Angular-momentum-dependent pseudopotentials

With plane-wave representation of wavefuntions the method for calculation of total energy has been described by Wendel and Martin (1978, 1979). Explicit expressions covering also nonlocal (angular momentum dependent) pseudopotentials and forces were given by Ihm, Zunger and Cohen (1979)- The method is described by Martin (this volume). 

For these norm-conserving pseudopotentials, a different potential needs to be applied on each orbital depending on its angular momentum. These pseudopotentials then have a semi-local form ... 

Several simple schemes have been formulated to extract ab initio ionic pseudopotentials from atomic calculations. The basic procedure is to generate a potential by inversion of the Kohn-Sham equation. Angular-momentum-dependent screened atomic pseudopotentials, V, are first constructed with the constraints that (1) the valence eigenvalues from the all-electron calculation... 

There are complicating issues in defmmg pseudopotentials, e.g. the pseudopotential in equation Al.3.78 is state dependent, orbitally dependent and the energy and spatial separations between valence and core electrons are sometimes not transparent. These are not insunnoimtable issues. The state dependence is usually weak and can be ignored. The orbital dependence requires different potentials for different angular momentum components. This can be incorporated via non-local operators. The distinction between valence and core states can be addressed by incorporating the core level in question as part of the valence shell. For... 

Pseudopotentials are usually derived from all-electron atomic calculations. The valence electron pseudopotential is then required to reproduce the behaviour and properties of the valence electrons in the full calculation. For example, the energy levels with the pseudopotential should be the same as for the all-electron calculation. In addition, the pseudopotential will often depend upon the orbital angular momentum of the wavefimction (i.e. for s, p, d, etc. orbitals) and it will be desired that the total valence electron density within the core radius equals that in the all-electron situation. Such pseudopotentials are... 

We will describe the calculation of the total energy and its relation to the band structure in the framework of Density Functional Theory (DFT in the following) see chapter 2. The reason is that this formulation has proven the most successful compromise between accuracy and efficiency for total-energy calculations in a very wide range of solids. In the following we will also adopt the pseudopotential method for describing the ionic cores, which, as we discussed in chapter 2, allows for an efficient treatment of the valence electrons only. Furthermore, we will assume that the ionic pseudopotentials are the same for all electronic states in the atom, an approximation known as the local pseudopotential this will help keep the discussion simple. In realistic calculations the pseudopotential typically depends on the angular momentum of the atomic state it represents, which is known as a non-local pseudopotential . 

---