Jahn Teller active mode

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... 

The symmetry argument actually goes beyond the above deterniination of the symmetries of Jahn-Teller active modes, the coefficients of the matrix element expansions in different coordinates are also symmetry determined. Consider, for simplicity, an electronic state of symmetiy in an even-electron molecule with a single threefold axis of symmetry, and choose a representation in which two complex electronic components, e ) = 1/v ( ca) i cb)), and two degenerate complex nuclear coordinate combinations Q = re " each have character T under the C3 operation, where x — The bras e have character x. Since the Hamiltonian operator is totally symmetric, the diagonal matrix elements e H e ) are totally symmetric, while the characters of the off-diagonal elements ezf H e ) are x. Since x = 1, it follows that an expansion of the complex Hamiltonian matrix to quadratic terms in Q. takes the form... 

Suppose now that A) and B) belong to an electronic representation I ,. Since H is totally symmetric, Eq. (6) implies that the matrix elements (A II TB) belong to the representation of symmetrized or anti-symmetrized products of the bras (A with the kets 7 A). However, the set TA) is, however, simply a reordering of the set ( A). Hence, the symmetry of the matrix elements in the even- and odd-electron cases is given, respectively, by the symmetrized [Ye x Te] and antisymmetrized Ff x I parts of the direct product of I , with itself. A final consideration is that coordinates belonging to the totally symmetric representation, To, cannot break any symmetry determined degeneracy. The symmetries of the Jahn-Teller active modes are therefore given by... 

Franke KJ, Schulze G, Pascual JI (2010) Excitation of Jahn-Teller active modes during electron transport through single C60 molecules on metal surfaces. J Phys Chem Lett 1 ... 

Fig. 9. Orbital pattern of singly occupied molecular orbital and Jahn-Teller active mode. Both SOMOs are antisymmetric with respect to the cr plane. For corannulene (top) the tangential direction along the pseudorotational path at the minimum corresponds to the antisymmetric vibration, while, for coronene (bottom), the symmetric vibration corresponds to the tangential direction at the minimum.

Fig. 10. The Jahn-Teller active modes in the U08 cubic cage. Only one of three components (Qxy) is shown for the f2g(l) and r2g(2) modes.

Figure 12.1 The two compounds of the sg Jahn-Teller active mode.

During the last decade it has been observed that the excited state of optical centers is often strongly distorted due to the Jahn-Teller effect. This puts the simple configurational coordinate model with the breathing mode as a coordinate in severe doubt. Also, it now becomes clear that the Stokes shift is in many cases due to a relaxation via a Jahn-Teller active mode. Let us illustrate these statements by several examples that originate from three types of centers, viz., transition... 

In molecules with a fourfold axis, e.g., those belonging to the group D4, the Jahn-Teller active modes are ft, and 62 Since these modes are nondegenerate, they cannot lift the vibronic degeneracy, so that all vibronic... 

A different series of organic systems whose Raman spectra and R Ps have been studied in some detail (lijima et al., 1975 Muramatsu et al., 1977 M. Asselin and H. G. Bernstein, unpublished) are ions of the type C 0, where n = 4, 5, 6, belonging to the point groups D /, Dsi, and respectively. These ions show evidence of Jahn Teller effects. Thus in 404 , the Jahn-Teller active modes big and hj show strong Raman activity when approaching resonance with the allowed Ajg - E ( r7r ) transition. Similarly in CsOs", resonance with the A l i(7r7r ) transition leads to high activity of Jahn-Teller active e 2 modes, which have intensities comparable... 

However, the e -electronic ground state of the benzene ion is degenerate and its w e function can be perturbed by certain vibrations of the molecule. This leads to Jahn-Teller active modes which are distinguished by angular momenta j = 1/2 and j = 3/2. This will allow further transitions from the vibrationless groundstate of to vibrational levels of j = 1/2 in the ion. 

Fig. 4. The X E band in the photoelectron spectrum of allene. (a) Calculated spectrum including two (r)4+ e) Jahn-Teller active modes, (b) Experimental high-resolution spectrum (reproduced from Ref. 100).

The situation described above applies to complexes packed in a crystal lattice. We have recently found that the situation is drastically different for the Cu -(2,2 6, 6"-terpyridine)2 complex in glassy matrices of frozen ethanol or frozen ethanol-dichloromethane mixtures and the same complex isolated in a matrix of crystalline ethanol. In these cases, only small amplitude librations along the Jahn-Teller active mode are observable below the melting point of ethanol. Above the melting point, averaging of the g and hyperfine tensor by the Jahn-Teller effect is complete [45]. 

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