractional ions with clear derivative discontinuities for all cases. The p-shell form reproduces accurate correlation-free energies of B-F and Al-Cl. We further test LEXX plus correlation energy calculations on fractional ions of C and F and again we find both derivative discontinuities and good agreement with exact results.
Gould, Tim; Dobson, John F
2013-01-07
By exploiting freedoms in the definitions of "correlation," "exchange," and "Hartree" physics in ensemble systems, we better generalise the notion of "exact exchange" (EXX) to systems with fractional occupations of the frontier orbitals, arising in the dissociation limit of some molecules. We introduce the linear EXX ("LEXX") theory whose pair distribution and energy are explicitly piecewise linear in the occupations f(i)(σ). We provide explicit expressions for these functions for frontier s and p shells. Used in an optimised effective potential (OEP) approach the LEXX yields energies bounded by the piecewise linear "ensemble EXX" (EEXX) energy and standard fractional optimised EXX energy: E(EEXX) ≤ E(LEXX) ≤ E(EXX). Analysis of the LEXX explains the success of standard OEP methods for diatoms at large spacing, and why they can fail when both spins are allowed to be non-integer so that "ghost" Hartree interactions appear between opposite spin electrons in the usual formula. The energy E(LEXX) contains a cancellation term for the spin ghost case. It is evaluated for H, Li, and Na fractional ions with clear derivative discontinuities for all cases. The p-shell form reproduces accurate correlation-free energies of B-F and Al-Cl. We further test LEXX plus correlation energy calculations on fractional ions of C and F and again we find both derivative discontinuities and good agreement with exact results.
Small, David W.; Sundstrom, Eric J.; Head-Gordon, Martin [Department of Chemistry, University of California, Berkeley, California 94720, USA and Chemical Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720 (United States)
2015-01-14
Restricted Hartree Fock using complex-valued orbitals (cRHF) is studied. We introduce an orbital pairing theorem, with which we obtain a concise connection between cRHF and real-valued RHF, and use it to uncover the close relationship between cRHF, unrestricted Hartree Fock, and generalized valence bond perfect pairing. This enables an intuition for cRHF, contrasting with the generally unintuitive nature of complex orbitals. We also describe an efficient computer implementation of cRHF and its corresponding stability analysis. By applying cRHF to the Be + H{sub 2} insertion reaction, a Woodward-Hoffmann violating reaction, and a symmetry-driven conical intersection, we demonstrate in genuine molecular systems that cRHF is capable of removing certain potential energy surface singularities that plague real-valued RHF and related methods. This complements earlier work that showed this capability in a model system. We also describe how cRHF is the preferred RHF method for certain radicaloid systems like singlet oxygen and antiaromatic molecules. For singlet O{sub 2}, we show that standard methods fail even at the equilibrium geometry. An implication of this work is that, regardless of their individual efficacies, cRHF solutions to the HF equations are fairly commonplace.
BDD Minimization for Approximate Computing
Soeken, Mathias; Grosse, Daniel; Chandrasekharan, Arun; Drechsler, Rolf
2016-01-01
We present Approximate BDD Minimization (ABM) as a problem that has application in approximate computing. Given a BDD representation of a multi-output Boolean function, ABM asks whether there exists another function that has a smaller BDD representation but meets a threshold w.r.t. an error metric. We present operators to derive approximated functions and present algorithms to exactly compute the error metrics directly on the BDD representation. An experimental evaluation demonstrates the app...
Tree wavelet approximations with applications
XU Yuesheng; ZOU Qingsong
2005-01-01
We construct a tree wavelet approximation by using a constructive greedy scheme(CGS). We define a function class which contains the functions whose piecewise polynomial approximations generated by the CGS have a prescribed global convergence rate and establish embedding properties of this class. We provide sufficient conditions on a tree index set and on bi-orthogonal wavelet bases which ensure optimal order of convergence for the wavelet approximations encoded on the tree index set using the bi-orthogonal wavelet bases. We then show that if we use the tree index set associated with the partition generated by the CGS to encode a wavelet approximation, it gives optimal order of convergence.
Diophantine approximation and automorphic spectrum
Ghosh, Anish; Nevo, Amos
2010-01-01
The present paper establishes qunatitative estimates on the rate of diophantine approximation in homogeneous varieties of semisimple algebraic groups. The estimates established generalize and improve previous ones, and are sharp in a number of cases. We show that the rate of diophantine approximation is controlled by the spectrum of the automorphic representation, and is thus subject to the generalised Ramanujan conjectures.
Some results in Diophantine approximation
the basic concepts on which the papers build. Among other it introduces metric Diophantine approximation, Mahler’s approach on algebraic approximation, the Hausdorff measure, and properties of the formal Laurent series over Fq. The introduction ends with a discussion on Mahler’s problem when considered...
Beyond the random phase approximation
Olsen, Thomas; Thygesen, Kristian S.
2013-01-01
We assess the performance of a recently proposed renormalized adiabatic local density approximation (rALDA) for ab initio calculations of electronic correlation energies in solids and molecules. The method is an extension of the random phase approximation (RPA) derived from time-dependent density...
Uniform approximation by (quantum) polynomials
Drucker, A.; de Wolf, R.
2011-01-01
We show that quantum algorithms can be used to re-prove a classical theorem in approximation theory, Jackson's Theorem, which gives a nearly-optimal quantitative version of Weierstrass's Theorem on uniform approximation of continuous functions by polynomials. We provide two proofs, based respectivel
Chushnyakova, M. V.; Bhattacharya, R.; Gontchar, I. I.
2014-07-01
Background: In our previous paper [Gontchar et al., Phys. Rev. C 89, 034601 (2014), 10.1103/PhysRevC.89.034601] we have calculated the capture (fusion) excitation functions for several reactions with O16,Si28, and S32 nuclei as the projectiles and Zr92,Sm144, and Pb208 nuclei as the targets. These calculations were performed by using our fluctuation-dissipation trajectory model based on the double-folding approach with the density-dependent M3Y NN forces that include the finite range exchange part. For the nuclear matter density the Hartree-Fock approach with the SKP coefficient set that includes the tensor interaction was applied. It was found that for most of the reactions induced by O16 the calculated cross sections cannot be brought into agreement with the data. This suggested that the deviation in the calculated nuclear density for O16 from the experimental one was crucial. Method: The SKX parameter set is used to obtain the nuclear densities. Reactions with C12 and S36 as the projectiles and Pb204 as the target are included in the analysis in addition to those of the previous paper. Only data that correspond to the collision energy Ec.m.>1.1UB0 (UB0 is the s-wave fusion barrier height) are included in the analysis. The radial friction strength KR is used as the individual adjustable parameter for each reaction. Results: For all 13 reactions (91 points) it is possible to reach an agreement with the experimental fusion cross sections within 10%. Only at ten points does the deviation exceed 5%. The value of KR, which provides the best agreement with the data in general, decreases as the system gets heavier in accord with the previous paper [Gontchar et al., Phys. Rev. C 89, 034601 (2014), 10.1103/PhysRevC.89.034601]. A universal analytical approximation for the dependence of KR upon the Coulomb barrier height is found. Conclusions: The developed model is able to reproduce the above-barrier portion of the fusion excitation function within 5% with a probability of
Global approximation of convex functions
Azagra, D
2011-01-01
We show that for every (not necessarily bounded) open convex subset $U$ of $\\R^n$, every (not necessarily Lipschitz or strongly) convex function $f:U\\to\\R$ can be approximated by real analytic convex functions, uniformly on all of $U$. In doing so we provide a technique which transfers results on uniform approximation on bounded sets to results on uniform approximation on unbounded sets, in such a way that not only convexity and $C^k$ smoothness, but also local Lipschitz constants, minimizers, order, and strict or strong convexity, are preserved. This transfer method is quite general and it can also be used to obtain new results on approximation of convex functions defined on Riemannian manifolds or Banach spaces. We also provide a characterization of the class of convex functions which can be uniformly approximated on $\\R^n$ by strongly convex functions.
Approximate circuits for increased reliability
Hamlet, Jason R.; Mayo, Jackson R.
2015-08-18
Embodiments of the invention describe a Boolean circuit having a voter circuit and a plurality of approximate circuits each based, at least in part, on a reference circuit. The approximate circuits are each to generate one or more output signals based on values of received input signals. The voter circuit is to receive the one or more output signals generated by each of the approximate circuits, and is to output one or more signals corresponding to a majority value of the received signals. At least some of the approximate circuits are to generate an output value different than the reference circuit for one or more input signal values; however, for each possible input signal value, the majority values of the one or more output signals generated by the approximate circuits and received by the voter circuit correspond to output signal result values of the reference circuit.
Approximate circuits for increased reliability
Hamlet, Jason R.; Mayo, Jackson R.
2015-12-22
Embodiments of the invention describe a Boolean circuit having a voter circuit and a plurality of approximate circuits each based, at least in part, on a reference circuit. The approximate circuits are each to generate one or more output signals based on values of received input signals. The voter circuit is to receive the one or more output signals generated by each of the approximate circuits, and is to output one or more signals corresponding to a majority value of the received signals. At least some of the approximate circuits are to generate an output value different than the reference circuit for one or more input signal values; however, for each possible input signal value, the majority values of the one or more output signals generated by the approximate circuits and received by the voter circuit correspond to output signal result values of the reference circuit.
Schimeczek, C.; Engel, D.; Wunner, G.
2012-07-01
Our previously published code for calculating energies and bound-bound transitions of medium-Z elements at neutron star magnetic field strengths [D. Engel, M. Klews, G. Wunner, Comput. Phys. Comm. 180 (2009) 302-311] was based on the adiabatic approximation. It assumes a complete decoupling of the (fast) gyration of the electrons under the action of the magnetic field and the (slow) bound motion along the field under the action of the Coulomb forces. For the single-particle orbitals this implied that each is a product of a Landau state and an (unknown) longitudinal wave function whose B-spline coefficients were determined self-consistently by solving the Hartree-Fock equations for the many-electron problem on a finite-element grid. In the present code we go beyond the adiabatic approximation, by allowing the transverse part of each orbital to be a superposition of Landau states, while assuming that the longitudinal part can be approximated by the same wave function in each Landau level. Inserting this ansatz into the energy variational principle leads to a system of coupled equations in which the B-spline coefficients depend on the weights of the individual Landau states, and vice versa, and which therefore has to be solved in a doubly self-consistent manner. The extended ansatz takes into account the back-reaction of the Coulomb motion of the electrons along the field direction on their motion in the plane perpendicular to the field, an effect which cannot be captured by the adiabatic approximation. The new code allows for the inclusion of up to 8 Landau levels. This reduces the relative error of energy values as compared to the adiabatic approximation results by typically a factor of three (1/3 of the original error), and yields accurate results also in regions of lower neutron star magnetic field strengths where the adiabatic approximation fails. Further improvements in the code are a more sophisticated choice of the initial wave functions, which takes into
Kalinowski, Jaroslaw; Wennmohs, Frank; Neese, Frank
2017-07-11
A resolution of identity based implementation of the Hartree-Fock method on graphical processing units (GPUs) is presented that is capable of handling basis functions with arbitrary angular momentum. For practical reasons, only functions up to (ff|f) angular momentum are presently calculated on the GPU, thus leaving the calculation of higher angular momenta integrals on the CPU of the hybrid CPU-GPU environment. Speedups of up to a factor of 30 are demonstrated relative to state-of-the-art serial and parallel CPU implementations. Benchmark calculations with over 3500 contracted basis functions (def2-SVP or def2-TZVP basis sets) are reported. The presented implementation supports all devices with OpenCL support and is capable of utilizing multiple GPU cards over either MPI or OpenCL itself.
Thiele, Robert; Son, Sang-Kil [Center for Free-Electron Laser Science, DESY, 22607 Hamburg (Germany); Ziaja, Beata [Center for Free-Electron Laser Science, DESY, 22607 Hamburg (Germany); Institute of Nuclear Physics, Polish Academy of Sciences, Radzikowskiego 152, 31-342 Krakow (Poland); Santra, Robin [Center for Free-Electron Laser Science, DESY, 22607 Hamburg (Germany); Department of Physics, University of Hamburg, 20355 Hamburg (Germany)
2013-07-01
X-ray free-electron lasers (XFELs) are a promising tool for the structural determination of macro- and biomolecules, using coherent diffractive imaging. During imaging, the intense XFEL pulses also efficiently ionize the molecules, so it is important to estimate how the charged environment within the molecule modifies atomic properties, in comparison to the case of an isolated atom. Here, we apply the XATOM toolkit to obtain predictions on the modified ionization thresholds and rates of some photoinduced processes in carbon. The Hartree-Fock-Slater model is extended to include the electron screening and ion correlation effects, induced by external charges. With this extended model, we obtain predictions on modifications of orbital energies, photoabsorption cross sections, Auger decay rates, fluorescence emission rates, and atomic scattering factors as a function of the density and temperature of the surrounding charges. Our results have implications for the studies of dynamics within XFEL irradiated samples, in particular for those dedicated to coherent diffraction imaging.
Greenman, Loren; Haxton, Daniel J; McCurdy, C William
2016-01-01
We have verified a mechanism for Raman excitation of atoms through continuum levels previously obtained by quantum optimal control using the multi-configurational time-dependent Hartree-Fock (MCTDHF) method. This mechanism, which was obtained at the time-dependent configuration interaction singles (TDCIS) level of theory, involves sequentially exciting an atom from the ground state to an intermediate core-hole state using a long pump pulse, and then transferring this population to the target Raman state with a shorter Stokes pulse. This process represents the first step in a multidimensional x-ray spectroscopy scheme that will provide a local probe of valence electronic correlations. Although at the optimal pulse intensities at the TDCIS level of theory the MCTDHF method predicts multiple ionization of the atom, at slightly lower intensities (reduced by a factor of about 4) the TDCIS mechanism is shown to hold qualitatively. Quantitatively, the MCTDHF populations are reduced from the TDCIS calculations by a f...
Carmona-Novillo, E.; Campos-Martínez, J.; Hernández, M. I.; Roncero, O.; Villarreal, P.; Delgado Barrio, G.
In this work we explore the application of a time-dependent Hartree (TDH) scheme to study the vibrational predissociation of Ne2I2 van der Waals clusters. The present approach is based on equations of motion extracted from the usual variational principle where the Hamiltonian has been previously represented in a set of diatomic vibrational states. The procedure leads to a set of coupled equations for the different modes on each diabatic state with, however, explicit separation between those modes. The application on a problem that inherently requires long-time propagation is shown to be successful. Calculated lifetimes compare well with previous calculations as well as with available experimental data. A more detailed mechanism, as the breath of the angular mode on the different vibrational channels, is better described.
Zhang, Ying; Meng, Jie
2010-01-01
The neutron pair correlation in nuclei near the neutron drip-line is investigated using the selfconsistent continuum Skyrme-Hartree-Fock-Bogoliubov theory formulated with the coordinate-space Green's function technique. Numerical analysis is performed for even-even N = 86 isotones in the Mo-Sn region, where the 3p3/2 and 3p1/2 orbits lying near the Fermi energy are either weakly bound or unbound. The quasiparticle states originating from the l = 1 orbits form resonances with large widths, which are due to the low barrier height and the strong continuum coupling caused by the pair potential. Analyzing in detail the pairing properties and roles of the quasiparticle resonances, we found that the l = 1 broad quasiparticle resonances persist to feel the pair potential and contribute to the pair correlation even when their widths are comparable with the resonance energy.
Jansík, Branislav; Høst, Stinne; Johansson, Mikael P; Olsen, Jeppe; Jørgensen, Poul; Helgaker, Trygve
2009-07-21
A hierarchical optimisation strategy has been introduced for minimising the Hartree-Fock/Kohn-Sham energy, consisting of three levels (3L): an atom-in-a-molecule optimisation, a valence-basis molecular optimisation, and a full-basis molecular optimisation. The density matrix formed at one level is used as a starting density matrix at the next level with no loss of information. To ensure a fast and reliable convergence to a minimum, the augmented Roothaan-Hall (ARH) algorithm is used in both the valence-basis and full-basis molecular optimisations. The performance of the ARH-3L method is compared with standard optimisation algorithms. Both for efficiency and reliability, we recommend to use the ARH-3L algorithm.
Zhang, Lin-Feng; Xia, Xue-Wei
2016-05-01
The α-decay energies (Q α ) are systematically investigated with the nuclear masses for 10 ⩽ Z ⩽ 120 isotopes obtained by the relativistic continuum Hartree-Bogoliubov (RCHB) theory with the covariant density functional PC-PK1, and compared with available experimental values. It is found that the α-decay energies deduced from the RCHB results present a similar pattern to those from available experiments. Owing to the large predicted Q α values (⩾ 4 MeV), many undiscovered heavy nuclei in the proton-rich side and super-heavy nuclei may have large possibilities for α-decay. The influence of nuclear shell structure on α-decay energies is also analysed. Supported by Major State 973 Program of China (2013CB834400), National Natural Science Foundation of China (11175002, 11335002, 11375015, 11461141002), Research Fund for the Doctoral Program of Higher Education (20110001110087) and National Undergraduate Innovation Training Programs of Peking University.
Markó, Gergely; Szép, Zsolt
2012-01-01
We study the phase transition of a real scalar phi^4 theory in the two-loop Phi-derivable approximation using the imaginary time formalism, extending our previous (analytical) discussion of the Hartree approximation. We combine Fast Fourier Transform algorithms and accelerated Matsubara sums in order to achieve a high accuracy. Our results confirm and complete earlier ones obtained in the real time formalism [1] but which were less accurate due to the integration in Minkowski space and the discretization of the spectral density function. We also provide a complete and explicit discussion of the renormalization of the two-loop Phi-derivable approximation at finite temperature, both in the symmetric and in the broken phase, which was already used in the real-time approach, but never published. Our main result is that the two-loop Phi-derivable approximation suffices to cure the problem of the Hartree approximation regarding the order of the transition: the transition is of the second order type, as expected on ...
{beta}-decay rates of r-process nuclei in the relativistic quasiparticle random phase approximation
Niksic, T.; Marketin, T.; Vretenar, D. [Zagreb Univ. (Croatia). Faculty of Science, Physics Dept.; Paar, N. [Technische Univ. Darmstadt (Germany). Inst. fuer Kernphysik; Ring, P. [Technische Univ. Muenchen, Garching (Germany). Physik-Department
2004-12-08
The fully consistent relativistic proton-neutron quasiparticle random phase approximation (PN-RQRPA) is employed in the calculation of {beta}-decay half-lives of neutron-rich nuclei in the N{approx}50 and N{approx}82 regions. A new density-dependent effective interaction, with an enhanced value of the nucleon effective mass, is used in relativistic Hartree-Bogolyubov calculation of nuclear ground states and in the particle-hole channel of the PN-RQRPA. The finite range Gogny D1S interaction is employed in the T=1 pairing channel, and the model also includes a proton-neutron particle-particle interaction. The theoretical half-lives reproduce the experimental data for the Fe, Zn, Cd, and Te isotopic chains, but overestimate the lifetimes of Ni isotopes and predict a stable {sup 132}Sn. (orig.)
Symmetry-broken local-density approximation for one-dimensional systems
Rogers, Fergus J M; Loos, Pierre-François
2016-01-01
Within density-functional theory, the local-density approximation (LDA) correlation functional is typically built by fitting the difference between the near-exact and Hartree-Fock (HF) energies of the uniform electron gas (UEG), together with analytic perturbative results from the high- and low-density regimes. Near-exact energies are obtained by performing accurate diffusion Monte Carlo calculations, while HF energies are usually assumed to be the Fermi fluid HF energy. However, it has been known since the seminal work of Overhauser that one can obtain lower, symmetry-broken (SB) HF energies at any density. Here, we have computed the SBHF energies of the one-dimensional UEG and constructed a SB version of the LDA (SBLDA) from the results. We compare the performance of the LDA and SBLDA functionals when applied to one-dimensional systems, including atoms and molecules. Generalization to higher dimensions is also discussed.
Ramos, E.; Silva-Valencia, J.; Franco, R.; Siqueira, E. C.; Figueira, M. S.
2015-11-01
We study the spin-current Seebeck effect through an immersed gate defined quantum dot, employing the U-finite atomic method for the single impurity Anderson model. Our description qualitatively confirms some of the results obtained by an earlier Hartree-Fock work, but as our calculation includes the Kondo effect, some new features will appear in the spin-current Seebeck effect S, which as a function of the gate voltage present an oscillatory shape. At intermediate temperatures, our results show a three zero structure and at low temperatures, our results are governed by the emergence of the Kondo peak in the transmittance, which defines the behavior of the shape of the S coefficient as a function of the parameters of the model. The oscillatory behavior obtained by the Hartree-Fock approximation reproduces the shape obtained by us in a non-interacting system (U=0). The S sign is sensitive to different polarization of the quantum dot, and as a consequence the device could be employed to experimentally detect the polarization states of the system. Our results also confirm that the large increase of S upon increasing U, obtained by the mean field approximation, is correct only for low temperatures. We also discuss the role of the Kondo peak in defining the behavior of the spin thermopower at low temperatures.
