Finite nucleus model

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]. 

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. 

The point charge model is sufficiently accurate if one is interested in valence properties of atoms and molecules, however, more realistic finite nucleus models may be used instead. In recent years a Gaussian nuclear charge distribution (Visser et al. 1987),... 

A practical advantage of the finite-nucleus model is that extremely high exponents of the one-particle basis functions are avoided. Since for quantities of chemical interest it is not very important which nuclear model is actually used, the Gaussian charge distribution is often applied, being the most convenient choice. 

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). 

K — Z /c in Hartree atomic units). Consequently, the standard Coulomb model for the electron-nucleus attraction can only be employed for atoms with Z < c (see dashed line in Figure 6.5) we may study any atom theoretically if we employ a finite-nucleus model. Figure 6.5 presents the resulting ground-state energies. 

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 this last section, we shall present some examples of results of numerical atomic structure calculations to demonstrate properties of radial functions and the magnitude of specific effects like the choice of the finite nucleus model or the inclusion of the Breit operator. It should be emphasized that the reliability of numerical calculations is solely governed by the affordable length of the Cl expansion of the many-electron wave function since the numerical solution techniques allow us to determine spinors with almost arbitrary accuracy. Expansions with many tens of thousands of CSFs can be routinely handled (with the basis-set techniques of chapter 10, expansions of billions of CSFs are feasible via subspace iteration techniques [372]). 

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. 

M. Abe, T. Suzuki, Y. Fujii, M. Hada. An ab initio study based on a finite nucleus model for isotope fractionation in the U(111)-U(IV) exchange reaction system. /. Chem. Phys., 128 (2008) 144309. 

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