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

Physical Review A

Calculation of optimal generalized antisymmetrized geminal-power (projected—Bardeen-Cooper-Schrieffer) functions and their associated excitation spectrum

Brian Weiner, Osvaldo Goscinski

Published Dec 1, 1980• 50 citations
Publisher

Abstract

The generalized antisymmetrized geminal-power (GAGP) function, shown to be the ground state for the oneparticle-hole propagator, can be computed using the rather simple properties of its reduced density matrices and the properties of unitary transformations. The practical procedure involves limited storage notwithstanding the rather large number of configurations with which the GAGP function can be associated. Using the stability properties previously derived for this function, we construct the self-consistent one-particle-hole propagator.

DOI
10.1103/physreva.22.2374

Related papers

Excitation operators associated with antisymmetrized geminal-power states

Brian Weiner, Osvaldo Goscinski

Various expressions for antisymmetrized geminal-power (AGP) states are reviewed. With the use of a simple relationship for creation and annihilation operators of BCS (Bardeen, Cooper, and Schrieffer)-type states, such operators are characterized for AGP and GAGP (generalized AGP) schemes. The effect of degeneracy in the first-order reduced density matrix on these operators is displayed.

Physical Review A•Jan 1, 1983•29 citations

Geminal-based configuration interaction

Thomas M. Henderson, Gustavo E. Scuseria

The antisymmetrized geminal power (AGP) wave function has a long history and considerable conceptual appeal, but in many situations its accuracy is wanting. Here, we consider a form of configuration interaction (CI) based upon the AGP wave function and take advantage of its killing operators to construct an excitation manifold. Our geminal CI reduces to standard single-determinant-based CI in the limit in which AGP reduces to a single determinant. It substantially improves upon AGP in the reduced BCS Hamiltonian, which serves as a prototype for the kinds of strong pairing correlations relevant in Bardeen-Cooper-Schrieffer-style superconductivity. Moreover, our geminal CI naturally generalizes to add correlation to more general geminal-based wave functions than AGP.

arXiv:1906.11361•The Journal of Chemical Physics•Aug 2, 2019•47 citations

The Role of Algebraic Formulations of Approximate Green's Functions for Systems With a Finite Number of Electrons

Osvaldo Goscinski, Brian Weiner

Some of the difficulties encountered in the approximation of Green's functions are reviewed with particular emphasis on the self-consistent polarization propagator. Some new basic theorems are presented, involving a general procedure, which lead to both a derivation and alternate prescriptions for the computations. As demonstrated by Linderberg and Öhrn the ground state is an AGP (antisymmetrized geminal power). An energy optimized AGP ( E AGP ) in a Brueckner orbital representation can be used to generate a manifold of excited states| K ⟩ = O K + |AGP⟩, O K |AGP⟩ = 0 with the excitation operators, O + K , and killers, O K , being combinations of particle-hole and hole-particle operators. The SCPP can be expressed then as ⟨⟨ X, Y ⟩⟩ E = - ⟨AGP| XT 0 (- E + E AGP ) Y + YT 0 ( E + E AGP ) X |AGP⟩ in terms of the resolvent T 0 ( ) = P ( - PHP ) -1 with P spanned by the states| K ⟩⟨ K |, in the sense of Löwdin. Our results establish a natural relationship between hermiticity, completeness, non-redundance of operator manifolds with stationarity and killer conditions.

Physica Scripta•Jan 1, 1980•44 citations

Structure of Fermion Density Matrices. II. Antisymmetrized Geminal Powers

Alan Coleman

Functions which can be expressed as an antisymmetrized power of a single two-particle function (a geminal) occur in the BCS ansatz. They constitute a comparatively tractable generalization of a single Slater determinant. The second-order reduced density matrix and bounds on its eigenvalues are obtained for the most general such antisymmetrized geminal power (AGP). These eigenvalues are physically of great importance, being analogs, in an analysis of the wavefunction into pair states, of the one-particle occupation numbers of the independent-particle model. A necessary and sufficient condition is given that an alleged first-order reduced density matrix be derivable from an N-particle AGP function. It is shown that only for exceptional functions (called extreme) is the generating geminal of the AGP function an eigenfunction of the 2-matrix. The extreme 2-matrix is diagonalized explicitly and provides a vivid example of the fact that one-particle occupation numbers alone are unable to convey certain information of decisive significance about the wavefunction. The extreme 2-matrix satisfies the assumption basic to current theories of superconductivity.

Journal of Mathematical Physics•Sep 1, 1965•269 citations

Excitation spectrum associated with a generalized antisymmetrized geminal power ground state

Osvaldo Goscinski, Brian Weiner

Self-consistent polarization-propagator approximations (i.e., those based on a proper vacuum) are compared to conventional time-dependent Hartree-Fock --- random-phaseapproximation ones, with special attention to the fulfillment of the $f$-sum rule. The effect on the properties of the consistent propagator, based on a general-antisymmetrizedgeminal-power vacuum, when degeneracies occurring in the first-order reduced-density matrix of this vacuum are examined. The foregoing analysis shows that low-lying electronic states can be expected to become degenerate with the vacuum in conjunction with superphenomena in extended systems and oscillator-strength anomalies in finite ones.

