Spherical shell charge density distribution

As Fig. 12 shows, the inner shell electrons of the alkaline ions behave classically like a polarizable spherical charge-density distribution. Therefore it seemed promising to apply a "frozen-core approximation in this case 194>. In this formalism all those orbitals which are not assumed to undergo larger changes in shape are not involved in the variational procedure. The orthogonality requirement is... 

Additionally, from Fig. 29 one sees that, if, as proposed by Frost 42), a spherical gaussian function is a fair representation of the distribution of charge within an electride ion, there should he, as found by Slater 97>, a very good correlation, and in many cases practically an equality, between the atomic radii. . . and the calculated radius of maximum radial charge density in the outermost shell of the atom". 

We now discuss the analysis of the x-ray intensities. The atoms of the C6o molecule are placed at the vertices of a truncated icosahedron. - The x-ray structure factor is given by the Fourier transform of the electronic charge density this can be factored into an atomic carbon form factor times the Fourier transform of a thin shell of radius R modulated by the angular distribution of the atoms. For a molecule with icosahedral symmetry, the leading terms in a spherical-harmonic expansion of the charge density are Koo(fl) (the spherically symmetric contribution) and KfimCn), where ft denotes polar and azimuthal coordinates. The corresponding terms in the molecular form factor are proportional to SS ° (q)ac jo(qR)ss n(qR)/qR and... 

This work W,ie(r) is path-independent for the symmetrical density systems noted previously since the V x = 0 for these cases. It is important to note, however, that the corresponding Fermi-Coulomb hole charge distribution Pxc(r r ) which gives rise to the field need not possess the same symmetry for arbitrary electron position. For example, in either closed shell atoms or open-shell atoms in the central-field approximation for which the density is spherically symmetric, the Fermi-Coulomb hole is not, the only exception being when the electron is at the nucleus. 

There is a continuing interest in exploring possible relationships between the shell structures of atoms and their electronic density distributions [31-39]. In this respect, considerable attention has focused upon the radial density function, D(r) = 4nr p(r), which goes through a series of maxima and minima with increasing radial distance from the nucleus [6,31-36,40], [p(r) is the electronic density function since atomic charge distributions are spherically symmetric... 

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... 

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