Vibronic coupling first order

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. 

Fig. 35. Schematic representation of the first-order potential energy surface for T0e vibronic coupling. The two components of the eg vibrational mode are shown on the left.

Thus while the Jahn-Teller theorem is generally invoked to account for the distorted octahedral geometries of the copper(II) ion, in terms of the first-order vibronic coupling the extension of the coupling to some second-order effects also allows some rationalization of the tetrahedral and five-coordinate geometries of the copper(II) ion.1063-1067... 

First order terms in Eq. (1) due to vibronic coupling may in general give rise to changes of the electronic wavefunctions. It can be easily seen that eigenfunctions (q) of the zero-order Hamiltonian H(q,0) may be intermixed by first order perturbation yielding... 

Except for first-order perturbation theory, we did not use any approximation in obtaining equations (4) and (5). These results are based upon the assumption that the vibronic coupling part of the Hamiltonian, //JT, has the same high symmetry as... 

Calculations of the second-order RFs are much more complicated than first-order RFs as they generally involve coupling to an infinite set of excited vibronic states for which details are often unknown. They can therefore only be calculated exactly in a pure adiabatic case, such as that found in the T <%> e system [1] in which the electronic states are not mixed. In other cases, approximations have to be made. Even so, as excited states with higher energies are included, the energy denominator increases whilst the overlaps between states located in different wells in the lowest adiabatic potential energy surface (APES) decrease. We note also that numerical approaches often mask the underlying physics whilst analytical methods such as... 

In the non-CT radiationless transition the change in electronic charge interacts with the nuclei in a similar maimer both before and after the transition. Two types of processes can be identified internal conversion processes in which the transition is between spin states of the same multiplicity and intersystem crossing process in which the transition is between states of different spin multiplicity. For non-CT internal conversion processes the full BO (Bom—Oppenheimer) adiabatic wave-functions for the supramolecular complex are used as the zero-order basis [42-44]. The perturbations that cause the transition are the vibronic coupling between the nuclear and electron motions. These are just the terms that are neglected in the BO approximation [45]. The terms are expanded (normally to first order) in the normal vibrational coordinates of the nuclei as is customarily done for optical vibronic transitions. Thus one obtains Eq. 61b for cases when only one normal mode couples the two states... 

According to (42), the T2 mode is not JT active (in first order) in E states. However, the matrix elements of the SO operator with non-relativistic electronic wave functions vanish for a state in symmetry. It is then essential to take account of the leading nonvanishing terms in the Taylor expansion of the matrix elements of the SO operator. As shown below, these are of first order of vibrational displacements of T2 symmetry, which implies the existence of a purely relativistic E xT JT effect [34]. Linear E xT vibronic coupling is not accounted for by the JT selection rules [2] it is, therefore, a novel type of JT effect. 

---