Generalized nonorthogonal matrix elements: Unifying Wick’s theorem and the Slater–Condon rules

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.

Analyzing the Behavior of Spin Phases in External Magnetic Fields by Means of Spin-Constrained States

“…This disordered phase quantifies where Löwdin’s dilemma matters and offers a unique perspective on the physics of the intermediate coupling regime as it straddles the limiting cases of collinear spin phases . We note that as the CGHF states effectively span the entire disordered phase, they could provide an excellent starting point for post-Hartree–Fock methods aimed at gaining more electron correlation or regaining spin symmetry. − …”

Methods to Calculate Electronic Excited-State Dynamics for Molecules on Large Metal Clusters with Many States: Ensuring Fast Overlap Calculations and a Robust Choice of Phase

“…Rather than rotate one set of orbitals (as in method #1), as shown by Sundstrom and Head-Gordon, the better approach is to rotate both sets of orbitals (those at the current and previous geometries) so as to generate a fully biorthogonal basis set. Such an approach has been used previously for several electronic structure calculations ,,, and significantly generalized by Burton . Here, we apply this technique to calculate the singly-excited state overlap matrix for use in non-adiabatic dynamics.…”

“…Such an approach has been used previously for several electronic structure calculations 25,31,34,35 and significantly generalized by Burton. 36 Here, we apply this technique to calculate the singly-excited state overlap matrix for use in nonadiabatic dynamics.…”

Fast and Accurate Computation of Nonadiabatic Coupling Matrix Elements Using the Truncated Leibniz Formula and Mixed-Reference Spin-Flip Time-Dependent Density Functional Theory

“…For example, obtaining the OI in connection with the single-excitation methods, such as TDDFT and SF-TDDFT, typically requires a computational effort approximately scaling as O ( n occ 5 n vir 2 ) with the number of occupied ( n occ ) and virtual ( n vir ) molecular orbitals (MOs), , or briefly O ( n 7 ) with the number of total MOs ( n ). Based on Wick’s theorem, a new method scaling approximately as O ( n 4 ) was recently proposed by Burton …”

“…or briefly O(n 7 ) with the number of total MOs (n). Based on Wick's theorem, a new method scaling approximately as O(n 4 ) was recently proposed by Burton 24. …”