Ewald Summation - Dipole Term

Dipole Term

The electrostatic energy of a polar crystal (i.e., a crystal with a net dipole in the unit cell) is conditionally convergent, i.e., depends on the order of the summation. For example, if the dipole-dipole interactions of a central unit cell with unit cells located on an ever-increasing cube, the energy converges to a different value than if the interaction energies had been summed spherically. Roughly speaking, this conditional convergence arises because (1) the number of interacting dipoles on a shell of radius grows like ; (2) the strength of a single dipole-dipole interaction falls like ; and (3) the mathematical summation diverges.

This somewhat surprising result can be reconciled with the finite energy of real crystals because such crystals are not infinite, i.e., have a particular boundary. More specifically, the boundary of a polar crystal has an effective surface charge density on its surface where is the surface normal vector and represents the net dipole moment per volume. The interaction energy of the dipole in a central unit cell with that surface charge density can be written

where and are the net dipole moment and volume of the unit cell, is an infinitesimal area on the crystal surface and is the vector from the central unit cell to the infinitesimal area. This formula results from integrating the energy where represents the infinitesimal electric field generated by an infinitesimal surface charge (Coulomb's law)

The negative sign derives from the definition of, which points towards the charge, not away from it.