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Computational Methods

  In order to calculate the matrix tex2html_wrap_inline3496 , the overlaps between atomic orbitals centred on different atomic sites must be calculated. If the atomic orbitals tex2html_wrap_inline3498 are defined centred on the origin,


where tex2html_wrap_inline3500 is the vector separating the centres of the two orbitals.

Now Equation gif 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, tex2html_wrap_inline3502 , may be written


where tex2html_wrap_inline3504 is a lattice vector, and tex2html_wrap_inline3506 is a normalised single atomic orbital centred on the origin.

The FT of tex2html_wrap_inline3502 is therefore given by


where tex2html_wrap_inline2531 is a reciprocal lattice vector.

Now, tex2html_wrap_inline3514 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 tex2html_wrap_inline3516 for s, p and d orbitals and an outline of the method for their calculation are given in Section gif.

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 (gif) and (gif) may be written


where tex2html_wrap_inline3518 , which may be calculated in a time O( tex2html_wrap_inline3520 ). Furthermore, tex2html_wrap_inline3522 may also be calculated in the same order of time, hence tex2html_wrap_inline3524 may also be calculated in a time O( tex2html_wrap_inline3520 ). The other computationally expensive calculation is the inversion of tex2html_wrap_inline3496 which is also an O( tex2html_wrap_inline3520 ) 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.

next up previous contents
Next: Orbital FT Kernels Up: Implementation of Population Analysis Previous: Techniques

Matthew Segall
Wed Sep 24 12:24:18 BST 1997