\emph{Ab initio} polarization calculations for large systems using a Wannier center method

The ionic (${\bf P}_{\rm ion}$) and electronic (${\bf P}_{e}$) terms in the electric polarization in solids are defined only modulo a quantum that connects different polarization branches and depends on the unit-cell dimensions. As the system cell size increases (especially perpendicular to ${\bf P}$) the quantum shrinks, becoming comparable to the expected polarization values, thus making the identification of the physically relevant branch unclear. To overcome this indeterminacy, a common approach involves multiple calculations of ${\bf P}$ along an adiabatic insulating path, so that $\Delta {\bf P}$ $\ll$ the quantum at every step. We remove the need for these multiple evaluations by using a representation of ${\bf P}_{e}$ in terms of the centers of Wannier functions. In this intuitive picture, the arbitrariness in ${\bf P}_{e}$ is removed by the implicit requirement that the position of the Wannier center move on a continuous path along with its corresponding ion or bond as the system is deformed from the initial to the final state. Our procedure for computing ${\bf P}$ consists of associating each Wannier center with a particular atom or bond in the unit cell, and then obtaining ${\bf P}_{\rm ion}$ and ${\bf P}_{e}$ through simple summations. We compare the results from this method to the Berry-phase technique using multiple evaluations for the case of supercell calculations of $V_{\rm Pb}$-$V_{\rm O}$ divacancies in PbTiO$_3$. The same polarization values are obtained, with differences below 0.75 $\mu$C/cm$^2$ associated with different $k$-point samplings.