Ewald summation and reaction field methods for potentials with atomic charges, dipoles, and polarizabilities

The Ewald summation technique and the reaction field method have been generalized to potentials with atomic charges, dipole moments, and anisotropic polarizabilities. These are two common methods to treat long-range interactions in molecular simulations. Expressions for the potential energy, the electrostatic potential, the electrostatic field, the electrostatic field gradient, the force, and the virial are given, allowing for the calculation of long-range contributions to these properties within the Ewald summation or reaction field methods. We have compared numerical results using the Ewald summation under vacuum conditions with those from direct summations for a number of simple systems and found a complete agreement within the numerical precision with the exception of trivial shifts of the potential. The expressions given will facilitate the use of polarizable models in molecular simulations and hence improving our understanding of condensed matter.