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 than their orthogonal counterparts. While several different approaches have been developed, these are predominantly derived from the first-quantized generalized 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-quantized 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 generalized 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 generalized 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 formulas with reduced scaling, promising significant computational acceleration for methods that rely on such terms.
Highlights
Matrix elements between nonorthogonal Slater determinants are increasingly common in emerging electronic structure methods
Intuitive derivations and efficient implementations of nonorthogonal matrix elements are becoming increasingly important for the development of nonorthogonal configuration interaction methods and inter-state coupling terms for state-specific excited state wave functions
For overlap terms and one-body operators, these excited nonorthogonal matrix elements can be expressed in terms of one-body intermediate terms that can be precomputed and stored for a given pair of reference determinants
Summary
Matrix elements between nonorthogonal Slater determinants are increasingly common in emerging electronic structure methods. The nonorthogonal Wick’s theorem could allow these matrix elements to be evaluated using only biorthogonal reference orbitals, but until now, this requires the reference determinants to have a strictly non-zero overlap In this contribution, we derive an entirely generalized nonorthogonal form of Wick’s theorem that applies to any pair of determinants with nonorthogonal orbitals, even if the overall determinants have a zero overlap. We show how evaluating intermediates for a given pair of determinants can reduce the scaling of overlap and one-body coupling terms between excited configurations to O(1) These particular nonorthogonal matrix elements become almost as straightforward as the orthogonal Slater–Condon rules or Wick’s theorem, promising considerable acceleration for methods that rely on such terms. VI, we extend our framework to the matrix elements between excited configurations and show how O(1) scaling can be achieved for overlap and one-body operators
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