Ewald summation of electrostatic multipole interactions up to the quadrupolar level

Ewald summation expressions are derived for the energy, interatomic forces, pressure tensor, electric fields, and electric field gradients in a periodic computer simulation system of molecules with charges, induced dipoles, and quadrupoles. The full expressions including quadrupolar multipole interactions are given; those involving dipoles are included for completeness. The expressions apply not only to molecules with permanent multipoles, but also to systems of polarizable ions where the multipoles are induced by the interionic interactions. The expressions which allow for this generalization consider the ions to be spherically polarizable, but include the consequences of the coupling between dipolar and quadrupolar induction. The phonon dispersion curves of MgO and the structural lattice constants of $Al2O3$ in its corundum phase are discussed as specific examples. In simulations with a dipole- and quadrupole-polarizable interaction model, a full Ewald summation of quadrupolar interactions is shown to have a significant effect.