Efficient computation of Coulomb and exchange integrals for multi-million atom nanostructures

Atomistic modeling of nanostructures such as quantum dots or nanowires often involves numbers of atoms reaching and even exceeding well beyond 1 million. Such a large quantity of atoms presents a very complex computational challenge especially at a stage of many-body calculation where numerous Coulomb matrix elements need to be calculated. Here we present a practical solution to this problem by performing calculations in the momentum space and utilizing fast Fourier transform combined with a memory-efficient way to compute the convolution that overcomes the problem of spurious interactions with quasi-charges from other super-cells. Finally, calculation of multiple integrals is optimized by reducing the problem to finding a minimum vertex cover of a graph. All these algorithms are implemented and presented here in a self-contained and highly parallelized computer program named Coulombo. Coulombo demonstrates quasi-linear scaling of computational time of Coulomb matrix elements with respect to the number of points in the computational box and, at the same time, significantly reduced memory demand. The proposed solution can have potential applications not only in the realm of nano-physics, but could be applied to other mesoscopic simulations or large-scale quantum chemistry problems.