In order to calculate the matrix 
, the overlaps between atomic
orbitals centred on different atomic sites must be calculated. If the
atomic orbitals 
are defined centred on the origin,
where 
is the vector separating the centres of the two
orbitals.
Now Equation 
is in the form of a convolution, therefore
and
The PW eigenstates are also represented in reciprocal space, therefore it will be efficient to calculate the Fourier Transform (FT) of the atomic orbitals and calculate the overlap matrix elements in reciprocal space.
As the calculations are performed assuming periodic boundary
conditions, 
, may be written
where 
is a lattice vector, and 
is a normalised
single atomic orbital centred on the origin.
The FT of is therefore given by
where 
is a reciprocal lattice vector.
Now, 
may be written in terms of a radial function and
a spherical harmonic
where the radial wavefunction is normalised such that
The angular integrations of the FT may be performed analytically such that
where
The kernels 
for s, p and d orbitals and an outline of the
method for their calculation are given in Section .
For each k-point, the FTs of the orbitals are calculated for each of the reciprocal lattice points lying within the sphere defined by the cut off energy of the PW basis set.
It should be noted that Equations () and () may
be written
where 
, which may be calculated in a
time O( 
). Furthermore, 
may also be calculated in the same order of time, hence 
may also be calculated in a time
O( ). The other computationally expensive calculation is
the inversion of which is also an O( )
operation. Thus the entire calculation scales with system size with the
same order as the original PW calculation, although the actual cost is
significantly less as there is no need to iterate to achieve self
consistency.
