Spherical shell configurations

In column IV there are listed the G2 values calculated from the parameters proposed by von Stackelberg. As an approximation to the configuration of the chlorine molecules spherical shells of uniform density were placed at the centers of the eight cavities formed by the oxygen atoms. It is seen that the calculated values are not compatible with... 

As a parametrization scheme this model is a typical intermediate between the crude spherical parametrization of the B and C parameters and the complete (and therefore impractical) set of the tetragonal repulsion integrals [31]. A similar strategy was used before by Koide and Pryce [32] in their treatment of octahedral d5-complexes they showed that quasi-degenerate terms of the half-filled d-shell configuration could be splitted by introducing a covalency difference between the t2g and eg orbitals. 

If we restrict the system to the collinear eZe configuration (i.e., a = ti), the corresponding TCM is topologically equivalent to a spherical shell with four holes. 

The obtained configuration is a thick spherical shell (see Fig 24), and the convergence to this geometry is quick. Also dependence on initial configuration must be weak, because the final result does not have strong dependence on gene samples (a) or (b) as explained later. Furthermore, the embedded results are significantly different from the purely random data. 

Table 2.4 shows some examples of the alternative spherical shell description of these electronic configurations. 

De Leeuw et al. show that a result can be found by adding the sums in a sequence of spherical shells. This corresponds to packing together replications of the central cell to form an infinite sphere. The bare potential (12) is then replaced by the effective interaction < >tpbc(12) and the configurational energy is calculated by the usual Ml procedure. De Leeuw et al. show that for their particular summation, <1>xpbc(12) is given by... 

Apparently the detailed configuration of the atomic arrangement in simple metal clusters does not seem to play an important role in the study of their physical properties. The spherical jellium model is very successful in correlating the prominent features of the ionization potential and also the main features of the mass spectra. However, there is also evidence of some features that the spherical assumption is unable to explain. Whenever a top-shell is not completely filled (N 2, 8, 20, 40, 58, 92,...) the electronic density becomes non-spherical, which in turn leads to an ellipsoidal distortion of the ionic background. This Jahn-Teller-type distortion, similar to those observed for molecules and nuclei, leads to a splitting of all spherical shells into spheroidal sub-shells [41]. Ellipsoidal clusters are prevalent for open-shell configurations. 

Our first many-electron example is argon. Argon is a noble gas with the closed shell configuration ls 2s 2p 3s 3p , so its ground-state is spherical. In Fig. 6.5 we plot the electron density for this atom as a function of the distance to the nucleus. The function n r) decays monotonically, with very little structure, and is therefore not a very elucidative quantity to behold. However, if we choose to represent r n(r), we can clearly identify the shell structure of the atom Three maxima, corresponding to the center of the three shells,... 

The shapes used, so far, for PCRVs have varied from a cyhnder bounded by two inverted, non-prestressed, hemispherical domes, torospherical domes to spherical shells. The current trend in PCRV configuration is the use of thick-walled cylinders, the ends of which are closed by fiat slabs known as caps. The boilers and circulators are either housed within the main cavity or within the thickness of the walls the latter is known as a multi-cavity-type vessel. This study is concerned with such vessels also apart from other shapes. 

When the curvature of the earth has to be taken into account it is necessary to drill a hole, to scale, in the spherical shell. The soap film must be prevented from entering this region. In the most general case all the minimum configurations have to be obtained and the absolute minimum determined. 

---