Finite nucleus potential

As can be seen from the table, the effect on the total energy is notable when switching from the singular potential of a point nucleus to a finite-nucleus potential. The finite-nucleus potentials, however, not differ very much. Most important is the effect on relative energies. Here we may note that the effect... 

The electron—nucleus potential is not quite trivial. For larger atoms the radial functions that are large at the nucleus are affected by the finite charge distribution of the nucleus. It is sufficient to use the potential for a uniform charge distribution of radius R... 

It is well known that for heavy atoms the effect of the finite nucleus charge distribution has to be taken into account (among other effects) in order to describe the electronic structure of the system correctly (see e.g. (36,37)). As a preliminary step in the search for the effect of the finite nuclei on the properties of molecules the potential energy curve of the Th 73+ has been calculated for point-like and finite nuclei models (Table 5). For finite nuclei the Fermi charge distribution with the standard value of the skin thickness parameter was adopted (t = 2.30 fm) (38,39). 

For the nucleus-electron interaction the Coulomb potential, either for a point nucleus or a finite nucleus, is used directly. The relativistic contributions to the interaction operator are obtained approximately, using the Pauli approximation. They are... 

For finite nucleus models with a well-defined nuclear size parameter R, beyond which the nuclear charge density is exactly( ) zero and the nuclear potential is exactly( ) given as —Z/r (r > R), the energy shifts can be obtained directly from the matching condition for the logarithmic derivatives L (r) = / r)/P [r) of the radial functions in the inner... 

A remark should be made here with respect to the generation and adjustment of the widely used effective core potentials (ECP, or pseudopotentials) [85] in standard non-relativistic quantum chemical calculations for atoms and molecules. The ECP, which is an effective one-electron operator, allows one to avoid the explicit treatment of the atomic cores (valence-only calculations) and, more important in the present context, to include easily the major scalar relativistic effects in a formally non-relativistic approach. In general, the parameters entering the expression for the ECP are adjusted to data obtained from numerical atomic reference calculations. For heavy and superheavy elements, these reference calculations should be performed not with the PNC, but with a finite nucleus model instead [86]. The reader is referred to e.g. [87-89], where the two-parameter Fermi-type model was used in the adjustment of energy-conserving pseudopotentials. 

The study of the electronic structure of diatomic species, which can nowadays be done most accurately with two-dimensional numerical finite difference techniques, both in the non-relativistic [90,91] and the relativistic framework [92-94], is still almost completely restricted to point-like representations of the atomic nuclei. An extension to allow the use of finite nucleus models (Gauss-type and Fermi-type model) in Hartree-Fock calculations has been made only very recently [95]. This extension faces the problem that different coordinate systems must be combined, the spherical one used to describe the charge density distribution p r) and the electrostatic potential V(r) of each of the two nuclei, and the prolate ellipsoidal one used to solve the electronic structure problem. 

Figure 1. ROPM exchange-only potentials for neutral Hg Selfconsistent Coulomb (C), Coulomb-Breit (C-l-B) and fully transverse (C-t-T) results in comparison with nonrelativistic limit (NR). At r = 10 Bohr one finds P = (3jr /j) /(ffic) =2.9 (for a finite nucleus).

We have already seen above that the choice of a point-like atomic nucleus limits the Dirac theory to atoms with a nuclear charge number Z < c, i.e., Zmax 137. A nonsingular electron-nucleus potential energy operator allows us to overcome this limit if an atomic nucleus of finite size is used. In relativistic electronic structure calculations on atoms — and thus also for calculations on molecules — it turned out that the effect of different finite-nucleus models on the total energy is comparable but distinct from the energy of a point-like nucleus (compare also section 9.8.4). 

We approach the effect of finite nuclear charge models from a formal perspective and introduce a general electron-nucleus potential energy Vnuc, which may be expanded in terms of a Taylor series around the origin. 

It can easily be seen that in contrast to the point-nucleus case, there exist no singularities in the electron-electron interaction potentials in the case of a finite nucleus. Hence, an analytical expression for the electron-electron interaction potentials at the origin can be determined for a finite nucleus, and for a point nucleus as far as shells with k, = 1 are concerned. 

Because of the implicit use of the first derivative of the coefficient functions, the Simpson-type discretization method may be problematic if piecewise-defined potential terms are present (for some finite-nucleus models such as the homogeneous charge distribution of Eq. (6.151)). For such special cases, it would be more advantageous to use a scheme which employs a diagonal matrix representation of the effective potential (cf. Ref. [492] for details). 

In section 6.9 we already introduced finite-size models of the atomic nucleus and analyzed their effect on the eigenstates of the Dirac hydrogen atom. This analysis has been extended in the previous sections to the many-electron case. It turned out that neither the electron-electron interaction potential functions nor the inhomogeneities affect the short-range behavior of the shell functions already obtained for the one-electron case. Table 9.5 now provides the total electronic energies calculated for the hydrogen atom and some neutral many-electron atoms obtained for different nuclear potentials provided by Visscher and Dyall [439], who also provided a list of recommended finite-nucleus model parameters recommended for use in calculations in order to make computed results comparable. 

For a point nucleus, vq = Z, v, = 0, i > 0. The form of the expansion for a finite nucleus can be derived by considering the potential due to a distributed charge, but in general we have vq = V2 = 0 for finite nuclei. Inserting these expansions into the radial Dirac equation gives... 

The exponents y and q are obtained by equating the lowest terms in these expressions. Here also the point nucleus and the finite nucleus cases must be considered separately because they have different expansions of the potential. 

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