Electronic and vibrational motions

Flowever, transition metal complexes do absorb in the visible region, giving them a characteristic colour. Flow can this happen if the transitions are forbidden The answer is that interaction may occur between the motion of the electrons and vibrational motions so that some vibronic transitions are allowed (see Section 7.3.4.2b). 

The coupling of electronic and vibrational motions is studied by two canonical transformations, namely, normal coordinate transformation and momentum transformation on molecular Hamiltonian. It is shown that by these transformations we can pass from crude approximation to adiabatic approximation and then to non-adiahatic (diabatic) Hamiltonian. This leads to renormalized fermions and renotmahzed diabatic phonons. Simple calculations on H2, HD, and D2 systems are performed and compared with previous approaches. Finally, the problem of reducing diabatic Hamiltonian to adiabatic and crude adiabatic is discussed in the broader context of electronic quasi-degeneracy. 

Non-adiabatic molecular Hamiltonian. Canonical transformation conpUng 383 electronic and vibrational motions... 

Interaction between electronic and vibrational motions in a molecular entity. 

As a consequence of the transformation, the equation of motion depends on three extra coordinates which describe the orientation in space of the rotating local system. Furthermore, there are additional terms in the Hamiltonian which represent uncoupled momenta of the nuclear and electronic motion and moment of inertia of the molecule. In general, the Hamiltonian has a structure which allows for separation of electronic and vibrational motions. The separation of rotations however is not obvious. Following the standard scheme of the various contributions to the energy, one may assume that the momentum and angular momentum of internal motions vanish. Thus, the Hamiltonian is simplified to the following form. 

The complete wave function of a molecule is called the rovibronic wave function. In the simplest approximation, the rovibronic function is a product of rotational, vibrational, and electronic functions. For certain applications, the rotational motion is first neglected, and the vibrational and electronic motions are treated together. The rotational motion is then taken into account. The wave function for electronic and vibrational motion is called the vibronic wave function. Just as we separately classified the electronic and vibrational wave functions according to their symmetries, we can do the same for the vibronic functions. In the simplest approximation, the vibronic wave function is a product of electronic and vibrational wave functions, and we can thus readily determine its symmetry. For example, if the electronic state is an e2 state and the vibrational state is a state, then the vibronic wave function is... 

The concept of a potential energy curve or surface is reasonably useful when the coupling between electronic and vibrational motion is small. The potential energy function gives the electronic energy of the system for arbitrary fixed positions of the nuclei. 

Conclusions of more general validity emerge by dispensing with the assumption that electronic and vibrational motions are independent. In this approximation one considers the joint vibronic (irrational -electronic) motions of the electrons, leading to a further set of selection rules. These vibronic rules are automatically satisfied for transitions 12 ... 

We take into account that (1,01 (f)l 1,0) = 0.). In systems with continuous phonon spectrum, the function under this integral differs essentially from zero only for t — t2 cr-1 < cS-1. For such a small time difference, electronic and vibrational motions are only weakly correlated this allows one to use the following decoupling of the electronic and the vibrational degrees of freedom ... 

This work has shown that ah initio calculations can assist in the explanation of many diatomic spectra, Of course, there are limitations to the methods described here. The treatment of electronic and vibrational motions independently may not always be valid. Furthermore, the use of virtual orbitals and crude potential functions may provide a very poor representation of the interacting states. Refinement of the model may be necessary in cases such as NO where the lack of information on excited states may be responsible for the poor answers. 

Along with the electronic interactions (double exchange, HDVV exchange, etc.) the coupling of electronic and vibrational motions (vibronic coupling) plays a crucial role in MV systems. One of the main characteristics of MV compounds is the... 

The problem has not been resolved analytically. Thirunamachandran and I showed that in special cases answers can be given. If we suppose that both electronic and vibrational motions are represented as simple harmonic vibrations, and the coupling between them given a sufficiently simple form, then the full Hamiltonian can be solved exactly to find energies and eigenfunctions. These exact solutions can be compared with those found in the adiabatic approximation with non-adiabatic corrections. 

In addition, in the absence of coupling between the electronic and vibrational motions it becomes... 

We shall here discuss only the nonadiabatic effects due to the conical intersection between two adiabatic electronic states. By therefore considering only the electronic and vibrational motion, the molecular Schrddinger equation... 

The linear coupling between electronic and vibrational motions leads to a potential surface which is called the mexican hat (Fig. 2). The upper and lower potential surfaces have energies ... 

---