Efficient Calculation of Derivatives of Integrals in a Basis of Non-Separable Gaussians Through Exploitation of Sparsity

TL;DR

This paper presents an efficient symbolic computational method for calculating derivatives of integrals over non-separable Gaussian basis functions, significantly improving energy gradient calculations in quantum simulations.

Contribution

It introduces a novel symbolic and automated approach exploiting sparsity for derivatives of Gaussian integrals, implemented in the Crystal code for enhanced efficiency.

Findings

Significantly improved computational efficiency over previous methods.

Successful application to geometry optimization, frequency, and elastic tensor calculations.

Validated on diverse materials including metals, semiconductors, and metal-organic frameworks.

Abstract

A computational procedure is developed for the efficient calculation of derivatives of integrals over non-separable Gaussian-type basis functions, used for the evaluation of gradients of the total energy in quantum-mechanical simulations. The approach, based on symbolic computation with computer algebra systems and automated generation of optimized subroutines, takes full advantage of sparsity and is here applied to first energy derivatives with respect to nuclear displacements and lattice parameters of molecules and materials. The implementation in the \textsc{Crystal} code is presented and the considerably improved computational efficiency over the previous implementation is illustrated. To this purpose, three different tasks involving the use of analytical forces are considered: i) geometry optimization; ii) harmonic frequency calculation; iii) elastic tensor calculation. Three test case materials are selected as representatives of different classes: i) a metallic 2D model of the Cu (111) surface; ii) a wide-gap semiconductor ZnO crystal, with a wurtzite-type structure; and iii) a porous metal-organic crystal, namely the ZIF-8 Zinc-imidazolate framework. Finally, it is argued that the present symbolic approach is particularly amenable to generalizations, and its potential application to other derivatives is sketched.