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

## TL;DR

A new graph-based algorithm efficiently computes matrix elements for quantum chemistry calculations using configuration state functions.

## Contribution

A general and efficient graph-based algorithm for computing matrix elements between configuration state functions.

## Key findings

- The algorithm achieves machine-level precision in numerical tests.
- It outperforms explicit determinant expansion by several orders of magnitude.
- The method enables new possibilities for CSF-based quantum chemistry implementations.

## Abstract

We present a graph-based algorithm for computing matrix
elements
of arbitrary second-quantized operators between configuration state
functions (CSFs) defined in a genealogical scheme. Unlike Slater determinants,
CSFs are spin-adapted and offer a more compact representation of many-electron
wave functions, but their use in quantum chemical methods is often
hindered by the complexity of evaluating matrix elements. Our approach
leverages a graphical representation to efficiently encode the expansion
of CSFs in terms of Slater determinants without explicitly constructing
the full expansion. The algorithm applies operator sequences directly
to the graph and computes overlaps via graph traversal, yielding matrix
elements, and is completely general for any operator sequence. Numerical
tests demonstrate that the method achieves machine-level precision
and outperforms explicit determinant expansion by several orders of
magnitude. This framework opens new possibilities for CSF-based implementations
in selected and stochastic configuration interaction methods.