Thermodynamics and phase transition of the O(N) model from the two-loop Phi-derivable approximation
Markó, Gergely; Szép, Zsolt
2013-01-01
We discuss the thermodynamics of the O(N) model across the corresponding phase transition using the two-loop Phi-derivable approximation of the effective potential and compare our results to those obtained in the literature within the Hartree-Fock approximation. In particular, we find that in the chiral limit the transition is of the second order, whereas it was found to be of the first order in the Hartree-Fock case. These features are manifest at the level of the thermodynamical observables. We also compute the thermal sigma and pion masses from the curvature of the effective potential. In the chiral limit, this guarantees that the Goldstone theorem is obeyed in the broken phase. A realistic parametrization of the model in the N=4 case, based on the vacuum values of the curvature masses, shows that a sigma mass of around 450 MeV can be obtained. The equations are renormalized after extending our previous results for the N=1 case by means of the general procedure described in [J. Berges et al., Annals Phys. ...
Dziedzic, J; Hill, Q; Skylaris, C-K
2013-12-07
We present a method for the calculation of four-centre two-electron repulsion integrals in terms of localised non-orthogonal generalised Wannier functions (NGWFs). Our method has been implemented in the ONETEP program and is used to compute the Hartree-Fock exchange energy component of Hartree-Fock and Density Functional Theory (DFT) calculations with hybrid exchange-correlation functionals. As the NGWFs are optimised in situ in terms of a systematically improvable basis set which is equivalent to plane waves, it is possible to achieve large basis set accuracy in routine calculations. The spatial localisation of the NGWFs allows us to exploit the exponential decay of the density matrix in systems with a band gap in order to compute the exchange energy with a computational effort that increases linearly with the number of atoms. We describe the implementation of this approach in the ONETEP program for linear-scaling first principles quantum mechanical calculations. We present extensive numerical validation of all the steps in our method. Furthermore, we find excellent agreement in energies and structures for a wide variety of molecules when comparing with other codes. We use our method to perform calculations with the B3LYP exchange-correlation functional for models of myoglobin systems bound with O2 and CO ligands and confirm that the same qualitative behaviour is obtained as when the same myoglobin models are studied with the DFT+U approach which is also available in ONETEP. Finally, we confirm the linear-scaling capability of our method by performing calculations on polyethylene and polyacetylene chains of increasing length.
Rytov approximation in electron scattering
Krehl, Jonas; Lubk, Axel
2017-06-01
In this work we introduce the Rytov approximation in the scope of high-energy electron scattering with the motivation of developing better linear models for electron scattering. Such linear models play an important role in tomography and similar reconstruction techniques. Conventional linear models, such as the phase grating approximation, have reached their limits in current and foreseeable applications, most importantly in achieving three-dimensional atomic resolution using electron holographic tomography. The Rytov approximation incorporates propagation effects which are the most pressing limitation of conventional models. While predominately used in the weak-scattering regime of light microscopy, we show that the Rytov approximation can give reasonable results in the inherently strong-scattering regime of transmission electron microscopy.
Rollout sampling approximate policy iteration
Dimitrakakis, C.; Lagoudakis, M.G.
2008-01-01
Several researchers have recently investigated the connection between reinforcement learning and classification. We are motivated by proposals of approximate policy iteration schemes without value functions, which focus on policy representation using classifiers and address policy learning as a
Approximate common divisors via lattices
Cohn, Henry
2011-01-01
We analyze the multivariate generalization of Howgrave-Graham's algorithm for the approximate common divisor problem. In the m-variable case with modulus N and approximate common divisor of size N^beta, this improves the size of the error tolerated from N^(beta^2) to N^(beta^((m+1)/m)), under a commonly used heuristic assumption. This gives a more detailed analysis of the hardness assumption underlying the recent fully homomorphic cryptosystem of van Dijk, Gentry, Halevi, and Vaikuntanathan. While these results do not challenge the suggested parameters, a 2^sqrt(n) approximation algorithm for lattice basis reduction in n dimensions could be used to break these parameters. We have implemented our algorithm, and it performs better in practice than the theoretical analysis suggests. Our results fit into a broader context of analogies between cryptanalysis and coding theory. The multivariate approximate common divisor problem is the number-theoretic analogue of noisy multivariate polynomial interpolation, and we ...
Approximate Implicitization Using Linear Algebra
Oliver J. D. Barrowclough
2012-01-01
Full Text Available We consider a family of algorithms for approximate implicitization of rational parametric curves and surfaces. The main approximation tool in all of the approaches is the singular value decomposition, and they are therefore well suited to floating-point implementation in computer-aided geometric design (CAGD systems. We unify the approaches under the names of commonly known polynomial basis functions and consider various theoretical and practical aspects of the algorithms. We offer new methods for a least squares approach to approximate implicitization using orthogonal polynomials, which tend to be faster and more numerically stable than some existing algorithms. We propose several simple propositions relating the properties of the polynomial bases to their implicit approximation properties.
Binary nucleation beyond capillarity approximation
Kalikmanov, V.I.
2010-01-01
Large discrepancies between binary classical nucleation theory (BCNT) and experiments result from adsorption effects and inability of BCNT, based on the phenomenological capillarity approximation, to treat small clusters. We propose a model aimed at eliminating both of these deficiencies. Adsorption
Weighted approximation with varying weight
Totik, Vilmos
1994-01-01
A new construction is given for approximating a logarithmic potential by a discrete one. This yields a new approach to approximation with weighted polynomials of the form w"n"(" "= uppercase)P"n"(" "= uppercase). The new technique settles several open problems, and it leads to a simple proof for the strong asymptotics on some L p(uppercase) extremal problems on the real line with exponential weights, which, for the case p=2, are equivalent to power- type asymptotics for the leading coefficients of the corresponding orthogonal polynomials. The method is also modified toyield (in a sense) uniformly good approximation on the whole support. This allows one to deduce strong asymptotics in some L p(uppercase) extremal problems with varying weights. Applications are given, relating to fast decreasing polynomials, asymptotic behavior of orthogonal polynomials and multipoint Pade approximation. The approach is potential-theoretic, but the text is self-contained.
Nonlinear approximation with redundant dictionaries
Borup, Lasse; Nielsen, M.; Gribonval, R.
2005-01-01
In this paper we study nonlinear approximation and data representation with redundant function dictionaries. In particular, approximation with redundant wavelet bi-frame systems is studied in detail. Several results for orthonormal wavelets are generalized to the redundant case. In general......, for a wavelet bi-frame system the approximation properties are limited by the number of vanishing moments of the system. In some cases this can be overcome by oversampling, but at a price of replacing the canonical expansion by another linear expansion. Moreover, for special non-oversampled wavelet bi-frames we...... can obtain good approximation properties not restricted by the number of vanishing moments, but again without using the canonical expansion....
Mathematical algorithms for approximate reasoning
Murphy, John H.; Chay, Seung C.; Downs, Mary M.
1988-01-01
Most state of the art expert system environments contain a single and often ad hoc strategy for approximate reasoning. Some environments provide facilities to program the approximate reasoning algorithms. However, the next generation of expert systems should have an environment which contain a choice of several mathematical algorithms for approximate reasoning. To meet the need for validatable and verifiable coding, the expert system environment must no longer depend upon ad hoc reasoning techniques but instead must include mathematically rigorous techniques for approximate reasoning. Popular approximate reasoning techniques are reviewed, including: certainty factors, belief measures, Bayesian probabilities, fuzzy logic, and Shafer-Dempster techniques for reasoning. A group of mathematically rigorous algorithms for approximate reasoning are focused on that could form the basis of a next generation expert system environment. These algorithms are based upon the axioms of set theory and probability theory. To separate these algorithms for approximate reasoning various conditions of mutual exclusivity and independence are imposed upon the assertions. Approximate reasoning algorithms presented include: reasoning with statistically independent assertions, reasoning with mutually exclusive assertions, reasoning with assertions that exhibit minimum overlay within the state space, reasoning with assertions that exhibit maximum overlay within the state space (i.e. fuzzy logic), pessimistic reasoning (i.e. worst case analysis), optimistic reasoning (i.e. best case analysis), and reasoning with assertions with absolutely no knowledge of the possible dependency among the assertions. A robust environment for expert system construction should include the two modes of inference: modus ponens and modus tollens. Modus ponens inference is based upon reasoning towards the conclusion in a statement of logical implication, whereas modus tollens inference is based upon reasoning away
Twisted inhomogeneous Diophantine approximation and badly approximable sets
Harrap, Stephen
2010-01-01
For any real pair i, j geq 0 with i+j=1 let Bad(i, j) denote the set of (i, j)-badly approximable pairs. That is, Bad(i, j) consists of irrational vectors x:=(x_1, x_2) in R^2 for which there exists a positive constant c(x) such that max {||qx_1||^(-i), ||qx_2||^(-j)} > c(x)/q for all q in N. Building on a result of Kurzweil, a new characterization of the set Bad(i, j) in terms of `well-approximable' vectors in the area of `twisted' inhomogeneous Diophantine approximation is established. In addition, it is shown that Bad^x(i, j), the `twisted' inhomogeneous analogue of Bad(i, j), has full Hausdorff dimension 2 when x is chosen from the set Bad(i, j).
Yang, Weitao; Mori-Sánchez, Paula; Cohen, Aron J
2013-09-14
The exact conditions for density functionals and density matrix functionals in terms of fractional charges and fractional spins are known, and their violation in commonly used functionals has been shown to be the root of many major failures in practical applications. However, approximate functionals are designed for physical systems with integer charges and spins, not in terms of the fractional variables. Here we develop a general framework for extending approximate density functionals and many-electron theory to fractional-charge and fractional-spin systems. Our development allows for the fractional extension of any approximate theory that is a functional of G(0), the one-electron Green's function of the non-interacting reference system. The extension to fractional charge and fractional spin systems is based on the ensemble average of the basic variable, G(0). We demonstrate the fractional extension for the following theories: (1) any explicit functional of the one-electron density, such as the local density approximation and generalized gradient approximations; (2) any explicit functional of the one-electron density matrix of the non-interacting reference system, such as the exact exchange functional (or Hartree-Fock theory) and hybrid functionals; (3) many-body perturbation theory; and (4) random-phase approximations. A general rule for such an extension has also been derived through scaling the orbitals and should be useful for functionals where the link to the Green's function is not obvious. The development thus enables the examination of approximate theories against known exact conditions on the fractional variables and the analysis of their failures in chemical and physical applications in terms of violations of exact conditions of the energy functionals. The present work should facilitate the calculation of chemical potentials and fundamental bandgaps with approximate functionals and many-electron theories through the energy derivatives with respect to the
Reinforcement Learning via AIXI Approximation
Veness, Joel; Hutter, Marcus; Silver, David
2010-01-01
This paper introduces a principled approach for the design of a scalable general reinforcement learning agent. This approach is based on a direct approximation of AIXI, a Bayesian optimality notion for general reinforcement learning agents. Previously, it has been unclear whether the theory of AIXI could motivate the design of practical algorithms. We answer this hitherto open question in the affirmative, by providing the first computationally feasible approximation to the AIXI agent. To develop our approximation, we introduce a Monte Carlo Tree Search algorithm along with an agent-specific extension of the Context Tree Weighting algorithm. Empirically, we present a set of encouraging results on a number of stochastic, unknown, and partially observable domains.
Approximate Matching of Hierarchial Data
Augsten, Nikolaus
The goal of this thesis is to design, develop, and evaluate new methods for the approximate matching of hierarchical data represented as labeled trees. In approximate matching scenarios two items should be matched if they are similar. Computing the similarity between labeled trees is hard...... as in addition to the data values also the structure must be considered. A well-known measure for comparing trees is the tree edit distance. It is computationally expensive and leads to a prohibitively high run time. Our solution for the approximate matching of hierarchical data are pq-grams. The pq...... formally proof that the pq-gram index can be incrementally updated based on the log of edit operations without reconstructing intermediate tree versions. The incremental update is independent of the data size and scales to a large number of changes in the data. We introduce windowed pq...
Concept Approximation between Fuzzy Ontologies
无
2006-01-01
Fuzzy ontologies are efficient tools to handle fuzzy and uncertain knowledge on the semantic web; but there are heterogeneity problems when gaining interoperability among different fuzzy ontologies. This paper uses concept approximation between fuzzy ontologies based on instances to solve the heterogeneity problems. It firstly proposes an instance selection technology based on instance clustering and weighting to unify the fuzzy interpretation of different ontologies and reduce the number of instances to increase the efficiency. Then the paper resolves the problem of computing the approximations of concepts into the problem of computing the least upper approximations of atom concepts. It optimizes the search strategies by extending atom concept sets and defining the least upper bounds of concepts to reduce the searching space of the problem. An efficient algorithm for searching the least upper bounds of concept is given.
Approximating Graphic TSP by Matchings
Mömke, Tobias
2011-01-01
We present a framework for approximating the metric TSP based on a novel use of matchings. Traditionally, matchings have been used to add edges in order to make a given graph Eulerian, whereas our approach also allows for the removal of certain edges leading to a decreased cost. For the TSP on graphic metrics (graph-TSP), the approach yields a 1.461-approximation algorithm with respect to the Held-Karp lower bound. For graph-TSP restricted to a class of graphs that contains degree three bounded and claw-free graphs, we show that the integrality gap of the Held-Karp relaxation matches the conjectured ratio 4/3. The framework allows for generalizations in a natural way and also leads to a 1.586-approximation algorithm for the traveling salesman path problem on graphic metrics where the start and end vertices are prespecified.
Diophantine approximation and Dirichlet series
Queffélec, Hervé
2013-01-01
This self-contained book will benefit beginners as well as researchers. It is devoted to Diophantine approximation, the analytic theory of Dirichlet series, and some connections between these two domains, which often occur through the Kronecker approximation theorem. Accordingly, the book is divided into seven chapters, the first three of which present tools from commutative harmonic analysis, including a sharp form of the uncertainty principle, ergodic theory and Diophantine approximation to be used in the sequel. A presentation of continued fraction expansions, including the mixing property of the Gauss map, is given. Chapters four and five present the general theory of Dirichlet series, with classes of examples connected to continued fractions, the famous Bohr point of view, and then the use of random Dirichlet series to produce non-trivial extremal examples, including sharp forms of the Bohnenblust-Hille theorem. Chapter six deals with Hardy-Dirichlet spaces, which are new and useful Banach spaces of anal...
Approximation methods in probability theory
Čekanavičius, Vydas
2016-01-01
This book presents a wide range of well-known and less common methods used for estimating the accuracy of probabilistic approximations, including the Esseen type inversion formulas, the Stein method as well as the methods of convolutions and triangle function. Emphasising the correct usage of the methods presented, each step required for the proofs is examined in detail. As a result, this textbook provides valuable tools for proving approximation theorems. While Approximation Methods in Probability Theory will appeal to everyone interested in limit theorems of probability theory, the book is particularly aimed at graduate students who have completed a standard intermediate course in probability theory. Furthermore, experienced researchers wanting to enlarge their toolkit will also find this book useful.
Approximate Sparse Regularized Hyperspectral Unmixing
Chengzhi Deng
2014-01-01
Full Text Available Sparse regression based unmixing has been recently proposed to estimate the abundance of materials present in hyperspectral image pixel. In this paper, a novel sparse unmixing optimization model based on approximate sparsity, namely, approximate sparse unmixing (ASU, is firstly proposed to perform the unmixing task for hyperspectral remote sensing imagery. And then, a variable splitting and augmented Lagrangian algorithm is introduced to tackle the optimization problem. In ASU, approximate sparsity is used as a regularizer for sparse unmixing, which is sparser than l1 regularizer and much easier to be solved than l0 regularizer. Three simulated and one real hyperspectral images were used to evaluate the performance of the proposed algorithm in comparison to l1 regularizer. Experimental results demonstrate that the proposed algorithm is more effective and accurate for hyperspectral unmixing than state-of-the-art l1 regularizer.
Transfinite Approximation of Hindman's Theorem
Beiglböck, Mathias
2010-01-01
Hindman's Theorem states that in any finite coloring of the integers, there is an infinite set all of whose finite sums belong to the same color. This is much stronger than the corresponding finite form, stating that in any finite coloring of the integers there are arbitrarily long finite sets with the same property. We extend the finite form of Hindman's Theorem to a "transfinite" version for each countable ordinal, and show that Hindman's Theorem is equivalent to the appropriate transfinite approximation holding for every countable ordinal. We then give a proof of Hindman's Theorem by directly proving these transfinite approximations.
Tree wavelet approximations with applications
无
2005-01-01
[1]Baraniuk, R. G., DeVore, R. A., Kyriazis, G., Yu, X. M., Near best tree approximation, Adv. Comput. Math.,2002, 16: 357-373.[2]Cohen, A., Dahmen, W., Daubechies, I., DeVore, R., Tree approximation and optimal encoding, Appl. Comput.Harmonic Anal., 2001, 11: 192-226.[3]Dahmen, W., Schneider, R., Xu, Y., Nonlinear functionals of wavelet expansions-adaptive reconstruction and fast evaluation, Numer. Math., 2000, 86: 49-101.[4]DeVore, R. A., Nonlinear approximation, Acta Numer., 1998, 7: 51-150.[5]Davis, G., Mallat, S., Avellaneda, M., Adaptive greedy approximations, Const. Approx., 1997, 13: 57-98.[6]DeVore, R. A., Temlyakov, V. N., Some remarks on greedy algorithms, Adv. Comput. Math., 1996, 5: 173-187.[7]Kashin, B. S., Temlyakov, V. N., Best m-term approximations and the entropy of sets in the space L1, Mat.Zametki (in Russian), 1994, 56: 57-86.[8]Temlyakov, V. N., The best m-term approximation and greedy algorithms, Adv. Comput. Math., 1998, 8:249-265.[9]Temlyakov, V. N., Greedy algorithm and m-term trigonometric approximation, Constr. Approx., 1998, 14:569-587.[10]Hutchinson, J. E., Fractals and self similarity, Indiana. Univ. Math. J., 1981, 30: 713-747.[11]Binev, P., Dahmen, W., DeVore, R. A., Petruchev, P., Approximation classes for adaptive methods, Serdica Math.J., 2002, 28: 1001-1026.[12]Gilbarg, D., Trudinger, N. S., Elliptic Partial Differential Equations of Second Order, Berlin: Springer-Verlag,1983.[13]Ciarlet, P. G., The Finite Element Method for Elliptic Problems, New York: North Holland, 1978.[14]Birman, M. S., Solomiak, M. Z., Piecewise polynomial approximation of functions of the class Wαp, Math. Sb.,1967, 73: 295-317.[15]DeVore, R. A., Lorentz, G. G., Constructive Approximation, New York: Springer-Verlag, 1993.[16]DeVore, R. A., Popov, V., Interpolation of Besov spaces, Trans. Amer. Math. Soc., 1988, 305: 397-414.[17]Devore, R., Jawerth, B., Popov, V., Compression of wavelet decompositions, Amer. J. Math., 1992, 114: 737-785.[18]Storozhenko, E
WKB Approximation in Noncommutative Gravity
Maja Buric
2007-12-01
Full Text Available We consider the quasi-commutative approximation to a noncommutative geometry defined as a generalization of the moving frame formalism. The relation which exists between noncommutativity and geometry is used to study the properties of the high-frequency waves on the flat background.
Approximation properties of haplotype tagging
Dreiseitl Stephan
2006-01-01
Full Text Available Abstract Background Single nucleotide polymorphisms (SNPs are locations at which the genomic sequences of population members differ. Since these differences are known to follow patterns, disease association studies are facilitated by identifying SNPs that allow the unique identification of such patterns. This process, known as haplotype tagging, is formulated as a combinatorial optimization problem and analyzed in terms of complexity and approximation properties. Results It is shown that the tagging problem is NP-hard but approximable within 1 + ln((n2 - n/2 for n haplotypes but not approximable within (1 - ε ln(n/2 for any ε > 0 unless NP ⊂ DTIME(nlog log n. A simple, very easily implementable algorithm that exhibits the above upper bound on solution quality is presented. This algorithm has running time O((2m - p + 1 ≤ O(m(n2 - n/2 where p ≤ min(n, m for n haplotypes of size m. As we show that the approximation bound is asymptotically tight, the algorithm presented is optimal with respect to this asymptotic bound. Conclusion The haplotype tagging problem is hard, but approachable with a fast, practical, and surprisingly simple algorithm that cannot be significantly improved upon on a single processor machine. Hence, significant improvement in computatational efforts expended can only be expected if the computational effort is distributed and done in parallel.
Truthful approximations to range voting
Filos-Ratsika, Aris; Miltersen, Peter Bro
We consider the fundamental mechanism design problem of approximate social welfare maximization under general cardinal preferences on a finite number of alternatives and without money. The well-known range voting scheme can be thought of as a non-truthful mechanism for exact social welfare...
Approximate Reasoning with Fuzzy Booleans
Broek, van den P.M.; Noppen, J.A.R.
2004-01-01
This paper introduces, in analogy to the concept of fuzzy numbers, the concept of fuzzy booleans, and examines approximate reasoning with the compositional rule of inference using fuzzy booleans. It is shown that each set of fuzzy rules is equivalent to a set of fuzzy rules with singleton crisp ante
Ultrafast Approximation for Phylogenetic Bootstrap
Bui Quang Minh, [No Value; Nguyen, Thi; von Haeseler, Arndt
2013-01-01
Nonparametric bootstrap has been a widely used tool in phylogenetic analysis to assess the clade support of phylogenetic trees. However, with the rapidly growing amount of data, this task remains a computational bottleneck. Recently, approximation methods such as the RAxML rapid bootstrap (RBS) and
On badly approximable complex numbers
Esdahl-Schou, Rune; Kristensen, S.
