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

International Journal of Quantum Chemistry

The properties of n‐electron wave functions built up with two kinds of geminals

Chengbu Liu, Conghao Deng, Haiquan Hu, Beiyan Jin

Published Apr 20, 1992• 4 citations
Publisher

Abstract

Abstract The properties of the n‐electron wave functions equation image with symplectic symmetries have been discussed. These functions are built up with two kinds of geminals. The well‐known antisymmetrized geminal power (AGP) function is only a special case of them. The first‐ and second‐order reduced density matrices and the natural expansions of these functions have been given, and the uniform expressions for system energies in the states corresponding to these functions have been deduced. It has been shown that the special case equation image is probably another consistent ground state for the particle‐hole propegator or random‐phase approximation (RPA) besides the AGP function.

DOI
10.1002/qua.560420207

Related papers

The AGP model for fermion systems

Alan Coleman

A mathematically rigorous treatment of the antisymmetrized geminal power (AGP) model is given. The model is advocated as an extremely flexible tool for studying a wide variety of fermion systems, characterized by a single Slater determinant, on the one hand, or Yang's most highly correlated wave function, on the other. When the first-order reduced density matrix of the actual system has eigenvalues which are evenly degenerate, the model can treat one-particle properties exactly. A formula is obtained for the precise energy of an AGP state in the Born-Oppenheimer approximation. An estimate is given of the error which occurs when the second-order density matrix of an AGP state is approximated by the widely used expression involving the so-called anomalous Green's function. © 1997 John Wiley & Sons, Inc.

International Journal of Quantum Chemistry•Jan 1, 1997•42 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

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

A solvable model with an extreme AGP ground state: Relationships among fermion pairs, pairons, and natural spin geminals

Everett G. Larson

A model many-fermion Hamiltonian is presented for which the ground state is asymptotically an Antisymmetrized Geminal Powers (AGP) wave function with largest possible greatest eigenvalue for its two-particle reduced density matrix. Closed analytical expressions and plane-wave expansions are presented for the generating geminal of the AGP ground state and for its one-particle reduced density matrix. The natural orbitals for this generating geminal are plane waves. The generating geminal shows intensely local character in its intracule and corresponds to the formation of a quasi-boson from two fermions. One may appropriately modify this generating geminal to introduce zero occupation numbers of its one-particle reduced density matrix and to make all the nonzero occupation numbers of its one-particle reduced density matrix equal, thus making this geminal a generator of an extreme AGP wave function, with an extreme large eigenvalue for its two-particle reduced density matrix. Closed analytical expressions are also given for this modified geminal and for its one-particle reduced density matrix. The modified generating geminal develops anomolously long “tails” in the intracule function. Nevertheless, both generating geminals introduced here reduce to a renormalized Dirac δ-function of the intracule coordinate, in the appropriate limit. In this limit, both of these generating geminals produce an absolute extreme AGP wave function with the absolute extreme value, N/2, for the Löwdin-normalized occupation number of the generating geminal, and vanishing occupation numbers for all other natural geminals, in the two-particle reduced density matrix of this wavefunction. A demonstration is given showing the formation of quasi-bosons in this ground state and showing their relationship to the generating geminal and to the other natural geminals of the two-particle reduced density matrix of the extreme AGP wave function. The similarities and differences of the features of this model and the accepted models of the superconducting ground state of electrons in metals, and the superfluid ground state of liquid He4 are mentioned.

International Journal of Quantum Chemistry•Mar 10, 1986•1 citation

Transformation to a geminal basis and stationary conditions for the exact wave function therein

Lasse Kragh Sørensen

Abstract We show the transformation from a one-particle basis to a geminal basis, transformations between different geminal bases demonstrate the Lie algebra of a geminal basis. From the basis transformations, we express both the wave function and Hamiltonian in the geminal basis. The necessary and sufficient conditions of the exact wave function expanded in a geminal basis are shown to be a Brillouin theorem of geminals. The variational optimization of the geminals in the antisymmetrized geminal power (AGP), antisymmetrized product of geminals (APG) and the full geminal product (FGP) wave function ansätze are discussed. We show that using a geminal replacement operator to describe geminal rotations introduce both primary and secondary rotations. The secondary rotations rotate two geminals in the reference at the same time due to the composite boson nature of geminals. Due to the completeness of the FGP, where all possible geminal combinations are present, the FGP is exact. The number of parameters in the FGP scale exponentially with the number of particles, like the full configuration interaction (FCI). Truncation in the FGP expansion can give compact representations of the wave function since the reference function in the FGP can be either the AGP or APG wave function.

arXiv:1910.06633•Theoretical Chemistry Accounts•Jun 18, 2024•0 citations

Coherent state approach to electron nuclear dynamics with an antisymmetrized geminal power state

Brian Weiner, Erik Deumens, Yngve Öhrn

A formulation of the complete dynamics of electrons and nuclei is presented. The dynamical equations are derived using the time-dependent variational principle (TDVP). The approximate electronic state vectors are antisymmetrized geminal power (AGP) states parameterized as projected coherent states, while the nuclei are treated as classical point particles. This leads to a formulation of time-dependent AGP theory that generalizes time-dependent Hartree–Fock (TDHF) theory and explicitly includes the dynamics of the nuclei. The linear approximation to the evolution equations (the classical harmonic approximation) which corresponds to a generalized random phase approximation (RPA) based on an AGP electronic reference state and which explicitly includes the dynamics of the nuclei, is studied and presented in this paper. The equations are formulated in terms of the primitive nonorthogonal electronic atomic basis thus avoiding any transformation to orthonormal molecular orbitals during the evolution.

