Nuclear charge distribution Gaussian

For spherically symmetric nuclear charge distribution (Gaussian, Fermi, or point nucleus), the electric field at a point r outside the nucleus can be evaluated from Gauss law as... 

With this notation, it is easy to derive a formula for the electron-nuclear interaction due to nucleus A , assuming a Gaussian nuclear charge distribution given by (110). The formula... 

O. Visser, P. J. C. Aerts, D. Hegarty, W. C. Nieuwpoort, The use of Gaussian nuclear charge distributions for the calculation of relativistic electronic wavefunctions using basis set expansions, Chem. Phys. Lett. 134 (1987) 34-38. 

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),... 

Figure 6.6 Comparison of ground-state energies E[glZ scaled by I7 obtained tor hydrogen-iike atoms from Schrodinger quantum mechanics (horizontal line on top at -0.5 hartree), from Dirac theory with a Couiomb potential from a point-like nucleus (dashed line) and from Dirac theory with a finite nuclear charge distribution of Gaussian form (thin black line). The highest energy of the positronic continuum states, -2meC, appears as a thick black line, which is bent because of the l/Z scaling.

Figure 6.7 This figure comprises the radial functions shown already in Figure 6.1, in Hartree atomic units, pius those obtained for Gaussian nuclear charge distribution for the one-electron atom with Z = 170. Oniy in the case of a finite nucleus, quantum mechanical states of atoms with nuciear charges Z > c are defined (c 137.037 in Hartree atomic units).

The effect of the Breit interaction on the wave function can be conveniently studied by comparing radial moments (r) of shells calculated with Dirac-Coulomb and Dirac-Coulomb-Breit Hamiltonians. The effect is not very large as can be seen for Li- and Be-like ions in Figure 9.4. From the plot we note that the effect of the Breit interaction on the radial functions is small, but increases linearly with the nuclear charge number Z. Moreover, the four different models to describe the positive nuclear charge distribution (point-like, exponential, Gaussian shaped and Fermi) can hardly be distinguished. 

At least inside the nucleus, P and Q are essentially Gaussian in shape. This means that in a method using a Gaussian basis set a nuclear charge distribution with a finite radius is preferred to a point nucleus the basis then has the right behavior at the origin, and the demands on the basis are smaller because of the cutoff in the potential (Visser et al. 1987, Ishikawa et al. 1985). 

The evaluation of the spin-orbit integrals now reduces to the same form as the one-electron spin-orbit integrals with a finite Gaussian nuclear charge distribution. 

The Sum-of-Gaussians expansion was introduced by Sick [67] as a model-independent way to describe the nuclear charge density distribution, and is an expansion in terms of symmetrized Gauss-type functions (see also Fig. 6) ... 

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. 

Many model potentials (pnuc f) have been used [131] but two have become most important in electronic structure calculations. These are the homogeneous and the Gaussian charge distributions. The homogeneously or uniformly charged sphere is a simple model for the finite size of the nucleus. It is piecewise defined, because the positive charge distribution is confined in a sphere of radius R. The total nuclear charge -f-Ze is uniformly distributed over the nuclear volume 4 rR /3,... 

The Gaussian charge density distribution is continuously defined, meaning that the nuclear charge is not exactly zero even at a large distance from the center of the nucleus. The Gaussian charge density distribution. 

---