Generation of Pseudopotentials

The generation of a pseudopotential involves several choices the number of spinors to be included in the valence space and the method for determining the pseudospinors and pseudopotentials. The partitioning between the core and the valence space determines 

The two principal approaches for generating pseudopotentials are inversion methods and fitting methods. The inversion methods rely on having a radial function without nodes. Then, the SCF equation for the pseudospinor can be inverted to obtain a pseudopotential  

This inversion is only valid if the pseudospinor is nodeless otherwise singularities would arise in the pseudopotential. The inversion is usually done on a radial grid with tabulated potentials and pseudospinors. The resulting pseudopotential is then fitted to 

The sum over p usually includes only the values 0, 1, and 2. This form of the pseudopotential contains a pseudo-angular-momentum term that prevents the penetration of the pseudospinors to the nucleus, which also helps prevent core spinors from mixing 

In some of the early studies, the pseudospinors were obtained by minimizing a function involving the kinetic energy (Kahn et al. 1976). The shape-consistent pseudopotentials of Hay and Wadt (Hay and Wadt 1985, Wadt and Hay 1985) and Christiansen, Ermler, and coworkers (Pacios and Christiansen 1985, Hurley et al. 1986, La John et al. 1987, Ross et al. 1990, 1994, Ermler et al. 1991) are obtained by fitting a polynomial function to the core tail with the requirements that it have no nodes and the minimum number of inflection points and must match the derivatives to the order of the polynomial at the join point. While this procedure guarantees the smoothness of the function, especially after the pseudospinor is expanded in a Gaussian basis set, the choice of the join point must be made with care. If it is too far out, the results can be unsatisfactory, as was found for the 6p elements (Wildman et al. 1997). 

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Within the density functional theory (DFT), several schemes for generation of pseudopotentials were developed. Some of them construct pseudopotentials for pseudoorbitals derived from atomic calculations [29] - [31], while the others make use [32] - [36] of parameterized analytical pseudopotentials. In a specific implementation of the numerical integration for solving the DFT one-electron equations, named Discrete-Variational Method (DVM) [37]- [41], one does not need to fit pseudoorbitals or pseudopotentials by any analytical functions, because the matrix elements of an effective Hamiltonian can be computed directly with either analytical or numerical basis set (or a mixed one). 

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