Modern Polarization Theory

Calculating the change in dipole moment per unit cell under PBC's, is a nontrivial task. In general one cannot define it as the first moment of the induced change in charge density $\delta \left( {\bf r} \right)$, through

$$\Delta {\bf P}= \frac{1}{\Omega_{0}} \int_{\Omega_{0}} {\bf r} \delta \left({\bf r} \right) d^{3}r$$ (A.1)

without introducing a dependency on the shape of $\Omega_{0}$, the chosen unit cell (see for instance Ref. [1]).

Recently King-Smith and Vanderbilt [2], building on the work of Resta [3], showed that the electronic contribution to the difference in polarization $\Delta {\bf P}_{e}$, due to a finite adiabatic change in the Hamiltonian of a system, can be identified as a geometric quantum phase or Berry phase of the valence wave functions. We will briefly summarize the essential results (for a review of geometric quantum phases in polarization theory see Refs. [4] and [5]).