Non-linear Core Correction

The success of pseudopotentials in density functional calculations relies on two assumptions. The transferability of the core electrons to different environments and the linearization of the exchange and correlation energy. The second assumption is only valid if the frozen core electrons and the valence state do not overlap. However, if there is significant overlap between core and valence densities, the linearization will lead to reduced transferability and systematic errors. The most straightforward remedy is to include "semi-core states" in addition to the valence shell, i.e. one more inner shell (which is from a chemical viewpoint an inert ``core level'') is treated explicitely. This approach, however, leads to quite hard pseudopotentials which call for high plane wave cutoffs. Alternatively, it was proposed to treat the non-linear parts of the exchange and correlation energy $E_{\rm xc}$ explicitely [103]. This idea does not lead to an increase of the cutoff but ameliorates the above-mentioned problems quite a bit. To achieve this, $E_{\rm xc}$ is calculated not from the valence density $n({\bf R})$ alone, but from a modified density

$$\tilde n({\bf R}) = n({\bf R}) + \tilde n_{\rm core}({\bf R}) \enspace ,$$ (199)

where $\tilde n_{core}({\bf R})$ denotes a density that is equal to the core density of the atomic reference state in the region of overlap with the valence density

$$\tilde n_{\rm core}(r) = n_{\rm core}(r) r > r_0 \enspace ;$$ (200)

with the vanishing valence density inside $r_0$ . Close to the nuclei a model density is chosen in order to reduce the cutoff for the plane wave expansion. Finally, the two densities and their derivatives are matched at $r_0$ . This procedure leads to a modified total energy, where $E_{\rm xc}$ is replace by

$$E_{\rm xc} = E_{\rm xc}(n + \tilde n_{\rm core}) \enspace ,$$ (201)

and the corresponding potential is

$$V_{\rm xc} = V_{\rm xc}(n + \tilde n_{\rm core}) \enspace .$$ (202)

The sum of all modified core densities

$$\tilde n_{core}({\bf G}) = \sum_{I} \tilde n^{I}_{core}({\bf G}) S_{I}({\bf G})$$ (203)

depends on the nuclear positions, leading to a new contribution to the forces

$${\partial E_{\rm xc} \over \partial {\bf R}_{I,s}} = - \Omega \su... ...ar_{\rm xc}({\bf G}) \tilde n^{I}_{\rm core}({\bf G}) S_{I}({\bf G}) \enspace .$$ (204)

The method of the non-linear core correction dramatically improves results on systems with alkali and transition metal atoms. For practical applications, one should keep in mind that the non-linear core correction should only be applied together with pseudopotentials that were generated using the same energy expression.