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arXiv:2007.03671 · The Journal of Chemical Physics

Correlating the antisymmetrized geminal power wave function

Thomas M. Henderson, Gustavo E. Scuseria

Published Aug 25, 2020• 42 citations
arXivHTMLPDF

Abstract

Strong pairing correlations are responsible for superconductivity and off-diagonal long-range order in the two-particle density matrix. The antisymmetrized geminal power wave function was championed many years ago as the simplest model that can provide a reasonable qualitative description for these correlations without breaking number symmetry. The fact remains, however, that the antisymmetrized geminal power is not generally quantitatively accurate in all correlation regimes. In this work, we discuss how we might use this wave function as a reference state for a more sophisticated correlation technique such as configuration interaction, coupled cluster theory, or the random phase approximation.

DOI
10.1063/5.0021144

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Strong pairing correlations are responsible for superconductivity and off-diagonal long-range order in the two-particle density matrix. The antisymmetrized geminal power wave function was championed many years ago as the simplest model that can provide a reasonable qualitative description for these correlations without breaking number symmetry. The fact remains, however, that the antisymmetrized geminal power is not generally quantitatively accurate in all correlation regimes. In this work, we discuss how we might use this wave function as a reference state for a more sophisticated correlation technique such as configuration interaction, coupled cluster theory, or the random phase approximation.
In this review, we discuss the recent progress in developing geminal-based theories for challenging problems in quantum chemistry. Specifically, we focus on the antisymmetrized geminal power, generalized valence bond, antisymmetrized product of strongly orthogonal geminals, singlet-type orthogonal geminals, the antisymmetric product of 1-reference orbital geminal, also known as the pair coupled cluster doubles ansatz, and geminals constructed from Richardson-Gaudin states. Furthermore, we review various corrections to account for the missing dynamical correlation effects in geminal models and possible extensions to target electronically excited states and open-shell species. Finally, we discuss some numerical examples and present-day challenges for geminal-based models, including a quantitative and qualitative analysis of wave functions, and software availability.
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.
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.
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.
The antisymmetrized geminal power (AGP), a wave function equivalent to number-projected Hartree-Fock-Bogoliubov (HFB), and number-projected Bardeen-Cooper-Schrieffer (BCS) when working in the paired (natural orbitals) basis, has proven to be an excellent reference for strong pairing interactions. Several correlation methods have also been applied on top of AGP. In this work, we show how AGP can also be applied to spin systems by simply basing its formulation on a spin su(2)su(2)su(2) algebra. We here implement spin AGP and spin AGP-based correlation techniques and benchmark them on the XXZ and ${\mathrm{J}}_{1}\text{\ensuremath{-}}{\mathrm{J}}_{2}$ Heisenberg models, both in one and two dimensions. Our results indicate that spin AGP is a promising starting point for modeling spin systems.
Strong electron correlation in materials is explored within a class of model Hamiltonians that treat only pair interactions between electrons. The model is unique among typical spin Hamiltonians in that it does not have an effective mean-field reference wave function. The ground-state wave functions from all Hamiltonians in the model have the same one-electron reduced density matrix (1-RDM); consequently, one-electron theories such as the Hartree–Fock and density functional theories are inapplicable. In contrast, the ground-state two-electron reduced density matrix (2-RDM) has a one-to-one mapping to the ground-state wave function. For a range of lattices including linear, ladder, and square topologies, we variationally compute the 2-RDM subject to constraints, known as N -representability conditions, that are necessary for the 2-RDM to represent an N -electron ensemble density matrix. We find that for all model Hamiltonians the 2-RDM is accurately computed as long as the D, Q, and G N -representability conditions are supplemented with the T 2 condition. Energies, orbital and geminal occupation numbers, and correlation functions are computed. The model has applications to superconductivity as well as more general pairing phenomena in electronic systems. Effective methods are needed to treat strong electron correlation arising in the study of materials including the study of high-temperature superconductivity and phase transitions.