Real space electrostatics for multipoles. I. Development of methods

Abstract

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.

Real space electrostatics for multipoles. II. Comparisons with the Ewald sum

“…The mathematical details of the SP, GSF, and TSF methods have been discussed in detail in Paper I of this series. 47 Here we briefly review the new methods and describe their essential features.…”

“…47 The 1/r Coulombic kernel for the interactions between point charges can be replaced by the complementary error function erfc(αr)/r to accelerate convergence, where α is a damping parameter with units of inverse distance. Altering the value of α is equivalent to changing the width of Gaussian charge distributions that replace each point charge, as Coulomb integrals with Gaussian charge distributions produce complementary error functions when truncated at a finite distance.…”

“…These methods extend the DSF and Wolf methods originally developed for charge-charge interactions and generalize them for higher order multipoles. The detailed mathematical development of these methods has been presented in Paper I of this series, 47 while this work covers the testing of energies, forces, torques, and energy conservation properties of the methods in realistic simulation environments. In all cases, the methods are compared with the reference method, a full multipolar Ewald treatment.…”

Long-range electrostatic corrections in multipolar/polarizable QM/MM simulations

“…While the higher-order moments are neglected in most of this discussion, these long-range methods can readily be extended to include the full multipole–multipole and polarizable interactions [28, 29]. …”

“…The LREC approach uses a combination of a smoothing function and the minimum image convention to scale the electrostatic interactions such that the potential and forces smoothly decrease to zero at a finite cutoff radius. While energy and force shifting approaches also smoothly truncate the potential at a finite radius [16, 28, 29, 35, 36, 44, 49], LREC has an exceptionally simple implementation in the QM/MM Hamiltonian. In the LREC approach, the external MM monopoles are scaled based on their distance from the QM region, which does not require modifications to the matrix elements or post-SCF corrections.…”