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International Journal of Quantum Chemistry

The AGP model for fermion systems

Alan Coleman

Published Jan 1, 1997• 42 citations
Publisher

Abstract

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.

DOI
10.1002/(sici)1097-461x(1997)63:1<23::aid-qua5>3.0.co;2-4

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Structure of Fermion Density Matrices. II. Antisymmetrized Geminal Powers

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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.

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

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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.

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Density matrix embedding theory (DMET) has emerged as a powerful tool for performing wave function-in-wave function embedding for strongly correlated systems. In traditional DMET, an accurate calculation is performed on a small impurity embedded in a mean field bath. Here, we extend the original DMET equations to account for correlation in the bath via an antisymmetrized geminal power (AGP) wave function. The resulting formalism has a number of advantages. First, it allows one to properly treat the weak correlation limit of independent pairs, which DMET is unable to do with a mean-field bath. Second, it associates a size extensive correlation energy with a given density matrix (for the models tested), which AGP by itself is incapable of providing. Third, it provides a reasonable description of charge redistribution in strongly correlated but non-periodic systems. Thus, AGP-DMET appears to be a good starting point for describing electron correlation in molecules, which are aperiodic and possess both strong and weak electron correlation.

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Four-body correlation embedded in antisymmetrized geminal power wave function

Airi Kawasaki, Osamu Sugino

We extend the Coleman's antisymmetrized geminal power (AGP) to develop a wave function theory that can incorporate up to four-body correlation in a region of strong correlation. To facilitate the variational determination of the wave function, the total energy is rewritten in terms of the traces of geminals. This novel trace formula is applied to a simple model system consisting of one dimensional Hubbard ring with a site of strong correlation. Our scheme significantly improves the result obtained by the AGP-configuration interaction scheme of Uemura et al. and also achieves more efficient compression of the degrees of freedom of the wave function. We regard the result as a step toward a first-principles wave function theory for a strongly correlated point defect or adsorbate embedded in an AGP-based mean-field medium.

arXiv:1609.01438•The Journal of Chemical Physics•Dec 28, 2016•9 citations
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.
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.
Density matrix embedding theory (DMET) has emerged as a powerful tool for performing wave function-in-wave function embedding for strongly correlated systems. In traditional DMET, an accurate calculation is performed on a small impurity embedded in a mean field bath. Here, we extend the original DMET equations to account for correlation in the bath via an antisymmetrized geminal power (AGP) wave function. The resulting formalism has a number of advantages. First, it allows one to properly treat the weak correlation limit of independent pairs, which DMET is unable to do with a mean-field bath. Second, it associates a size extensive correlation energy with a given density matrix (for the models tested), which AGP by itself is incapable of providing. Third, it provides a reasonable description of charge redistribution in strongly correlated but non-periodic systems. Thus, AGP-DMET appears to be a good starting point for describing electron correlation in molecules, which are aperiodic and possess both strong and weak electron correlation.
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.
We extend the Coleman's antisymmetrized geminal power (AGP) to develop a wave function theory that can incorporate up to four-body correlation in a region of strong correlation. To facilitate the variational determination of the wave function, the total energy is rewritten in terms of the traces of geminals. This novel trace formula is applied to a simple model system consisting of one dimensional Hubbard ring with a site of strong correlation. Our scheme significantly improves the result obtained by the AGP-configuration interaction scheme of Uemura et al. and also achieves more efficient compression of the degrees of freedom of the wave function. We regard the result as a step toward a first-principles wave function theory for a strongly correlated point defect or adsorbate embedded in an AGP-based mean-field medium.