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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

$\displaystyle \Delta {\bf P}=\Delta {\bf P}_{\text{ion}}+ \Delta {\bf P}_{e}$ (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

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

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


next up previous
Next: Bibliography Up: Modern Polarization Theory Previous: Computational Aspects.
Georg Kresse
2001-03-23