Pseudopotentials in the Plane Wave Basis

With the methods described in the last section we are able to construct pseudopotentials for states $l = s, p, d, f$ by using reference configurations that are either the ground state of the atom or of an ion, or excited states. In principle higher angular momentum states could also be generated but there physical significance is questionable. In a solid or molecular environment there will be wavefunction components of all angular momentum character at each atom. The general form of a pseudopotential is

$$V_{\rm pp}({\bf r}, {\bf r}\prime)= \sum_{l=0}^{\infty} \sum_{m=-l}^l V^l(r) P^{lm}(\omega) \enspace ,$$ (174)

where $P^{lm}(\omega)$ is a projector on angular momentum functions. A good approximation is to use

$$V^l(r) = V^c(r) \enspace \enspace \enspace \enspace l > l_{\rm max} \enspace .$$ (175)

With this approximation one can rewrite

$$V_{\rm pp}({\bf r}, {\bf r}\prime) = \sum_{L}^{\infty} V^c(r) P^{lm}(\omega) + \sum_{L}^{\infty} \left[ V^l(r) - V^c(r) \right] P^{lm}(\omega)$$

$$= V^c(r) \sum_{L}^{\infty} P^{lm}(\omega) + \sum_{L}^{\infty} \delta V^l(r) P^{lm}(\omega)$$

$$= V^c(r) + \sum_{L}^{l_{\rm max}} \delta V^l(r) P^{lm}(\omega) \enspace ,$$ (176)

where the combined index $L=\{l,m\}$ has been used. The pseudopotential is now separated into two parts; the local or core pseudopotential $V^c(r)$ and the non-local pseudopotentials $\delta V^l(r) P^{lm}(\omega)$ . The pseudopotentials of this type are also called semilocal, as they are local in the radial coordinate and the nonlocality is restricted to the angular part.

The contribution of the local pseudopotential to the total energy in a Kohn-Sham calculation is of the form

$$E_{\rm local} = \int V^c({\bf r}) n({\bf r}) d{\bf r} \enspace .$$ (177)

It can easily be calculated together with the other local potentials. The non-local part needs special consideration as the operator in the plane wave basis has no simple structure in real or reciprocal space. There are two approximations that can be used to calculate this contribution to the energy. One is based on numerical integration and the other on a projection on a local basis set.