The most general form of a non-local pseudopotential is
where are spherical harmonics and is the angular momentum component of the pseudopotential acting on the wavefunction. If there are plane waves in the expansion of the wavefunction at each k-point and there are k-points then the evaluation of will require projectors of the above form to be calculated for each angular momentum component l.
This can be seen as follows: the crystal potential is obtained by placing a pseudopotential for each species at each site in the lattice. The structure factor incorporates the crystal symmetry, hence
where the summation index is over ionic species and the structure factor for each species is . The total ion-electron energy is then
It can be seen that this gives an inseparable double sum over and . 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 . 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 is an arbitrary local potential, are the pseudoatom wavefunctions and is defined by
where 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 and becomes separable. This allows the expression to be evaluated in only calculations. As a result the pseudopotential part of the calculation now scales linearly with the size of basis set.