Calculating the Born effective charge tensor.

When calculating the electronic contribution to column $\beta$ of the Born effective charge tensor $Z^{*}_{\kappa,\alpha \beta}$, we let $\lambda$ parameterize the change in the Kohn-Sham potential $V^{\left(\lambda\right)}_{KS}$ due to the displacement $u$ of atom $\kappa$ in direction $\beta$.

Because $N_{k_{\perp}}J$, the number of $k$-points in the Brillouin zone sampling used to evaluate Eq. (7) tends to be quite large (typically $N_{k_{\perp}}J=16\times10$), and because this equation will have to be evaluated six times to establish ${\bf P}_{e}^{\left(\lambda_{1}\right)}$ and ${\bf P}_{e}^{\left(\lambda_{2}\right)}$, it is computationally advantageous to generate $V^{\left(\lambda_{1}\right)}_{KS}$ and $V^{\left(\lambda_{2}\right)}_{KS}$, the self-consistent Kohn-Sham potentials for respectively the undistorted and the distorted unit cell, using some adequate but less extensive Monkhorst-Pack sampling of the Brillouin zone. These potentials are kept fixed in the subsequent calculations of the wave functions $\psi^{\left(\lambda_{1}\right)}_{n{\bf k}_{j}}$ and $\psi^{\left(\lambda_{2}\right)}_{n{\bf k}_{j}}$, for all $k$-points ${\bf k}_{j}={\bf k}_{\perp}+ j{\bf G}_{\parallel}/J,\;\;j\in[0,J-1], \;\;{\bf k}_{\perp}\in{\bf k}_{\perp}\text{-mesh}$.

The total difference in polarization between the distorted ( $\lambda_{2}$) and undistorted ( $\lambda_{1}$) structures is

$$\Delta {\bf P}=\Delta {\bf P}_{\text{ion}}+ \Delta {\bf P}_{e}$$ (A.13)

with the electronic contribution to the difference in polarization $\Delta {\bf P}_{e}$ given by Eq. (4), and the ionic or core contribution by

$$\Delta {\bf P}_{\text{ion}}={\vert e\vert Z_{\kappa}u \over \Omega_{0}}$$ (A.14)

where in the context of a pseudopotential calculation, $Z_{\kappa}$ is the valence atomic number of pseudoatom $\kappa$.

Once $\Delta{\bf P}$ is known, column $\beta$ of the Born effective charge tensor is found from

$$Z^{*}_{\kappa,\alpha \beta}={\Omega_{0}\over\vert e\vert} {(\Delta{\bf P})_{\alpha}\over u}$$ (A.15)