Potential Mexican hat

Figure 6. Mexican-hat potential energy surface of AuCk, after Ref 14.

A quantitative treatment of the Jahn-Teller effect is more challenging (46). A major issue is that many theoretical models explicitly or implicitly assume the Bom—Oppenheimer approximation which, for octahedral Cu(II) systems in the vibronic coupling regime, cannot be correct (46,51). Hitchman and co-workers solved the vibronic Hamiltonian in order to model the temperature dependence of the molecular structure and the attendant spectroscopic properties, notably EPR spectra (52). Others, including us, take a more simphstic approach (53,54) but, in either case, a similar Mexican hat potential energy description of the principal features of the Jahn-Teller effect in homoleptic Cu(II) complexes emerges (Fig. 13). 

Fig. 13. Top Schematic representation of the two components of the Jahn-Teller-active vibrational mode for the E e Jahn-Teller coupling problem for octahedral d9 Cu(II) complexes. Bottom Resulting first-order Mexican hat potential energy surface for showing the Jahn-Teller radius, p, and the first-order Jahn-Teller stabilization energy, Ejt.

Figure 10. (a) The Qe and Qe components of the g vibrational mode, (b) the Mexican hat potential energy surface for copper(II) in an octahedral ligand field, and (c) a cross-section through the warped... 

The prototype of the IT surface is the celebrated Mexican hat potential, which describes the effect of the twofold-degenerate cubic or trigonal E state. A typical example is the Eg ground state of octahedral Cu + complexes, with (t2g) eg9 configuration. The JT-active mode in this case is restricted to an eg mode, corresponding to the symmetrized square. 

Fig. 6.1 The Mexican hat potential-energy surface of the fi x e linear JT problem. The nuclear displacement coordinates are the tetragonal elongation, Qe, and the orthorhombic in-plane distortion,...

Fig. 2. Vibronic E e JT spectra for two different values of Eyp, Eq. (6). The electronic transition is from a nondegenerate, JT-undistorted state to the JT-distorted final state as indicated schematically in the lowest panel of the figure. Here the Mexican hat potential energy surfaces are represented roughly by the double cone, with the upper-cone vibrational levels indicated by the three straight lines (for Ejt = 400ta).

In graphic representation, a Mexican-hat potential with warping is obtained (Fig. 5). 

Figure 25. Illustrative view of the scalar Higgs potential (Eq. 24), the Mexican hat, added to the vacuum to explain the origin of the nonzero mass of bosons. At each point along the bottom of the trough the symmetry is spontaneously broken and the Higgs field has a fixed nonzero value, the same in all these points, reahzing the continues degeneracy. The Higgs potential coincides with the Mexican-hat potential of the ITE problem E e in Fig. 2, except for the conical intersection ai.cp =cp =(i in the latter.

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