Orthogonalized plane waves

This method is due to Herring, 1940. The idea is to build valence states using plane waves that are orthogonal to the core states. the cores states are treated as known, generally taken from tight-binding calculations using atomic orbitals.

The OPW state is constructed by orthogonalizing the wave-function with respect to the core states:

$$\vert\phi^{OPW}_{\bf k+K}\rangle = \vert{\bf K+k}\rangle - \sum_c \langle \phi_c\vert{\bf k+K}\rangle \vert\phi_c\rangle.$$ (217)

where the sum runs over all core states with Bloch vector ${\bf k}$.

The orthogonalized plane waves satisfy a Scrödinger equation similar to (216), but with the modified potential:

$$V_{OPW} =V+\sum_c(\epsilon - \epsilon_c)\vert\phi_c\rangle\langle \phi_c\vert$$ (218)