Real space electrostatics for multipoles. I. Development of methods.

We have extended the original damped-shifted force (DSF) electrostatic kernel and have been able to derive three new electrostatic potentials for higher-order multipoles that are based on truncated Taylor expansions around the cutoff radius. These include a shifted potential (SP) that generalizes the Wolf method for point multipoles, and Taylor-shifted force (TSF) and gradient-shifted force (GSF) potentials that are both generalizations of DSF electrostatics for multipoles. We find that each of the distinct orientational contributions requires a separate radial function to ensure that pairwise energies, forces, and torques all vanish at the cutoff radius. In this paper, we present energy, force, and torque expressions for the new models, and compare these real-space interaction models to exact results for ordered arrays of multipoles. We find that the GSF and SP methods converge rapidly to the correct lattice energies for ordered dipolar and quadrupolar arrays, while the TSF is too severe an approximation to provide accurate convergence to lattice energies. Because real-space methods can be made to scale linearly with system size, SP and GSF are attractive options for large Monte Carlo and molecular dynamics simulations, respectively.