Introduction: Hartree-Fock

The non-local Fock exchange energy, $E_x$ (using orbitals in real space) can be written as

$$E_{\mathrm{x}}= -\frac{e^2}{2}\sum_{{\bf k}n,{\bf q}m} f_{{\bf k}... ...hi_{{\bf k}n}({\bf r}')\phi_{{\bf q}m}({\bf r})} {\vert {\bf r}-{\bf r}' \vert}$$ (6.59)

with $\{\phi_{{\bf k}n}({\bf r})\}$ being the set of one-electron Bloch states of the system, and $\{f_{{\bf k}n}\}$ the corresponding set of (possibly fractional) occupational numbers. The sums over $\bf k$ and $\bf q$ run over all $k$-points chosen to sample the Brillouin zone (BZ), whereas the sums over $m$ and $n$ run over all bands at these $k$-points.

The corresponding non-local Fock potential is given by

$$V_x\left({\bf r},{\bf r}'\right)= -\frac{e^2}{2}\sum_{{\bf q}m}f_... ...)u_{{\bf q}m}({\bf r})} {\vert {\bf r}-{\bf r}' \vert} e^{i{\bf q}\cdot{\bf r}}$$ (6.60)

where $u_{{\bf q}m}({\bf r})$ is the cell periodic part of the Bloch state, $\phi_{{\bf q}n}({\bf r})$, at $k$-point, $\bf q$, with band index $m$.

Using the decomposition of the Bloch states, $\phi_{{\bf q}m}$, in plane waves,

$$\phi_{m{\bf q}}({\bf r})= \frac{1}{\sqrt{\Omega}} \sum_{\bf G}C_{m{\bf q}}({\bf G})e^{i({\bf q}+{\bf G}) \cdot {\bf r}}$$ (6.61)

Equ. (6.60) can be rewritten as

$$V_x\left({\bf r},{\bf r}'\right)= \sum_{\bf k}\sum_{{\bf G}{\bf G... ...} V_{\bf k}\left( {\bf G},{\bf G}'\right) e^{-i({\bf k}+{\bf G}')\cdot{\bf r'}}$$ (6.62)

where

$$V_{\bf k}\left( {\bf G},{\bf G}'\right)= \langle {\bf k}+{\bf G} ... ... G}'') C_{m{\bf q}}({\bf G}-{\bf G}'')} {\vert{\bf k}-{\bf q}+{\bf G}''\vert^2}$$ (6.63)

is the representation of the Fock potential in reciprocal space.

Note: For a comprehensive description of the implementation of the Fock-exchange operator within the PAW formalism see Ref. [92]