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The non-local Fock exchange energy, (using orbitals in real space) can be written as
being the set of one-electron
Bloch states of the system, and
set of (possibly fractional) occupational numbers.
The sums over and run over all -points chosen to sample the
Brillouin zone (BZ), whereas the sums over and run over all bands at these
The corresponding non-local Fock potential is given by
is the cell periodic part of the Bloch state,
, at -point, , with band index .
Using the decomposition of the Bloch states,
, in plane waves,
Equ. (6.58) can be rewritten as
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. 
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