Almost spherical cages

A Quantum Chemistry of Highly Symmetrical Molecules and Free-Space Clusters, Plus Almost Spherical Cages of C and B Atoms... 

Almost spherical cages of B and of C atoms Hartree-Fock studies and beyond... 

By the last two assumptions the theory, strictly speaking, is only applicable to the monatomic gases A, Kr, Xe, to a somewhat lesser extent to the almost spherical molecules CH4, CF4, SFe, and perhaps to nonpolar diatomic molecules. The rotation of even slightly nonspherical molecules like Q2 and N2 will not be free in the entire cavity when such a molecule comes close to the wall of its cage it will have to orient itself parallel to this wall. Furthermore, some of the cavities are somewhat oblate (cf. Section I.B), and thus the rotation of relatively large, oblong molecules may be seriously... 

Almost all mesostructures reported far have been governed by the geometrical packing of surfactants. The mesopores are near-cylindrical channels or near-spherical cages. Here is an exception. KSW-2[156] has a mesostructure of rectangular arrangements of square or lozenge-shaped 1-D channels. KSW-2 was synthesized by mild acid treatment (pH value... 

Amovilli et al. [24] essentially replaced the use of TF statistical theory in March [1] with HF theory for the fullerenes C50, Qq, C70, and Cg4 with almost spherical C cages. At this point, it is relevant to make a brief digression to relate to the lower dimensionality example of planar ring clusters. The work of Amovilli and March [36] is, essentially, the two-dimensional analogue of the TF self-consistent field treatment of March [1],... 

Attention is then shifted to almost spherical C and B cages. As shown for C cages C50, Cgo, C70, and Cg4 by Amovilli et al. [24], is proportional to 4n, where n is the number of C atoms. Our arguments here suggest that the force constant k is either independent of or very insensitive to the value of n. For B cages, as Amovilli and March [4] demonstrated, R = 0.471- /nA, having the same n dependence as for the four C cages discussed above. 

Abstract. The 1/Z expansion will first be used to discuss the scaling properties of the ground-state energy of heavy (non-relativistic) neutral atoms with atomic number Z. The question will be addressed as to what order in Z electron correlation first enters the expansion. The density functional theory (DFT) invoked above will be utilized then to treat, but now inevitably more approximately, the correlation energy in a variety of molecules. Finally, recent studies at Hartree-Fock level on almost spherical B and C cages will be reviewed. For buckminsterfullerene, the role of electron correlation will then be assessed using the Hubbard Hamiltonian, as in the study of Flocke et al. 

In the third area, having discussed above results with explicit account of electron correlation, recent work on almost spherical B cages at Hartree-Fock level will be summarized, following the study of Amovilli and March [8] on B2k stnd with k going from 15 to 27, some contact being made... 

Following this discussion of correlation energy in polyatomic molecules, we shall return below to calculations at Hartree-Fock level, with particular reference to almost spherical B and C cages. 

This latter assumption was relaxed in the investigation of almost spherical B cages by Amovilli and March [8], and the Thomas-Fermi approximation for the TT-electrons is replaced by all-electron Hartree-Fock calculations. 

Figure 4- Equilibrium radius R of almost spherical B ca es from all-electron Hartree-Fock calculations versus where n is the number of B atoms in a cage (Redrawn from ref. [8])...

In very recent work, Amovilli et al [9] have carried out Hartree-Fock calculations on four almost-spherical C cages. Only for one of these cages, namely buckminster fullerene, is the structure of the lowest isomer known (European football). For the other three cages, namely Cso, C70 and Cg4, the lowest isomers are still not known. For Csa therefore, the classification of isomers by Mandopoulos and Fowler [36] has been utilized. The one chosen (albeit with inevitable arbitrariness) is that of the highest symmetry [Fig. 1(a) of Amovilli et al [9]]. For C70 similar symmetry considerations led them to a definite nuclear structure [Fig. 1(b) of Ref. [9]]. The framework adopted for C50 [see also Schmalz et al [37]] is given in Fig. 1(c) of Ref. [9]. 

The study of B and C cages, having almost spherical symmetry, has then been reported [8], [9]. These investigations were at Hartree-Fock level, and aspects of electron correlation in the particular case of buckminsterfullerene have been pointed out, which emerge from the study of Flocke et al [10] using the Hubbard Hamiltonian. For this special case of Ceo, Hartree-Fock studies are largely vindicated, but the same should not be assumed necessarily to apply to C50, C70 and Cs4 also treated at Hartree-Fock level. 

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