Generalised Nonorthogonal Matrix Elements: Unifying Wick's Theorem and the Slater-Condon Rules

TL;DR

This paper introduces a universal method to evaluate nonorthogonal matrix elements in electronic structure calculations by unifying Wick's theorem and the Slater-Condon rules, enabling efficient computation for all determinant pairs.

Contribution

It derives a generalised nonorthogonal Wick's theorem applicable to all determinant pairs, unifying two key approaches and improving computational efficiency.

Findings

Unified Wick's theorem and Slater-Condon rules for nonorthogonal determinants.

Derived efficient formulas for overlap and one-body operators.

Potential for significant acceleration in electronic structure methods.

Abstract

Matrix elements between nonorthogonal Slater determinants represent an essential component of many emerging electronic structure methods. However, evaluating nonorthogonal matrix elements is conceptually and computationally harder then their orthogonal counterparts. While several different approaches have been developed, these are predominantly derived from the first-quantised generalised Slater-Condon rules and usually require biorthogonal occupied orbitals to be computed for each matrix element. For coupling terms between nonorthogonal excited configurations, a second-quantised approach such as the nonorthogonal Wick's theorem is more desirable, but this fails when the two reference determinants have a zero many-body overlap. In this contribution, we derive an entirely generalised extension to the nonorthogonal Wick's theorem that is applicable to all pairs of determinants with nonorthogonal orbitals. Our approach creates a universal methodology for evaluating any nonorthogonal matrix element and allows Wick's theorem and the generalised Slater-Condon rules to be unified for the first time. Furthermore, we present a simple well-defined protocol for deriving arbitrary coupling terms between nonorthogonal excited configurations. In the case of overlap and one-body operators, this protocol recovers efficient formulae with reduced scaling, promising significant computational acceleration for methods that rely on such terms.