Jahn-Teller activity

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... 

A particularly interesting case is when a set of hydrogens which are chemically equivalent in the unionized molecule become inequivalent in the positive ion. Obvious examples are Jahn-Teller active molecules, but the same phenomenon may be found also in Jahn-Teller inactive systems. Since deuteration fcr practical reasons must be done before ionization, it may happen that a single deuterated molecule may produce several inequivalent isomers of the radical cation, e.g., upon irradiation. This will obviously influence the recorded ESR spectrum. 

We have previously in a number of papers [1-5] investigated these effects ft -both the Jahn-Teller inactive molecule n-butane [1] and the Jahn-Teller active molecules ethane, cyclopropane, and cyclohexane [2-5]. The choice of systems was largely dictated by the availability of experimental results [5-8]. New experiments being performed on selectively deuterated benzene have motivated a closer theoretical study of this system, and a first presentation of these investigations is given in the present paper. 

Why is the structure square-pyramidal It has been shown that a diamagnetic d6 ML5 complex distorts away from the Jahn-Teller active trigonal bipyramidal structure.20 Two more stable structures are possible a square pyramid (T) and a distorted trigonal bipyramid (Y). Theoretical studies21 have shown that the T and Y structures are very close in energy and that the preference for one over the other comes from a subtle balance of a and it properties of the ligands. 

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... 

The above results mainly apply to the Longuet-Higgins E x e problem, but this historical survey would be incomplete without reference to early work on the much more challenging problems posed by threefold or higher electronic degeneracies in molecules with tetrahedral or octahedral symmetry [3]. For example, tetrahedral species, with electronic symmetry T or T2, have at least five Jahn-Teller active vibrations belonging to the representations E and T with individual coordinates (Qa,Qb) and (Qx. Qx. Q ) say. The linear terms in the nine Hamiltonian matrix elements were shown in 1957 [3] to be... 

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 ... 

Figure 4.19 Model structures of (a) incommensurate and (b) commensurate phases of K2Pb[Cu(N02)6]. The displacement pattern of Jahn-Teller active phonons is shown by arrows. In (a) the phonon mode has wave-vector k — (0.425, 0.425, 0) and in (b), wave-vector of the phonon mode is k = (, j, ). (After Yamada, 1977.)...

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.

Fig. 13. A schematic diagram of the potential energy surface for a doubly degenerate Jahn-Teller active electronic state.

Figure 5. One of the adiabatic energy surfaces along the Q2 Q3 coordinates belonging to the Jahn—Teller active eg mode calculated for a parameter set appropriate to SeClg2-. Reproduced with permission from Ref. 12. Copyright 1980, North-Holland Publishing Company.

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. 7. The bond orders for P4 in the C2v symmetry and one of the Jahn-Teller active distortions.

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.

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