We show that the set of complex numbers which are badly approximable by ratios of elements of , where has maximal Hausdorff dimension. In addition, the intersection of these sets is shown to have maximal dimension. The results remain true when the sets in question are intersected with a suitably...
Rational approximation of vertical segments
Salazar Celis, Oliver; Cuyt, Annie; Verdonk, Brigitte
2007-08-01
In many applications, observations are prone to imprecise measurements. When constructing a model based on such data, an approximation rather than an interpolation approach is needed. Very often a least squares approximation is used. Here we follow a different approach. A natural way for dealing with uncertainty in the data is by means of an uncertainty interval. We assume that the uncertainty in the independent variables is negligible and that for each observation an uncertainty interval can be given which contains the (unknown) exact value. To approximate such data we look for functions which intersect all uncertainty intervals. In the past this problem has been studied for polynomials, or more generally for functions which are linear in the unknown coefficients. Here we study the problem for a particular class of functions which are nonlinear in the unknown coefficients, namely rational functions. We show how to reduce the problem to a quadratic programming problem with a strictly convex objective function, yielding a unique rational function which intersects all uncertainty intervals and satisfies some additional properties. Compared to rational least squares approximation which reduces to a nonlinear optimization problem where the objective function may have many local minima, this makes the new approach attractive.
Approximation on the complex sphere
Alsaud, Huda; Kushpel, Alexander; Levesley, Jeremy
2012-01-01
We develop new elements of harmonic analysis on the complex sphere on the basis of which Bernstein's, Jackson's and Kolmogorov's inequalities are established. We apply these results to get order sharp estimates of $m$-term approximations. The results obtained is a synthesis of new results on classical orthogonal polynomials, harmonic analysis on manifolds and geometric properties of Euclidean spaces.
On badly approximable complex numbers
Esdahl-Schou, Rune; Kristensen, S.
We show that the set of complex numbers which are badly approximable by ratios of elements of , where has maximal Hausdorff dimension. In addition, the intersection of these sets is shown to have maximal dimension. The results remain true when the sets in question are intersected with a suitably...
Pythagorean Approximations and Continued Fractions
Peralta, Javier
2008-01-01
In this article, we will show that the Pythagorean approximations of [the square root of] 2 coincide with those achieved in the 16th century by means of continued fractions. Assuming this fact and the known relation that connects the Fibonacci sequence with the golden section, we shall establish a procedure to obtain sequences of rational numbers…
Approximation of Surfaces by Cylinders
Randrup, Thomas
1998-01-01
We present a new method for approximation of a given surface by a cylinder surface. It is a constructive geometric method, leading to a monorail representation of the cylinder surface. By use of a weighted Gaussian image of the given surface, we determine a projection plane. In the orthogonal...
Approximate Reanalysis in Topology Optimization
Amir, Oded; Bendsøe, Martin P.; Sigmund, Ole
2009-01-01
In the nested approach to structural optimization, most of the computational effort is invested in the solution of the finite element analysis equations. In this study, the integration of an approximate reanalysis procedure into the framework of topology optimization of continuum structures...
Low Rank Approximation in $G_0W_0$ Approximation
Shao, Meiyue; Yang, Chao; Liu, Fang; da Jornada, Felipe H; Deslippe, Jack; Louie, Steven G
2016-01-01
The single particle energies obtained in a Kohn--Sham density functional theory (DFT) calculation are generally known to be poor approximations to electron excitation energies that are measured in transport, tunneling and spectroscopic experiments such as photo-emission spectroscopy. The correction to these energies can be obtained from the poles of a single particle Green's function derived from a many-body perturbation theory. From a computational perspective, the accuracy and efficiency of such an approach depends on how a self energy term that properly accounts for dynamic screening of electrons is approximated. The $G_0W_0$ approximation is a widely used technique in which the self energy is expressed as the convolution of a non-interacting Green's function ($G_0$) and a screened Coulomb interaction ($W_0$) in the frequency domain. The computational cost associated with such a convolution is high due to the high complexity of evaluating $W_0$ at multiple frequencies. In this paper, we discuss how the cos...
$L^2$ Analysis of the Multi-Configuration Time-Dependent Hartree-Fock Equations
Mauser, Norbert J
2010-01-01
The multiconfiguration methods are widely used by quantum physicists and chemists for numerical approximation of the many electron Schr\\"odinger equation. Recently, first mathematically rigorous results were obtained on the time-dependent models, e.g. short-in-time well-posedness in the Sobolev space $H^2$ for bounded interactions (C. Lubichand O. Koch} with initial data in $H^2$, in the energy space for Coulomb interactions with initial data in the same space (Trabelsi, Bardos et al.}, as well as global well-posedness under a sufficient condition on the energy of the initial data (Bardos et al.). The present contribution extends the analysis by setting an $L^2$ theory for the MCTDHF for general interactions including the Coulomb case. This kind of results is also the theoretical foundation of ad-hoc methods used in numerical calculation when modification ("regularization") of the density matrix destroys the conservation of energy property, but keeps invariant the mass.
Approximate Inference for Wireless Communications
Hansen, Morten
This thesis investigates signal processing techniques for wireless communication receivers. The aim is to improve the performance or reduce the computationally complexity of these, where the primary focus area is cellular systems such as Global System for Mobile communications (GSM) (and extensions...... complexity can potentially lead to limited power consumption, which translates into longer battery life-time in the handsets. The scope of the thesis is more specifically to investigate approximate (nearoptimal) detection methods that can reduce the computationally complexity significantly compared...... to the optimal one, which usually requires an unacceptable high complexity. Some of the treated approximate methods are based on QL-factorization of the channel matrix. In the work presented in this thesis it is proven how the QL-factorization of frequency-selective channels asymptotically provides the minimum...
Validity of the eikonal approximation
Kabat, D
1992-01-01
We summarize results on the reliability of the eikonal approximation in obtaining the high energy behavior of a two particle forward scattering amplitude. Reliability depends on the spin of the exchanged field. For scalar fields the eikonal fails at eighth order in perturbation theory, when it misses the leading behavior of the exchange-type diagrams. In a vector theory the eikonal gets the exchange diagrams correctly, but fails by ignoring certain non-exchange graphs which dominate the asymptotic behavior of the full amplitude. For spin--2 tensor fields the eikonal captures the leading behavior of each order in perturbation theory, but the sum of eikonal terms is subdominant to graphs neglected by the approximation. We also comment on the eikonal for Yang-Mills vector exchange, where the additional complexities of the non-abelian theory may be absorbed into Regge-type modifications of the gauge boson propagators.
Approximate Privacy: Foundations and Quantification
Feigenbaum, Joan; Schapira, Michael
2009-01-01
Increasing use of computers and networks in business, government, recreation, and almost all aspects of daily life has led to a proliferation of online sensitive data about individuals and organizations. Consequently, concern about the privacy of these data has become a top priority, particularly those data that are created and used in electronic commerce. There have been many formulations of privacy and, unfortunately, many negative results about the feasibility of maintaining privacy of sensitive data in realistic networked environments. We formulate communication-complexity-based definitions, both worst-case and average-case, of a problem's privacy-approximation ratio. We use our definitions to investigate the extent to which approximate privacy is achievable in two standard problems: the second-price Vickrey auction and the millionaires problem of Yao. For both the second-price Vickrey auction and the millionaires problem, we show that not only is perfect privacy impossible or infeasibly costly to achieve...
Validity of the Eikonal Approximation
Kabat, Daniel
1992-01-01
We summarize results on the reliability of the eikonal approximation in obtaining the high energy behavior of a two particle forward scattering amplitude. Reliability depends on the spin of the exchanged field. For scalar fields the eikonal fails at eighth order in perturbation theory, when it misses the leading behavior of the exchange-type diagrams. In a vector theory the eikonal gets the exchange diagrams correctly, but fails by ignoring certain non-exchange graphs which dominate the asymp...
Buica, Gabriela; 10.1016/j.jqsrt.2007.05.004
2013-01-01
We theoretically study multiphoton ionization of Mg in the circularly as well as the linearly polarized laser fields. Specifically two-, three-, and four-photon ionization cross sections from the ground and first excited states are calculated as a function of photon energy. Calculations are performed using the frozen-core Hartree-Fock and also the model potential approaches and the results are compared. We find that the model potential approach provide results as good as or even slightly better than those by the frozen-core Hartree-Fock approach. We also report the relative ratios of the ionization cross sections by the circularly and linearly polarized laser fields as a function of photon energy, which exhibit clear effects of electron correlations.
Ground state properties of even-even and odd Nd,Ce and Sm isotopes in Hartree-Fock-Bogoliubov method
Bassem, Younes El
2015-01-01
In this work, we have studied ground-state properties of both even-even and odd Nd isotopes within Hartree-Fock-Bogoliubov method with SLy5 Skyrme force in which the pairing strength has been generalized with a new proposed formula. We calculated bind- ing energies, two-neutron separation energies, quadrupole deformation, charge, neutron and proton radii. Similar calculations have been carried out for Ce and Sm in order to verify the validity of our pairing strength formula. The results have been compared with available experimental data, the results of Hartree-Fock-Bogoliubov calculations based on the D1S Gogny effective nucleon-nucleon interaction and predictions of some nuclear models such as Finite Range Droplet Model (FRDM) and Relativistic Mean Field (RMF) theory.
Approximate Counting of Graphical Realizations.
Erdős, Péter L; Kiss, Sándor Z; Miklós, István; Soukup, Lajos
2015-01-01
In 1999 Kannan, Tetali and Vempala proposed a MCMC method to uniformly sample all possible realizations of a given graphical degree sequence and conjectured its rapidly mixing nature. Recently their conjecture was proved affirmative for regular graphs (by Cooper, Dyer and Greenhill, 2007), for regular directed graphs (by Greenhill, 2011) and for half-regular bipartite graphs (by Miklós, Erdős and Soukup, 2013). Several heuristics on counting the number of possible realizations exist (via sampling processes), and while they work well in practice, so far no approximation guarantees exist for such an approach. This paper is the first to develop a method for counting realizations with provable approximation guarantee. In fact, we solve a slightly more general problem; besides the graphical degree sequence a small set of forbidden edges is also given. We show that for the general problem (which contains the Greenhill problem and the Miklós, Erdős and Soukup problem as special cases) the derived MCMC process is rapidly mixing. Further, we show that this new problem is self-reducible therefore it provides a fully polynomial randomized approximation scheme (a.k.a. FPRAS) for counting of all realizations.
Approximate Counting of Graphical Realizations.
Péter L Erdős
Full Text Available In 1999 Kannan, Tetali and Vempala proposed a MCMC method to uniformly sample all possible realizations of a given graphical degree sequence and conjectured its rapidly mixing nature. Recently their conjecture was proved affirmative for regular graphs (by Cooper, Dyer and Greenhill, 2007, for regular directed graphs (by Greenhill, 2011 and for half-regular bipartite graphs (by Miklós, Erdős and Soukup, 2013. Several heuristics on counting the number of possible realizations exist (via sampling processes, and while they work well in practice, so far no approximation guarantees exist for such an approach. This paper is the first to develop a method for counting realizations with provable approximation guarantee. In fact, we solve a slightly more general problem; besides the graphical degree sequence a small set of forbidden edges is also given. We show that for the general problem (which contains the Greenhill problem and the Miklós, Erdős and Soukup problem as special cases the derived MCMC process is rapidly mixing. Further, we show that this new problem is self-reducible therefore it provides a fully polynomial randomized approximation scheme (a.k.a. FPRAS for counting of all realizations.
Many Faces of Boussinesq Approximations
Vladimirov, Vladimir A
2016-01-01
The \\emph{equations of Boussinesq approximation} (EBA) for an incompressible and inhomogeneous in density fluid are analyzed from a viewpoint of the asymptotic theory. A systematic scaling shows that there is an infinite number of related asymptotic models. We have divided them into three classes: `poor', `reasonable' and `good' Boussinesq approximations. Each model can be characterized by two parameters $q$ and $k$, where $q =1, 2, 3, \\dots$ and $k=0, \\pm 1, \\pm 2,\\dots$. Parameter $q$ is related to the `quality' of approximation, while $k$ gives us an infinite set of possible scales of velocity, time, viscosity, \\emph{etc.} Increasing $q$ improves the quality of a model, but narrows the limits of its applicability. Parameter $k$ allows us to vary the scales of time, velocity and viscosity and gives us the possibility to consider any initial and boundary conditions. In general, we discover and classify a rich variety of possibilities and restrictions, which are hidden behind the routine use of the Boussinesq...
Rabilloud, Franck
2014-10-14
Absorption spectra of Ag20 and Ag55(q) (q = +1, -3) nanoclusters are investigated in the framework of the time-dependent density functional theory in order to analyse the role of the d electrons in plasmon-like band of silver clusters. The description of the plasmon-like band from calculations using density functionals containing an amount of Hartree-Fock exchange at long range, namely, hybrid and range-separated hybrid (RSH) density functionals, is in good agreement with the classical interpretation of the plasmon-like structure as a collective excitation of valence s-electrons. In contrast, using local or semi-local exchange functionals (generalized gradient approximations (GGAs) or meta-GGAs) leads to a strong overestimation of the role of d electrons in the plasmon-like band. The semi-local asymptotically corrected model potentials also describe the plasmon as mainly associated to d electrons, though calculated spectra are in fairly good agreement with those calculated using the RSH scheme. Our analysis shows that a portion of non-local exchange modifies the description of the plasmon-like band.
Vendrell, Oriol; Meyer, Hans-Dieter
2011-01-28
The multilayer multiconfiguration time-dependent Hartree (ML-MCTDH) method is discussed and a fully general implementation for any number of layers based on the recursive ML-MCTDH algorithm given by Manthe [J. Chem. Phys. 128, 164116 (2008)] is presented. The method is applied first to a generalized Henon-Heiles (HH) hamiltonian. For 6D HH the overhead of ML-MCTDH makes the method slower than MCTDH, but for 18D HH ML-MCTDH starts to be competitive. We report as well 1458D simulations of the HH hamiltonian using a seven-layer scheme. The photoabsorption spectrum of pyrazine computed with the 24D hamiltonian of Raab et al. [J. Chem. Phys. 110, 936 (1999)] provides a realistic molecular test case for the method. Quick and small ML-MCTDH calculations needing a fraction of the time and resources of reference MCTDH calculations provide already spectra with all the correct features. Accepting slightly larger deviations, the calculation can be accelerated to take only 7 min. When pushing the method toward convergence, results of similar quality than the best available MCTDH benchmark, which is based on a wavepacket with 4.6×10(7)time-dependent coefficients, are obtained with a much more compact wavefunction consisting of only 4.5×10(5) coefficients and requiring a shorter computation time.
Kato, Tsuyoshi; Ide, Yoshihiro; Yamanouchi, Kaoru [Department of Chemistry, School of Science, The University of Tokyo, 7-3-1, Hongo Bunkyo-ku, Tokyo, 113-0033 (Japan)
2015-12-31
We first calculate the ground-state molecular wave function of 1D model H{sub 2} molecule by solving the coupled equations of motion formulated in the extended multi-configuration time-dependent Hartree-Fock (MCTDHF) method by the imaginary time propagation. From the comparisons with the results obtained by the Born-Huang (BH) expansion method as well as with the exact wave function, we observe that the memory size required in the extended MCTDHF method is about two orders of magnitude smaller than in the BH expansion method to achieve the same accuracy for the total energy. Second, in order to provide a theoretical means to understand dynamical behavior of the wave function, we propose to define effective adiabatic potential functions and compare them with the conventional adiabatic electronic potentials, although the notion of the adiabatic potentials is not used in the extended MCTDHF approach. From the comparison, we conclude that by calculating the effective potentials we may be able to predict the energy differences among electronic states even for a time-dependent system, e.g., time-dependent excitation energies, which would be difficult to be estimated within the BH expansion approach.
Fasshauer, Elke
2015-01-01
We report on an implementation of the multiconfigurational time-dependent Hartree method (MCTDH) for spin-polarized fermions (MCTDHF). Our approach is based on a mapping for opera- tors in Fock space that allows a compact and efficient application of the Hamiltonian and solution of the MCTDHF equations of motion. Our implementation extends, builds on and exploits the recursive implementation of MCTDH for bosons (R-MCTDHB) package. Together with R-MCTDHB, the present implementation of MCTDHF forms the MCTDH-X package. We benchmark the accuracy of the algorithm with the harmonic interaction model and a time-dependent generalization thereof. These models consider parabolically trapped particles that interact through a harmonic interaction potential. We demonstrate, that MCTDHF is capable of solving the time-dependent many-fermion Schr\\"odinger equation to an in principle arbitrary degree of precision and can hence yield numerically exact results even in the case of Hamiltonians with time-dependent one-body and t...
Choi, Sunghwan; Kwon, Oh-Kyoung; Kim, Jaewook; Kim, Woo Youn
2016-09-15
We investigated the performance of heterogeneous computing with graphics processing units (GPUs) and many integrated core (MIC) with 20 CPU cores (20×CPU). As a practical example toward large scale electronic structure calculations using grid-based methods, we evaluated the Hartree potentials of silver nanoparticles with various sizes (3.1, 3.7, 4.9, 6.1, and 6.9 nm) via a direct integral method supported by the sinc basis set. The so-called work stealing scheduler was used for efficient heterogeneous computing via the balanced dynamic distribution of workloads between all processors on a given architecture without any prior information on their individual performances. 20×CPU + 1GPU was up to ∼1.5 and ∼3.1 times faster than 1GPU and 20×CPU, respectively. 20×CPU + 2GPU was ∼4.3 times faster than 20×CPU. The performance enhancement by CPU + MIC was considerably lower than expected because of the large initialization overhead of MIC, although its theoretical performance is similar with that of CPU + GPU. © 2016 Wiley Periodicals, Inc.
Wang, Hao
2014-07-01
The metal-insulator transition of VO2 so far has evaded an accurate description by density functional theory. The screened hybrid functional of Heyd, Scuseria and Ernzerhof leads to reasonable solutions for both the low-temperature monoclinic and high-temperature rutile phases only if spin polarization is excluded from the calculations. We explore whether a satisfactory agreement with experiment can be achieved by tuning the fraction of Hartree Fock exchange (α) in the density functional. It is found that two branches of locally stable solutions exist for the rutile phase for 12.5%≤α≤20%. One is metallic and has the correct stability as compared to the monoclinic phase, the other is insulating with lower energy than the metallic branch. We discuss these observations based on the V 3d orbital occupations and conclude that α=10% is the best possible choice for spin-polarized VO2 calculations. © 2014 Elsevier B.V. All rights reserved.
Lewis, Cannada A; Valeev, Edward F
2015-01-01
Clustered Low Rank (CLR) framework for block-sparse and block-low-rank tensor representation and computation is described. The CLR framework depends on 2 parameters that control precision: one controlling the CLR block rank truncation and another that controls screening of small contributions in arithmetic operations on CLR tensors. As these parameters approach zero CLR representation and arithmetic become exact. There are no other ad-hoc heuristics, such as domains. Use of the CLR format for the order-2 and order-3 tensors that appear in the context of density fitting (DF) evaluation of the Hartree-Fock (exact) exchange significantly reduced the storage and computational complexities below their standard $\\mathcal{O}(N^3)$ and $\\mathcal{O}(N^4)$ figures. Even for relatively small systems and realistic basis sets CLR-based DF HF becomes more efficient than the standard DF approach, and significantly more efficient than the conventional non-DF HF, while negligibly affecting molecular energies and properties.
$\\it{Ab}$ $\\it{initio}$ nuclear many-body perturbation calculations in the Hartree-Fock basis
Hu, Baishan; Sun, Zhonghao; Vary, James P; Li, Tong
2016-01-01
Starting from realistic nuclear forces, the chiral N$^3$LO and JISP16, we have applied many-body perturbation theory (MBPT) to the structure of closed-shell nuclei, $^4$He and $^{16}$O. The two-body N$^3$LO interaction is softened by a similarity renormalization group transformation while JISP16 is adopted without renormalization. The MBPT calculations are performed within the Hartree-Fock (HF) bases. The angular momentum coupled scheme is used, which can reduce the computational task. Corrections up to the third order in energy and up to the second order in radius are evaluated. Higher-order corrections in the HF basis are small relative to the leading-order perturbative result. Using the anti-symmetrized Goldstone diagram expansions of the wave function, we directly correct the one-body density for the calculation of the radius, rather than calculate corrections to the occupation propabilities of single-particle orbits as found in other treatments. We compare our results with other methods where available a...
Brorsen, Kurt R.; Sirjoosingh, Andrew; Pak, Michael V.; Hammes-Schiffer, Sharon
2015-06-01
The nuclear electronic orbital (NEO) reduced explicitly correlated Hartree-Fock (RXCHF) approach couples select electronic orbitals to the nuclear orbital via Gaussian-type geminal functions. This approach is extended to enable the use of a restricted basis set for the explicitly correlated electronic orbitals and an open-shell treatment for the other electronic orbitals. The working equations are derived and the implementation is discussed for both extensions. The RXCHF method with a restricted basis set is applied to HCN and FHF- and is shown to agree quantitatively with results from RXCHF calculations with a full basis set. The number of many-particle integrals that must be calculated for these two molecules is reduced by over an order of magnitude with essentially no loss in accuracy, and the reduction factor will increase substantially for larger systems. Typically, the computational cost of RXCHF calculations with restricted basis sets will scale in terms of the number of basis functions centered on the quantum nucleus and the covalently bonded neighbor(s). In addition, the RXCHF method with an odd number of electrons that are not explicitly correlated to the nuclear orbital is implemented using a restricted open-shell formalism for these electrons. This method is applied to HCN+, and the nuclear densities are in qualitative agreement with grid-based calculations. Future work will focus on the significance of nonadiabatic effects in molecular systems and the further enhancement of the NEO-RXCHF approach to accurately describe such effects.
