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Kleinman-Bylander Pseudopotentials

The most general form of a non-local pseudopotential is

  eqnarray656

where tex2html_wrap_inline5706 are spherical harmonics and tex2html_wrap_inline5708 is the tex2html_wrap_inline5710 angular momentum component of the pseudopotential acting on the wavefunction. If there are tex2html_wrap_inline5712 plane waves in the expansion of the wavefunction at each k-point and there are tex2html_wrap_inline5716 k-points then the evaluation of tex2html_wrap_inline5720 will require tex2html_wrap_inline5722 projectors of the above form to be calculated for each angular momentum component l.

This can be seen as follows[44]: the crystal potential tex2html_wrap_inline5726 is obtained by placing a pseudopotential for each species at each site in the lattice. The structure factor incorporates the crystal symmetry, hence

  eqnarray672

where the summation index is over ionic species and the structure factor for each species is tex2html_wrap_inline5728 . The total ion-electron energy is then

  eqnarray688

It can be seen that this gives an inseparable double sum over tex2html_wrap_inline5536 and tex2html_wrap_inline5732 . Evaluation of the ion-electron contribution to the total energy therefore scales as the square of the number of plane waves used in the expansion. This is computationally inefficient and will severly limit the size of any calculation.

A more efficient way of evaluating this contribution is due to Kleinman and Bylander [45]. By expressing the pseudopotential in a different form they were able to split the double sum into a product of two single sums. The Kleinman-Bylander pseudopotential has the form

 

where tex2html_wrap_inline5734 is an arbitrary local potential, tex2html_wrap_inline5736 are the pseudoatom wavefunctions and tex2html_wrap_inline5738 is defined by

  eqnarray714

where tex2html_wrap_inline5740 is the l angular momentum component of a non-local pseudopotential.

To evaluate the electron-ion interaction using this form of the pseudopotential it is found that if the pseudo wavefunction is expanded in plane waves the double sum over tex2html_wrap_inline5536 and tex2html_wrap_inline5732 becomes separable. This allows the expression to be evaluated in only tex2html_wrap_inline5748 calculations. As a result the pseudopotential part of the calculation now scales linearly with the size of basis set.


next up previous
Next: The Car-Parrinello Method Up: Ion-Electron Interactions Previous: Norm Conserving Pseudopotentials

Stewart Clark
Thu Oct 31 19:32:00 GMT 1996