Angular Gausslets

TL;DR

This paper introduces generalized gausslets on the sphere combined with radial gausslets to create an atom-centered basis for electronic structure calculations, enabling efficient DMRG methods and accurate energy computations for atoms.

Contribution

It extends gausslet basis constructions to three dimensions with spherical harmonics, facilitating systematic convergence and efficient DMRG calculations for atomic systems.

Findings

Systematic convergence with increasing angular resolution.

Accurate energy calculations for the Be atom within 0.1 mH of the basis set limit.

Development of DMRG methods with compact MPOs and variance extrapolation.

Abstract

Gausslets are one of the few basis constructions for electronic structure that combine locality, orthonormality, variable resolution, and an accurate diagonal approximation for the electron-electron interaction, but the original construction is tied to one dimension. Radial gausslets extended this idea to atoms while leaving the angular degrees of freedom in spherical harmonics, so the atomic interaction remained only partially diagonal in the combined basis. Here we introduce generalized gausslets on the sphere and combine them shell by shell with radial gausslets to form an atom-centered basis in which the electron-electron interaction takes a two-index integral-diagonal form. The angular basis starts from localized spherical Gaussians and uses injection to make a low-$\ell$ spherical-harmonic subspace exact. Tests of the kinetic spectrum, low-$\ell$ Coulomb matrix elements, spherium, first-row Hartree--Fock calculations, and He exact diagonalization show systematic convergence with increasing angular resolution. We also develop DMRG methods for this basis, including compact MPOs, correlated small-space starting states, Givens-rotation transfers between nearby angular sizes, and embedded sampled variance extrapolation (ESVE). We show that this combination of ingredients can be used to solve the Be atom, with extrapolations in the number of angular functions but with fixed radial resolution, to within about 0.1 mH of the complete basis set limit exact energy. This shows that DMRG calculations of first row atoms which include both static and accurate dynamic correlation on the same footing are feasible.