Jahn-Teller modes

To estimate the vibronic coupling constant, we performed CF calculations for the UOg cluster distorted along the eg, f2g( 1) and t2g(2) Jahn-Teller modes (Fig. 10). A similar approach was previously applied for calculations of vibronic coupling constants in a series of UXg octahedral complexes (X = F, Cl, Br, and I) and in the U(NCS)g cubic complex in compound (NEt4)4U(NCS)8 [14]. 

For each mode, the amplitude of the distortion corresponds to the actual displacement of oxygen atoms, 0.014 A. Then the vibronic coupling constants C were calculated from the splitting energy AE of the ground T5 level. 

The largest vibronic coupling constants are found for the eg and f2g(2) modes, while for the f2g( 1) mode it is about one order of magnitude smaller. From these data we can also estimate the JT stabilization energy per U atom, 10 cm-1 for eg, 8 cm-1 for tyjl), and only 0.5 cm-1 for 2g(l)- All these energies are smaller than the thermal energy at the transition point, TN /kH 20 cm 1. 

Instead, the actual distortion of the UOg cage in U02 just corresponds to the trigonal JT minimum of the f2g(l) mode, which has the vibronic coupling constant being more than 20 times smaller than those for the eg or t2g(2) modes. In fact, the distortion shown in Fig. 11 can be represented by the sum (Q[ly] + of 

The fact that the phase transition in UO2 has the first-order character and the ordered magnetic moment of 1.74 /ab is considerably lower than the paramagnetic moment (about 3 /ab) is qualitatively consistent with the ratio of the strength of the bilinear and biquadratic parts of the effective spin Hamiltonian (8) of the 5f2-5f2 

---

Figure f. Temperature dependence of peak position of Mn-O stretching modes. Upper (lower) panel for the breathing mode at 620 cm-1 (Jahn-Teller mode at 490 cm-1). Arrows indicate the magnetic ordering temperature and the lines are guides to the eye. 

This case corresponds to the fifth possibility in (29) which does not occur in lower dimensions and exemplifies a partial overlap between the Jahn-Teller modes and the normal modes. In order to destroy the symmetry of this level one would need modes of type f[ and Fp. A glance at the normal modes of the hyperoctahedron... 

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... 

If the states are degenerate rather than of different symmetry, the model Hamiltonian becomes the Jahn-Teller model Hamiltonian. For example, in many point groups D and so a doubly degenerate electronic state can interact with a doubly degenerate vibrational mode. In this, the x e Jahn-Teller effect the first-order Hamiltonian is then [65]... 

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). 

The cyclopentadienyl radical and the cyclopentadienyl cation are two well-known Jahn-Teller problems The traditional Jahn-Teller heatment starts at the D k symmetry, and looks for the normal modes that reduce the symmetry by first-01 second-order vibronic coupling. A Longuet-Higgins treatment will search for anchors that may be used to form the proper loop. The coordinates relevant to this approach are reaction coordinates. 

The versatile binding modes of the Cu2+ ion with coordination number from four to six due to Jahn-Teller distortion is one of the important reasons for the diverse structures of the Cu-Ln amino acid complexes. In contrast, other transition metal ions prefer the octahedral mode. For the divalent ions Co2+, Ni2+, and Zn2+, only two distinct structures were observed one is a heptanuclear octahedral [LnM6] cluster compound, and the other is also heptanuclear but with a trigonal-prismatic structure. 

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 ... 

---