Dubey, Archana; Badu, S. R.; Scheicher, R. H.; Sahoo, N.; Pink, R. H.; Schulte, A.; Saha, H. P.; Chow, Lee; Nagamine, K.; Das, T. P.
2008-03-01
The observation of paramagnetic susceptibility [1] in Oxy-Hb from measurements over a broad temperature range has stimulated interest in the occurrence of a low-lying excited triplet state close to the ground singlet state of Oxy-Hb. An earlier theoretical investigation [2] has shown the existence of such a triplet state providing support to the interpretation of the susceptibility data [1]. Support for the low-lying excited triplet state has been augmented recently [3] from microscopic relaxation rate measurements for muon attached to the heme group of Oxy-Hb. We are studying by first principles Hartree-Fock procedure the energies and the electronic wave functions of the ground and triplet states and the quantitative theoretical prediction of muon magnetic hyperfine interaction in room temperature μSR measurements on Oxy-Hb. Results will be presented for hyperfine interactions of muon and other nuclei in Oxy-Hb [1] M.Cerdonio etal. Proc. Nat. Acad. Sci USA 75, 4916(1978). [2] Zalek S. Herman and Gilda H Loew JACS 102, 1815(1980).[ 3] K. Nagamine etal Proc. Jpn. Acad.Ser.B 83,120(2007).
Xu Sun; You-song Gu; Xue-qiang Wang; Yue Zhang
2012-01-01
The electronic properties and stability of Li-doped ZnO with various defects have been studied by calculating the electronic structures and defect formation energies via first-principles calculations using hybrid Hartree-Fock and density functional methods.The results from formation energy calculations show that Li pair complexes have the lowest formation energy in most circumstances and they consume most of the Li content in Li doped ZnO,which make the p-type conductance hard to obtain.The formation of Li pair complexes is the main obstacle to realize p-type conductance in Li doped ZnO.However,the formation energy of Lizn decreases as environment changes from Zn-rich to O-rich and becomes more stable than that of Li-pair complexes at highly O-rich environment.Therefore,p-type conductance can be obtained by Li doped ZnO grown or post annealed in oxygen rich atmosphere.
Saravanan, S. P.; Sankar, A.; Parimala, K.
2017-01-01
The complete structural and vibrational analysis of the 2,5-Difluoronitrobenzene (DNB) was carried out by Hartree-Fock (HF) and density functional theory (DFT) method (B3LYP) with 6-311++G (d,p) basis set. The fundamental vibrations are assigned on the basis of the potential energy distribution (PED) of the vibrational modes calculated with scaled quantum mechanics (SQM) method. Using the time-dependent density functional theory (TD-DFT) method, electronic absorption spectra of the title compound have been predicted and a good agreement with the experimental ones is determined. 13C and 1H NMR spectra were recorded and chemical shifts of the molecule were calculated using the gauge independent atomic orbital (GIAO) method. The hyperconjugative interaction energy (E(2)) and electron densities of donor (i) and acceptor (j) bonds were calculated using natural bond orbital (NBO) analysis. In addition, molecular electrostatic potential (MEP) and atomic charges were calculated using B3LYP/6-311++G (d,p) level of theory. Moreover, thermodynamic properties (heat capacities, entropy, enthalpy and Gibb's free energy) of the title compound at different temperatures were calculated.
Zhao, Jie; Zhao, En-Guang; Zhou, Shan-Gui
2016-01-01
We develop a multidimensionally-constrained relativistic Hartree-Bogoliubov (MDC-RHB) model in which the pairing correlations are taken into account by making the Bogoliubov transformation. In this model, the nuclear shape is assumed to be invariant under the reversion of $x$ and $y$ axes, i.e., the intrinsic symmetry group is $V_4$ and all shape degrees of freedom $\\beta_{\\lambda\\mu}$ with even $\\mu$ are included self-consistently. The RHB equation is solved in an axially deformed harmonic oscillator basis. A separable pairing force of finite range is adopted in the MDC-RHB model. The potential energy curves of neutron-rich even-even Zr isotopes are calculated. The ground state shapes of $^{108-112}$Zr are predicted to be tetrahedral with both functionals DD-PC1 and PC-PK1 and $^{106}$Zr is also predicted to have a tetrahedral ground state with the functional PC-PK1. The tetrahedral ground states are caused by large energy gaps at $Z=40$ and $N=70$ when $\\beta_{32}$ deformation is included. Although the incl...
Augspurger, Joseph D.; Dykstra, Clifford E.
1993-08-01
Molecular Sternheimer shielding constants, γ, the proportionality constants relating the electric field gradient at a quadrupolar nucleus to an external electric field gradient are usually introduced phenomenologically. In this report, we take a comprehensive view of the sensitivity of the electric field gradient at a nucleus to arbitrary external electrical potentials and we show how the response can be obtained from analytically determined properties via derivative Hartree-Fock theory. From application of this ab initio technique, values have been obtained for the first and second order changes in nuclear quadrupole coupling with respect to external fields and field gradients, as well as nearby ideal multipole moments, for HCN and HCl. These values have been used to evaluate the change in the nuclear quadrupole coupling for several weakly bound complexes and to provide a nonempirical approach to relative effects on Sternheimer shielding. In weak molecular complexes, the effect of uniform fields can be as sizable as the effect of external field gradients in the overall change in nuclear quadrupole coupling, and so the underlying issue of convergence of multipolar expansions is considered over a range of geometries. This is important for structural interpretations of both nuclear magnetic resonance (NMR) and microwave data, and a simple formula, representing a practical point of truncation, is presented for quadrupole coupling analysis.
A revised electronic Hessian for approximate time-dependent density functional theory.
Ziegler, Tom; Seth, Michael; Krykunov, Mykhaylo; Autschbach, Jochen
2008-11-14
Time-dependent density functional theory (TD-DFT) at the generalized gradient level of approximation (GGA) has shown systematic errors in the calculated excitation energies. This is especially the case for energies representing electron transitions between two separated regions of space or between orbitals of different spatial extents. It will be shown that these limitations can be attributed to the electronic ground state Hessian G(GGA). Specifically, we shall demonstrate that the Hessian G(GGA) can be used to describe changes in energy due to small perturbations of the electron density (Deltarho), but it should not be applied to one-electron excitations involving the density rearrangement (Deltarho) of a full electron charge. This is in contrast to Hartree-Fock theory where G(HF) has a trust region that is accurate for both small perturbations and one-electron excitations. The large trust radius of G(HF) can be traced back to the complete cancellation of Coulomb and exchange terms in Hartree-Fock (HF) theory representing self-interaction (complete self-interaction cancellation, CSIC). On the other hand, it is shown that the small trust radius for G(GGA) can be attributed to the fact that CSIC is assumed for GGA in the derivation of G(GGA) although GGA (and many other approximate DFT schemes) exhibits incomplete self-interaction cancellation (ISIC). It is further shown that one can derive a new matrix G(R-DFT) with the same trust region as G(HF) by taking terms due to ISIC properly into account. Further, with TD-DFT based on G(R-DFT), energies for state-to-state transitions represented by a one-electron excitation (psi(i)-->psi(a)) are approximately calculated as DeltaE(ai). Here DeltaE(ai) is the energy difference between the ground state Kohn-Sham Slater determinant and the energy of a Kohn-Sham Slater determinant where psi(i) has been replaced by psi(a). We make use of the new Hessian in two numerical applications involving charge-transfer excitations. It is
Chang, Yao-Wen; Jin, Bih-Yaw
2012-01-14
Many-body perturbation theory is used to investigate the effect of π-electron correlations on the quasi-particle band structures of conjugated polymers at the level of the Pariser-Parr-Pople model. The self-consistent GW approximation with vertex corrections to both the self-energy and the polarization in Hedin's equations is employed in order to eliminate self-interaction errors and include the effects of electron-hole attraction in screening processes. The dynamic inverse dielectric function is constructed from the generalized plasmon-pole approximation with the static dressed polarization given by the coupled-perturbed Hartree-Fock equation. The bandgaps of trans-polyacetylene, trans-polyphenylenevinylene and poly(para)phenylene are calculated by both the Hartree-Fock and GW approximation, and a lowering of bandgaps due to electron correlations is found. We conclude that both dielectric screening and vertex corrections are important for calculating the quasi-particle bandgaps of conjugated polymers.
Rollout Sampling Approximate Policy Iteration
Dimitrakakis, Christos
2008-01-01
Several researchers have recently investigated the connection between reinforcement learning and classification. We are motivated by proposals of approximate policy iteration schemes without value functions which focus on policy representation using classifiers and address policy learning as a supervised learning problem. This paper proposes variants of an improved policy iteration scheme which addresses the core sampling problem in evaluating a policy through simulation as a multi-armed bandit machine. The resulting algorithm offers comparable performance to the previous algorithm achieved, however, with significantly less computational effort. An order of magnitude improvement is demonstrated experimentally in two standard reinforcement learning domains: inverted pendulum and mountain-car.
Approximate Deconvolution Reduced Order Modeling
Xie, Xuping; Wang, Zhu; Iliescu, Traian
2015-01-01
This paper proposes a large eddy simulation reduced order model(LES-ROM) framework for the numerical simulation of realistic flows. In this LES-ROM framework, the proper orthogonal decomposition(POD) is used to define the ROM basis and a POD differential filter is used to define the large ROM structures. An approximate deconvolution(AD) approach is used to solve the ROM closure problem and develop a new AD-ROM. This AD-ROM is tested in the numerical simulation of the one-dimensional Burgers equation with a small diffusion coefficient(10^{-3})
Approximation for Bayesian Ability Estimation.
1987-02-18
posterior pdfs of ande are given by p(-[Y) p(F) F P((y lei’ j)P )d. SiiJ i (4) a r~d p(e Iy) - p(t0) 1 J i P(Yij ei, (5) As shown in Tsutakawa and Lin...inverse A Hessian of the log of (27) with respect to , evaulatedat a Then, under regularity conditions, the marginal posterior pdf of O is...two-way contingency tables. Journal of Educational Statistics, 11, 33-56. Lindley, D.V. (1980). Approximate Bayesian methods. Trabajos Estadistica , 31
Plasma Physics Approximations in Ares
Managan, R. A.
2015-01-08
Lee & More derived analytic forms for the transport properties of a plasma. Many hydro-codes use their formulae for electrical and thermal conductivity. The coefficients are complex functions of Fermi-Dirac integrals, F_{n}( μ/θ ), the chemical potential, μ or ζ = ln(1+e^{ μ/θ} ), and the temperature, θ = kT. Since these formulae are expensive to compute, rational function approximations were fit to them. Approximations are also used to find the chemical potential, either μ or ζ . The fits use ζ as the independent variable instead of μ/θ . New fits are provided for A^{α} (ζ ),A^{β} (ζ ), ζ, f(ζ ) = (1 + e^{-μ/θ})F_{1/2}(μ/θ), F_{1/2}'/F_{1/2}, F_{c}^{α}, and F_{c}^{β}. In each case the relative error of the fit is minimized since the functions can vary by many orders of magnitude. The new fits are designed to exactly preserve the limiting values in the non-degenerate and highly degenerate limits or as ζ→ 0 or ∞. The original fits due to Lee & More and George Zimmerman are presented for comparison.
Rational approximations to fluid properties
Kincaid, J. M.
1990-05-01
The purpose of this report is to summarize some results that were presented at the Spring AIChE meeting in Orlando, Florida (20 March 1990). We report on recent attempts to develop a systematic method, based on the technique of rational approximation, for creating mathematical models of real-fluid equations of state and related properties. Equation-of-state models for real fluids are usually created by selecting a function tilde p(T,rho) that contains a set of parameters (gamma sub i); the (gamma sub i) is chosen such that tilde p(T,rho) provides a good fit to the experimental data. (Here p is the pressure, T the temperature and rho is the density). In most cases, a nonlinear least-squares numerical method is used to determine (gamma sub i). There are several drawbacks to this method: one has essentially to guess what tilde p(T,rho) should be; the critical region is seldom fit very well and nonlinear numerical methods are time consuming and sometimes not very stable. The rational approximation approach we describe may eliminate all of these drawbacks. In particular, it lets the data choose the function tilde p(T,rho) and its numerical implementation involves only linear algorithms.
Many-Body Approximations in the sd-Shell Sandbox
Sen'kov, R A; Brown, B A; Luo, Y L; Zelevinsky, V G
2008-01-01
A new theoretical approach is presented that combines the Hartree-Fock variational scheme with the exact solution of the pairing problem in the finite orbital space. Using this formulation in the sd-space as an example, we show that the exact pairing significantly improves the results for the ground state energy
Stoitsov, M. V.; Schunck, N.; Kortelainen, M.; Michel, N.; Nam, H.; Olsen, E.; Sarich, J.; Wild, S.
2013-06-01
We describe the new version 2.00d of the code HFBTHO that solves the nuclear Skyrme-Hartree-Fock (HF) or Skyrme-Hartree-Fock-Bogoliubov (HFB) problem by using the cylindrical transformed deformed harmonic oscillator basis. In the new version, we have implemented the following features: (i) the modified Broyden method for non-linear problems, (ii) optional breaking of reflection symmetry, (iii) calculation of axial multipole moments, (iv) finite temperature formalism for the HFB method, (v) linear constraint method based on the approximation of the Random Phase Approximation (RPA) matrix for multi-constraint calculations, (vi) blocking of quasi-particles in the Equal Filling Approximation (EFA), (vii) framework for generalized energy density with arbitrary density-dependences, and (viii) shared memory parallelism via OpenMP pragmas. Program summaryProgram title: HFBTHO v2.00d Catalog identifier: ADUI_v2_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADUI_v2_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: GNU General Public License version 3 No. of lines in distributed program, including test data, etc.: 167228 No. of bytes in distributed program, including test data, etc.: 2672156 Distribution format: tar.gz Programming language: FORTRAN-95. Computer: Intel Pentium-III, Intel Xeon, AMD-Athlon, AMD-Opteron, Cray XT5, Cray XE6. Operating system: UNIX, LINUX, WindowsXP. RAM: 200 Mwords Word size: 8 bits Classification: 17.22. Does the new version supercede the previous version?: Yes Catalog identifier of previous version: ADUI_v1_0 Journal reference of previous version: Comput. Phys. Comm. 167 (2005) 43 Nature of problem: The solution of self-consistent mean-field equations for weakly-bound paired nuclei requires a correct description of the asymptotic properties of nuclear quasi-particle wave functions. In the present implementation, this is achieved by using the single-particle wave functions
Dodgson's Rule Approximations and Absurdity
McCabe-Dansted, John C
2010-01-01
With the Dodgson rule, cloning the electorate can change the winner, which Young (1977) considers an "absurdity". Removing this absurdity results in a new rule (Fishburn, 1977) for which we can compute the winner in polynomial time (Rothe et al., 2003), unlike the traditional Dodgson rule. We call this rule DC and introduce two new related rules (DR and D&). Dodgson did not explicitly propose the "Dodgson rule" (Tideman, 1987); we argue that DC and DR are better realizations of the principle behind the Dodgson rule than the traditional Dodgson rule. These rules, especially D&, are also effective approximations to the traditional Dodgson's rule. We show that, unlike the rules we have considered previously, the DC, DR and D& scores differ from the Dodgson score by no more than a fixed amount given a fixed number of alternatives, and thus these new rules converge to Dodgson under any reasonable assumption on voter behaviour, including the Impartial Anonymous Culture assumption.
Approximation by double Walsh polynomials
Ferenc Móricz
1992-01-01
Full Text Available We study the rate of approximation by rectangular partial sums, Cesàro means, and de la Vallée Poussin means of double Walsh-Fourier series of a function in a homogeneous Banach space X. In particular, X may be Lp(I2, where 1≦p<∞ and I2=[0,1×[0,1, or CW(I2, the latter being the collection of uniformly W-continuous functions on I2. We extend the results by Watari, Fine, Yano, Jastrebova, Bljumin, Esfahanizadeh and Siddiqi from univariate to multivariate cases. As by-products, we deduce sufficient conditions for convergence in Lp(I2-norm and uniform convergence on I2 as well as characterizations of Lipschitz classes of functions. At the end, we raise three problems.
Interplay of approximate planning strategies.
Huys, Quentin J M; Lally, Níall; Faulkner, Paul; Eshel, Neir; Seifritz, Erich; Gershman, Samuel J; Dayan, Peter; Roiser, Jonathan P
2015-03-10
Humans routinely formulate plans in domains so complex that even the most powerful computers are taxed. To do so, they seem to avail themselves of many strategies and heuristics that efficiently simplify, approximate, and hierarchically decompose hard tasks into simpler subtasks. Theoretical and cognitive research has revealed several such strategies; however, little is known about their establishment, interaction, and efficiency. Here, we use model-based behavioral analysis to provide a detailed examination of the performance of human subjects in a moderately deep planning task. We find that subjects exploit the structure of the domain to establish subgoals in a way that achieves a nearly maximal reduction in the cost of computing values of choices, but then combine partial searches with greedy local steps to solve subtasks, and maladaptively prune the decision trees of subtasks in a reflexive manner upon encountering salient losses. Subjects come idiosyncratically to favor particular sequences of actions to achieve subgoals, creating novel complex actions or "options."
Approximate reduction of dynamical systems
Tabuada, Paulo; Julius, Agung; Pappas, George J
2007-01-01
The reduction of dynamical systems has a rich history, with many important applications related to stability, control and verification. Reduction of nonlinear systems is typically performed in an exact manner - as is the case with mechanical systems with symmetry--which, unfortunately, limits the type of systems to which it can be applied. The goal of this paper is to consider a more general form of reduction, termed approximate reduction, in order to extend the class of systems that can be reduced. Using notions related to incremental stability, we give conditions on when a dynamical system can be projected to a lower dimensional space while providing hard bounds on the induced errors, i.e., when it is behaviorally similar to a dynamical system on a lower dimensional space. These concepts are illustrated on a series of examples.
Diophantine approximations and Diophantine equations
Schmidt, Wolfgang M
1991-01-01
"This book by a leading researcher and masterly expositor of the subject studies diophantine approximations to algebraic numbers and their applications to diophantine equations. The methods are classical, and the results stressed can be obtained without much background in algebraic geometry. In particular, Thue equations, norm form equations and S-unit equations, with emphasis on recent explicit bounds on the number of solutions, are included. The book will be useful for graduate students and researchers." (L'Enseignement Mathematique) "The rich Bibliography includes more than hundred references. The book is easy to read, it may be a useful piece of reading not only for experts but for students as well." Acta Scientiarum Mathematicarum
Truthful approximations to range voting
Filos-Ratsika, Aris; Miltersen, Peter Bro
We consider the fundamental mechanism design problem of approximate social welfare maximization under general cardinal preferences on a finite number of alternatives and without money. The well-known range voting scheme can be thought of as a non-truthful mechanism for exact social welfare...... maximization in this setting. With m being the number of alternatives, we exhibit a randomized truthful-in-expectation ordinal mechanism implementing an outcome whose expected social welfare is at least an Omega(m^{-3/4}) fraction of the social welfare of the socially optimal alternative. On the other hand, we...... show that for sufficiently many agents and any truthful-in-expectation ordinal mechanism, there is a valuation profile where the mechanism achieves at most an O(m^{-{2/3}) fraction of the optimal social welfare in expectation. We get tighter bounds for the natural special case of m = 3...
Approximation of Surfaces by Cylinders
Randrup, Thomas
1998-01-01
We present a new method for approximation of a given surface by a cylinder surface. It is a constructive geometric method, leading to a monorail representation of the cylinder surface. By use of a weighted Gaussian image of the given surface, we determine a projection plane. In the orthogonal...... projection of the surface onto this plane, a reference curve is determined by use of methods for thinning of binary images. Finally, the cylinder surface is constructed as follows: the directrix of the cylinder surface is determined by a least squares method minimizing the distance to the points...... in the projection within a tolerance given by the reference curve, and the rulings are lines perpendicular to the projection plane. Application of the method in ship design is given....
Analytical approximations for spiral waves
Löber, Jakob, E-mail: jakob@physik.tu-berlin.de; Engel, Harald [Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstrasse 36, EW 7-1, 10623 Berlin (Germany)
2013-12-15
We propose a non-perturbative attempt to solve the kinematic equations for spiral waves in excitable media. From the eikonal equation for the wave front we derive an implicit analytical relation between rotation frequency Ω and core radius R{sub 0}. For free, rigidly rotating spiral waves our analytical prediction is in good agreement with numerical solutions of the linear eikonal equation not only for very large but also for intermediate and small values of the core radius. An equivalent Ω(R{sub +}) dependence improves the result by Keener and Tyson for spiral waves pinned to a circular defect of radius R{sub +} with Neumann boundaries at the periphery. Simultaneously, analytical approximations for the shape of free and pinned spirals are given. We discuss the reasons why the ansatz fails to correctly describe the dependence of the rotation frequency on the excitability of the medium.
