Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Generalized Approximative Procedures in the Many Body Problem, III.

I. Kelson

Published Oct 1, 1975• 0 citations
OpenAlex

Abstract

The generalized Hartree-Fock approach leads to computational problems which need special treatment. In particular, expansion methods are proposed for the calculation of matrix elements between states in different single particle representations, rotated or deformed relative to each other. Special numerical examples are carried out, as well as a definite application: the projection of the lowest state in a rotational band, using the discrete point configuration method.

Related papers

Generalized Hartree-Fock Approximation for the Calculation of Collective States of a Finite Many-Particle System

Arthur K. Kerman, Abraham Klein

A finite many-particle system can have collective states for which the off-diagonal matrix elements of certain one-particle operators are of the same order of magnitude as the diagonal elements. In such cases it is suggested that the random-phase approximation is in need of generalization. Examples are the uniform translational motion of any system and the rotational motion of deformed nuclei. The generalization is suggested after a review and critical analysis of the Hartree-Fock approximation. The model single-particle wave functions of the latter are replaced by wave functions in a space labeled both by the particle variables and by the quantum numbers of the collective motion. These generalized amplitudes are defined field-theoretically, and a self-consistent scheme for their calculation is obtained from the equations of motion. In addition to the self-consistent potential defined in the enlarged space, the energies of the excited states also turn out to be given by a natural self-consistency requirement. The new calculational scheme is first applied to a systematic restudy of the random-phase approximation where the self-consistency requirement on the energies has previously been overlooked. As a first characteristic application we obtain without "pushing" the total mass of a system in uniform translation, and a reinterpretation of the Hartree-Fock average field.

Physical Review•Nov 1, 1963•121 citations

An ab-initio many-body method for electronic structure calculations of solids. I. Description of the method

I. Schnell, G. Czycholl, Robert C. Albers

We propose a new, alternative method for ab-initio calculations of the electronic structure of solids, which has been specifically adapted to treat many-body effects in a more rigorous way than many existing ab-initio methods. We start from a standard band-structure calculation for an effective one-particle Hamiltonian approximately describing the material of interest. This yields a suitable set of one-particle basis functions, from which well localized Wannier functions can be constructed using a method proposed by Marzari and Vanderbilt. Within this (minimal) basis of localized Wannier functions the matrix elements of the non-interacting (one-particle) Hamiltonian as well as the Coulomb matrix elements can be calculated. The result is a many-body Hamiltonian in second quantization with parameters determined from first principles calculations for the material of interest. The Hamiltonian is in the form of a multi-band Hamiltonian in second quantization (a kind of extended, multi-band Hubbard model) such that all the standard many-body methods can be applied. We explicitly show how this approach can be solved in the simplest many-body approximation, the mean-field Hartree-Fock approximation (HFA), which takes into account exact exchange and corrects for self-interaction effects.

arXiv:cond-mat/0304643•Apr 28, 2003•0 citations

A new representation of the many body wave function and its application as a post Hartree-Fock energy variation method

Wataru Uemura

In this paper, we introduce a new representation of many body electron wave function and a few calculation results of the ground state energies of many body systems using that representation, which is systematically better than the Hartree-Fock approximation.

arXiv:1102.4909•Feb 24, 2011•0 citations

Quantum Theory of Many-Particle Systems. III. Extension of the Hartree-Fock Scheme to Include Degenerate Systems and Correlation Effects

Per‐Olov Löwdin

In treating a system of $N$ antisymmetric particles, it is shown that, if the total Hamiltonian ${\mathcal{H}}_{\mathrm{op}}$ is degenerate, the eigenstates of the operator used for classifying the corresponding degenerate states may be selected by means of a "projection operator" $\mathcal{O}$. If the total wave function is approximated by such a projection of a single determinant, the description of the system may be reduced to the ordinary Hartree-Fock scheme treating this determinant, if the original Hamiltonian is replaced by a complete Hamiltonian ${\ensuremath{\Omega}}_{\mathrm{op}}={\mathcal{O}}^{\ifmmode\dagger\else\textdagger\fi{}}{\mathcal{H}}_{\mathrm{op}}\mathcal{O}$ containing also many-particle interactions. This approach corresponds to a "fixed" configurational interaction, but the scheme has preserved the physical simplicity and visuality of the Hartree-Fock approximation. The idea of "doubly filled" orbitals is abandoned, and the orbitals associated with different spins will automatically try to arrange themselves in such a way that particles having antiparallel spins will tend to avoid each other due to their mutual repulsion.

Physical Review•Mar 15, 1955•1,140 citations

Method for Simplifying Calculations Involving Products of Operators in Many-Particle Systems

Harry J. Lipkin

In calculations for quantum mechanical systems containing many particles, expressions involving matrix elements of products of operators are frequently encountered. This is of particular interest in the calculation of nuclear moments of inertia. The evaluation of such expressions by conventional methods is laborious when antisymmetric wave functions are used. A method is proposed which greatly simplifies the calculation of these product expressions.

