  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.
