Fast adaptive discontinuous basis sets for electronic structure

TL;DR

This work develops a discontinuous Galerkin framework to build adaptive, discontinuous basis sets for electronic structure calculations, combining atom-centered Gaussians with polynomials and solving mean-field HF/DFT problems efficiently on a DG mesh. By introducing an auxiliary Poisson-grid solver and adaptive multigrid preconditioning, the approach achieves chemical accuracy with modest basis sizes and favorable sparsity, enabling scalable HF/DFT computations. Key contributions include a complete DG discretization with SIP laplacian, an adaptive basis construction and truncation strategy, efficient Coulomb integral handling via Gaussian-sum and Fourier expansions, and open-source software. The results on H$_2$, LiH, H$_2$O, and C$_6$H$_6$ validate accuracy gains and scalability, suggesting a practical path toward systematically improvable, structured adaptive basis sets for electronic structure theory.

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. 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.

Figures (16)

• Figure 1: Schematic of free space mesh generation. A bounding box is placed around atom centres, shown in black, and carved up into uniform cells. Elements on domain boundaries are extended to infinity on sides with no neighbours.
• Figure 2: Penalty parameter $\sigma$ in hexahedral elements using tensor product basis functions can be reduced to the one-dimensional case. The problem is projected onto the dimension normal to the shared interface between two elements, in this case the $x$-direction. The penalty from the one-dimensional reduction can be used directly for the original three-dimensional case.
• Figure 3: DG polynomial and Gaussian basis functions. Polynomial basis functions (left) on infinite domains are truncated to have support only within original mesh bounding box. Gaussian basis functions (right) are not truncated but are split across elements on which they have significant mass and duplicated.
• Figure 4: Schematic of adaptive mesh construction for solving Hartree and Fock exchange Poisson problems in two dimensions. The auxiliary mesh $\mathcal{S}$ with elements $\{ J_m \}$ (right) is constructed from the original DG mesh for the orbitals $\mathcal{T}$ with elements $\{ K_m \}$ (left) by truncating the domain to a rectangular region where $|\eta_i| > 10^{-9}$. The new auxiliary mesh $\mathcal{S}$ is then adaptively refined around sharp regions of the densities.
• Figure 5: Visualisation of element refinement for auxiliary mesh $\mathcal{S}$. Elements with roughly equal side lengths are refined uniformly into 8 smaller boxes, otherwise elements are not divided in dimensions where the edge length is below half that of those in the other dimensions.