Non-local pseudopotential

The non-local pseudopotential energy is given by

$$E_{\mathrm{ps,NL}} = \int {\mathrm d}{\bf r}~{\mathrm d}{\bf... ...\bf r}) = 2 K^{\alpha \beta} V_{{\mathrm {NL}},\beta \alpha} .$$ (7.18)

The matrix elements of the non-local pseudopotential in the representation of the support functions $V_{{\mathrm {NL}},\alpha \beta}$ are calculated by summing over all ions whose cores overlap the support regions of $\phi_{\alpha}$ and $\phi_{\beta}$, and using the method described in section 5.6.2 to calculate the spherical-wave basis function matrix elements ${\cal V}_{\alpha , n \ell m ;\beta , n' \ell' m'}$ analytically. The result is therefore

$$V_{{\mathrm {NL}},\alpha \beta} = \sum_{n \ell m, n' \ell' m... ...pha , n \ell m ;\beta , n' \ell' m'} c^{n' \ell' m'}_{(\beta)}$$ (7.19)

which is of exactly the same form as the kinetic energy, so that in practice the basis function matrix elements for the kinetic energy and non-local pseudopotential are summed and the two contributions to the energy combined.