An efficient basis set representation for calculating electrons in molecules

TL;DR

This paper introduces a scalable, accurate basis set method using sinc functions on a Cartesian grid for electronic structure calculations in molecules, achieving high precision with reduced computational complexity.

Contribution

It generalizes a previous method to provide a simple, accurate, and scalable basis set representation that simplifies Coulomb matrix calculations and improves computational efficiency.

Findings

Achieves 1 kcal/mol precision for valence energies

Reduces two-electron integral calculations to O(N log N) operations

Produces superior energies and properties compared to variational methods

Abstract

The method of McCurdy, Baertschy, and Rescigno, J. Phys. B, 37, R137 (2004) is generalized to obtain a straightforward, surprisingly accurate, and scalable numerical representation for calculating the electronic wave functions of molecules. It uses a basis set of product sinc functions arrayed on a Cartesian grid, and yields 1 kcal/mol precision for valence transition energies with a grid resolution of approximately 0.1 bohr. The Coulomb matrix elements are replaced with matrix elements obtained from the kinetic energy operator. A resolution-of-the-identity approximation renders the primitive one- and two-electron matrix elements diagonal; in other words, the Coulomb operator is local with respect to the grid indices. The calculation of contracted two-electron matrix elements among orbitals requires only O(N log(N)) multiplication operations, not O(N^4), where N is the number of basis functions; N = n^3 on cubic grids. The representation not only is numerically expedient, but also produces energies and properties superior to those calculated variationally. Absolute energies, absorption cross sections, transition energies, and ionization potentials are reported for one- (He^+, H_2^+ ), two- (H_2, He), ten- (CH_4) and 56-electron (C_8H_8) systems.