Generalized nonorthogonal matrix elements: Unifying Wick's theorem and the Slater-Condon rules

Area of Science:

• Quantum Chemistry
• Computational Physics
• Theoretical Chemistry

Background:

• Evaluating matrix elements between nonorthogonal Slater determinants is crucial for advanced electronic structure methods.
• Current methods often rely on first-quantized rules and require biorthogonal orbitals, posing computational challenges.
• Existing second-quantized approaches like nonorthogonal Wick's theorem fail for determinants with zero overlap.

Purpose of the Study:

• To develop a generalized extension of the nonorthogonal Wick's theorem.
• To unify Wick's theorem and generalized Slater-Condon rules for nonorthogonal determinants.
• To provide a protocol for deriving coupling terms in nonorthogonal excited configurations.

Main Methods:

• Derivation of a generalized nonorthogonal Wick's theorem applicable to all determinant pairs.
• Development of a protocol for arbitrary coupling terms between nonorthogonal excited configurations.
• Application of the protocol to overlap and one-body operators.

Main Results:

• A universal methodology for evaluating any nonorthogonal matrix element.
• Unification of Wick's theorem and generalized Slater-Condon rules.
• Efficient formulas with reduced scaling for overlap and one-body operators.

Conclusions:

• The generalized nonorthogonal Wick's theorem provides a unified and broadly applicable approach.
• The derived protocol offers significant computational acceleration for electronic structure methods.
• This work advances the efficiency and scope of quantum chemical calculations.