Fast adaptive discontinuous basis sets for electronic structure

TL;DR

This paper introduces a flexible discontinuous Galerkin framework for adaptive basis sets in electronic structure calculations, enabling efficient, accurate, and scalable Hartree-Fock and DFT computations with structured sparsity.

Contribution

It develops a novel DG-based method for constructing adaptive basis sets that support discontinuity, structured sparsity, and efficient multigrid preconditioning for electronic structure calculations.

Findings

Achieves chemical accuracy with modest basis sizes.

Offers improved computational scalability over traditional basis sets.

Supports adaptive multigrid preconditioning for efficient eigensolvers.

Abstract

We develop a discontinuous Galerkin (DG) framework for automatically constructing adaptive basis sets for electronic structure calculations. By allowing basis functions to be discontinuous across element interfaces, our approach supports flexible combinations of atom-centered and polynomial basis sets, maintains favorable numerical conditioning, and induces structured sparsity of the one- and two-electron integrals, which we compute using specialised numerical integration strategies. In addition, we introduce a simple post-processing procedure to obtain continuous solutions if desired. We also introduce multigrid-preconditioned Poisson solvers that enable fast algorithms for both Hartree-Fock (HF) and density functional theory (DFT) calculations within our DG basis sets. Moreover, these basis sets naturally support adaptive multigrid preconditioning for the linear eigensolvers employed within the self-consistent field iteration for HF and DFT. Numerical experiments for HF and DFT demonstrate that our approach achieves chemical accuracy with modest basis sizes that compare favorably to the sizes of ordinary GTO basis sets achieving similar accuracy, while offering additional structured sparsity and improved computational scalability in the size-extensive limit. The framework thus provides a flexible route toward the construction of systematically improvable and structured adaptive basis sets for electronic structure theory.