A Graph-Based Algorithm for Computing Matrix Elements of Arbitrary Operators between Configuration State Functions

Area of Science:

• Quantum Chemistry
• Computational Physics
• Theoretical Chemistry

Background:

• Configuration state functions (CSFs) offer a compact representation of many-electron wave functions.
• Evaluating matrix elements for CSFs is computationally challenging in quantum chemical methods.
• Existing methods often rely on explicit Slater determinant expansions, which can be inefficient.

Purpose of the Study:

• To develop a novel graph-based algorithm for computing matrix elements between CSFs.
• To overcome the computational complexity associated with CSF matrix element evaluation.
• To provide a general framework applicable to various quantum chemical methods.

Main Methods:

• A graph-based representation is used to encode CSF expansions without explicit construction.
• Operator sequences are applied directly to the graphical representation.
• Matrix elements are computed via graph traversal and overlap calculations.

Main Results:

• The algorithm achieves machine-level precision in matrix element computations.
• The graph-based approach demonstrates superior performance, outperforming explicit determinant expansion by orders of magnitude.
• The method is general and applicable to any operator sequence.

Conclusions:

• The developed graph-based algorithm provides an efficient and precise method for calculating CSF matrix elements.
• This framework facilitates the implementation of CSF-based approaches in advanced quantum chemistry techniques like selected and stochastic configuration interaction.
• The study enhances computational efficiency in electronic structure calculations.