A linear-scaling spectral-element method for computing electrostatic potentials

Abstract

A new linear-scaling method is presented for the fast numerical evaluation of the electronic Coulomb potential. Our approach uses a simple real-space partitioning of the system into cubic cells and a spectral-element representation of the density in a tensorial basis of high-order Chebyshev polynomials. Electrostatic interactions between non-neighboring cells are described using the fast multipole method. The remaining near-field interactions are computed in the tensorial basis as a sum of differential contributions by exploiting the numerical low-rank separability of the Coulomb operator. The method is applicable to arbitrary charge densities, avoids the Poisson equation, and does not involve the solution of any systems of linear equations. Above all, an adaptive resolution of the Chebyshev basis in each cell facilitates the accurate and efficient treatment of molecular systems. We demonstrate the performance of our implementation for quantum chemistry with benchmark calculations on the noble gas atoms, long-chain alkanes, and diamond fragments. We conclude that the spectral-element method can be a competitive tool for the accurate computation of electrostatic potentials in large-scale molecular systems.