Phillips-Kleinman approach

Therefore, if the pseudo wave function is normalized, the true wave function is not. In the Phillips-Kleinman approach, the density is given by ... 

In the Reporters pseudopotential calculations31 a different approach is taken by replacing the Phillips-Kleinman term with an operator,... 

One not so obvious problem with the shape-consistent REP formalism (or any nodeless pseudoorbital approach) is that some molecular properties are determined primarily by the electron density in the core region (some molecular moments, Breit corrections, etc.) and cannot be computed directly from the valence-only wave function. For Phillips-Kleinman (21) types of wave functions, Daasch et al. (52) have shown that the core electron density can be approximated quite accurately by adding in the atomic core orbitals and then Schmidt orthogonalizing the valence orbitals to the core. This new set of orbitals (core plus orthogonalized valence) is a reasonable approximation to the all-electron set and can be used to compute the desired properties. This will not work for the shape-consistent case because / from Eq. (18) cannot be accurately described in terms of the core orbitals alone. On the other hand, it is clear from that equation that the corelike portion of the valence orbitals could be reintroduced by adding in fy (53),... 

This method is probably as accurate as some other simple pseudopotential approaches. However, there appear to be some difficulties in improving it to the standard of some of the recent pseudopotential calculations. Attempts to use larger than minimal basis sets required the inclusion of a Phillips-Kleinman term in addition to the orthogonality procedure in order to prevent collapse of the valence orbitals into the core space. Thus in calculations on AlaQ, Vincait had to include not only the A1 3s and Cl 3j and 3p shells but also the A12p and Cl 2s and 2p shells explicitly in the valence-electron basis in order to obtain good results. Consequently this calculation was not substantially less expensive in computing time than an equivalent all-electron calculation. 

Explicit inclusion of relativistic effects in valence-only calculations has been by far less frequently attempted. Datta, Ewig and van Wazer [135] used a Phillips-Kleinman PP in a study of PbO, whereas Ishikawa and Malli [136] tested PPs of semilocal form in four-component atomic DHF finite difference calculations. This work was extended by Dolg [137] to four-component molecular DHF calculations with a subsequent correlation treatment. In addition a complicated form of Vcv based on the Foldy-Wouthuysen transformation [138] was derived by Pyper [139] and applied in atomic calculations [140]. For all these approaches the computational effort is significantly higher than for the implicit treatment of relativity, and the gain of computational accuracy is not obvious at all. 

The meaning of the concept of norm-conservation can be understood from the Phillips-Kleinman type of approach, discussed so far. From the relation between the wave function and the pseudo wave function, one obviously obtains ... 

Early work based on a statistical treatment of the core electrons was published by Hellmaim (1935) and Gombas (1935) for molecular and solid-state physics, respectively. Quantum-mechanical justifications in the framework of the Hartree and Hartree-Fock theories were given by Fenyes (1943) and Szepfalusy (1955, 1956), respectively. The first derivation of the pseudopotential approach within the Hartree-Fock formalism which came to general attention is due to Phillips and Kleinman (1959) and was later generalized by Hazi and Rice (1968). The accuracy and limitations of the pseudopotential approach... 

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