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Introduction: HartreeFock
The nonlocal Fock exchange energy, (using orbitals in real space) can be written as

(6.57) 
with
being the set of oneelectron
Bloch states of the system, and
the corresponding
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
points.
The corresponding nonlocal Fock potential is given by

(6.58) 
where
is the cell periodic part of the Bloch state,
, at point, , with band index .
Using the decomposition of the Bloch states,
, in plane waves,

(6.59) 
Equ. (6.58) can be rewritten as

(6.60) 
where

(6.61) 
is the representation of the Fock potential in reciprocal space.
Note: For a comprehensive description of the implementation of the
Fockexchange operator within the PAW formalism see Ref. [92]
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