Model nucleus

At the restoration stage, a one-center expansion in the spherical harmonics with numerical radial parts is most appropriate both for orbitals (spinors) and for the description of external interactions with respect to the core regions of a considered molecule. In the scope of the discussed two-step methods for the electronic structure calculation of a molecule, finite nucleus models and quantum electrodynamic terms including, in particular, two-electron Breit interaction may be taken into account without problems [67]. 

The results concerning the allowance for the / -source molecular structure (Kaplan et al., 1982,1983) were used in reduction of the new experimental data obtained by the ITEP group. Table I (Boris et al., 1983) shows the most probable values of the neutrino rest mass obtained by data reduction using the models of a bare nucleus, of a tritium atom, and of a real valine molecule. The reduction of the new experimental data within the bare-nucleus model did not lead to a nonzero rest mass of the neutrino. The allowance for the electron structure of valine gives mvc2 = 33 eV.2 Thus, the quantum chemical calculations proved to be a necessary step in the reduction of the data concerning the properties of an elementary particle. 

The order of perturbation at which various levels of excitation first arise is illustrated in Figure 11 for three different reference functions. In Figure 11(a), the Hartree-Fock orbitals are used to form the reference function, in Figure 11(b) the bare-nucleus model is used in zero-order, while in Figure 11(c) Brueckner orbitals are used to construct the reference function. 

Whereas the probability of a mononuclear reaction (i.e. for radioactive transmutation) is given by the decay constant 2, two probabilities are decisive in the compound nucleus model the probability that the projectile x will react with the nuclide A (the first step of reaction (8.5)) and the probability that the nuclide B is produced (the second step of reaction (8.5)). 

Most low-energy nuclear reactions proceed via formation of a compound nucleus (eq. (8.5)). In the compound nucleus model that was proposed in 1936 by Bohr it is assumed that the energy of the incident particle and its binding energy are distributed evenly or nearly evenly to all nucleons of the target nucleus. The excitation energy of the compound nucleus is... 

Gaussian functions are appropriate functions for electronic structure calculations not only because of the widely recognized fact that they lead to molecular integrals which can be evaluated efficiently and accurately but also because such functions do not introduce a cusp into the approximation for the wave function at a physically inappropriate point when off-nuclei functions are employed. Furthermore, Gaussian functions are suitable for the description of wave functions in the vicinity of nuclei once the point nucleus model is abandoned in favour of a more realistic finite nucleus model. 

The eigenvalue problems, defined above for the radial functions through Eqs. (113) to (116), reduce to homogeneous problems in the case of one-electron atoms, since the various terms A (r) and j(r) are zero. Closed-form solutions are well known for the one-electron atom with a point-like nucleus, both in the non-relativistic and the relativistic framework, but do not exist for the large majority of finite nucleus models. A determination of the energy eigenvalue for a bound state i of the one-electron atom with a finite nucleus, is then possible in two ways, either by perturbation... 

This distribution function is obtained straightforwardly from the known analytic expressions for the radial functions. Finally, the energy shift from first-order perturbation theory is obtained, for any finite nucleus model, in terms of a series expansion in X = 2ZR, where i is a model-specific radial nuclear size parameter, as... 

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... 

Kiuc( ) continuous at the matching radius. The inner solution can be calculated, for any such finite nucleus model and for an axbitrary state i, from a short-range series expansion, whereas the outer solution is known as the irregular solution of the Coulomb problem of the hydrogen-like atom with a point-like nucleus, both in the non-relativistic and the relativistic case. The logarithmic derivative for the outer solution, is very sensi-... 

As long as one is interested only in the total energy of the atomic electron system, the change from the simple but unrealistic PNC to a roughly realistic FNC is much more important than finer details due to variation of the finite nucleus model. This can be seen also from a recently published comparative study on numerical Dirac-Hartree-Fock calculations for... 

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. 

Since the introduction of the Gauss-t3q>e finite nucleus model (see Sect. 4.4) into relativistic quantum chemistry by Visser et al. [96], the combination of this model with Gauss-type basis functions has been most widely distributed, see, e.g., the programs developed by Dyall et al. [97], by Viss-cher et al. [98,99] (MOLFDIR), and by Saue et al. [100] (DIRAC). 

Nuclear attraction integrals for the combination of the homogeneous finite nucleus model (see Sect. 4.3) with Gauss-type basis functions were implemented by Ishikawa et al. [101], by Matsuoka et al. [102,103], and by dementi et al. [104-106]. 

None of these implementations can be used to study effects due to variar tion of the finite nucleus model, due to their limitation to a single finite nucleus model. Of course, it is unlikely that such variations lead to significant changes in the chemical behaviour of atoms and molecules, e.g., reaction enthalpies, valence electronic charge density distribution etc. However, the finer details of the electron distribution in the vicinity of heavy atomic nuclei will be more sensitive to the variation of the finite nucleus model, but this is clearly a field in the area of atomic and nuclear physics. 

D. Andrae, M. Reiher, J. Hinze, A comparative study of finite nucleus models for low-l dng states of few-electron high-Z atoms, Chem. Phys. Lett. 320 (2000) 457-468. 

O. Matsuoka, Basis-set expansion method for relativistic atoms in the atomic-nucleus model of a finite sphere of constant electric field, Chem. Phys. Lett. 172 (1990) 175-179. 

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