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
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
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.