On quantum and approximate privacy
Klauck, H
2001-01-01
This paper studies privacy in communication complexity. The focus is on quantum versions of the model and on protocols with only approximate privacy against honest players. We show that the privacy loss (the minimum divulged information) in computing a function can be decreased exponentially by using quantum protocols, while the class of privately computable functions (i.e., those with privacy loss 0) is not increased by quantum protocols. Quantum communication combined with small information leakage on the other hand makes certain functions computable (almost) privately which are not computable using quantum communication without leakage or using classical communication with leakage. We also give an example of an exponential reduction of the communication complexity of a function by allowing a privacy loss of o(1) instead of privacy loss 0.
IONIS: Approximate atomic photoionization intensities
Heinäsmäki, Sami
2012-02-01
A program to compute relative atomic photoionization cross sections is presented. The code applies the output of the multiconfiguration Dirac-Fock method for atoms in the single active electron scheme, by computing the overlap of the bound electron states in the initial and final states. The contribution from the single-particle ionization matrix elements is assumed to be the same for each final state. This method gives rather accurate relative ionization probabilities provided the single-electron ionization matrix elements do not depend strongly on energy in the region considered. The method is especially suited for open shell atoms where electronic correlation in the ionic states is large. Program summaryProgram title: IONIS Catalogue identifier: AEKK_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEKK_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 1149 No. of bytes in distributed program, including test data, etc.: 12 877 Distribution format: tar.gz Programming language: Fortran 95 Computer: Workstations Operating system: GNU/Linux, Unix Classification: 2.2, 2.5 Nature of problem: Photoionization intensities for atoms. Solution method: The code applies the output of the multiconfiguration Dirac-Fock codes Grasp92 [1] or Grasp2K [2], to compute approximate photoionization intensities. The intensity is computed within the one-electron transition approximation and by assuming that the sum of the single-particle ionization probabilities is the same for all final ionic states. Restrictions: The program gives nonzero intensities for those transitions where only one electron is removed from the initial configuration(s). Shake-type many-electron transitions are not computed. The ionized shell must be closed in the initial state. Running time: Few seconds for a
Approximate analytic solutions to the NPDD: Short exposure approximations
Close, Ciara E.; Sheridan, John T.
2014-04-01
There have been many attempts to accurately describe the photochemical processes that take places in photopolymer materials. As the models have become more accurate, solving them has become more numerically intensive and more 'opaque'. Recent models incorporate the major photochemical reactions taking place as well as the diffusion effects resulting from the photo-polymerisation process, and have accurately described these processes in a number of different materials. It is our aim to develop accessible mathematical expressions which provide physical insights and simple quantitative predictions of practical value to material designers and users. In this paper, starting with the Non-Local Photo-Polymerisation Driven Diffusion (NPDD) model coupled integro-differential equations, we first simplify these equations and validate the accuracy of the resulting approximate model. This new set of governing equations are then used to produce accurate analytic solutions (polynomials) describing the evolution of the monomer and polymer concentrations, and the grating refractive index modulation, in the case of short low intensity sinusoidal exposures. The physical significance of the results and their consequences for holographic data storage (HDS) are then discussed.
Potential energy curves for Mo2: multi-component symmetry-projected Hartree-Fock and beyond
Bytautas, Laimutis; Jiménez-Hoyos, Carlos A.; Rodríguez-Guzmán, R.; Scuseria, Gustavo E.
2014-07-01
The molybdenum dimer is an example of a transition metal system with a formal sextuple bond that constitutes a challenging case for ab initio quantum chemistry methods. In particular, the complex binding pattern in the Mo2 molecule requires a high-quality description of non-dynamic and dynamic electron correlation in order to yield the correct shape of the potential energy curve. The present study examines the performance of a recently implemented multi-component symmetry projected Hartree-Fock (HF) approach. In this work, the spin and spatial symmetries of a trial wavefunction written in terms of non-orthogonal Slater determinants are deliberately broken and then restored in a variation-after-projection framework. The resulting symmetry-projected HF wavefunctions, which possess well-defined quantum numbers, can account for static and some dynamic correlations. A single symmetry-projected configuration in a D∞hS-UHF or a D∞hKS-UHF framework offers a reasonable description of the potential energy curve of Mo2, though the binding energy is too small for the former. Our multi-component strategy offers a way to improve on the single configuration result in a systematic way towards the exact wavefunction: in the def2-TZVP basis set considered in this study, a 7-determinant multi-component D∞hS-UHF approach yields a bond length of 2.01 Å, in good agreement with experimental results, while the predicted binding energy is 39.2 mhartree. The results of this exploratory study suggest that a multi-component symmetry-projected HF stategy is a promising alternative in a high-accuracy description of the electronic structure of challenging systems. We also present and discuss some benchmark calculations based on the CEEIS-FCI (correlation energy extrapolation by intrinsic scaling - full configuration interaction) method for selected geometries.
Goriely, S.; Chamel, N.; Pearson, J. M.
2016-03-01
Extending our earlier work, a new family of three Hartree-Fock-Bogoliubov (HFB) mass models, labeled HFB-30, HFB-31, and HFB-32, is presented, along with their underlying interactions, BSk30, BSk31, and BSk32, respectively. The principle new feature is a purely phenomenological pairing term that depends on the density gradient. This enables us to have a bulk pairing term that is fitted to realistic nuclear-matter calculations in which for the first time the self-energy corrections are included, while the behavior of the nucleon effective masses in asymmetric homogeneous nuclear matter is significantly improved. Furthermore, in the particle-hole channel all the highly realistic constraints of our earlier work are retained. In particular, the unconventional Skyrme forces containing t4 and t5 terms are still constrained to fit realistic equations of state of neutron matter stiff enough to support the massive neutron stars PSR J1614-2230 and PSR J0348+0432. All unphysical long-wavelength spin and spin-isospin instabilities of nuclear matter, including the unphysical transition to a polarized state in neutron-star matter, are eliminated. Our three interactions are characterized by values of the symmetry coefficient J of 30, 31, and 32 MeV, respectively. The best fit to the database of 2353 nuclear masses is found for model HFB-31 (J =31 MeV ) with a model error of 0.561 MeV. This model also fits the charge-radius data with an root-mean-square error of 0.027 fm.
The second Born approximation for the double ionization of N2 by electron impact
Lamy, P.; Dal Cappello, C.; Charpentier, I.; Ruiz-Lopez, M. F.; Hervieux, P. A.
2016-07-01
In their (e,3e) and (e,3-1e) experiments of the double ionization (DI) of the outermost orbital of N2, Li et al (2012 J. Phys. B: At. Mol. Opt. Phys. 45 135201) recently showed that the process is largely dominated by a two-step-2 mechanism, which is a double interaction of the incident electron with the target. From a theoretical point of view, this should entail the use of the second Born approximation. In the past, very few theoretical calculations had been carried out this way because it requires a difficult numerical triple integration. We propose here to take into account the second Born approximation for the DI of N2 by using the closure approximation. The initial state is described by a single-center wave function derived from the usual multi-center wave function obtained in the self-consistent-field Hartree-Fock method using the linear combination of atomic orbitals-molecular orbital (LCAO-MO) approximation. The final state describes the interaction between each of the ejected electrons and the target by a Coulomb wave and the interaction between the two ejected electrons with the use of the Gamow factor. We calculate differential cross sections using the same kinematic conditions as Li et al (intermediate incident energy about 600 eV) for (e,3e) and (e,3-1e) DI of N2. The results show that the model does not allow a shift of the variation of the four-fold differential cross section near the momentum transfer to be obtained nor its opposite when we include the contribution given by the second Born approximation, as in (e,3-1e) experiments.
Randomized approximate nearest neighbors algorithm.
Jones, Peter Wilcox; Osipov, Andrei; Rokhlin, Vladimir
2011-09-20
We present a randomized algorithm for the approximate nearest neighbor problem in d-dimensional Euclidean space. Given N points {x(j)} in R(d), the algorithm attempts to find k nearest neighbors for each of x(j), where k is a user-specified integer parameter. The algorithm is iterative, and its running time requirements are proportional to T·N·(d·(log d) + k·(d + log k)·(log N)) + N·k(2)·(d + log k), with T the number of iterations performed. The memory requirements of the procedure are of the order N·(d + k). A by-product of the scheme is a data structure, permitting a rapid search for the k nearest neighbors among {x(j)} for an arbitrary point x ∈ R(d). The cost of each such query is proportional to T·(d·(log d) + log(N/k)·k·(d + log k)), and the memory requirements for the requisite data structure are of the order N·(d + k) + T·(d + N). The algorithm utilizes random rotations and a basic divide-and-conquer scheme, followed by a local graph search. We analyze the scheme's behavior for certain types of distributions of {x(j)} and illustrate its performance via several numerical examples.
GUSEINOV I.Israfil; AKSU Hüseyin
2008-01-01
@@ Using formulae for one-and two-electron integrals of Coulomb interaction potential fk(r)=r-k with non-integer indices k established by one of the authors with the help of complete orthonormal sets of Ψa-exponential-type orbitals(a=1,0,-1,-2,…),we perform the calculations for isoelectronic series of the He atom containing nuclear charges from 2 to 10,where k=1-μ(-1＜μ＜0).For this purpose we have used the dogble-zeta approximation,the configuration interaction and coupled-cluster methods employing the integer-n Slater-type orbitals as basis sets.It is demonstrated that the results of calculations obtained are better than the numerical Hartree-Fock values.
Sarriguren, P
2013-01-01
Electron-capture rates at different density and temperature conditions are evaluated for a set of pf-shell nuclei representative of the constituents in presupernova formations. The nuclear structure part of the problem is described within a quasiparticle random-phase approximation based on a deformed Skyrme Hartree-Fock selfconsistent mean field with pairing correlations and residual interactions in particle-hole and particle-particle channels. The energy distributions of the Gamow-Teller strength are evaluated and compared to benchmark shell-model calculations and experimental data extracted from charge-exchange reactions. The model dependence of the weak rates are discussed and the various sensitivities to both density and temperature are analyzed.
Obtaining exact value by approximate computations
Jing-zhong ZHANG; Yong FENG
2007-01-01
Numerical approximate computations can solve large and complex problems fast. They have the advantage of high efficiency. However they only give approximate results, whereas we need exact results in some fields. There is a gap between approximate computations and exact results.In this paper, we build a bridge by which exact results can be obtained by numerical approximate computations.
Fuzzy Set Approximations in Fuzzy Formal Contexts
Mingwen Shao; Shiqing Fan
2006-01-01
In this paper, a kind of multi-level formal concept is introduced. Based on the proposed multi-level formal concept, we present a pair of rough fuzzy set approximations within fuzzy formal contexts. By the proposed rough fuzzy set approximations, we can approximate a fuzzy set according to different precision level. We discuss the properties of the proposed approximation operators in detail.
Obtaining exact value by approximate computations
2007-01-01
Numerical approximate computations can solve large and complex problems fast.They have the advantage of high efficiency.However they only give approximate results,whereas we need exact results in some fields.There is a gap between approximate computations and exact results. In this paper,we build a bridge by which exact results can be obtained by numerical approximate computations.
Nonlinear approximation with dictionaries I. Direct estimates
Gribonval, Rémi; Nielsen, Morten
2004-01-01
We study various approximation classes associated with m-term approximation by elements from a (possibly) redundant dictionary in a Banach space. The standard approximation class associated with the best m-term approximation is compared to new classes defined by considering m-term approximation...... with algorithmic constraints: thresholding and Chebychev approximation classes are studied, respectively. We consider embeddings of the Jackson type (direct estimates) of sparsity spaces into the mentioned approximation classes. General direct estimates are based on the geometry of the Banach space, and we prove...
Nonlinear approximation with dictionaries, I: Direct estimates
Gribonval, Rémi; Nielsen, Morten
We study various approximation classes associated with $m$-term approximation by elements from a (possibly redundant) dictionary in a Banach space. The standard approximation class associated with the best $m$-term approximation is compared to new classes defined by considering $m......$-term approximation with algorithmic constraints: thresholding and Chebychev approximation classes are studied respectively. We consider embeddings of the Jackson type (direct estimates) of sparsity spaces into the mentioned approximation classes. General direct estimates are based on the geometry of the Banach space...
Schunck, N.; Dobaczewski, J.; McDonnell, J.; Satuła, W.; Sheikh, J. A.; Staszczak, A.; Stoitsov, M.; Toivanen, P.
2012-01-01
We describe the new version (v2.49t) of the code HFODD which solves the nuclear Skyrme-Hartree-Fock (HF) or Skyrme-Hartree-Fock-Bogolyubov (HFB) problem by using the Cartesian deformed harmonic-oscillator basis. In the new version, we have implemented the following physics features: (i) the isospin mixing and projection, (ii) the finite-temperature formalism for the HFB and HF + BCS methods, (iii) the Lipkin translational energy correction method, (iv) the calculation of the shell correction. A number of specific numerical methods have also been implemented in order to deal with large-scale multi-constraint calculations and hardware limitations: (i) the two-basis method for the HFB method, (ii) the Augmented Lagrangian Method (ALM) for multi-constraint calculations, (iii) the linear constraint method based on the approximation of the RPA matrix for multi-constraint calculations, (iv) an interface with the axial and parity-conserving Skyrme-HFB code HFBTHO, (v) the mixing of the HF or HFB matrix elements instead of the HF fields. Special care has been paid to using the code on massively parallel leadership class computers. For this purpose, the following features are now available with this version: (i) the Message Passing Interface (MPI) framework, (ii) scalable input data routines, (iii) multi-threading via OpenMP pragmas, (iv) parallel diagonalization of the HFB matrix in the simplex-breaking case using the ScaLAPACK library. Finally, several little significant errors of the previous published version were corrected. New version program summaryProgram title:HFODD (v2.49t) Catalogue identifier: ADFL_v3_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADFL_v3_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: GNU General Public Licence v3 No. of lines in distributed program, including test data, etc.: 190 614 No. of bytes in distributed program, including test data, etc.: 985 898 Distribution
Dyall, Kenneth G.
1992-01-01
The effect of omission of two-electron integrals involving basis functions for the small component of the wavefunction on the eigenvalue spectrum in the Dirac-Hartree-Fock problem is studied. From an analysis of the Fock matrix it is shown that omission of these integrals moves the negative-energy states down, not up. Their complete omission does not give rise to intruder states. The appearance of intruder states occurs when only some of the core integrals are omitted, due to the nature of particular contraction schemes used for the core basis functions. Use of radially localized functions rather than atomic functions alleviates the intruder state problem.
Forte, G; March, N H; Pucci, R
2014-01-01
The Hartree-Fock (HF) method, supplemented by low-order Moller-Plesset (MP2) perturbation theory, has been utilized to predict the nuclear geometry, assuming planarity, of a low-lying isomer of the free space cluster BOSi$_2$. The planar structure found at equilibrium geometry is shown to be stable against small amplitude molecular vibrations. Finally, some brief comments are made on the possible relevance of the above free-space cluster geometry to the known B-O defects which limit the improvement of minority carrier lifetime in a form of p-type silicon.
Forte, G.; Angilella, G. G. N.; March, N. H.; Pucci, R.
2014-07-01
The Hartree-Fock (HF) method, supplemented by low-order Møller-Plesset (MP2) perturbation theory, has been utilized to predict the nuclear geometry, assuming planarity, of a low-lying isomer of the free space cluster BOSi2. The planar structure found at equilibrium geometry is shown to be stable against small amplitude molecular vibrations. Finally, some brief comments are made on the possible relevance of the above free-space cluster geometry to the known B-O defects which limit the improvement of minority carrier lifetime in a form of p-type silicon.
Maschio, Lorenzo; Kirtman, Bernard; Rérat, Michel; Orlando, Roberto; Dovesi, Roberto
2013-10-28
We present a fully analytical formulation for calculating Raman intensities of crystalline periodic systems using a local basis set. Numerical differentiation with respect to atomic coordinates and with respect to wavevectors is entirely avoided as is the determination of crystal orbital coefficient derivatives with respect to nuclear displacements. Instead, our method utilizes the orbital energy-weighted density matrix and is based on the self-consistent solution of first- and second-order Coupled Perturbed Hartree-Fock/Kohn-Sham equations for the electronic response to external electric fields at the equilibrium geometry. This method has also been implemented in the Crystal program, which uses a Gaussian type basis set.
Y. Sajeev
2015-08-01
Full Text Available The equation-of-motion coupled cluster (EOMCC method based on the excited state Hartree-Fock (ESHF solutions is shown to be appropriate for computing the entire ground state potential energy curves of strongly correlated higher-order bonds. The new approach is best illustrated for the homolytic dissociation of higher-order bonds in molecules. The required multireference character of the true ground state wavefunction is introduced through the linear excitation operator of the EOMCC method. Even at the singles and doubles level of cluster excitation truncation, the nonparallelity error of the ground state potential energy curve from the ESHF based EOMCC method is small.
Meng, Qingyong; Meyer, Hans-Dieter
2017-05-01
To study the scattering of CO off a movable Cu(100) surface, extensive multilayer multiconfiguration time-dependent Hartree (ML-MCTDH) calculations are performed based on the SAP [R. Marquardt et al., J. Chem. Phys. 132, 074108 (2010)] potential energy surface in conjunction with a recently developed expansion model [Q. Meng and H.-D. Meyer, J. Chem. Phys. 143, 164310 (2015)] for including lattice motion. The surface vibration potential is constructed by a sum of Morse potentials where the parameters are determined by simulating the vibrational energies of a clean Cu(100) surface. Having constructed the total Hamiltonian, extensive dynamical calculations in both time-independent and time-dependent schemes are performed. Two-layer MCTDH (i.e., normal MCTDH) block-improved-relaxations (time-independent scheme) show that increasing the number of included surface vibrational dimensions lets the vibrational energies of CO/Cu(100) decrease for the frustrated translation (T mode), which is of low energy but increase those of the frustrated rotation (R mode) and the CO-Cu stretch (S mode), whose vibrational energies are larger than the energies of the in-plane surface vibrations (˜79 cm-1). This energy-shifting behavior was predicted and discussed by a simple model in our previous publication [Q. Meng and H.-D. Meyer, J. Chem. Phys. 143, 164310 (2015)]. By the flux analysis of the MCTDH/ML-MCTDH propagated wave packets, we calculated the sticking probabilities for the X + 0D, X + 1D, X + 3D, X + 5D, and X + 15D systems, where "X" stands for the used dimensionality of the CO/rigid-surface system and the second entry denotes the number of surface degrees of freedom included. From these sticking probabilities, the X + 5D/15D calculations predict a slower decrease of sticking with increasing energy as compared to the sticking of the X + 0D/1D/3D calculations. This is because the translational energy of CO is more easily transferred to surface vibrations, when the vibrational
APPROXIMATE SAMPLING THEOREM FOR BIVARIATE CONTINUOUS FUNCTION
杨守志; 程正兴; 唐远炎
2003-01-01
An approximate solution of the refinement equation was given by its mask, and the approximate sampling theorem for bivariate continuous function was proved by applying the approximate solution. The approximate sampling function defined uniquely by the mask of the refinement equation is the approximate solution of the equation, a piece-wise linear function, and posseses an explicit computation formula. Therefore the mask of the refinement equation is selected according to one' s requirement, so that one may controll the decay speed of the approximate sampling function.
Bernstein-type approximations of smooth functions
Andrea Pallini
2007-10-01
Full Text Available The Bernstein-type approximation for smooth functions is proposed and studied. We propose the Bernstein-type approximation with definitions that directly apply the binomial distribution and the multivariate binomial distribution. The Bernstein-type approximations generalize the corresponding Bernstein polynomials, by considering definitions that depend on a convenient approximation coefficient in linear kernels. In the Bernstein-type approximations, we study the uniform convergence and the degree of approximation. The Bernstein-type estimators of smooth functions of population means are also proposed and studied.
Jemai, M
2004-07-01
In the present thesis we have applied the self consistent random phase approximation (SCRPA) to the Hubbard model with a small number of sites (a chain of 2, 4, 6,... sites). Earlier SCRPA had produced very good results in other models like the pairing model of Richardson. It was therefore interesting to see what kind of results the method is able to produce in the case of a more complex model like the Hubbard model. To our great satisfaction the case of two sites with two electrons (half-filling) is solved exactly by the SCRPA. This may seem a little trivial but the fact is that other respectable approximations like 'GW' or the approach with the Gutzwiller wave function yield results still far from exact. With this promising starting point, the case of 6 sites at half filling was considered next. For that case, evidently, SCRPA does not any longer give exact results. However, they are still excellent for a wide range of values of the coupling constant U, covering for instance the phase transition region towards a state with non zero magnetisation. We consider this as a good success of the theory. Non the less the case of 4 sites (a plaquette), as indeed all cases with 4n sites at half filling, turned out to have a problem because of degeneracies at the Hartree Fock level. A generalisation of the present method, including in addition to the pairs, quadruples of Fermions operators (called second RPA) is proposed to also include exactly the plaquette case in our approach. This is therefore a very interesting perspective of the present work. (author)
He, Liming; Zhu, Yunxia; Zhang, Meng; Tu, Yaoquan
2011-11-01
We present a new second-order representation of the relativistic Hartree-Fock equation, which can be solved by the standard Hartree-Fock technique. An alternative reduction for the magnetic part of the Breit interaction is presented in an explicit expression. A corresponding program has been developed, which improves significantly the scaled linear mesh introduced by Herman and Skillman. The structures for a number of atoms and ions are calculated and the agreement of our results with those published is excellent. We evaluate the fine-structure intervals of nd(n = 3-40) Rydberg series for sodium. The inverted fine-structure splitting values are obtained directly as the differences of eigenvalues obtained from a self-consistent field procedure. Taking into account the Gaunt effect enables the accuracy of the calculation to be substantially improved. The complete treatments reproduce very well the inverted fine structures along the Rydberg series and the relative difference between the present results and the experiments does not exceed 4.4%.
