Jahn-Teller active vibrations

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... 

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.

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... 

The symmetry of the normal mode of vibration that can take the molecule out of the degenerate electronic state will have to be such as to satisfy Eq. (6-7). The direct product of E with itself (see Table 6-11) reduces to A + A 2 + E. The molecule has three normal modes of vibration [(3 x 3) - 6 = 3], and their symmetry species are A + E. A totally symmetric normal mode, A, does not reduce the molecular symmetry (this is the symmetric stretching mode), and thus the only possibility is a vibration of E symmetry. This matches one of the irreducible representations of the direct product E E therefore, this normal mode of vibration is capable of reducing th eZ)3/, symmetry of the H3 molecule. These types of vibrations are called Jahn-Teller active vibrations. 

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]. 

The Jahn-Teller active vibrations for the E2u electronic ground state are those with Blg and B2g symmetry. The Blg vibration distorts the molecule in a structure with alternating bond lengths. As the mirror plane m runs through the carbon atoms in the 1 and 5 positions of the COT anion (Figure 6), a static distortion induced by a Blg vibration is impossible. The B2g vibration moves the carbon atoms alternately inward and outward. Such a distortion is in agreement with the local symmetry. 

These types of vibrations are called Jahn-Teller active vibrations. 

Because there are many excellent reviews covering the theory of electronic Eg ground states, the degeneracy of which is lifted by the Jahn-Teller active vibrational Eg mode (Fig. 2) we will present only a short outline of those aspects which are needed in the following chapters. 

Tl Jahn-Teller-active vibration u, has a groundstate frequency of 608 cm, whereas the frequencies of ue other modes lie more or less above 1100 cm. So the assignment of the step at 650 cm to the... 

Some authors, on the other hand /, applied the group theoretical approach to this problem it is based on the symmetry properties of the Jahn-Teller active vibrational mode. Unfortunately, these studies have also been restricted to the first-order perturbation theory. This section exploits an entirely different treatment. The correlation among symmetry point groups is investigated for individual Jahn-Teller active systems from the viewpoint of the pertinent electronic state. 

Restriction to the so-called Jahn-Teller active vibrations yields the distorted geometries 03d and O41, in accordance with earlier investigations the present approach is more general. 

There is one result of the general theory which should be mentioned. It can be shown that in a regular octahedron the Jahn-Teller effect only operates by way of vibrations of symmetry when the electronic state is "Eg (the value of n is irrelevant because the Jahn-Teller theorem applies only to space functions, not spin). For electronic states of either or "T2g symmetries then the Jahn-Teller effect operates through vibrations of either 6g or t2g symmetries a vibration of the latter symmetry is shown in Fig. 8.11. Of course, the orbital degeneracy in an octahedral complex may be relieved by distortions other than those shown in Figs. 8.8 and 8.11. However, in such cases we may conclude that whatever is responsible for the distortion it is not the Jahn-Teller effect. In particular, all of the Jahn-Teller-active vibrations of an octahedron carry the g suffix and this means that they cannot give rise to a distortion which destroys the centre of symmetry of an octahedron. Distorted octahedral complexes which lack a centre of symmetry cannot owe their distortion to the operation of the Jahn-Teller effect. This account of the Jahn-Teller effect indicates why it is of little importance when the t2g orbitals are unequally occupied. Occupation of these orbitals... 

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