Harmonic theory

More advanced mathematical aspects of the graph-theoretical models for aromaticity are given in other references [36, 48, 49]. Some alternative methods, beyond the scope of this chapter, for the study of aromaticity in deltahedral molecules include tensor surface harmonic theory [51-53] and the topological solutions of non-linear field theory related to the Skyrmions of nuclear physics [54]. 

The bonding in gold cluster molecules has been interpreted using free electron models based on Stone s tensor surface harmonic theory [48, 49]. High similarity has... 

The remainder of this article is largely concerned with describing how some of the above observations can be rationalized using Stone s Tensor Surface Harmonic theory, and with the further imphcations of this model for dynamical processes such as cluster rearrangements. The number of example systems and electron count rationalizations will be kept relatively small in favor of explaining the theoretical foundations that underhe the method. Tables of examples and more detailed analyses of the various cases may be found elsewhere. ... 

Spherical-Shell Technique, Tensor Surface Harmonic Theory... 

III. Harmonic and Quasi-harmonic Theories of Lattice Dynamics. 149... 

Depending on the character of the molecular motions, one can distinguish several physical situations. In most cases, the molecules are trapped in relatively deep potential wells. Then, they perform small translational and orientational oscillations around well-defined equilibrium positions and orientations. Such motions are reasonably well described by the harmonic approximation. The collective vibrational excitations of the crystal, which are considered as a set of harmonic oscillators, are called phonons. Those phonons that represent pure angular oscillations, or libra-tions, are called librons. For some properties it turns out to be necessary to look at the effects of anharmonicities. Anharmonic corrections to the harmonic model can be made by perturbation theory or by the self-consistent phonon method. These methods, which are summarized in Section III under the name quasi-harmonic theories, can be considered to be the standard tools in lattice dynamics calculations, in addition to the harmonic model. They are only applicable in the case of fairly small amplitude motions. Only the simple harmonic approximation is widely used the calculation of anharmonic corrections is often hard in practice. For detailed descriptions of these methods, we refer the reader to the books and reviews by Maradudin et al. (1968, 1971, 1974), Cochran and Cowley (1967), Barron and Klein (1974), Birman (1974), Wallace (1972), and Cali-fano et al. (1981). 

Lattice dynamics calculations on the plastic /3-nitrogen phase are relatively scarce because, obviously, the standard (quasi-) harmonic theory cannot be applied to this phase. Classical Monte Carlo calculations have been made by Gibbons and Klein (1974) and Mandell (1974) on a face-centered cubic (a-nitrogen) lattice of 108 N2 molecules, while Mandell has also studied a 32-molecule system and a system of 96 N2 molecules on a hexagonal close-packed (/3-nitrogen) lattice. Gibbons and Klein used 12-6... 

For the reactions of medium-sized molecules we have the following Lindemann-Hinshelwood theory, RRK theory. Slater s harmonic theory, RRKM theory, phase-space theory, absolute reaction rate theory, quasi-equilibrium theory, and several others. All of those are grouped under the umbrella of "transition state theory" (Robinson Holbrook, 1972 Forst, 1973). Among these theories, some are regarded as "inaccurate" or "outdated." But several rivals remain as viable alternatives on which to base a theoretical study of a reaction system, at least as far as Joiunal referees are concerned. 

Application of Tensor Surface Harmonic Theory to 3- and 4-Connected 68... 

Several years before the formulation of the Tensor Surface Harmonic Theory, King presented a graph theoretical treatment of bonding in borane clusters563. It is instructive to compare this with Stone s methodology. 

A complementary approach is the Tensor Surface Harmonic Theory [19], based on the linear combination of atomic orbitals (LCAO) model, which explicitly incorporates the atomic positions. A set of atomic cores on the surface of a sphere are considered, and a basis set of s atomic orbitals used. If only these s orbitals are used, then the results are identical to the spherical jellium model. The three most stable orbitals are respectively 1,3 and 5 fold degenerate, leading to closed shells at 2, 8 and 18 electrons. 

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