Jahn-Teller degeneracy

In general, at least three anchors are required as the basis for the loop, since the motion around a point requires two independent coordinates. However, symmetry sometimes requires a greater number of anchors. A well-known case is the Jahn-Teller degeneracy of perfect pentagons, heptagons, and so on, which will be covered in Section V. Another special case arises when the electronic wave function of one of the anchors is an out-of-phase combination of two spin-paired structures. One of the vibrational modes of the stable molecule in this anchor serves as the out-of-phase coordinate, and the loop is constructed of only two anchors (see Fig. 12). 

Figure 23. A Loiigiiet-Higgms loop around the Jahn-Teller degeneracy of CPDR at D h symmetry, preserve and Qinv n aje the phase-inverting and phase-preserving coordinates that define the loop.

A final comment concerns the presence of other conical intersections near the central one. They are enclosed by loops consisting of two Ai (type-VI) and one A[ (type-V) species, as depicted in Figure 32. This is a phase-inverting ip loop. Thus, the main Jahn-Teller degeneracy is sunounded by five further degeneracies, arranged in a symmetrical fashion. 

In our final case study we shall focus on the photoinduced mer to fac isomerization reaction recently observed by Harkins and Peters (109). We have investigated this system in detail (110), and it provides a nice example of a general conical intersection in an inorganic photochemical problem, i.e., one that is not imposed by symmetry via a Jahn-Teller degeneracy. 

The UPS of the dimagnetic tetra-amides of Ti, Zr, and Hf can also be interpreted according to Fig. 29. However, there are two less valence electrons for the Group IVB amides than for Mo(NR2)4 hence the b, metal orbital is unoccupied. All the Group IVB amide spectra exhibit splitting of the 2if ionic state (Table XXXIV), presumably owing to the relief of Jahn-Teller degeneracy. 

Judd, B.R. (1977) Jahn-Teller degeneracies of Thorson and Moffitt. 

Aspects of the Jahn-Teller symmetry argument will be relevant in later sections. Suppose that the electronic states aie n-fold degenerate, with symmetry at some symmetiical nuclear configuration Qq. The fundamental question concerns the symmetry of the nuclear coordinates that can split the degeneracy linearly in Q — Qo, in other words those that appeal linearly in Taylor series for the matrix elements A H B). Since the bras (/1 and kets B) both transform as and H are totally symmetric, it would appear at first sight that the Jahn-Teller active modes must have symmetry Fg = F x F. There... 

This is the central Jahn-Teller [4,5] result. Three important riders should be noted. First, Fg = 0 for spin-degenerate systems, because F, x F = Fo. This is a particular example of the fact that Kramer s degeneracies, aiising from spin alone can only be broken by magnetic fields, in the presence of which H and T no longer commute. Second, a detailed study of the molecular point groups reveals that all degenerate nonlinear polyatomics, except those with Kramer s... 

In molecular physics, the topological aspect has met its analogue in the Jahn-Teller effect [47,157] and, indeed, in any situation where a degeneracy of electronic states is encountered. The phase change was discussed from various viewpoints in [144,158-161] and [163]. 

Non-adiabatic coupling is also termed vibronic coupling as the resulting breakdown of the adiabatic picture is due to coupling between the nuclear and electi onic motion. A well-known special case of vibronic coupling is the Jahn-Teller effect [14,164-168], in which a symmetrical molecule in a doubly degenerate electronic state will spontaneously distort so as to break the symmetry and remove the degeneracy. 

Accepting the Longuet-Higgins rule as the basis for the search of conical intersection, it is necessary to look for the appropriate loop. The -type degeneracy of a Jahn-Teller system is removed by a nonsymmetric motion. 

As shown in Figure 27, an in-phase combination of type-V structures leads to another A] symmetry structures (type-VI), which is expected to be stabilized by allyl cation-type resonance. However, calculation shows that the two shuctures are isoenergetic. The electronic wave function preserves its phase when tr ansported through a complete loop around the degeneracy shown in Figure 25, so that no conical intersection (or an even number of conical intersections) should be enclosed in it. This is obviously in contrast with the Jahn-Teller theorem, that predicts splitting into A and states. 

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