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Subsections
A quantity which is frequently required is the overlap matrix
defined by

(7.3) 
The overlap matrix elements between the sphericalwave basis functions can
be calculated analytically (section 5.4),
and are denoted
where

(7.4) 
so that the overlap matrix elements are given by

(7.5) 
recalling that the support functions may be assumed real in the case of
point Brillouin zone sampling.
The kinetic energy of the noninteracting KohnSham system is given by

(7.6) 
in which

(7.7) 
are the matrix elements of the kinetic energy operator in the representation
of the support functions. Since all of the matrix elements between the
sphericalwave basis functions can be calculated analytically,

(7.8) 
where denotes matrix elements of the kinetic energy
operator between sphericalwave basis functions.
The Hartree and exchangecorrelation terms are calculated by
determining the electronic density on a realspace
grid , and Fast Fourier Transforms (FFTs) are used to transform between
real and reciprocalspace^{7.1}to obtain
. The Hartree energy is then given by

(7.9) 
where
is the volume of the supercell and the
(infinite) term is omitted because the system is charge neutral
overall. This term is therefore cancelled by similar terms in the ionion
and electronion interaction energies. The Hartree potential in realspace is
given by

(7.10) 
but is calculated in reciprocalspace as

(7.11) 
and then transformed back into realspace by a FFT.
Having calculated the electron density on the grid points, the exchangecorrelation energy is obtained by summing over those grid points

(7.12) 
in the local density approximation. is the volume of the supercell divided by the number of grid points. The exchangecorrelation potential is
similarly calculated at each grid point as

(7.13) 
In practice, the values of
and
are tabulated for various values of
the electronic density and then interpolated during the calculation.
Like the Hartree potential, the local pseudopotential is also calculated in
reciprocalspace as

(7.14) 
where the summation is over ionic species ,
is the local pseudopotential for an isolated ion of species in
reciprocalspace and
is the structure factor for
species defined by

(7.15) 
where the sum is over all ions of species with positions
.
We note that in general the calculation of the structure factor is
an operation, but since it only has to be calculated once
for each atomic configuration, it is not a limiting factor of the overall
calculation at this stage. Within the quantum chemistry community,
work on generalised multipole expansions and new algorithms
[156,157,158,159,160,161,162]
has led to the development of methods to calculate Coulomb
interaction matrix elements which scale linearly with systemsize.
The local pseudopotential energy can be calculated in reciprocalspace as
where
is the pseudopotential core energy, and
the number of ions of species . The matrix elements
are defined by

(7.17) 
The Hartree potential and local pseudopotential can be summed and then
transformed back
together into realspace and added to the exchangecorrelation potential to
obtain the local part of the KohnSham potential in realspace.
We note that the FFT is not strictly an operation but an
operation
(where is some small number which depends upon the prime factors of the
number of grid points), but in practice (section 9.2)
this scaling is not observed.
The nonlocal pseudopotential energy is given by

(7.18) 
The matrix elements of the nonlocal pseudopotential in the representation of
the support functions
are calculated by
summing over all ions whose cores overlap the support regions of
and , and using the method described in
section 5.6.2 to calculate the sphericalwave basis function
matrix elements
analytically. The result is therefore

(7.19) 
which is of exactly the same form as the kinetic energy, so that in practice
the basis function matrix elements for the kinetic energy and nonlocal
pseudopotential are summed and the two contributions to the energy combined.
Next: 7.2 Energy gradients
Up: 7. Computational implementation
Previous: 7. Computational implementation
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Peter Haynes