Efficient and scalable electrostatics via spherical grids and treecode summation

Evaluation of noncovalent electrostatic interactions is the dominant bottleneck in classical molecular dynamics simulations, and evaluation of Coulombic matrix elements similarly limits quantum mechanical self-consistent field calculations. These difficulties are a result of the Coulomb operator's slow decay, which necessitates the evaluation of large numbers of interactions. In this work, we use a combination of cubature techniques to factorize the Coulomb operator and devise a hierarchical summation scheme, arriving at a novel technique that requires O(N⁡log(N)) effort to evaluate electrostatic interactions. The factorization may be made arbitrarily accurate, allowing full control between computational expense and accuracy. By avoiding the fast Fourier transform to evaluate terms, the resulting algorithm bears a resemblance to that of the fast multipole method and offers many opportunities for highly scalable parallel implementations.