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.

We report on tests of the shifted potential (SP), gradient shifted force (GSF), and Taylor shifted force (TSF) real-space methods for multipole interactions developed in Paper I of this series, using the multipolar Ewald sum as a reference method. The tests were carried out in a variety of condensed-phase environments designed to test up to quadrupole-quadrupole interactions. Comparisons of the energy differences between configurations, molecular forces, and torques were used to analyze how well the real-space models perform relative to the more computationally expensive Ewald treatment. We have also investigated the energy conservation, structural, and dynamical properties of the new methods in molecular dynamics simulations. The SP method shows excellent agreement with configurational energy differences, forces, and torques, and would be suitable for use in Monte Carlo calculations. Of the two new shifted-force methods, the GSF approach shows the best agreement with Ewald-derived energies, forces, and torques and also exhibits energy conservation properties that make it an excellent choice for efficient computation of electrostatic interactions in molecular dynamics simulations. Both SP and GSF are able to reproduce structural and dynamical properties in the liquid models with excellent fidelity.

In Papers I and II, we developed new shifted potential, gradient shifted force, and Taylor shifted force real-space methods for multipole interactions in condensed phase simulations. Here, we discuss the dielectric properties of fluids that emerge from simulations using these methods. Most electrostatic methods (including the Ewald sum) require correction to the conducting boundary fluctuation formula for the static dielectric constants, and we discuss the derivation of these corrections for the new real space methods. For quadrupolar fluids, the analogous material property is the quadrupolar susceptibility. As in the dipolar case, the fluctuation formula for the quadrupolar susceptibility has corrections that depend on the electrostatic method being utilized. One of the most important effects measured by both the static dielectric and quadrupolar susceptibility is the ability to screen charges embedded in the fluid. We use potentials of mean force between solvated ions to discuss how geometric factors can lead to distance-dependent screening in both quadrupolar and dipolar fluids.