Calculation of polarization

The polarization of a crystal is its dipole moment pc divided by its volume Vc . This quantity is only defined to within a constant because its value will depend on surface structure. However, a change in polarization is given uniquely by DP5Dpc /Vc for large crystals. Martin has shown that one cannot unambiguously compute a change in polarization from changes in charge density that occur within a unit cell volume of the crystal, because the result, in general, will depend on the particular unit cell volume selected. This posed a problem for band-structure methods that went unsolved until King-Smith and Vanderbilt cast the change in polarization in terms of a current density, which is ultimately determined by integrations of quantities over the Brillouin zone and a Berry phase. While this was an important advance for band-structure methods, unfortunately it also has been used ~incorrectly! to imply that more direct approaches based on overlapping atomiclike densities are fundamentally flawed. Examples of overlapping-atom models vary in their levels of sophistication, from methods that calculate atomiclike densities from a density-functional approach, to early models of lattice dynamics based on this principle. Following Resta, we call such models Clasius-Mossetti ~CM! type models, and use the term ‘‘extreme’’ CM-type models to identify systems whose atomiclike charges are so localized that they are nonoverlapping. On the other hand, Souza et al. state that the CM approximation itself implies nonoverlapping charge distributions. We believe the definition of CM-type models offered in Sec. II is probably closer to the original idea leading to the CM equation. According to Scaife the concept of isolated charges was introduced by Faraday to ensure the insulating property of the medium. In other words, the charges in a CM-type model are isolated from each other in the sense that they would be in an insulator—they do not move from site to site. The argument that CM-type models are fundamentally flawed is simple. It goes as follows: ~1! Martin has shown that polarization defined in terms of the charge density in a unit cell is ambiguous. ~2! This is true for any model in which the charge density is nonzero everywhere in the unit cell. ~In other words, the argument would not apply to extreme CM-type models.! ~3! Thus any realistic CM-type