Kleinman and Bylander Form of the Pseudo-potential

The semi-local form of the pseudo-potentials described above leads to a complicated evaluation of their action on a wave-fimction 

Unfortunately, the last integral must be calculated for each r. In a plane-wave expansion, this involves the product of an A pw x ATpw matrix with the vector 

The semi-local potential can be rewritten in a form that separates long and short range components. The long range component is local, and corresponds to the Coulomb tail. Choosing an arbitrary angular momentum component (usually the most repulsive one) and defining 

Kleinman and Bylander [35] suggested that the non-local part of (6.74) are written as a separable potential, thus transforming the semi-local potential into a truly non-local pseudo-potential. If r)Yim r) denotes 

The KB separable form has, however, some disadvantages, leading sometimes to solutions with nodal surfaces that are lower in energy than solutions with no nodes [75,76]. These (ghost) states are an artifact of the KB procedure. To eliminate them one can use a different component of the pseudopotential as the local part of the KB form or choose a different set of core radii for the pseudo-potential generation. As a rule of thumb, the local component of the KB form should be the most repulsive pseudo-potential component. For example, for the Cu potential of Fig. 6.4, the choice of / = 2 as the local component leads to a ghost state, but choosing instead I = 0 remedies the problem. 

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