Physical Review A•Feb 1, 1982•12 citations

The antisymmetrized geminal power approximation to the excitation propagator

Brian Weiner, Henry A. Kurtz

Abstract A consistent propagator approximation, denoted as the excitation propagator, is introduced. This propagator describes excitations between N‐particle states and its approximation has properties required of consistent random phase approximation schemes. Several properties of this propagator are explored when based on a generalized antisymmetrized geminal power wavefunction. How singularities in the metric occur and how to remove them is discussed in detail. The excitation propagator is also contrasted with the principal (polarization) propagator.

International Journal of Quantum Chemistry•Jun 1, 1985•5 citations
The generalized antisymmetrized geminal-power (GAGP) function, shown to be the ground state for the oneparticle-hole propagator, can be computed using the rather simple properties of its reduced density matrices and the properties of unitary transformations. The practical procedure involves limited storage notwithstanding the rather large number of configurations with which the GAGP function can be associated. Using the stability properties previously derived for this function, we construct the self-consistent one-particle-hole propagator.
The antisymmetrized geminal power (AGP) wave function has a long history and considerable conceptual appeal, but in many situations its accuracy is wanting. Here, we consider a form of configuration interaction (CI) based upon the AGP wave function and take advantage of its killing operators to construct an excitation manifold. Our geminal CI reduces to standard single-determinant-based CI in the limit in which AGP reduces to a single determinant. It substantially improves upon AGP in the reduced BCS Hamiltonian, which serves as a prototype for the kinds of strong pairing correlations relevant in Bardeen-Cooper-Schrieffer-style superconductivity. Moreover, our geminal CI naturally generalizes to add correlation to more general geminal-based wave functions than AGP.
Some of the difficulties encountered in the approximation of Green's functions are reviewed with particular emphasis on the self-consistent polarization propagator. Some new basic theorems are presented, involving a general procedure, which lead to both a derivation and alternate prescriptions for the computations. As demonstrated by Linderberg and Öhrn the ground state is an AGP (antisymmetrized geminal power). An energy optimized AGP ( E AGP ) in a Brueckner orbital representation can be used to generate a manifold of excited states| K ⟩ = O K + |AGP⟩, O K |AGP⟩ = 0 with the excitation operators, O + K , and killers, O K , being combinations of particle-hole and hole-particle operators. The SCPP can be expressed then as ⟨⟨ X, Y ⟩⟩ E = - ⟨AGP| XT 0 (- E + E AGP ) Y + YT 0 ( E + E AGP ) X |AGP⟩ in terms of the resolvent T 0 ( ) = P ( - PHP ) -1 with P spanned by the states| K ⟩⟨ K |, in the sense of Löwdin. Our results establish a natural relationship between hermiticity, completeness, non-redundance of operator manifolds with stationarity and killer conditions.
Functions which can be expressed as an antisymmetrized power of a single two-particle function (a geminal) occur in the BCS ansatz. They constitute a comparatively tractable generalization of a single Slater determinant. The second-order reduced density matrix and bounds on its eigenvalues are obtained for the most general such antisymmetrized geminal power (AGP). These eigenvalues are physically of great importance, being analogs, in an analysis of the wavefunction into pair states, of the one-particle occupation numbers of the independent-particle model. A necessary and sufficient condition is given that an alleged first-order reduced density matrix be derivable from an N-particle AGP function. It is shown that only for exceptional functions (called extreme) is the generating geminal of the AGP function an eigenfunction of the 2-matrix. The extreme 2-matrix is diagonalized explicitly and provides a vivid example of the fact that one-particle occupation numbers alone are unable to convey certain information of decisive significance about the wavefunction. The extreme 2-matrix satisfies the assumption basic to current theories of superconductivity.
Self-consistent polarization-propagator approximations (i.e., those based on a proper vacuum) are compared to conventional time-dependent Hartree-Fock --- random-phaseapproximation ones, with special attention to the fulfillment of the fff-sum rule. The effect on the properties of the consistent propagator, based on a general-antisymmetrizedgeminal-power vacuum, when degeneracies occurring in the first-order reduced-density matrix of this vacuum are examined. The foregoing analysis shows that low-lying electronic states can be expected to become degenerate with the vacuum in conjunction with superphenomena in extended systems and oscillator-strength anomalies in finite ones.
Abstract A consistent propagator approximation, denoted as the excitation propagator, is introduced. This propagator describes excitations between N‐particle states and its approximation has properties required of consistent random phase approximation schemes. Several properties of this propagator are explored when based on a generalized antisymmetrized geminal power wavefunction. How singularities in the metric occur and how to remove them is discussed in detail. The excitation propagator is also contrasted with the principal (polarization) propagator.