Pseudopotentials

Electrons in matter can be broadly categorised into two types - core electrons, which are strongly localised in the closed inner atomic shells, and valence electrons, which exist outside the core. Unfortunately, a plane-wave basis set is generally not suitable for describing electron wavefunctions since a prohibitively large number would be required to accurately describe the oscillations in the core regions which maintain orthogonality between valence and core electrons. As a result, all-electron plane-wave calculations demand a huge computational expense that is simply not practical. However, by realising that the electronic structure of the core-electrons remains largely unchanged in different chemical environments, and is also of minimal interest generally, the problems relating to the core-electrons can be overcome by use of the pseudopotential approximation [25,26,27]

The pseudopotential approximation replaces the strong ionic potential $v_{\hbox{{ion}}}({\bf r})$ in the core region, by a weaker pseudopotential $v_{\hbox {{ion}}}^{\hbox {{PS}}}({\bf r})$. The corresponding set of pseudo-wavefunctions $\psi^{\hbox{{PS}}}({\bf r})$ and the all-electron wavefunctions $\psi^{\hbox{{AE}}}({\bf r})$ are identical outside a chosen cutoff radius $r_c$ and so exhibit the same scattering properties, but $\psi^{\hbox{{PS}}}({\bf r})$ does not possess the nodal structure that cause the oscillations inside $r_c$, which means they can now be described with a reasonable number of plane-waves. A schematic illustration of the pseudopotential concept is shown in Fig. 1.1.

Figure 1.1: Schematic illustration of the pseudopotential concept. The solid lines show the all-electron wavefunction, $\Psi^{\hbox{{AE}}}({\bf r})$, and ionic potential, $v_{\hbox{{ion}}}^{\hbox{{AE}}}({\bf r})$, while the dashed lines show the corresponding pseudo-wavefunction, $\Psi^{\hbox{{PS}}}({\bf r})$, given by the pseudopotential, $v_{\hbox{{ion}}}^{\hbox{{PS}}}({\bf r})$. All quantities are shown as a function of distance, ${\bf r}$, from the atomic nucleus. The cutoff radius $r_c$ marks the point beyond which the all-electron and pseudo quantities become identical.