Theory

The electrons in an atom can be divided into two types --- core electrons and valence electrons. The core electrons are tightly bound to the nucleus, while the valence electrons are more extended. A working definition for core electrons is that they are the ones which play no part in the interactions between atoms, while the valence electrons dictate most of the properties of the material. It is common to make the frozen core approximation; the core electrons are constrained not to differ from their free atomic nature when placed in the solid state environment. This reduces the number of electronic degrees of freedom in an all-electron calculation. It is a very good approximation. A different, but physically related, approach is taken in the pseudopotential approximation. Since, in an all-electron calculation, the valence electron wavefunctions must be orthogonal to the core wavefunctions they necessarily have strong oscillations in the region near the nucleus (see the all-electron wavefunction in Figure~\ref{fig:wvfn}). Given that a planewave basis set is to be used to describe the wavefunctions, these strong oscillations are undesirable --- requiring many plane waves for an accurate description. Further, these oscillations are of very little consequence for the electronic structure in the solid, since they occur close to the nucleus. In the pseudopotential approach only the valence electrons are explicitly considered, the effects of the core electrons being integrated within a new ionic potential. The valence wavefunctions need no longer be orthogonal to the core states, and so the orthogonality oscillations disappear; hence far fewer plane waves are required to describe the valence wavefunctions. Numerous schemes to produce optimally soft pseudopotentials have been developed. Common choices are the norm-conserving potentials due to Troullier and Martins1 and Vanderbilt's ultrasoft scheme2.

  1. N. Troullier and José Luís Martins. Efficient pseudopotentials for plane-wave calculations. Phys. Rev. B, 43:1993–2006, Jan 1991. doi:10.1103/PhysRevB.43.1993. ↩

  2. David Vanderbilt. Soft self-consistent pseudopotentials in a generalized eigenvalue formalism. Phys. Rev. B, 41:7892–7895, Apr 1990. doi:10.1103/PhysRevB.41.7892. ↩