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Introduction: Hartree-Fock

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

$\displaystyle E_{\mathrm{x}}= -\frac{e^2}{2}\sum_{{\bf k}n,{\bf q}m} f_{{\bf k}...
...hi_{{\bf k}n}({\bf r}')\phi_{{\bf q}m}({\bf r})} {\vert {\bf r}-{\bf r}' \vert}$ (6.57)

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

$\displaystyle V_x\left({\bf r},{\bf r}'\right)= -\frac{e^2}{2}\sum_{{\bf q}m}f_...
...)u_{{\bf q}m}({\bf r})} {\vert {\bf r}-{\bf r}' \vert} e^{i{\bf q}\cdot{\bf r}}$ (6.58)

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,

$\displaystyle \phi_{m{\bf q}}({\bf r})= \frac{1}{\sqrt{\Omega}} \sum_{\bf G}C_{m{\bf q}}({\bf G})e^{i({\bf q}+{\bf G}) \cdot {\bf r}}$ (6.59)

Equ. (6.58) can be rewritten as

$\displaystyle V_x\left({\bf r},{\bf r}'\right)= \sum_{\bf k}\sum_{{\bf G}{\bf G...
...} V_{\bf k}\left( {\bf G},{\bf G}'\right) e^{-i({\bf k}+{\bf G}')\cdot{\bf r'}}$ (6.60)

where

$\displaystyle V_{\bf k}\left( {\bf G},{\bf G}'\right)= \langle {\bf k}+{\bf G} ...
... G}'') C_{m{\bf q}}({\bf G}-{\bf G}'')} {\vert{\bf k}-{\bf q}+{\bf G}''\vert^2}$ (6.61)

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]


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