# Fast evaluation of solid harmonic Gaussian integrals for local resolution-of-the-identity methods and range-separated hybrid functionals

## TL;DR

This paper introduces a fast integral evaluation scheme for solid harmonic Gaussian functions, significantly improving computational efficiency for local resolution-of-the-identity methods and range-separated hybrid functionals.

## Contribution

The authors develop a novel integral scheme that outperforms traditional Cartesian Gaussian-based methods, especially for large angular momenta and contracted basis sets.

## Key findings

- Speed-up of up to three orders of magnitude over OS scheme.
- Enhanced efficiency for large angular momenta and contracted basis sets.
- Effective evaluation of non-local potentials in hybrid functionals.

## Abstract

An integral scheme for the efficient evaluation of two-center integrals over contracted solid harmonic Gaussian functions is presented. Integral expressions are derived for local operators that depend on the position vector of one of the two Gaussian centers. These expressions are then used to derive the formula for three-index overlap integrals where two of the three Gaussians are located at the same center. The efficient evaluation of the latter is essential for local resolution-of-the-identity techniques that employ an overlap metric. We compare the performance of our integral scheme to the widely used Cartesian Gaussian-based method of Obara and Saika (OS). Non-local interaction potentials such as standard Coulomb, modified Coulomb and Gaussian-type operators, that occur in range-separated hybrid functionals, are also included in the performance tests. The speed-up with respect to the OS scheme is up to three orders of magnitude for both, integrals and their derivatives. In particular, our method is increasingly efficient for large angular momenta and highly contracted basis sets.