Nuclear charge distribution point

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. 

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... 

Considering point nuclei this energy shift was calculated by Beier and Soff [56]. Schneider et al. presented an improved calculation for extended nuclear charge distributions [57], which results in -0.72 eV for the lsj/2-state of uranium. Currently research is in progress to overcome the Uehling approximation in this contribution. 

The global effect of an applied external field on a molecule involves distortions both in the electronic charge distribution and in the nuclear charge distribution the latter leads to the so-called vibrational contribution to the (hyper)polarizabilities. As said above, the analysis of the vibrational components reveals the presence of the distinct components, the curvar ture related to the effect of the field vibrational motion and including the zero point vibrational correction (ZPV) (see above), and the nuclear relaxation (nr) originates from the shift of the equilibrium geometry induced by the field. 

Wheeler 2 has pointed out that further information about the nucleus can be obtained from more precise measurements of the X-ray spectrum especially of the fine structure that should be present. The things that one can study include nuclear quadrupole moments, the non-uniformity of the nuclear charge distribution, and nuclear polarizability. 

Let s start with the case of imiformly (isotropic) nuclear charge distribution in volume. Yet, we have to consider the situations r R. The last situation is immediately recognized from the Coulomb law on the basis that all charge sources behind the action horizon may be treated as point like sources ... 

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.

At this point it is useful to summarize some of the features of the solutions for point and finite nuclear charge distributions. 

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). 

If a nuclear charge distribution with a finite radius is to be used, the question of the functional form of the distribution must be raised. The point nuclear model was simple now we have to consider some form of the nuclear charge distribution that bears some relation to experimentally determined distributions. The Coulomb potential at a point r from a charge distribution p (r) is... 

For a molecule with a continuous electron charge distribution and nuclear point charges, the expression becomes ... 

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