# Numerical Integration of Slater Basis Functions Over Prolate Spheroidal Grids

## TL;DR

This paper introduces a new numerical integration method using prolate spheroidal grids to improve accuracy in electronic structure simulations with large basis sets.

## Contribution

The novel use of prolate spheroidal grids enables accurate integration for larger basis sets in polyatomic systems.

## Key findings

- The new grid reduces integration errors by ~3 orders of magnitude compared to Becke partitioning.
- The method is tested on polyatomic systems and shows reliability for correlated electronic structure computations.

## Abstract

Slater basis functions have desirable properties that can improve electronic structure simulations, but improved numerical integration methods are needed. This work builds upon the SlaterGPU library for the evaluation of Hamiltonian matrix elements in the resolution‐of‐the‐identity approximation. In particular, a prolate spheroidal grid will provide sufficient integral accuracy to employ larger basis sets (quadruple‐zeta and greater) in practical computations involving polyatomics. To integrate 3‐center Coulomb and nuclear attraction terms, an improved grid representation around the third center is introduced. The RMSEs of the integral quantities are evaluated and compared to the previous numerical integration method used in SlaterGPU (Becke partitioning), resulting in a ~3 order of magnitude reduction in the error for 2‐center integral quantities. The procedure is generally applicable to polyatomic systems, GPU accelerated for high performance computing, and tested on self‐consistent field and full configuration interaction wavefunctions. Results for a number of 3‐atom models as well as propanediyl (C3H6) demonstrate the reliability of the new integration scheme.

Atom‐centered Slater functions are integrated over a prolate spheroidal grid, allowing the use of larger basis sets in correlated electronic structure computations.