Surface harmonics

ImAi 11 ( ) represents the (21 + l)th derivative of the (n + i)th Laguerre polynomial and P m (cos 3) is Ferrer s associated Legendre function of the first kind, of degree l and order m. Ylm Zm thus constitutes a surface harmonic. The F s are in this form orthogonal and normalised with respect to unity, so that they fulfil the conditions... 

According to the aspherical-atom formalism proposed by Stewart [12], the one-electron density function is represented by an expansion in terms of rigid pseudoatoms, each formed by a core-invariant part and a deformable valence part. Spherical surface harmonics (multipoles) are employed to describe the directional properties of the deformable part. Our model consisted of two monopole (three for the sulfur atom), three dipole, five quadrupole, and seven octopole functions for each non-H atom. The generalised scattering factors (GSF) for the monopoles of these species were computed from the Hartree-Fockatomic functions tabulated by Clementi [14]. 

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... 

Vi = Y so that Ytm is the value of the solid harmonic at points on the surface of the unit sphere defined by the coordinates 8 and (p, and hence Y is called a surface harmonic of degree l. Surface harmonics are orthogonal on the surface of the unit sphere and not at r = 0, as commonly assumed in the definition of atomic orbitals. 

Not only is hybridization an artificial simulation without scientific foundation, but even the assumed "orbital shapes" that it relies upon, are gross distortions of actual electron density distributions. The density plot shown above, like all textbook caricatures of atomic orbitals, is a misrepresentation of the spherical surface harmonics that describe normal excitation modes of atomic charge distributions. These functions are defined in the surface of the charge-density function, as in Fig. 2.13, and not at r = 0, as shown in Figure 2.16. 

There is no quantum-mechanical evidence for the localization of electron pairs between atomic nuclei, and atomic orbitals, in so far as they correspond to spherical surface harmonics, have their nodal curves in the surface of the density sphere. Sets of real hybrid orbitals are physically undefined. To understand intramolecular interactions as a quantum phenomenon it is necessary to approach the problem with the minimum of assumptions and to state all essential assumptions clearly and precisely at the outset. 

An unbiased simulation may use a truncated basis set that represents the lowest complex surface harmonics of the atomic valence shell on a Born-Oppenheimer framework with the correct relative atomic masses. For small molecules, of less than about fifteen atoms, the nuclear framework could perhaps even be generated computationally without assumption. The required criterion is the optimal quenching of angular momentum vectors. The derivation of molecular structure by the angular-momentum criterion will be demonstrated qualitatively for some small molecules. 

Here, the following assumptions are made the radial function Ri r) is the same for all basis functions of the same ul quantum number , and its dependence on a shell quantum number n is of no consequence. The coefficients a/ describe the contribution of s, p, d,. .. character to the hybrid, and the bim govern the shape and orientation of that contribution. Sim are the real surface harmonics, defined in terms of the spherical harmonics (Y m). 

DFT = Density Functional Theory DSD = Diamond-Square-Diamond HOMO = Highest Occupied Molecular Orbital IR = Irreducible Representation LCAO = Linear Combination of Atomic Orbitals Ph = Phenyl PSEPT = Polyhedral Skeletal Electron Pair Theory SCF = Self-Consistent-Field SDDS = Square-Diamond, Diamond-Square TSH = Tensor Surface Harmonic. 

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. ... 

Transition metal clusters also have Axy and atomic orbitals, which are classified as 5-type in TSH theory. To represent the transformation properties of these orbitals, we use second derivatives of the spherical harmonics, that is, tensor spherical harmonics - hence the name of the theory. As for the vector surface harmonics, there are again both odd and even 5 cluster orbitals, denoted by L and L, respectively. Usually, both sets are completely filled in transition metal clusters, and we will not consider their properties in any detail in this review. However, the cases of partial occupation are important and have been described in previous articles. ... 

Table 3.12 The modulating coefficients, reduced to their simplest ratios, for the linear combinations over the uj[here and vj [here from equation 3.21 to form the group orbitals of the F3 triangle displayed in the 3rd and 4th columns of Figure 3.6 using the vector surface harmonics of Table 3.11.

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

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