Hybrid grid/basis set discretizations of the Schr\"odinger equation

TL;DR

This paper introduces gausslets, a new basis function for Schrödinger equation discretization that combines grid and basis set advantages, improving computational efficiency in electronic structure calculations.

Contribution

The paper proposes gausslets, a novel wavelet-like basis function composed of Gaussians, and introduces diagonal approximations to reduce computational scaling.

Findings

Gausslets are orthogonal, smooth, symmetric, and locally supported.

Gausslets can be integrated with traditional Gaussian bases.

Diagonal approximations significantly reduce computational cost.

Abstract

We present a new kind of basis function for discretizing the Schr\"odinger equation in electronic structure calculations, called a gausslet, which has wavelet-like features but is composed of a sum of Gaussians. Gausslets are placed on a grid and combine advantages of both grid and basis set approaches. They are orthogonal, infinitely smooth, symmetric, polynomially complete, and with a high degree of locality. Because they are formed from Gaussians, they are easily combined with traditional atom-centered Gaussian bases. We also introduce diagonal approximations which dramatically reduce the computational scaling of two-electron Coulomb terms in the Hamiltonian.