Efficient evaluation of two-center Gaussian integrals in periodic systems

TL;DR

This paper presents an efficient method for evaluating two-center Gaussian integrals in periodic systems by combining real and reciprocal space summations, significantly improving computational speed for various kernels.

Contribution

The authors develop a novel algorithm that partitions lattice summations using Poisson's formula, optimizing the calculation of Gaussian integrals in periodic systems.

Findings

Summation converges exponentially fast in both spaces.

Algorithm is only 5 to 15 times slower than molecular integrals.

Significant efficiency gains for highly contracted basis functions.

Abstract

By using Poisson's summation formula, we calculate periodic integrals over Gaussian basis functions by partitioning the lattice summations between the real and reciprocal space, where both sums converge exponentially fast with a large exponent. We demonstrate that the summation can be performed efficiently to calculate 2-center Gaussian integrals over various kernels including overlap, kinetic, and Coulomb. The summation in real space is performed using an efficient flavor of the McMurchie-Davidson Recurrence Relation (MDRR). The expressions for performing summation in the reciprocal space are also derived and implemented. The algorithm for reciprocal space summation allows us to reuse several terms and leads to significant improvement in efficiency when highly contracted basis functions with large exponents are used. We find that the resulting algorithm is only between a factor of 5 to 15 slower than that for molecular integrals, indicating the very small number of terms needed in both the real and reciprocal space summations. An outline of the algorithm for calculating 3-center Coulomb integrals is also provided.