Communication: practical and rigorous reduction of the many-electron quantum mechanical Coulomb problem to O(N(2/3)) storage.

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 processors with 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 2.8 Gbytes to less than 2.4 Mbytes.