Reduced density matrices of Richardson–Gaudin states in the Gaudin algebra basis

Abstract

Eigenvectors of the reduced Bardeen–Cooper–Schrieffer Hamiltonian have recently been employed as a variational wavefunction ansatz in quantum chemistry. This wavefunction is a mean-field of pairs of electrons (geminals). In this contribution, we report optimal expressions for their reduced density matrices in both the original physical basis and the basis of the Richardson–Gaudin pairs. Physical basis expressions were originally reported by Gorohovsky and Bettelheim [Phys. Rev. B 84, 224503 (2011)]. In each case, the expressions scale like O(N4), with the most expensive step being the solution of linear equations. Analytic gradients are also reported in the physical basis. These expressions are an important step toward practical mean-field methods to treat strongly correlated electrons.

Near-exact treatment of seniority-zero ground and excited states with a Richardson-Gaudin mean-field

“…The RDM elements for RG states have been written many times. 76,77,[81][82][83] RDM elements of any order can be evaluated with the derivatives of the rapidities with respect to the single-particle energies, which are obtained as solutions of the linear equations…”

Density Matrices of Seniority-Zero Geminal Wavefunctions

“…Recently [48][49][50][51] we have employed the eigenvectors of the reduced Bardeen-Cooper-Schrieffer (BCS) Hamiltonian 33,34 ĤBCS = 1 2…”

Construction of linearly independent non-orthogonal AGP states

“…[22][23][24] Exact ground states of the reduced BCS Hamiltonian have recently been appiled to the molecular Hamiltonian. [25][26][27] Although simplistic, this Hamiltonian shows non-trivial physics in the attractive regime, 28 which traditional quantum chemistry methods fail to describe. [28][29][30] In contrast, AGP and AGP-based methods are able to capture most of the correlation energies systematically.…”