Marcos, S [Departamento de FIsica Moderna, Universidad de Cantabria, E-39005 Santander (Spain); Savushkin, L N [Department of Physics, St Petersburg University for Telecommunications, 191065 St Petersburg (Russian Federation); Fomenko, V N [Department of Mathematics, St Petersburg University for Railway Engineering, 190031 St Petersburg (Russian Federation); Lopez-Quelle, M [Departamento de FIsica Aplicada, Universidad de Cantabria, E-39005 Santander (Spain); Niembro, R [Departamento de FIsica Moderna, Universidad de Cantabria, E-39005 Santander (Spain)
2004-06-01
An exact method is suggested to treat the nonlinear self-interactions (NLSI) in the relativistic Hartree-Fock (RHF) approach for nuclear systems. We consider here the NLSI constructed from the relativistic scalar nucleon densities including products of six and eight fermion fields. This type of NLSI corresponds to the zero-range limit of the standard cubic and quartic self-interactions of the scalar field. The method to treat the NLSI uses the Fierz transformation, which enables one to express the exchange (Fock) components in terms of the direct (Hartree) ones. The method is applied to nuclear matter and finite nuclei. It is shown that, in the RHF formalism, the NLSI, which are explicitly isovector-independent, generate scalar, vector and tensor nucleon self-energies with a strong isovector dependence. This strong isovector structure of the self-energies is due to the exchange terms of the RHF method. Calculations are carried out with a parametrization containing five free parameters. The model allows a description of both types of systems compatible with experimental data.
Martínez, E; Rincon, L
2002-01-01
Theoretical results of photoemission energy spectral of the atomic sulfur and of the SO sub 2 molecule, adsorbed over surfaces of Ni(110) and Ni(l l l) clusters, are reported in this work. Clusters with 11, 13, 15 and 17 atoms of Ni were used for the model. The calculations were done by Hartree-Fock method, and basis sets of type STO-NG and p-q1G (p3,6; q= 2,3; N= 3,6) were used. The ionization potentials (IP) were interpreted within the Koopmans Theorem. The results obtained for the IP of 1s, 2s and 2p orbitals are 2472.03 eV, 238.14 eV and 173.55 eV, respectively; while for the same orbitals of the sulfur in SO sub 2 these values are 2481.30 eV, 246.61 eV and 182.17 eV. The theoretical results were compared with experimental results reported in the references, and the error ranges are between 5 eV and 30 eV, in agreement with the standard for the Hartree-Fock method. (Author)
Applications of Discrepancy Theory in Multiobjective Approximation
Glaßer, Christian; Witek, Maximilian
2011-01-01
We apply a multi-color extension of the Beck-Fiala theorem to show that the multiobjective maximum traveling salesman problem is randomized 1/2-approximable on directed graphs and randomized 2/3-approximable on undirected graphs. Using the same technique we show that the multiobjective maximum satisfiablilty problem is 1/2-approximable.
Fractal Trigonometric Polynomials for Restricted Range Approximation
Chand, A. K. B.; Navascués, M. A.; Viswanathan, P.; Katiyar, S. K.
2016-05-01
One-sided approximation tackles the problem of approximation of a prescribed function by simple traditional functions such as polynomials or trigonometric functions that lie completely above or below it. In this paper, we use the concept of fractal interpolation function (FIF), precisely of fractal trigonometric polynomials, to construct one-sided uniform approximants for some classes of continuous functions.
Axiomatic Characterizations of IVF Rough Approximation Operators
Guangji Yu
2014-01-01
Full Text Available This paper is devoted to the study of axiomatic characterizations of IVF rough approximation operators. IVF approximation spaces are investigated. The fact that different IVF operators satisfy some axioms to guarantee the existence of different types of IVF relations which produce the same operators is proved and then IVF rough approximation operators are characterized by axioms.
Some relations between entropy and approximation numbers
郑志明
1999-01-01
A general result is obtained which relates the entropy numbers of compact maps on Hilbert space to its approximation numbers. Compared with previous works in this area, it is particularly convenient for dealing with the cases where the approximation numbers decay rapidly. A nice estimation between entropy and approximation numbers for noncompact maps is given.
Nonlinear approximation with dictionaries, I: Direct estimates
Gribonval, Rémi; Nielsen, Morten
$-term approximation with algorithmic constraints: thresholding and Chebychev approximation classes are studied respectively. We consider embeddings of the Jackson type (direct estimates) of sparsity spaces into the mentioned approximation classes. General direct estimates are based on the geometry of the Banach space...
Operator approximant problems arising from quantum theory
Maher, Philip J
2017-01-01
This book offers an account of a number of aspects of operator theory, mainly developed since the 1980s, whose problems have their roots in quantum theory. The research presented is in non-commutative operator approximation theory or, to use Halmos' terminology, in operator approximants. Focusing on the concept of approximants, this self-contained book is suitable for graduate courses.
Advanced Concepts and Methods of Approximate Reasoning
1989-12-01
and L. Valverde. On mode and implication in approximate reasoning. In M.M. Gupta, A. Kandel, W. Bandler , J.B. Kiszka, editors, Approximate Reasoning and...190, 1981. [43] E. Trillas and L. Valverde. On mode and implication in approximate reasoning. In M.M. Gupta, A. Kandel, W. Bandler , J.B. Kiszka
NONLINEAR APPROXIMATION WITH GENERAL WAVE PACKETS
L. Borup; M. Nielsen
2005-01-01
We study nonlinear approximation in the Triebel-Lizorkin spaces with dictionaries formed by dilating and translating one single function g. A general Jackson inequality is derived for best m-term approximation with such dictionaries. In some special cases where g has a special structure, a complete characterization of the approximation spaces is derived.
Approximate Nearest Neighbor Queries among Parallel Segments
Emiris, Ioannis Z.; Malamatos, Theocharis; Tsigaridas, Elias
2010-01-01
We develop a data structure for answering efficiently approximate nearest neighbor queries over a set of parallel segments in three dimensions. We connect this problem to approximate nearest neighbor searching under weight constraints and approximate nearest neighbor searching on historical data...
Nonlinear approximation with general wave packets
Borup, Lasse; Nielsen, Morten
2005-01-01
We study nonlinear approximation in the Triebel-Lizorkin spaces with dictionaries formed by dilating and translating one single function g. A general Jackson inequality is derived for best m-term approximation with such dictionaries. In some special cases where g has a special structure, a complete...... characterization of the approximation spaces is derived....
Nonlinear approximation with bi-framelets
Borup, Lasse; Nielsen, Morten; Gribonval, Rémi
2005-01-01
We study the approximation in Lebesgue spaces of wavelet bi-frame systems given by translations and dilations of a finite set of generators. A complete characterization of the approximation spaces associated with best m-term approximation of wavelet bi-framelet systems is given...
Approximation properties of fine hyperbolic graphs
Benyin Fu
2016-05-01
In this paper, we propose a definition of approximation property which is called the metric invariant translation approximation property for a countable discrete metric space. Moreover, we use the techniques of Ozawa’s to prove that a fine hyperbolic graph has the metric invariant translation approximation property.
Tsuchimochi, Takashi; Ten-No, Seiichiro
2017-04-11
We propose a size-consistent generalization of the recently developed spin-extended configuration interaction with singles and doubles (ECISD), where a CI wave function is explicitly spin-projected. The size-consistent effect is effectively incorporated by treating quadruples within the formulation of coupled electron pair approximation. As in coupled-cluster theory, quadruple excitations are approximated by a disconnected product of double excitations. Despite its conceptual similarity to the standard single-reference and multireference analogues, such a generalization requires careful derivation, as the spin-projected CI space is nonorthogonal and overcomplete. Although our methods generally yield better results than ECISD, size-consistency is only approximately retained because the action of a symmetry-projection operator is size-inconsistent. In this work, we focus on simple models where exclusion-principle-violating terms, which eliminate undesired contributions to the correlation effects, are either completely neglected or averaged. These models possess an orbital-invariant energy functional that is to be minimized by diagonalizing an energy-shifted effective Hamiltonian within the singles and doubles manifold. This allows for a straightforward generalization of the ECISD analytical gradients needed to determine molecular properties and geometric optimization. Given the multireference nature of the spin-projected Hartree-Fock method, the proposed approaches are expected to handle static correlation, unlike single-reference analogues. We critically assess the performance of our methods using dissociation curves of molecules, singlet-triplet splitting gaps, hyperfine coupling constants, and the chromium dimer. The size-consistency and size-extensivity of the methods are also discussed.
Effective restoration of dipole sum rules within the renormalized random-phase approximation
Hung, N Quang; Hao, T V Nhan; Phuc, L Tan
2016-01-01
The dipole excitations for calcium and zirconium isotopes are studied within the fully self-consistent Hartree-Fock mean field incorporated with the renormalized random-phase approximation (RRPA) using the Skyrme interaction SLy5. The RRPA takes into account the effect of ground-state correlations beyond RPA owing to the Pauli principle between the particle-hole pairs that form the RPA excitations as well as the correlations due to the particle-particle and hole-hole transitions, whose effects are treated here in an effective way. By comparing the RPA results with the RRPA ones, which are obtained for isoscalar (IS) and isovector (IV) dipole excitations in $^{48, 52, 58}$Ca and $^{90, 96, 110}$Zr, it is shown that ground-state correlations beyond the RPA reduce the IS transition strengths. They also shift up the energy of the lowest IV dipole state and slightly push down the peak energy of the IV giant dipole resonance. As the result, the energy-weighted sums of strengths of both IS and IV modes decrease, cau...
Resonant-state expansion Born Approximation
Doost, M B
2015-01-01
The Born Approximation is a fundamental formula in Physics, it allows the calculation of weak scattering via the Fourier transform of the scattering potential. I extend the Born Approximation by including in the formula the Fourier transform of a truncated basis of the infinite number of appropriately normalised resonant states. This extension of the Born Approximation is named the Resonant-State Expansion Born Approximation or RSE Born Approximation. The resonant-states of the system can be calculated using the recently discovered RSE perturbation theory for electrodynamics and normalised correctly to appear in spectral Green's functions via the flux volume normalisation.
Canonical Sets of Best L1-Approximation
Dimiter Dryanov
2012-01-01
Full Text Available In mathematics, the term approximation usually means either interpolation on a point set or approximation with respect to a given distance. There is a concept, which joins the two approaches together, and this is the concept of characterization of the best approximants via interpolation. It turns out that for some large classes of functions the best approximants with respect to a certain distance can be constructed by interpolation on a point set that does not depend on the choice of the function to be approximated. Such point sets are called canonical sets of best approximation. The present paper summarizes results on canonical sets of best L1-approximation with emphasis on multivariate interpolation and best L1-approximation by blending functions. The best L1-approximants are characterized as transfinite interpolants on canonical sets. The notion of a Haar-Chebyshev system in the multivariate case is discussed also. In this context, it is shown that some multivariate interpolation spaces share properties of univariate Haar-Chebyshev systems. We study also the problem of best one-sided multivariate L1-approximation by sums of univariate functions. Explicit constructions of best one-sided L1-approximants give rise to well-known and new inequalities.
Mapping moveout approximations in TI media
Stovas, Alexey
2013-11-21
Moveout approximations play a very important role in seismic modeling, inversion, and scanning for parameters in complex media. We developed a scheme to map one-way moveout approximations for transversely isotropic media with a vertical axis of symmetry (VTI), which is widely available, to the tilted case (TTI) by introducing the effective tilt angle. As a result, we obtained highly accurate TTI moveout equations analogous with their VTI counterparts. Our analysis showed that the most accurate approximation is obtained from the mapping of generalized approximation. The new moveout approximations allow for, as the examples demonstrate, accurate description of moveout in the TTI case even for vertical heterogeneity. The proposed moveout approximations can be easily used for inversion in a layered TTI medium because the parameters of these approximations explicitly depend on corresponding effective parameters in a layered VTI medium.
An Approximate Approach to Automatic Kernel Selection.
Ding, Lizhong; Liao, Shizhong
2016-02-02
Kernel selection is a fundamental problem of kernel-based learning algorithms. In this paper, we propose an approximate approach to automatic kernel selection for regression from the perspective of kernel matrix approximation. We first introduce multilevel circulant matrices into automatic kernel selection, and develop two approximate kernel selection algorithms by exploiting the computational virtues of multilevel circulant matrices. The complexity of the proposed algorithms is quasi-linear in the number of data points. Then, we prove an approximation error bound to measure the effect of the approximation in kernel matrices by multilevel circulant matrices on the hypothesis and further show that the approximate hypothesis produced with multilevel circulant matrices converges to the accurate hypothesis produced with kernel matrices. Experimental evaluations on benchmark datasets demonstrate the effectiveness of approximate kernel selection.
On Gakerkin approximations for the quasigeostrophic equations
Rocha, Cesar B; Grooms, Ian
2015-01-01
We study the representation of approximate solutions of the three-dimensional quasigeostrophic (QG) equations using Galerkin series with standard vertical modes. In particular, we show that standard modes are compatible with nonzero buoyancy at the surfaces and can be used to solve the Eady problem. We extend two existing Galerkin approaches (A and B) and develop a new Galerkin approximation (C). Approximation A, due to Flierl (1978), represents the streamfunction as a truncated Galerkin series and defines the potential vorticity (PV) that satisfies the inversion problem exactly. Approximation B, due to Tulloch and Smith (2009b), represents the PV as a truncated Galerkin series and calculates the streamfunction that satisfies the inversion problem exactly. Approximation C, the true Galerkin approximation for the QG equations, represents both streamfunction and PV as truncated Galerkin series, but does not satisfy the inversion equation exactly. The three approximations are fundamentally different unless the b...
Heuser, Johannes; Höfener, Sebastian
2017-10-15
We report the derivation and implementation of analytical nuclear gradients for excited states using time-dependent density functional theory using the Tamm-Dancoff approximation combined with uncoupled frozen-density embedding using density fitting. Explicit equations are presented and discussed. The implementation is able to treat singlet as well as triplet states and functionals using the local density approximation, the generalized gradient approximation, combinations with Hartree-Fock exchange (hybrids), and range-separated functionals such as CAM-B3LYP. The new method is benchmarked against supermolecule calculations in two case studies: The solvatochromic shift of the (vertical) fluorescence energy of 4-aminophthalimide on solvation, and the first local excitation of the benzonitrile dimer. Whereas for the 4-aminophthalimide-water complex deviations of about 0.2 eV are obtained to supermolecular calculations, for the benzonitrile dimer the maximum error for adiabatic excitation energies is below 0.01 eV due to a weak coupling of the subsystems. © 2017 Wiley Periodicals, Inc. © 2017 Wiley Periodicals, Inc.
Bučinský, Lukáš
2015-05-11
"Kramers pairs symmetry breaking" is evaluated at the 2-component (2c) Kramers unrestricted and/or general complex Hartree-Fock (GCHF) level of theory, and its analogy with "spin contamination" at the 1-component (1c) unrestricted Hartree-Fock (UHF) level of theory is emphasized. The GCHF "Kramers pairs symmetry breaking" evaluation is using the square of overlaps between the set of occupied spinorbitals with the projected set of Kramers pairs. In the same fashion, overlaps between α and β orbitals are used in the evaluation of "spin contamination" at the UHF level of theory. In this manner, UHF Š2 expectation value is made formally extended to the GCHF case. The directly evaluated GCHF expectation value of the Š2 operator is considered for completeness. It is found that the 2c GCHF Kramers pairs symmetry breaking has a very similar extent in comparison to the 1c UHF spin contamination. Thus higher excited states contributions to the 1c and 2c unrestricted wave functions of open shell systems have almost the same extent and physical consequences. Moreover, it is formally shown that a single determinant wave function in the restricted open shell Kramers case has the expectation value of K2 operator equal to the negative number of open shell electrons, while the eigenvalue of K2 for the series of simple systems (H, He, He*-triplet, Li and Li*-quartet) are found to be equal to minus the square of the number of open shell electrons. The concept of unpaired electron density is extended to the GCHF regime and compared to UHF and restricted open shell Hartree-Fock spin density. The "collinear" and "noncollinear" analogs of spin density at the GCHF level of theory are considered as well. Spin contamination and/or Kramers pairs symmetry breaking, spin populations and spin densities are considered for H2O+, Cl, HCl+, phenoxyl radical (C6H5O) as well as for Cu, Cu2+, Fe and the [OsCl5(1H-pyrazole)]- anion. The 1c and 2c unpaired electron density representation is found
Improving biconnectivity approximation via local optimization
Ka Wong Chong; Tak Wah Lam [Univ. of Hong Kong (Hong Kong)
1996-12-31
The problem of finding the minimum biconnected spanning subgraph of an undirected graph is NP-hard. A lot of effort has been made to find biconnected spanning subgraphs that approximate to the minimum one as close as possible. Recently, new polynomial-time (sequential) approximation algorithms have been devised to improve the approximation factor from 2 to 5/3 , then 3/2, while NC algorithms have also been known to achieve 7/4 + {epsilon}. This paper presents a new technique which can be used to further improve parallel approximation factors to 5/3 + {epsilon}. In the sequential context, the technique reveals an algorithm with a factor of {alpha} + 1/5, where a is the approximation factor of any 2-edge connectivity approximation algorithm.
Frankenstein's Glue: Transition functions for approximate solutions
Yunes, N
2006-01-01
Approximations are commonly employed to find approximate solutions to the Einstein equations. These solutions, however, are usually only valid in some specific spacetime region. A global solution can be constructed by gluing approximate solutions together, but this procedure is difficult because discontinuities can arise, leading to large violations of the Einstein equations. In this paper, we provide an attempt to formalize this gluing scheme by studying transition functions that join approximate solutions together. In particular, we propose certain sufficient conditions on these functions and proof that these conditions guarantee that the joined solution still satisfies the Einstein equations to the same order as the approximate ones. An example is also provided for a binary system of non-spinning black holes, where the approximate solutions are taken to be given by a post-Newtonian expansion and a perturbed Schwarzschild solution. For this specific case, we show that if the transition functions satisfy the...
Floating-Point $L^2$-Approximations
Brisebarre, Nicolas; Hanrot, Guillaume
2007-01-01
International audience; Computing good polynomial approximations to usual functions is an important topic for the computer evaluation of those functions. These approximations can be good under several criteria, the most desirable being probably that the relative error is as small as possible in the $L^{\\infty}$ sense, i.e. everywhere on the interval under study. In the present paper, we investigate a simpler criterion, the $L^2$ case. Though finding a best polynomial $L^2$-approximation with ...
Metric Diophantine approximation on homogeneous varieties
Ghosh, Anish; Nevo, Amos
2012-01-01
We develop the metric theory of Diophantine approximation on homogeneous varieties of semisimple algebraic groups and prove results analogous to the classical Khinchin and Jarnik theorems. In full generality our results establish simultaneous Diophantine approximation with respect to several completions, and Diophantine approximation over general number fields using S-algebraic integers. In several important examples, the metric results we obtain are optimal. The proof uses quantitative equidistribution properties of suitable averaging operators, which are derived from spectral bounds in automorphic representations.
Approximately liner phase IIR digital filter banks
J. D. Ćertić; M. D. Lutovac; L. D. Milić
2013-01-01
In this paper, uniform and nonuniform digital filter banks based on approximately linear phase IIR filters and frequency response masking technique (FRM) are presented. Both filter banks are realized as a connection of an interpolated half-band approximately linear phase IIR filter as a first stage of the FRM design and an appropriate number of masking filters. The masking filters are half-band IIR filters with an approximately linear phase. The resulting IIR filter banks are compared with li...
A Note on Generalized Approximation Property
Antara Bhar
2013-01-01
Full Text Available We introduce a notion of generalized approximation property, which we refer to as --AP possessed by a Banach space , corresponding to an arbitrary Banach sequence space and a convex subset of , the class of bounded linear operators on . This property includes approximation property studied by Grothendieck, -approximation property considered by Sinha and Karn and Delgado et al., and also approximation property studied by Lissitsin et al. We characterize a Banach space having --AP with the help of -compact operators, -nuclear operators, and quasi--nuclear operators. A particular case for ( has also been characterized.
Upper Bounds on Numerical Approximation Errors
Raahauge, Peter
2004-01-01
This paper suggests a method for determining rigorous upper bounds on approximationerrors of numerical solutions to infinite horizon dynamic programming models.Bounds are provided for approximations of the value function and the policyfunction as well as the derivatives of the value function....... The bounds apply to moregeneral problems than existing bounding methods do. For instance, since strict concavityis not required, linear models and piecewise linear approximations can bedealt with. Despite the generality, the bounds perform well in comparison with existingmethods even when applied...... to approximations of a standard (strictly concave)growth model.KEYWORDS: Numerical approximation errors, Bellman contractions, Error bounds...
