Spherical nuclear charge density distributions

Spherical nuclear charge density distributions In the following we restrict ourselves to the case of spherical charge density distributions. In consequence, this restriction also gives spherical potentials, so that Eqs. (4) and (16) reduce to 

The normalization condition for the charge density distribution now reduces to a radial integral, 

The solution to this problem is given by the Poisson integral, 

The integral representation in Eq. (32) shows that V r) is, in general, a continuous function, even when p(r) is not continuous. As a consequence, the electron-nucleus potential V (r) itself will be continuous and continuously differentiable, in general. In addition, we understand immediately 

A radial expectation value of the normalized charge density distribution function for an arbitrary function of the radius, f r), is obtainable from the general formula 

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This section contains detailed information for a few spherical nuclear charge density distribution models. The models included here are either frequently used in electronic structure calculations or are of importance due to their use for representing nuclear charge density distributions as obtained from experiment. For further details on these models, and for a detailed discussion of a large number of other models for nuclear charge density distributions, see [41]. The following general symbols will be used in this section ... 

Similar and additional other geometrical quantities for nuclear charge density distributions were defined by Myers [20], see also [21] and the comprehensive discussion in [22] which extends this subject to the case of non-spherical shapes. The most complete use of expectation values () (including the extension to arbitrary real powers of r) is made by the moment function M(p),... 

We can now summarize our discussion on nuclear structure as follows A stationary state of the atomic nucleus can be represented, in general, by a real-valued non-negative charge density distribution p r) (a scalar function of coordinates), and by a real-valued current density distribution j r) (a vector function of coordinates). The former can be expanded into a series with standard spherical harmonics i i(r) [ the unit vector r = r/r is equivalent to the angles Q = 9,(f)) ],... 

Therefore, one usually employs perturbation theory as discussed throughout this chapter and sticks to the point-charge model for atomic nuclei (or to some simplified spherically symmetric model density distribution) in quantum chemistry. The tiny effects of multipole moments of the nuclear charge density are then not included in the variational procedure for the determination of the electronic wave function. 

In some molecular crystals which contain only light elements bonding effects have been demonstrated using both x-ray and neutron diffraction. If there is any displacement of electrons relative to the nuclei the internuclear distances in the molecule are not necessarily given by the separation of the maxima in the charge density, and the nuclear positions need to be determined by neutron diffraction. If the problems associated with molecular vibrations can be overcome, a comparison of the measured electron distribution with that calculated for free atoms centered on the nuclear positions may reveal effects associated, for example, with bond formation and lone pairs. Figure 3 shows the experimental deviations from spherical symmetry for the charge density... 

In light atoms (for which Zfine-structure constant) relativistic effects are negligible. In such a case, only s electrons (having spherically symmetric distribution) have nonvanishing charge density at the place of the nucleus. Considering also the small size of the nucleus, the electronic density can be taken to be approximately constant over the nuclear volume, and therefore ... 

Since LaH and LuH differ by the 4fshells, a comparison of the two molecules is in order. Pyykko shows the plot of the spherically averaged radial electron densities (fig. 30). As seen from fig. 30, the LuH 6s electron distribution is shifted to the left with a maximum occurring to the left of the corresponding maximum of LaH. This means that the 6s orbital of Lu is contracted since the 4f shells incompletely shield the nucleus but the nuclear charge is increased by 14 for Lu compared to La. This is the effect of the lanthanide contraction. However, the LuH 5d electron distribution is more extended... 

In a free atom, the electron density distribution is spherical, and centred on the nucleus, with the number of electrons equal to the nuclear charge. So diagrams often represent an atom by a point at its nucleus, the atom bearing a net charge of zero. In a homonuclear diatomic molecule, bonding electrons are shared equally between the... 

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