Practical and Rigorous reduction of the Many-Electron Quantum-Mechanical Coulomb Problem to O(N^(2/3) Storage

TL;DR

This paper introduces a new method to drastically reduce the storage requirements for two-electron Coulomb integrals in quantum mechanics calculations, enabling efficient computations on limited-memory parallel systems.

Contribution

It presents a novel algorithm that reduces the storage of Coulomb integrals from O(N^{2}) to O(N^{2/3}) for separable basis functions, improving computational efficiency.

Findings

Storage requirement decreases from 2.8GB to 2.4MB for 9171 planewaves.

Algorithm enables fast reconstruction of integrals on parallel processors.

Applicable to large-scale many-electron quantum calculations.

Abstract

It is tacitly accepted that, for practical basis sets consisting of N functions, solution of the two-electron Coulomb problem in quantum mechanics requires storage of O(N^4) integrals in the small N limit. For localized functions, in the large N limit, or for planewaves, due to closure, the storage can be reduced to O(N^2) integrals. Here, it is shown that the storage can be further reduced to O(N^{2/3}) for separable basis functions. A practical algorithm, that uses standard one-dimensional Gaussian-quadrature sums, is demonstrated. The resulting algorithm allows for the simultaneous storage, or fast reconstruction, of any two-electron coulomb integral required for a many-electron calculation, on each and every processor of massively parallel computers even if such processors have very limited memory and disk space. For example, for calculations involving a basis of 9171 planewaves, the memory required to effectively store all coulomb integrals decreases from 2.8Gbytes to less than 2.4 Mbytes.