Journal of Mathematical Physics•Mar 1, 1994•17 citations
Abstract The properties of the n‐electron wave functions equation image with symplectic symmetries have been discussed. These functions are built up with two kinds of geminals. The well‐known antisymmetrized geminal power (AGP) function is only a special case of them. The first‐ and second‐order reduced density matrices and the natural expansions of these functions have been given, and the uniform expressions for system energies in the states corresponding to these functions have been deduced. It has been shown that the special case equation image is probably another consistent ground state for the particle‐hole propegator or random‐phase approximation (RPA) besides the AGP function.
A mathematically rigorous treatment of the antisymmetrized geminal power (AGP) model is given. The model is advocated as an extremely flexible tool for studying a wide variety of fermion systems, characterized by a single Slater determinant, on the one hand, or Yang's most highly correlated wave function, on the other. When the first-order reduced density matrix of the actual system has eigenvalues which are evenly degenerate, the model can treat one-particle properties exactly. A formula is obtained for the precise energy of an AGP state in the Born-Oppenheimer approximation. An estimate is given of the error which occurs when the second-order density matrix of an AGP state is approximated by the widely used expression involving the so-called anomalous Green's function. © 1997 John Wiley & Sons, Inc.
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.
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.
A model many-fermion Hamiltonian is presented for which the ground state is asymptotically an Antisymmetrized Geminal Powers (AGP) wave function with largest possible greatest eigenvalue for its two-particle reduced density matrix. Closed analytical expressions and plane-wave expansions are presented for the generating geminal of the AGP ground state and for its one-particle reduced density matrix. The natural orbitals for this generating geminal are plane waves. The generating geminal shows intensely local character in its intracule and corresponds to the formation of a quasi-boson from two fermions. One may appropriately modify this generating geminal to introduce zero occupation numbers of its one-particle reduced density matrix and to make all the nonzero occupation numbers of its one-particle reduced density matrix equal, thus making this geminal a generator of an extreme AGP wave function, with an extreme large eigenvalue for its two-particle reduced density matrix. Closed analytical expressions are also given for this modified geminal and for its one-particle reduced density matrix. The modified generating geminal develops anomolously long “tails” in the intracule function. Nevertheless, both generating geminals introduced here reduce to a renormalized Dirac δ-function of the intracule coordinate, in the appropriate limit. In this limit, both of these generating geminals produce an absolute extreme AGP wave function with the absolute extreme value, N/2, for the Löwdin-normalized occupation number of the generating geminal, and vanishing occupation numbers for all other natural geminals, in the two-particle reduced density matrix of this wavefunction. A demonstration is given showing the formation of quasi-bosons in this ground state and showing their relationship to the generating geminal and to the other natural geminals of the two-particle reduced density matrix of the extreme AGP wave function. The similarities and differences of the features of this model and the accepted models of the superconducting ground state of electrons in metals, and the superfluid ground state of liquid He4 are mentioned.
Abstract We show the transformation from a one-particle basis to a geminal basis, transformations between different geminal bases demonstrate the Lie algebra of a geminal basis. From the basis transformations, we express both the wave function and Hamiltonian in the geminal basis. The necessary and sufficient conditions of the exact wave function expanded in a geminal basis are shown to be a Brillouin theorem of geminals. The variational optimization of the geminals in the antisymmetrized geminal power (AGP), antisymmetrized product of geminals (APG) and the full geminal product (FGP) wave function ansätze are discussed. We show that using a geminal replacement operator to describe geminal rotations introduce both primary and secondary rotations. The secondary rotations rotate two geminals in the reference at the same time due to the composite boson nature of geminals. Due to the completeness of the FGP, where all possible geminal combinations are present, the FGP is exact. The number of parameters in the FGP scale exponentially with the number of particles, like the full configuration interaction (FCI). Truncation in the FGP expansion can give compact representations of the wave function since the reference function in the FGP can be either the AGP or APG wave function.
A formulation of the complete dynamics of electrons and nuclei is presented. The dynamical equations are derived using the time-dependent variational principle (TDVP). The approximate electronic state vectors are antisymmetrized geminal power (AGP) states parameterized as projected coherent states, while the nuclei are treated as classical point particles. This leads to a formulation of time-dependent AGP theory that generalizes time-dependent Hartree–Fock (TDHF) theory and explicitly includes the dynamics of the nuclei. The linear approximation to the evolution equations (the classical harmonic approximation) which corresponds to a generalized random phase approximation (RPA) based on an AGP electronic reference state and which explicitly includes the dynamics of the nuclei, is studied and presented in this paper. The equations are formulated in terms of the primitive nonorthogonal electronic atomic basis thus avoiding any transformation to orthonormal molecular orbitals during the evolution.