Theorem Jahn—Teller

Jahn-TeHer effect The Jahn-Teller theorem states that, when any degenerate electronic slate contains a number of electrons such that the degenerate orbitals are not completely filled, the geometry of the species will change so as to produce non-degenerate orbitals. Particularly applied to transition metal compounds where the state is Cu(II)... 

The stoi7 begins with studies of the molecular Jahn-Teller effect in the late 1950s [1-3]. The Jahn-Teller theorems themselves [4,5] are 20 years older and static Jahn-Teller distortions of elecbonically degenerate species were well known and understood. Geomebic phase is, however, a dynamic phenomenon, associated with nuclear motions in the vicinity of a so-called conical intersection between potential energy surfaces. 

V. Longuet-Higgins Loops and the Jahn-Teller Theorem... 

V. LONGUET-HIGGINS LOOPS AND THE JAHN-TELLER THEOREM... 

There is no analytic proof of the Jahn-Teller theorem. It was shown to be valid by considering all possible point groups one by one. The theorem is traditionally treated within perturbation theory The Hamiltonian is divided into three parts... 

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. 

Types-VI and -VII structures can be formed from the symmetric D f, structure by the vibration as shown in Figure 29, in accord with the Jahn-Teller theorem. 

Figure 31 shows the proposed Longuet-Higgins loop for the cyclopentadienyl cation. It uses the type-VI Ai anchors, with the type-VII B structures as transition states between them. This situation is completely analogous to that of the radical (Fig. 23). Since the loop is phase inverting, a conical intersection should be located at its center—as required by the Jahn-Teller theorem.

Appendix D Degenerate and Near-Degenerate Vibrational Levels Appendix E Adiabatic States in the Vicinity of a Conical Intersection 1, Jahn-Teller Theorem... 

Clearly, Eq. (E.12) shows that to a first approximation the elecbonic energy varies linearly with displacements in p, increasing for one component state while decreasing for the other. Thus, the potential minimum cannot be at p = 0. This is the statement of the Jahn-Teller theorem for a X3 molecule belonging to the D3 , point gioup. 

Jahn-Teller distortions cobalt and copper complexes, 2, 91 hydrates, 2, 308 Jahn-Teller effect, 5, 535 Jahn-Teller theorem, 1, 247 Jarosites... 

This last reference is for those who would like to see the Jahn-Teller theorem at a technical level. 

Geometric phase effect (GPE) (Continued) adiabatic states, conical intersections invariant operators, 735-737 Jahn-Teller theorem, 733-735 antilinear operator properties, 721-723 degenerate/near-degenerate vibration levels, 728-733... 

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