Molecular vibronic states

The eigenstates of h are molecular vibronic states B and J are respectively operators and undressed excitonic interactions, purely electronic. 

The polaron transformation, executed on the Hamiltonian (12.8)-( 12.10) was seen to yield a new Hamiltonian, Eq. (12.15), in which the interstate coupling is renormalized or dressed by an operator that shifts the position coordinates associated with the boson field. This transformation is well known in the solid-state physics literature, however in much of the chemical literature a similar end is achieved via a different route based on the Bom-Oppenheimer (BO) theory of molecular vibronic stmcture (Section 2.5). In the BO approximation, molecular vibronic states are of the form (/) (r,R)x ,v(R) where r and R denote electronic and nuclear coordinates, respectively, R) are eigenfunctions of the electronic Hamiltonian (with corresponding eigenvalues E r ) ) obtained at fixed nuclear coordinates R and... 

There are two extreme approaches to the description of vibroiiic states in crystals. The crystal states may be regarded as products of molecular vibronic states, or as linear combinations of vibronic site states. In the former case the electrons and nuclei in the unit cell couple independently, while for the latter description the intramolecular vibronic interactions are assumed to dominate. These two approaches are somewhat analogous to y and LS coupling of spin and orbital angular momentum in atoms and are called strong and weak coupling, respectively. The approach to the classification of states is different for each case. 

As we have seen in the previous sections a molecular state is split in the molecular crystal into a number of states that depends on the number of nontranslationally equivalent molecules in the unit cell. In the simplest case of weak coupling (Figure 10.7) each electronic and vibronic state is split, and two or more spectral lines replace the single molecular spectral line. If the molecular spectrum contains vibrationally induced intensity (see Eq. 8.40) then the experimental situation becomes more complex. The example illustrated in Figure 10.8 is again the C , two molecule per cell case. The molecular jB2u state can contain a bsc vibration (the molecular symmetry is assumed to be Dih for convenience) if the vibronic intensity can be transferred from a, Biu state (not shown in the diagram). The vibronic symmetry of the induced band is X bsg = Biu. The molecular vibronic state function is, from Eqs. 8.38-8.40,... 

Most of the experimental results on CJTE can be explained on the basis of molecular field theory. This is because the interaction between the electron strain and elastic strain is fairly long-range. Employing simple molecular field theory, expressions have been derived for the order parameter, transverse susceptibility, vibronic states, specific heat, and elastic constants. A detailed discussion of the theory and its applications may be found in the excellent review by Gehring Gehring (1975). In Fig. 4.23 various possible situations of different degrees of complexity that can arise in JT systems are presented. 

Pump-probe diffraction techniques offer exciting new ways to probe transient structures in molecular, nanoscale, and biological systems. For dilute systems, or very small targets, electron diffraction is the preferred tool, because the cross sections for scattering of electrons from molecules are very large. In our research we show that pump-probe electron diffraction is an excellent technique to probe the dynamics of chemical reactions in the rarified environment of jet expansions, and for probing the diffraction signatures of individually excited vibronic states. 

The wavefunction developed in the preceding subsection is, of course, only a first approximation to the true molecular continuum wavefunction. We have thus far considered only the nonvibrating molecule, and treated the resulting Born-Oppenheimer states as if they were the true eigenstates. In a real system vibrational motion of the nuclei, configuration interaction between vibronic states, etc., must be included in the description. 

R. Wallace, Chem. Phys., 37, 285 (1979). Vibronic State Symmetry, Selection Rules and Transition Probabilities for a Molecular Rearrangement Process. The Butadiene-Cyclobutene Rearrangement. 

Since the interfacial ET process is faster than the vibrational relaxation, one needs the transition rate from a single molecular vibronic level to the conduction band (or local states coupled to the adsorbed molecule) of the... 

Thus the hamiltonian (2.15) couples the electronic excitations to the vibrations by linear terms in and by quadratic terms in A. The molecular eigenstates of (2.15) are the vibronic states they are different from tensorial products of electronic excitations and undressed vibrations. Even for this simple intramolecular effect, we cannot, when moving to the crystal, consider excitonic and vibrational motions as independent. 

The vibronic exciton approximation restricts H to a subspace corresponding to a given vibronic molecular state. In this subspace the degeneracy of the localized vibronic states is lifted by the interactions JnmB Bm. Using the translational invariance, the eigenstates of the crystal are seen to be the vibronic excitons, or vibrons ... 

It is important to note that the Hamiltonian (2.120) contains the terms which produce both the adiabatic and non-adiabatic effects. In chapter 7 we shall show how the total Hamiltonian can be reduced to an effective Hamiltonian which operates only in the rotational subspace of a single vibronic state, the non-adiabatic effects being treated by perturbation theory and incorporated into the molecular parameters which define the effective Hamiltonian. Almost for the first time in this book, this introduces an extremely important concept and tool, outlined in chapter 1, the effective Hamiltonian. Observed spectra are analysed in terms of an appropriate effective Hamiltonian, and this process leads to the determination of the values of what are best called molecular parameters . An alternative terminology of molecular constants , often used, seems less appropriate. The quantitative interpretation of the molecular parameters is the link between experiment and electronic structure. 

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