Pseudospinors shape-consistent

All the above restoration schemes are called nonvariational as compared to the variational one-center restoration (VOCR, see below) procedure proposed in [79, 80]. Proper behavior of the molecular orbitals (four-component spinors) in atomic cores of molecules can be restored in the scope of a variational procedure if the molecular pseudoorbitals (two-component pseudospinors) match correctly the original orbitals (large components of bispinors) in the valence region after the molecular RECP calculation. As is demonstrated in [69, 44], this condition is rather correct when the shape-consistent RECP is involved to the molecular calculation with explicitly... 

The LECP RECPs are defined using shape-consistent, nodeless two-component pseudospinors [22] extracted from numerical two-component DF atomic spinors [23]. These pseudospinors are used to define RECPs that are expressed as radially local one-electron operators in the context of atomic yy-coupling,... 

This effect presents some serious problems for the development of pseudopotentials. A pseudopotential that depends critically on the shape of the pseudospinor and for which the results are sensitive to the valence occupation is of no value. The problem was overcome (Christiansen et al. 1979) by the definition of the so-called shape-consistent pseudospinors and the corresponding pseudopotentials. 

The shape-consistent pseudospinor is defined to be equal to the Hartree-Fock spinor outside a certain radius rc, and inside this radius it falls smoothly to zero. 

Alternatively, the shape-consistent pseudospinor can be written in terms of the Hartree-Fock spinor and a function that is confined inside the chosen radius rc,... 

The introduction of shape-consistent pseudospinors solved the problems in the pseudopotential that were caused by the use of the Philips-Kleinman pseudospinors. However, the other characteristics of pseudospinors that were discussed in the previous section still apply to shape-consistent pseudospinors. The inclusion of virtual spinors in the expansion of the core tail does not alter the conclusions drawn all but the lowest pseudospinor mix, and the eigenvalue spectrum is compressed. 

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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