The conventional Fock matrix calculation method from stored nonzero two-electron integrals was reformulated into the linear scaling Fock matrix calculation method from these integrals stored by using a data-compression technique and density difference prescreening method for preliminary evaluation of their contribution to the Fock matrix. This method substantially reduces numerical complexity of Fock matrix calculations compared to direct methods by eliminating recalculations of integrals on each iterative step. It was shown that the reformulated method possesses the linear scaling property with respect to problem size. This property follows from the proven theorem, which states that the total number of nonzero integrals scales asymptotically linearly with respect to the number of basis functions for large molecular systems. An analysis of Fock matrix calculation with density or density difference prescreening shows that its linear scaling property arises due to asymptotically linear scaling properties of the number of nonzero integrals and the linear scaling property of the number of leading matrix elements of the density matrix. The use of density and density difference prescreening in Fock matrix calculation enhances this property, while the criterion for exclusion of calculations with integrals giving small contributions to the Fock matrix transforms this method to a linear scaling one even at those number of basis functions where such a regime has not been reached for total number of nonzero integrals. The numerical calculations with the proposed method show that its linear scaling property begins in calculation with the number of basis functions from 2500 to 4500 in dependence on basis function type in molecular calculations.
A. V. Mitin (Sun,) studied this question.