TMB: Automatic differentiation and laplace approximation
Kristensen, Kasper; Nielsen, Anders; Berg, Casper Willestofte
2016-01-01
computations. The user defines the joint likelihood for the data and the random effects as a C++ template function, while all the other operations are done in R; e.g., reading in the data. The package evaluates and maximizes the Laplace approximation of the marginal likelihood where the random effects...... are automatically integrated out. This approximation, and its derivatives, are obtained using automatic differentiation (up to order three) of the joint likelihood. The computations are designed to be fast for problems with many random effects (approximate to 10(6)) and parameters (approximate to 10...
Kjærgaard, Thomas; Jørgensen, Poul; Thorvaldsen, Andreas;
2009-01-01
-orbital density-matrix based formulation of response theory and use London atomic orbitals to parametrize the magnetic field dependence. It yields a computational procedure which is both gauge-origin independent and suitable for linear-scaling at the level of time-dependent Hartree-Fock and density functional......A Lagrangian approach has been used to derive gauge-origin independent expressions for two properties that rationalize magneto-optical activity, namely the Verdet constant V(ω) of the Faraday effect and the B term of magnetic circular dichroism. The approach is expressed in terms of an atomic...... theory. The formulation includes a modified preconditioned conjugated gradient algorithm, which projects out the excited state component from the solution to the linear response equation. This is required when solving one of the response equations for the determination of the B term and divergence...
Ucun, Fatih; Sağlam, Adnan; Güçlü, Vesile
2007-06-01
The molecular structures, vibrational frequencies and corresponding vibrational assignments of xanthine and its methyl derivatives (caffeine and theobromine) have been calculated using ab initio Hartree-Fock (HF) and density functional theory (B3LYP) methods with 6-31G(d, p) basis set level. The calculations were utilized to the CS symmetries of the molecules. The obtained vibrational frequencies and optimised geometric parameters (bond lengths and bond angles) were seen to be well agreement with the experimental data. The used scale factors which have been obtained the ratio of the frequency values of the strongest peaks in the calculated and experimental spectra seem to cause the gained vibrations well corresponding to the experimental ones. Theoretical infrared intensities and Raman activities are also reported.
Akamatsu, Hirofumi; Kumagai, Yu; Oba, Fumiyasu; Fujita, Koji; Murakami, Hideo; Tanaka, Katsuhisa; Tanaka, Isao
2011-06-01
A superexchange mechanism between Eu2+ 4f spins via the 3d states of nonmagnetic Ti4+ ions is proposed through first-principles calculations based on a hybrid Hartree-Fock density functional approach to explain G-type antiferromagnetism in EuTiO3. This mechanism is supported by systematic calculations for related Eu2+-based perovskite oxides. In EuTiO3, the competition between the antiferromagnetic superexchange and an indirect ferromagnetic exchange via the Eu 5d states leads to a delicate balance between antiferromagnetic and ferromagnetic phases. The superexchange mechanism involving the Ti 3d states hints at the microscopic origin of the strong spin-lattice coupling in EuTiO3.
Ebata, Shuichiro; Inakura, Tsunenori
2014-01-01
Systematic investigations of the electric dipole (E1) modes of excitation are performed using the canonical-basis time-dependent Hartree-Fock-Bogoliubov (Cb-TDHFB) theory. The Cb-TDHFB is able to describe dynamical pairing correlations in excited states of nuclear systems. We apply the method to the real-time calculation of linear response in even-even nuclei with Skyrme functionals. Effects of shell structure, neutron skin, deformation, and neutron chemical potential (separation energy) are studied in a systematic way. This reveals a number of characteristic features of the low-energy E1 modes. We also find a universal behavior in the low-energy E1 modes for heavy neutron-rich isotopes, which suggests the emergence of decoupled E1 peaks beyond N = 82.
Gundra, Kondayya
2011-01-01
Pariser-Parr-Pople (P-P-P) model Hamiltonian is employed frequently to study the electronic structure and optical properties of $\\pi$-conjugated systems. In this paper we describe a Fortran 90 computer program which uses the P-P-P model Hamiltonian to solve the Hartree-Fock (HF) equation for infinitely long, one-dimensional, periodic, $\\pi$-electron systems. The code is capable of computing the band structure, as also the linear optical absorption spectrum, by using the tight-binding (TB) and the HF methods. Furthermore, using our program the user can solve the HF equation in the presence of a finite external electric field, thereby, allowing the simulation of gated systems. We apply our code to compute various properties of polymers such as $trans$-polyacetylene ($t$-PA), poly-\\emph{para}-phenylene (PPP), and armchair and zigzag graphene nanoribbons, in the infinite length limit.
Inakura, T.; Mizutori, S.; Yamagami, M.; Matsuyanagi, K. E-mail: ken@ruby.scphys.kyoto-u.ac.jp
2002-11-18
With the use of the symmetry-unrestricted cranked Skyrme-Hartree-Fock method in the three-dimensional coordinate-mesh representation, we have carried out a systematic theoretical search for the superdeformed and hyperdeformed rotational bands in the mass A=30-50 region. Along the N=Z line, we have found superdeformed solutions in {sup 32}S, {sup 36}Ar, {sup 40}Ca, {sup 44}Ti, and hyperdeformed solutions in {sup 36}Ar, {sup 40}Ca, {sup 44}Ti, {sup 48}Cr. The superdeformed band in {sup 40}Ca is found to be extremely soft against both the axially symmetric (Y{sub 30}) and asymmetric (Y{sub 31}) octupole deformations. An interesting role of symmetry breaking in the mean field is pointed out.
Inakura, T.; Yamagami, M.; Matsuyanagi, K. [Kyoto Univ., Dept. of Physics, Kyoto (Japan); Mizutori, S. [Kansai Women' s College, Dept. of Human Science, Kashiwara, Osaka (Japan)
2003-02-01
With the use of the symmetry-unrestricted cranked Skyrme-Hartree-Fock method in the three-dimensional coordinate-mesh representation, we have carried out a systematic theoretical search for the superdeformed and hyperdeformed rotational bands in the mass A=30-50 region. Along the N=Z line, we have found superdeformed solutions in {sup 32}S, {sup 36}Ar, {sup 40}Ca, {sup 44}Ti, and hyperdeformed solutions in {sup 36}Ar, {sup 40}Ca, {sup 44}Ti, {sup 48}Cr. The superdeformed band in {sup 40}Ca is found to be extremely soft against both the axially symmetric (Y{sub 30}) and asymmetric (Y{sub 31}) octupole deformations. An interesting role of symmetry breaking in the mean field is pointed out. (author)
Inakura, T.; Mizutori, S.; Yamagami, M.; Matsuyanagi, K.
2002-11-01
With the use of the symmetry-unrestricted cranked Skyrme-Hartree-Fock method in the three-dimensional coordinate-mesh representation, we have carried out a systematic theoretical search for the superdeformed and hyperdeformed rotational bands in the mass A=30-50 region. Along the N= Z line, we have found superdeformed solutions in 32S, 36Ar, 40Ca, 44Ti, and hyperdeformed solutions in 36Ar, 40Ca, 44Ti, 48Cr. The superdeformed band in 40Ca is found to be extremely soft against both the axially symmetric ( Y30) and asymmetric ( Y31) octupole deformations. An interesting role of symmetry breaking in the mean field is pointed out.
Joshi, Bhawani Datt; Srivastava, Anubha; Tandon, Poonam; Jain, Sudha
2011-11-01
Yohimbine hydrochloride (YHCl) is an aphrodisiac and promoted for erectile dysfunction, weight loss and depression. The optimized geometry, total energy, potential energy surface and vibrational wavenumbers of yohimbine hydrochloride have been determined using ab initio, Hartree-Fock (HF) and density functional theory (DFT/B3LYP) method with 6-311++G(d,p) basis set. A complete vibrational assignment is provided for the observed Raman and IR spectra of YHCl. The UV absorption spectrum was examined in ethanol solvent and compared with the calculated one in gas phase as well as in solvent environment (polarizable continuum model, PCM) using TD-DFT/6-31G basis set. These methods are proposed as a tool to be applied in the structural characterization of YHCl. The calculated highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) with frontier orbital gap are presented.
Inversion and approximation of Laplace transforms
Lear, W. M.
1980-01-01
A method of inverting Laplace transforms by using a set of orthonormal functions is reported. As a byproduct of the inversion, approximation of complicated Laplace transforms by a transform with a series of simple poles along the left half plane real axis is shown. The inversion and approximation process is simple enough to be put on a programmable hand calculator.
Computing Functions by Approximating the Input
Goldberg, Mayer
2012-01-01
In computing real-valued functions, it is ordinarily assumed that the input to the function is known, and it is the output that we need to approximate. In this work, we take the opposite approach: we show how to compute the values of some transcendental functions by approximating the input to these functions, and obtaining exact answers for their…
Non-Linear Approximation of Bayesian Update
Litvinenko, Alexander
2016-06-23
We develop a non-linear approximation of expensive Bayesian formula. This non-linear approximation is applied directly to Polynomial Chaos Coefficients. In this way, we avoid Monte Carlo sampling and sampling error. We can show that the famous Kalman Update formula is a particular case of this update.
Random Attractors of Stochastic Modified Boussinesq Approximation
郭春晓
2011-01-01
The Boussinesq approximation is a reasonable model to describe processes in body interior in planetary physics. We refer to [1] and [2] for a derivation of the Boussinesq approximation, and [3] for some related results of existence and uniqueness of solution.
Approximating a harmonizable isotropic random field
Randall J. Swift
2001-01-01
Full Text Available The class of harmonizable fields is a natural extension of the class of stationary fields. This paper considers a stochastic series approximation of a harmonizable isotropic random field. This approximation is useful for numerical simulation of such a field.
On approximating multi-criteria TSP
Manthey, Bodo; Albers, S.; Marion, J.-Y.
2009-01-01
We present approximation algorithms for almost all variants of the multi-criteria traveling salesman problem (TSP), whose performances are independent of the number $k$ of criteria and come close to the approximation ratios obtained for TSP with a single objective function. We present randomized app
Regression with Sparse Approximations of Data
Noorzad, Pardis; Sturm, Bob L.
2012-01-01
We propose sparse approximation weighted regression (SPARROW), a method for local estimation of the regression function that uses sparse approximation with a dictionary of measurements. SPARROW estimates the regression function at a point with a linear combination of a few regressands selected by...
A case where BO Approximation breaks down
无
2007-01-01
@@ The Bom-Oppenheimer (BO)Approximation is ubiquitous in molecular physics,quantum physics and quantum chemistry. However, CAS researchers recently observed a breakdown of the Approximation in the reaction of fluorine with deuterium atoms.The result has been published in the August 24 issue of Science.
Two Point Pade Approximants and Duality
Banks, Tom
2013-01-01
We propose the use of two point Pade approximants to find expressions valid uniformly in coupling constant for theories with both weak and strong coupling expansions. In particular, one can use these approximants in models with a strong/weak duality, when the symmetries do not determine exact expressions for some quantity.
Function Approximation Using Probabilistic Fuzzy Systems
J.H. van den Berg (Jan); U. Kaymak (Uzay); R.J. Almeida e Santos Nogueira (Rui Jorge)
2011-01-01
textabstractWe consider function approximation by fuzzy systems. Fuzzy systems are typically used for approximating deterministic functions, in which the stochastic uncertainty is ignored. We propose probabilistic fuzzy systems in which the probabilistic nature of uncertainty is taken into account.
Approximation of the Inverse -Frame Operator
M R Abdollahpour; A Najati
2011-05-01
In this paper, we introduce the concept of (strong) projection method for -frames which works for all conditional -Riesz frames. We also derive a method for approximation of the inverse -frame operator which is efficient for all -frames. We show how the inverse of -frame operator can be approximated as close as we like using finite-dimensional linear algebra.
Nonlinear approximation with dictionaries I. Direct estimates
Gribonval, Rémi; Nielsen, Morten
2004-01-01
with algorithmic constraints: thresholding and Chebychev approximation classes are studied, respectively. We consider embeddings of the Jackson type (direct estimates) of sparsity spaces into the mentioned approximation classes. General direct estimates are based on the geometry of the Banach space, and we prove...
Approximations for stop-loss reinsurance premiums
Reijnen, Rajko; Albers, Willem/Wim; Kallenberg, W.C.M.
2005-01-01
Various approximations of stop-loss reinsurance premiums are described in literature. For a wide variety of claim size distributions and retention levels, such approximations are compared in this paper to each other, as well as to a quantitative criterion. For the aggregate claims two models are use
Quirks of Stirling's Approximation
Macrae, Roderick M.; Allgeier, Benjamin M.
2013-01-01
Stirling's approximation to ln "n"! is typically introduced to physical chemistry students as a step in the derivation of the statistical expression for the entropy. However, naive application of this approximation leads to incorrect conclusions. In this article, the problem is first illustrated using a familiar "toy…
INVARIANT RANDOM APPROXIMATION IN NONCONVEX DOMAIN
R. Shrivastava
2012-05-01
Full Text Available Random fixed point results in the setup of compact and weakly compact domain of Banach spaces which is not necessary starshaped have been obtained in the present work. Invariant random approximation results have also been determined asits application. In this way, random version of invariant approximation results due toMukherjee and Som [13] and Singh [17] have been given.
Approximability and Parameterized Complexity of Minmax Values
Hansen, Kristoffer Arnsfelt; Hansen, Thomas Dueholm; Miltersen, Peter Bro;
2008-01-01
We consider approximating the minmax value of a multi player game in strategic form. Tightening recent bounds by Borgs et al., we observe that approximating the value with a precision of ε log n digits (for any constant ε > 0) is NP-hard, where n is the size of the game. On the other hand...
Hardness of approximation for strip packing
Adamaszek, Anna Maria; Kociumaka, Tomasz; Pilipczuk, Marcin
2017-01-01
[SODA 2016] have recently proposed a (1.4 + ϵ)-approximation algorithm for this variant, thus showing that strip packing with polynomially bounded data can be approximated better than when exponentially large values are allowed in the input. Their result has subsequently been improved to a (4/3 + ϵ...
Approximations for stop-loss reinsurance premiums
Reijnen, Rajko; Albers, Willem; Kallenberg, Wilbert C.M.
2005-01-01
Various approximations of stop-loss reinsurance premiums are described in literature. For a wide variety of claim size distributions and retention levels, such approximations are compared in this paper to each other, as well as to a quantitative criterion. For the aggregate claims two models are use
Approximations for stop-loss reinsurance premiums
Reijnen, R.; Albers, W.; Kallenberg, W.C.M.
2003-01-01
Various approximations of stop-loss reinsurance premiums are described in literature. For a wide variety of claim size distributions and retention levels, such approximations are compared in this paper to each other, as well as to a quantitative criterion. For the aggregate claims two models are use
Lifetime of the Nonlinear Geometric Optics Approximation
Binzer, Knud Andreas
The subject of the thesis is to study acertain approximation method for highly oscillatory solutions to nonlinear partial differential equations.......The subject of the thesis is to study acertain approximation method for highly oscillatory solutions to nonlinear partial differential equations....
Simple Lie groups without the approximation property
Haagerup, Uffe; de Laat, Tim
2013-01-01
For a locally compact group G, let A(G) denote its Fourier algebra, and let M0A(G) denote the space of completely bounded Fourier multipliers on G. The group G is said to have the Approximation Property (AP) if the constant function 1 can be approximated by a net in A(G) in the weak-∗ topology...
An improved proximity force approximation for electrostatics
Fosco, C D; Mazzitelli, F D
2012-01-01
A quite straightforward approximation for the electrostatic interaction between two perfectly conducting surfaces suggests itself when the distance between them is much smaller than the characteristic lengths associated to their shapes. Indeed, in the so called "proximity force approximation" the electrostatic force is evaluated by first dividing each surface into a set of small flat patches, and then adding up the forces due two opposite pairs, the contribution of which are approximated as due to pairs of parallel planes. This approximation has been widely and successfully applied to different contexts, ranging from nuclear physics to Casimir effect calculations. We present here an improvement on this approximation, based on a derivative expansion for the electrostatic energy contained between the surfaces. The results obtained could be useful to discuss the geometric dependence of the electrostatic force, and also as a convenient benchmark for numerical analyses of the tip-sample electrostatic interaction i...
Approximate Furthest Neighbor in High Dimensions
Pagh, Rasmus; Silvestri, Francesco; Sivertsen, Johan von Tangen;
2015-01-01
Much recent work has been devoted to approximate nearest neighbor queries. Motivated by applications in recommender systems, we consider approximate furthest neighbor (AFN) queries. We present a simple, fast, and highly practical data structure for answering AFN queries in high-dimensional Euclid......Much recent work has been devoted to approximate nearest neighbor queries. Motivated by applications in recommender systems, we consider approximate furthest neighbor (AFN) queries. We present a simple, fast, and highly practical data structure for answering AFN queries in high......-dimensional Euclidean space. We build on the technique of Indyk (SODA 2003), storing random projections to provide sublinear query time for AFN. However, we introduce a different query algorithm, improving on Indyk’s approximation factor and reducing the running time by a logarithmic factor. We also present a variation...
Trajectory averaging for stochastic approximation MCMC algorithms
Liang, Faming
2010-01-01
The subject of stochastic approximation was founded by Robbins and Monro [Ann. Math. Statist. 22 (1951) 400--407]. After five decades of continual development, it has developed into an important area in systems control and optimization, and it has also served as a prototype for the development of adaptive algorithms for on-line estimation and control of stochastic systems. Recently, it has been used in statistics with Markov chain Monte Carlo for solving maximum likelihood estimation problems and for general simulation and optimizations. In this paper, we first show that the trajectory averaging estimator is asymptotically efficient for the stochastic approximation MCMC (SAMCMC) algorithm under mild conditions, and then apply this result to the stochastic approximation Monte Carlo algorithm [Liang, Liu and Carroll J. Amer. Statist. Assoc. 102 (2007) 305--320]. The application of the trajectory averaging estimator to other stochastic approximation MCMC algorithms, for example, a stochastic approximation MLE al...
Approximating maximum clique with a Hopfield network.
Jagota, A
1995-01-01
In a graph, a clique is a set of vertices such that every pair is connected by an edge. MAX-CLIQUE is the optimization problem of finding the largest clique in a given graph and is NP-hard, even to approximate well. Several real-world and theory problems can be modeled as MAX-CLIQUE. In this paper, we efficiently approximate MAX-CLIQUE in a special case of the Hopfield network whose stable states are maximal cliques. We present several energy-descent optimizing dynamics; both discrete (deterministic and stochastic) and continuous. One of these emulates, as special cases, two well-known greedy algorithms for approximating MAX-CLIQUE. We report on detailed empirical comparisons on random graphs and on harder ones. Mean-field annealing, an efficient approximation to simulated annealing, and a stochastic dynamics are the narrow but clear winners. All dynamics approximate much better than one which emulates a "naive" greedy heuristic.
Frankenstein's glue: transition functions for approximate solutions
Yunes, Nicolás
2007-09-01
Approximations are commonly employed to find approximate solutions to the Einstein equations. These solutions, however, are usually only valid in some specific spacetime region. A global solution can be constructed by gluing approximate solutions together, but this procedure is difficult because discontinuities can arise, leading to large violations of the Einstein equations. In this paper, we provide an attempt to formalize this gluing scheme by studying transition functions that join approximate analytic solutions together. In particular, we propose certain sufficient conditions on these functions and prove that these conditions guarantee that the joined solution still satisfies the Einstein equations analytically to the same order as the approximate ones. An example is also provided for a binary system of non-spinning black holes, where the approximate solutions are taken to be given by a post-Newtonian expansion and a perturbed Schwarzschild solution. For this specific case, we show that if the transition functions satisfy the proposed conditions, then the joined solution does not contain any violations to the Einstein equations larger than those already inherent in the approximations. We further show that if these functions violate the proposed conditions, then the matter content of the spacetime is modified by the introduction of a matter shell, whose stress energy tensor depends on derivatives of these functions.
The tendon approximator device in traumatic injuries.
Forootan, Kamal S; Karimi, Hamid; Forootan, Nazilla-Sadat S
2015-01-01
Precise and tension-free approximation of two tendon endings is the key predictor of outcomes following tendon lacerations and repairs. We evaluate the efficacy of a new tendon approximator device in tendon laceration repairs. In a comparative study, we used our new tendon approximator device in 99 consecutive patients with laceration of 266 tendons who attend a university hospital and evaluated the operative time to repair the tendons, surgeons' satisfaction as well as patient's outcomes in a long-term follow-up. Data were compared with the data of control patients undergoing tendon repair by conventional method. Totally 266 tendons were repaired by approximator device and 199 tendons by conventional technique. 78.7% of patients in first group were male and 21.2% were female. In approximator group 38% of patients had secondary repair of cut tendons and 62% had primary repair. Patients were followed for a mean period of 3years (14-60 months). Time required for repair of each tendon was significantly reduced with the approximator device (2 min vs. 5.5 min, ptendon repair were identical in the two groups and were not significantly different. 1% of tendons in group A and 1.2% in group B had rupture that was not significantly different. The new nerve approximator device is cheap, feasible to use and reduces the time of tendon repair with sustained outcomes comparable to the conventional methods.
