Jahn-Teller active coordinate

Fig. 10 Snapshots of Cr(CO)5 wavepacket dynamics on the lowest and first excited adiabatic potential surfaces and right panels). The contours show the two-dimensional Jahn-Teller surface in the space of the (Qj, Q2) pair of Jahn-Teller active coordinates, shown to the left...

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... 

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... 

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.

For instance Cr(CO)6+ is formed only during LI. The time-dependent behavior of the ion yields of Cr(CO)6+ is presented in Fig. 13. Deconvolution of the time-dependent ion yield with the instrument function derived from the Xe+ signal provides a measure of the time constant (ij) of 12.5 0.05 fs for the LI level (Table 2). This represents the time it takes for the excited Cr(CO)6 to cross to the repulsive surface through the conical intersection close to the Franck-Condon state. At the Franck-Condon point with Oh symmetry, the only coordinates with nonzero slope are the totally symmetric alg M-C stretch or the Jahn-Teller-active vibrations which have eg or t2g symmetry [32], The time taken for a wavepacket to travel from any... 

Figure 18 Representation of the potential energy surfaces involved in photoionization from a nondegenerate ground state of a molecule to a Jahn-Teller active state of a molecular ion. The distortion coordinate is the Jahn-Teller active vibration. The vertical arrows represent the most probable transitions...

In octahedral symmetry, the copper(ll) ion has a electronic ground state due to the d electron configuration with the unpaired electron in an Cg a anti-bonding orbital. An exact octahedral geometry of six-coordinate copper(II) complexes is never realized due to a strong Jahn-Teller effect. The symmetry of the Jahn-Teller active vibration is eg, the non-totally symmetric part of the symmetric square [Eg Eg]. For a Cu(Il)Lg complex, the two components of the degenerate eg vibration are shown in Fig. 1 a [2]. 

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... 

Our model system will be a molecule with three electronic levels. The electronic ground state is taken to be nondegenerate. The two excited electronic states are Jahn-Teller active, being degenerate at the symmetrical nuclear configuration and coupled by distortion of the molecule away from the symmetrical configuration. The symmetric shape corresponds to values <7, = 2 0 of the two internal coordinates. We suppress... 

The increase in volume change for the Cr system is explained by the electronic change in the inner coordination sphere from the Jahn-Teller distorted coordination sphere of Cr to an octahedral one in Cr -R (56). Further increase for the Co system is due to change of the electronic configuration of high spin d for Co to low spin d for Co -R in this system (73). The largest volume of activation found in the Ni system is attributed to the change in coordination number of the central Ni ion from planar low spin d (Nicyclam) to octahedral d for the Ni-R complex (45). 

Structurally [Pd (983)2] " consists of a centrosymmetric tetragonally elongated six-coordinate complex, as expected for this Jahn-Teller active (low-spin d ) ion (Fig. 6) [132]. Axial Pd-S distances exceed equatorial ones by 0.18 A (Table 1). Oxidation of [Pd(9S3)2] shortens the axial Pd-S distance by over 0.4 A, but equatorial ones by only 0.04 A. In its electrochemical behavior the... 

Importantly, differences in the metal-Ugand bonding to the two tertiary amine donors have been shown to be of importance for the tuning of the spin state of the ferryl complexes (see Section 6.4) [13g] and the redox potentials and catalytic activities of Cu couples [21] the bispidine-derived geometry is particularly well suited for the Jahn-Teller active Cu" ion and has led to a rich Cu coordination chemistry [22] with interesting applications in bioinorganic model chemistry (hemocyanin [23], catecholase[24]), catalytic aziridination [21a,... 

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