Descent of symmetry

For example, the family of perovskite minerals and high-temperature ceramic superconductors exhibits this descent of symmetry, from the cubic "ideal" perovskite structure (space group Pm3m, the real mineral perovskite is orthorhombic, space group Pnma, with a fourfold larger unit cell than the ideal cubic one) to orthorhombic structures for the highest-critical... 

For isolated molecules a variety of approaches have proved useful in the interpretation of vibrational spectra. Firstly, a species may approximate to a symmetry higher than its actual. In such cases a correlation with-descent in symmetry from — the higher symmetry usually simplifies the interpretation of its spectra. Secondly, local group vibrations, essentially uncoupled from the vibration of other equivalent or near-equivalent groups, may occur. Thirdly, chemically distinct groups may couple... 

A combination of descent in symmetry (Ctv —> C3v —> Cs) and orbital overlap considerations have been utilized in order to assign the six lowest energy bands in the (L-/V )MoO(dithiolene) series. The inherently low oscillator strengths of the Mo ligand-field bands in oxo-molybdenum complexes (s < 100 M em ), coupled with the presence of low-energy LMCT excitations, result in ligand-field transitions that are difficult to observe in (L-/V3)MoO(dithiolene) compounds. As a result, LMCT bands are anticipated to dominate the absorption spectra of these complexes. 

A set of a-type atomic orbitals transform amongst themselves under point group operations like scalar quantities. Hence, to consttuct descent-in-symmetry-type LCAO cluster orbitals, we simply follow the same reasoning as for the linear and cychc polyenes, above, and define (unnormalized) cr-type cluster orbitals as... 

In general, this display shows how the set of harmonic polynomials for given f-value splits on Descent in Symmetry from the spherical group to the particular point group in question. The calculation is limited to -values less than or equal to 60. 

Table 3.5 Classification of the general spherical harmonics up to angular momentum level 4 by descent in symmetry, into their irreducible components for the molecular point groups Cj, Cs, C2, Cy, C4, C5, Ce, C7 and Cg.

Table 3.9 Classification of the kubic spherical harmonics np to angnlar momentnm level 4 by descent in symmetry, into their irredncible components for the molecniar point gronps Oh, O, Tj, Tjj and T.

The application of the method of step-by-step descent in symmetry to the D h parent symmetry group for all the possible degenerate electronic states may be seen in Scheme 4 [21]. Among its immediate subgroups, only the D2h one is JT stable. 

The method of step-by-step descent in symmetry indicates two descent paths to JT stable immediate subgroups of D2d (see Scheme 5d) [22]... 

Possible symmetry groups originating in JT symmetry descent of parent group with triple electron degeneracy ( Ti or electronic state for B4+ cluster) in (36) formally agree with the epikernel principle but it does not hold for the... 

Planar C2v structure with Ai electronic state (C model) cannot be explained by JT symmetry descent from parent group and must be explained by JT symmetry descent of parent D h symmetry group (see Fig. 4) with double electron degeneracy by the symmetry descent path (Scheme 6b)... 

The method of step-by-step symmetry descent does not explain the mechanisms that are responsible for JT distortions. Some opponents argue that its predictions are far too wide on account of selectivity ( all is possible ). On the other hand, this treatment is based exclusively on group theory and does not account for any approximations used in the recent solutions of Schrddinger equation. Chemical thermodynamics does not solve the problems of chemical kinetics but nobody demands to do it as well. Thus we cannot demand this theory to solve also the mechanistic problems despite the epikernel principle solves it. The problem of too wide predictions can be reduced by minimizing the numbers and lengths of symmetry descent paths (see the applications in this study). 

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