Entanglement in the Born-Oppenheimer Approximation
Izmaylov, Artur F
2016-01-01
The role of electron-nuclear entanglement on the validity of the Born-Oppenheimer (BO) approximation is investigated. While nonadiabatic couplings generally lead to entanglement and to a failure of the BO approximation, surprisingly the degree of electron-nuclear entanglement is found to be uncorrelated with the degree of validity of the BO approximation. This is because while the degree of entanglement of BO states is determined by their deviation from the corresponding states in the crude BO approximation, the accuracy of the BO approximation is dictated, instead, by the deviation of the BO states from the exact electron-nuclear states. In fact, in the context of a minimal avoided crossing model, extreme cases are identified where an adequate BO state is seen to be maximally entangled, and where the BO approximation fails but the associated BO state remains approximately unentangled. Further, the BO states are found to not preserve the entanglement properties of the exact electron-nuclear eigenstates, and t...
DIFFERENCE SCHEMES BASING ON COEFFICIENT APPROXIMATION
MOU Zong-ze; LONG Yong-xing; QU Wen-xiao
2005-01-01
In respect of variable coefficient differential equations, the equations of coefficient function approximation were more accurate than the coefficient to be frozen as a constant in every discrete subinterval. Usually, the difference schemes constructed based on Taylor expansion approximation of the solution do not suit the solution with sharp function.Introducing into local bases to be combined with coefficient function approximation, the difference can well depict more complex physical phenomena, for example, boundary layer as well as high oscillatory, with sharp behavior. The numerical test shows the method is more effective than the traditional one.
Approximate equivalence in von Neumann algebras
DING; Huiru; Don; Hadwin
2005-01-01
One formulation of D. Voiculescu's theorem on approximate unitary equivalence is that two unital representations π and ρ of a separable C*-algebra are approximately unitarily equivalent if and only if rank o π = rank o ρ. We study the analog when the ranges of π and ρ are contained in a von Neumann algebra R, the unitaries inducing the approximate equivalence must come from R, and "rank" is replaced with "R-rank" (defined as the Murray-von Neumann equivalence of the range projection).
Approximation of free-discontinuity problems
Braides, Andrea
1998-01-01
Functionals involving both volume and surface energies have a number of applications ranging from Computer Vision to Fracture Mechanics. In order to tackle numerical and dynamical problems linked to such functionals many approximations by functionals defined on smooth functions have been proposed (using high-order singular perturbations, finite-difference or non-local energies, etc.) The purpose of this book is to present a global approach to these approximations using the theory of gamma-convergence and of special functions of bounded variation. The book is directed to PhD students and researchers in calculus of variations, interested in approximation problems with possible applications.
Mathematical analysis, approximation theory and their applications
Gupta, Vijay
2016-01-01
Designed for graduate students, researchers, and engineers in mathematics, optimization, and economics, this self-contained volume presents theory, methods, and applications in mathematical analysis and approximation theory. Specific topics include: approximation of functions by linear positive operators with applications to computer aided geometric design, numerical analysis, optimization theory, and solutions of differential equations. Recent and significant developments in approximation theory, special functions and q-calculus along with their applications to mathematics, engineering, and social sciences are discussed and analyzed. Each chapter enriches the understanding of current research problems and theories in pure and applied research.
Regression with Sparse Approximations of Data
Noorzad, Pardis; Sturm, Bob L.
2012-01-01
We propose sparse approximation weighted regression (SPARROW), a method for local estimation of the regression function that uses sparse approximation with a dictionary of measurements. SPARROW estimates the regression function at a point with a linear combination of a few regressands selected...... by a sparse approximation of the point in terms of the regressors. We show SPARROW can be considered a variant of \\(k\\)-nearest neighbors regression (\\(k\\)-NNR), and more generally, local polynomial kernel regression. Unlike \\(k\\)-NNR, however, SPARROW can adapt the number of regressors to use based...
Orthorhombic rational approximants for decagonal quasicrystals
S Ranganathan; Anandh Subramaniam
2003-10-01
An important exercise in the study of rational approximants is to derive their metric, especially in relation to the corresponding quasicrystal or the underlying clusters. Kuo’s model has been the widely accepted model to calculate the metric of the decagonal approximants. Using an alternate model, the metric of the approximants and other complex structures with the icosahedral cluster are explained elsewhere. In this work a comparison is made between the two models bringing out their equivalence. Further, using the concept of average lattices, a modified model is proposed.
Approximation of the semi-infinite interval
A. McD. Mercer
1980-01-01
Full Text Available The approximation of a function f∈C[a,b] by Bernstein polynomials is well-known. It is based on the binomial distribution. O. Szasz has shown that there are analogous approximations on the interval [0,∞ based on the Poisson distribution. Recently R. Mohapatra has generalized Szasz' result to the case in which the approximating function is αe−ux∑k=N∞(uxkα+β−1Γ(kα+βf(kαuThe present note shows that these results are special cases of a Tauberian theorem for certain infinite series having positive coefficients.
Nissenbaum, Daniel; Lin, Hsin; Barbiellini, Bernardo; Bansil, Arun
2009-03-01
To study the performance of the Stochastic Gradient Approximation (SGA) for variational Quantum Monte Carlo methods, we have considered lithium nano-clusters [1] described by Hartree-Fock wavefunctions multiplied by two-body Jastrow factors with a single variational parameter b. Even when the system size increases, we have shown the feasibility of obtaining an accurate value of b that minimizes the energy without an explicit calculation of the energy itself. The present SGA algorithm is so efficient because an analytic gradient formula is used and because the statistical noise in the gradient is smaller than in the energy [2]. Interestingly, in this scheme the absolute value of the gradient is less important than the sign of the gradient. Work supported in part by U.S. DOE. [1] D. Nissenbaum et al., Phys. Rev. B 76, 033412 (2007). [2] A. Harju, J. Low. Temp. Phys. 140, 181 (2005).
Roper, Ian P E; Besley, Nicholas A
2016-03-21
The simulation of X-ray emission spectra of transition metal complexes with time-dependent density functional theory (TDDFT) is investigated. X-ray emission spectra can be computed within TDDFT in conjunction with the Tamm-Dancoff approximation by using a reference determinant with a vacancy in the relevant core orbital, and these calculations can be performed using the frozen orbital approximation or with the relaxation of the orbitals of the intermediate core-ionised state included. Both standard exchange-correlation functionals and functionals specifically designed for X-ray emission spectroscopy are studied, and it is shown that the computed spectral band profiles are sensitive to the exchange-correlation functional used. The computed intensities of the spectral bands can be rationalised by considering the metal p orbital character of the valence molecular orbitals. To compute X-ray emission spectra with the correct energy scale allowing a direct comparison with experiment requires the relaxation of the core-ionised state to be included and the use of specifically designed functionals with increased amounts of Hartree-Fock exchange in conjunction with high quality basis sets. A range-corrected functional with increased Hartree-Fock exchange in the short range provides transition energies close to experiment and spectral band profiles that have a similar accuracy to those from standard functionals.
An overview on Approximate Bayesian computation*
Baragatti Meïli
2014-01-01
Full Text Available Approximate Bayesian computation techniques, also called likelihood-free methods, are one of the most satisfactory approach to intractable likelihood problems. This overview presents recent results since its introduction about ten years ago in population genetics.
Trigonometric Approximations for Some Bessel Functions
Muhammad Taher Abuelma'atti
1999-01-01
Formulas are obtained for approximating the tabulated Bessel functions Jn(x), n = 0–9 in terms of trigonometric functions. These formulas can be easily integrated and differentiated and are convenient for personal computers and pocket calculators.
Low Rank Approximation Algorithms, Implementation, Applications
Markovsky, Ivan
2012-01-01
Matrix low-rank approximation is intimately related to data modelling; a problem that arises frequently in many different fields. Low Rank Approximation: Algorithms, Implementation, Applications is a comprehensive exposition of the theory, algorithms, and applications of structured low-rank approximation. Local optimization methods and effective suboptimal convex relaxations for Toeplitz, Hankel, and Sylvester structured problems are presented. A major part of the text is devoted to application of the theory. Applications described include: system and control theory: approximate realization, model reduction, output error, and errors-in-variables identification; signal processing: harmonic retrieval, sum-of-damped exponentials, finite impulse response modeling, and array processing; machine learning: multidimensional scaling and recommender system; computer vision: algebraic curve fitting and fundamental matrix estimation; bioinformatics for microarray data analysis; chemometrics for multivariate calibration; ...
Asynchronous stochastic approximation with differential inclusions
David S. Leslie
2012-01-01
Full Text Available The asymptotic pseudo-trajectory approach to stochastic approximation of Benaïm, Hofbauer and Sorin is extended for asynchronous stochastic approximations with a set-valued mean field. The asynchronicity of the process is incorporated into the mean field to produce convergence results which remain similar to those of an equivalent synchronous process. In addition, this allows many of the restrictive assumptions previously associated with asynchronous stochastic approximation to be removed. The framework is extended for a coupled asynchronous stochastic approximation process with set-valued mean fields. Two-timescales arguments are used here in a similar manner to the original work in this area by Borkar. The applicability of this approach is demonstrated through learning in a Markov decision process.
An approximate Expression for Viscosity of Nanosuspensions
Domostroeva, N G
2009-01-01
We consider liquid suspensions with dispersed nanoparticles. Using two-points Pade approximants and combining results of both hydrodynamic and molecular dynamics methods, we obtain the effective viscosity for any diameters of nanoparticles
On Approximating Four Covering and Packing Problems
Ashley, Mary; Berman, Piotr; Chaovalitwongse, Wanpracha; DasGupta, Bhaskar; Kao, Ming-Yang; 10.1016/j.jcss.2009.01.002
2011-01-01
In this paper, we consider approximability issues of the following four problems: triangle packing, full sibling reconstruction, maximum profit coverage and 2-coverage. All of them are generalized or specialized versions of set-cover and have applications in biology ranging from full-sibling reconstructions in wild populations to biomolecular clusterings; however, as this paper shows, their approximability properties differ considerably. Our inapproximability constant for the triangle packing problem improves upon the previous results; this is done by directly transforming the inapproximability gap of Haastad for the problem of maximizing the number of satisfied equations for a set of equations over GF(2) and is interesting in its own right. Our approximability results on the full siblings reconstruction problems answers questions originally posed by Berger-Wolf et al. and our results on the maximum profit coverage problem provides almost matching upper and lower bounds on the approximation ratio, answering a...
Approximations of solutions to retarded integrodifferential equations
Dhirendra Bahuguna
2004-11-01
Full Text Available In this paper we consider a retarded integrodifferential equation and prove existence, uniqueness and convergence of approximate solutions. We also give some examples to illustrate the applications of the abstract results.
Approximation methods in gravitational-radiation theory
Will, C. M.
1986-02-01
The observation of gravitational-radiation damping in the binary pulsar PSR 1913+16 and the ongoing experimental search for gravitational waves of extraterrestrial origin have made the theory of gravitational radiation an active branch of classical general relativity. In calculations of gravitational radiation, approximation methods play a crucial role. The author summarizes recent developments in two areas in which approximations are important: (1) the quadrupole approximation, which determines the energy flux and the radiation reaction forces in weak-field, slow-motion, source-within-the-near-zone systems such as the binary pulsar; and (2) the normal modes of oscillation of black holes, where the Wentzel-Kramers-Brillouin approximation gives accurate estimates of the complex frequencies of the modes.
APPROXIMATE DEVELOPMENTS FOR SURFACES OF REVOLUTION
Mădălina Roxana Buneci
2016-12-01
Full Text Available The purpose of this paper is provide a set of Maple procedures to construct approximate developments of a general surface of revolution generalizing the well-known gore method for sphere
Methods of Fourier analysis and approximation theory
Tikhonov, Sergey
2016-01-01
Different facets of interplay between harmonic analysis and approximation theory are covered in this volume. The topics included are Fourier analysis, function spaces, optimization theory, partial differential equations, and their links to modern developments in the approximation theory. The articles of this collection were originated from two events. The first event took place during the 9th ISAAC Congress in Krakow, Poland, 5th-9th August 2013, at the section “Approximation Theory and Fourier Analysis”. The second event was the conference on Fourier Analysis and Approximation Theory in the Centre de Recerca Matemàtica (CRM), Barcelona, during 4th-8th November 2013, organized by the editors of this volume. All articles selected to be part of this collection were carefully reviewed.
Seismic wave extrapolation using lowrank symbol approximation
Fomel, Sergey
2012-04-30
We consider the problem of constructing a wave extrapolation operator in a variable and possibly anisotropic medium. Our construction involves Fourier transforms in space combined with the help of a lowrank approximation of the space-wavenumber wave-propagator matrix. A lowrank approximation implies selecting a small set of representative spatial locations and a small set of representative wavenumbers. We present a mathematical derivation of this method, a description of the lowrank approximation algorithm and numerical examples that confirm the validity of the proposed approach. Wave extrapolation using lowrank approximation can be applied to seismic imaging by reverse-time migration in 3D heterogeneous isotropic or anisotropic media. © 2012 European Association of Geoscientists & Engineers.
Approximate Flavor Symmetry in Supersymmetric Model
Tao, Zhijian
1998-01-01
We investigate the maximal approximate flavor symmetry in the framework of generic minimal supersymmetric standard model. We consider the low energy effective theory of the flavor physics with all the possible operators included. Spontaneous flavor symmetry breaking leads to the approximate flavor symmetry in Yukawa sector and the supersymmetry breaking sector. Fermion mass and mixing hierachies are the results of the hierachy of the flavor symmetry breaking. It is found that in this theory i...
Pointwise approximation by elementary complete contractions
Magajna, Bojan
2009-01-01
A complete contraction on a C*-algebra A, which preserves all closed two sided ideals J, can be approximated pointwise by elementary complete contractions if and only if the induced map on the tensor product of B with A/J is contractive for every C*-algebra B, ideal J in A and C*-tensor norm on the tensor product. A lifting obstruction for such an approximation is also obtained.
Polynomial approximation of functions in Sobolev spaces
Dupont, T.; Scott, R.
1980-01-01
Constructive proofs and several generalizations of approximation results of J. H. Bramble and S. R. Hilbert are presented. Using an averaged Taylor series, we represent a function as a polynomial plus a remainder. The remainder can be manipulated in many ways to give different types of bounds. Approximation of functions in fractional order Sobolev spaces is treated as well as the usual integer order spaces and several nonstandard Sobolev-like spaces.
Parallel local approximation MCMC for expensive models
Conrad, Patrick; Davis, Andrew; Marzouk, Youssef; Pillai, Natesh; Smith, Aaron
2016-01-01
Performing Bayesian inference via Markov chain Monte Carlo (MCMC) can be exceedingly expensive when posterior evaluations invoke the evaluation of a computationally expensive model, such as a system of partial differential equations. In recent work [Conrad et al. JASA 2015, arXiv:1402.1694] we described a framework for constructing and refining local approximations of such models during an MCMC simulation. These posterior--adapted approximations harness regularity of the model to reduce the c...
The Actinide Transition Revisited by Gutzwiller Approximation
Xu, Wenhu; Lanata, Nicola; Yao, Yongxin; Kotliar, Gabriel
2015-03-01
We revisit the problem of the actinide transition using the Gutzwiller approximation (GA) in combination with the local density approximation (LDA). In particular, we compute the equilibrium volumes of the actinide series and reproduce the abrupt change of density found experimentally near plutonium as a function of the atomic number. We discuss how this behavior relates with the electron correlations in the 5 f states, the lattice structure, and the spin-orbit interaction. Our results are in good agreement with the experiments.
Intuitionistic Fuzzy Automaton for Approximate String Matching
K.M. Ravi
2014-03-01
Full Text Available This paper introduces an intuitionistic fuzzy automaton model for computing the similarity between pairs of strings. The model details the possible edit operations needed to transform any input (observed string into a target (pattern string by providing a membership and non-membership value between them. In the end, an algorithm is given for approximate string matching and the proposed model computes the similarity and dissimilarity between the pair of strings leading to better approximation.
Approximations for the Erlang Loss Function
Mejlbro, Leif
1998-01-01
Theoretically, at least three formulae are needed for arbitrarily good approximates of the Erlang Loss Function. In the paper, for convenience five formulae are presented guaranteeing a relative error <1E-2, and methods are indicated for improving this bound.......Theoretically, at least three formulae are needed for arbitrarily good approximates of the Erlang Loss Function. In the paper, for convenience five formulae are presented guaranteeing a relative error
Lattice quantum chromodynamics with approximately chiral fermions
Hierl, Dieter
2008-05-15
In this work we present Lattice QCD results obtained by approximately chiral fermions. We use the CI fermions in the quenched approximation to investigate the excited baryon spectrum and to search for the {theta}{sup +} pentaquark on the lattice. Furthermore we developed an algorithm for dynamical simulations using the FP action. Using FP fermions we calculate some LECs of chiral perturbation theory applying the epsilon expansion. (orig.)
Nonlinear approximation in alpha-modulation spaces
Borup, Lasse; Nielsen, Morten
2006-01-01
The α-modulation spaces are a family of spaces that contain the Besov and modulation spaces as special cases. In this paper we prove that brushlet bases can be constructed to form unconditional and even greedy bases for the α-modulation spaces. We study m -term nonlinear approximation with brushlet...... bases, and give complete characterizations of the associated approximation spaces in terms of α-modulation spaces....
On surface approximation using developable surfaces
Chen, H. Y.; Lee, I. K.; Leopoldseder, s.
1999-01-01
We introduce a method for approximating a given surface by a developable surface. It will be either a G(1) surface consisting of pieces of cones or cylinders of revolution or a G(r) NURBS developable surface. Our algorithm will also deal properly with the problems of reverse engineering and produce...... robust approximation of given scattered data. The presented technique can be applied in computer aided manufacturing, e.g. in shipbuilding. (C) 1999 Academic Press....
On surface approximation using developable surfaces
Chen, H. Y.; Lee, I. K.; Leopoldseder, S.
1998-01-01
We introduce a method for approximating a given surface by a developable surface. It will be either a G_1 surface consisting of pieces of cones or cylinders of revolution or a G_r NURBS developable surface. Our algorithm will also deal properly with the problems of reverse engineering and produce...... robust approximation of given scattered data. The presented technique can be applied in computer aided manufacturing, e.g. in shipbuilding....
Differential geometry of proteins. Helical approximations.
Louie, A H; Somorjai, R L
1983-07-25
We regard a protein molecule as a geometric object, and in a first approximation represent it as a regular parametrized space curve passing through its alpha-carbon atoms (the backbone). In an earlier paper we argued that the regular patterns of secondary structures of proteins (morphons) correspond to geodesics on minimal surfaces. In this paper we discuss methods of recognizing these morphons on space curves that represent the protein backbone conformation. The mathematical tool we employ is the differential geometry of curves and surfaces. We introduce a natural approximation of backbone space curves in terms of helical approximating elements and present a computer algorithm to implement the approximation. Simple recognition criteria are given for the various morphons of proteins. These are incorporated into our helical approximation algorithm, together with more non-local criteria for the recognition of beta-sheet topologies. The method and the algorithm are illustrated with several examples of representative proteins. Generalizations of the helical approximation method are considered and their possible implications for protein energetics are sketched.
Sadegh, Payman
1997-01-01
This paper deals with a projection algorithm for stochastic approximation using simultaneous perturbation gradient approximation for optimization under inequality constraints where no direct gradient of the loss function is available and the inequality constraints are given as explicit functions...
Ito, K.
1984-01-01
The stability and convergence properties of the Legendre-tau approximation for hereditary differential systems are analyzed. A charactristic equation is derived for the eigenvalues of the resulting approximate system. As a result of this derivation the uniform exponential stability of the solution semigroup is preserved under approximation. It is the key to obtaining the convergence of approximate solutions of the algebraic Riccati equation in trace norm.
Ito, K.
1985-01-01
The stability and convergence properties of the Legendre-tau approximation for hereditary differential systems are analyzed. A characteristic equation is derived for the eigenvalues of the resulting approximate system. As a result of this derivation the uniform exponential stability of the solution semigroup is preserved under approximation. It is the key to obtaining the convergence of approximate solutions of the algebraic Riccati equation in trace norm.
Kondayya, Gundra; Shukla, Alok
2012-03-01
Pariser-Parr-Pople (P-P-P) model Hamiltonian is employed frequently to study the electronic structure and optical properties of π-conjugated systems. In this paper we describe a Fortran 90 computer program which uses the P-P-P model Hamiltonian to solve the Hartree-Fock (HF) equation for infinitely long, one-dimensional, periodic, π-electron systems. The code is capable of computing the band structure, as also the linear optical absorption spectrum, by using the tight-binding and the HF methods. Furthermore, using our program the user can solve the HF equation in the presence of a finite external electric field, thereby, allowing the simulation of gated systems. We apply our code to compute various properties of polymers such as trans-polyacetylene, poly- para-phenylene, and armchair and zigzag graphene nanoribbons, in the infinite length limit. Program summaryProgram title: ppp_bulk.x Catalogue identifier: AEKW_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEKW_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 87 464 No. of bytes in distributed program, including test data, etc.: 2 046 933 Distribution format: tar.gz Programming language: Fortran 90 Computer: PCs and workstations Operating system: Linux, Code was developed and tested on various recent versions of 64-bit Fedora including Fedora 14 (kernel version 2.6.35.12-90). Classification: 7.3 External routines: This program needs to link with LAPACK/BLAS libraries compiled with the same compiler as the program. For the Intel Fortran Compiler we used the ACML library version 4.4.0, while for the gfortran compiler we used the libraries supplied with the Fedora distribution. Nature of problem: The electronic structure of one-dimensional periodic π-conjugated systems is an intense area of research at