An enhanced adaptive finite element method with linear scaling for 3D Hartree–Fock equation

The Hartree-Fock equation serves as a fundamental theoretical tool in quantum chemistry and condensed matter physics for describing the ground-state electronic structure of multi-electron systems. Despite its significance, the numerical solution faces challenges due to its complex mathematical structure. This paper presents an efficient adaptive finite element method for solving the 3D Hartree-Fock equation, incorporating three acceleration techniques: a multilevel correction scheme, a two-level nested self-consistent field iteration, and a density fitting strategy. The method aims to enhance computational efficiency, reduce memory consumption, and improve the parallel scalability. By reformulating the problem into a sequence of large-scale boundary value problems and small-scale eigenvalue problems, our algorithm drastically reduces computational load and storage demands. Notably, it achieves linear computational complexity O(NNk), with the coefficient depending on the number of wavefunctions N rather than the correction space dimension NH, representing a significant improvement over the O((N + NH)Nk) complexity in prior work [Xu et al., J. Chem. Phys. 164, 074111 (2026)] that proposed a simplified multilevel correction algorithm (yielding up to fivefold speedup in numerical simulations). Furthermore, the correction process is tailored for each wavefunction independently, endowing it with substantial potential for parallel computation. Numerical experiments conducted on various molecular systems demonstrate the superior precision, efficiency, and memory savings of the proposed algorithm compared to the direct adaptive finite element method.