Hafner-Schwarz model

Pacchioni has recently carried out calculations on the low-lying states of Sn2 and Pb2. This author gives the impression that he is the first to carry out a comparative ab initio Cl calculation on these systems. We would like to clarify this further. First, his calculation starts with the Hafner-Schwarz model potentials in comparison to our relativistic ab initio potentials derived from numerical Dirac-Fock solutions of the atoms. Pacchioni s calculations ignore spin-orbit interaction. Our calculations include spin-orbit interaction in a relativistic Cl scheme in comparison to the non-relativistic Cl of Pacchioni. Thus, he obtains a Z), approximately twice the experimental value which he corrects by a semi-empirical scheme to arrive at a value close to our calculated value with a relativistic Cl. Our calculations have clearly demonstrated the need to carry out an intermediate-coupling Cl calculation for Pbj as a result of large spin-orbit contamination. Calculations without spin-orbit, such as Pacchioni s, have little relationship to the real Pb2 molecule. 

Both steps, introduction of the pseudopotential and simplification of the Hamiltonian, may formally be carried out in any order, resulting in different expressions for the valence-only Hamiltonian (Hafner and Schwarz 1979, Pyper 1980a,b, 1981, Pyper and Marketos 1981). The derivation of pseudopotentials to be used within the Dirac-Hartree-Fock scheme has also been attempted (Ishikawa and Malli 1981, 1983, Dolg 1995). We note that the derivation of valence-only Hamiltonians used in pseudopotential theory is neither unique nor may be carried out without a multitude of approximations. Therefore, in practice a proper adjustment of free parameters in a suitably chosen model Hamiltonian seems to be much more important than a lengthy theoretical derivation. 

Hafner and Schwarz (1979) proposed quasi-relativistic model potentials expressed as... 

Hafner and Schwarz have proposed a quasi-relativistic model of potentials of the form in equations (44) and (45). 

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