Physical Review•Oct 15, 1957•1 citation

Toward extremes of angular momentum: Application of the Pfaffian algorithm in realistic calculations

Long-Jun Wang, Fangqi Chen, T. Mizusaki +2 more

In a calculation of rotated matrix elements with angular momentum projection, the generalized Wick's theorem may encounter a practical problem of combinatorial complexity when the configurations have more than four quasiparticles (qp's). The problem can be solved by employing the Pfaffian algorithm generally applicable to calculations of matrix elements for Hartree-Fock-Bogoliubov states with any number of qp's. This breakthrough in many-body techniques enables studies of high-spin states in a shell-model framework. As the first application of the Pfaffian algorithm, the configuration space of the projected shell model is expanded to include 6-qp states for both positive and negative parities. Taking $^{166}\mathrm{Hf}$ as an example, we show that 6-qp states become the main configuration of the yrast band beyond spin $I\ensuremath{\approx}34\ensuremath{\hbar}$, which explains the observed third back-bending in moment of inertia. Structures of multi-qp high-$K$ isomers in $^{176}\mathrm{Hf}$ are analyzed as another example.

arXiv:1407.3024•Physical Review C•Jul 21, 2014•36 citations
The generalized Hartree-Fock approach leads to computational problems which need special treatment. In particular, expansion methods are proposed for the calculation of matrix elements between states in different single particle representations, rotated or deformed relative to each other. Special numerical examples are carried out, as well as a definite application: the projection of the lowest state in a rotational band, using the discrete point configuration method.
A finite many-particle system can have collective states for which the off-diagonal matrix elements of certain one-particle operators are of the same order of magnitude as the diagonal elements. In such cases it is suggested that the random-phase approximation is in need of generalization. Examples are the uniform translational motion of any system and the rotational motion of deformed nuclei. The generalization is suggested after a review and critical analysis of the Hartree-Fock approximation. The model single-particle wave functions of the latter are replaced by wave functions in a space labeled both by the particle variables and by the quantum numbers of the collective motion. These generalized amplitudes are defined field-theoretically, and a self-consistent scheme for their calculation is obtained from the equations of motion. In addition to the self-consistent potential defined in the enlarged space, the energies of the excited states also turn out to be given by a natural self-consistency requirement. The new calculational scheme is first applied to a systematic restudy of the random-phase approximation where the self-consistency requirement on the energies has previously been overlooked. As a first characteristic application we obtain without "pushing" the total mass of a system in uniform translation, and a reinterpretation of the Hartree-Fock average field.
We propose a new, alternative method for ab-initio calculations of the electronic structure of solids, which has been specifically adapted to treat many-body effects in a more rigorous way than many existing ab-initio methods. We start from a standard band-structure calculation for an effective one-particle Hamiltonian approximately describing the material of interest. This yields a suitable set of one-particle basis functions, from which well localized Wannier functions can be constructed using a method proposed by Marzari and Vanderbilt. Within this (minimal) basis of localized Wannier functions the matrix elements of the non-interacting (one-particle) Hamiltonian as well as the Coulomb matrix elements can be calculated. The result is a many-body Hamiltonian in second quantization with parameters determined from first principles calculations for the material of interest. The Hamiltonian is in the form of a multi-band Hamiltonian in second quantization (a kind of extended, multi-band Hubbard model) such that all the standard many-body methods can be applied. We explicitly show how this approach can be solved in the simplest many-body approximation, the mean-field Hartree-Fock approximation (HFA), which takes into account exact exchange and corrects for self-interaction effects.
In treating a system of NNN antisymmetric particles, it is shown that, if the total Hamiltonian Hop{\mathcal{H}}_{\mathrm{op}}Hop​ is degenerate, the eigenstates of the operator used for classifying the corresponding degenerate states may be selected by means of a "projection operator" O\mathcal{O}O. If the total wave function is approximated by such a projection of a single determinant, the description of the system may be reduced to the ordinary Hartree-Fock scheme treating this determinant, if the original Hamiltonian is replaced by a complete Hamiltonian ${\ensuremath{\Omega}}_{\mathrm{op}}={\mathcal{O}}^{\ifmmode\dagger\else\textdagger\fi{}}{\mathcal{H}}_{\mathrm{op}}\mathcal{O}$ containing also many-particle interactions. This approach corresponds to a "fixed" configurational interaction, but the scheme has preserved the physical simplicity and visuality of the Hartree-Fock approximation. The idea of "doubly filled" orbitals is abandoned, and the orbitals associated with different spins will automatically try to arrange themselves in such a way that particles having antiparallel spins will tend to avoid each other due to their mutual repulsion.
In a calculation of rotated matrix elements with angular momentum projection, the generalized Wick's theorem may encounter a practical problem of combinatorial complexity when the configurations have more than four quasiparticles (qp's). The problem can be solved by employing the Pfaffian algorithm generally applicable to calculations of matrix elements for Hartree-Fock-Bogoliubov states with any number of qp's. This breakthrough in many-body techniques enables studies of high-spin states in a shell-model framework. As the first application of the Pfaffian algorithm, the configuration space of the projected shell model is expanded to include 6-qp states for both positive and negative parities. Taking 166Hf^{166}\mathrm{Hf}166Hf as an example, we show that 6-qp states become the main configuration of the yrast band beyond spin $I\ensuremath{\approx}34\ensuremath{\hbar}$, which explains the observed third back-bending in moment of inertia. Structures of multi-qp high-KKK isomers in 176Hf^{176}\mathrm{Hf}176Hf are analyzed as another example.