Quadratic vibronic coupling model

In concluding this subsection, we mention that both systems have been treated also by more elaborate calculations beyond the linear-plus-quadratic vibronic-coupling model. The vibronic dynamics of O3 has been... 

Let us start with the Hamiltonian for an N-mode system described by the quadratic vibronic coupling model. For a two state conical intersection in the diabatic representation it has the form... 

Fig. 2 Representative cuts through the potential energy surfaces of Bz+ (upperpanel or a) and its mono fluoro derivative, F-Bz+ (lowerpanel or b). The upper panel shows the results for the linear vibronic coupling model, while in the lower one the quadratic coupling terms are also included. In both panels the effective coordinate connects the centre of the Franck-Condon zone to the minimum of the intersection seam between the A and C states of F-Bz" ", and between the X and B states of the parent cation (within the subspace of JT active coordinates)...

A high symmetry of the molecule does not only help to (sometimes dramatically) reduce the number of parameters, it also provides a solid basis for the vibronic coupling model Hamiltonian. When the two interacting electronic states are of different symmetry (as assumed here), the interstate coupling must be an odd function of the couphng coordinate. Hence, there can be no constant or quadratic terms, only linear or bilinear ones are allowed. The vibronic coupling Hamiltonian was first derived by Cederbamn et and is more fully described in a review article by Kbppel et and in Chapter 7 of this book. 

The accuracy of the linear vibronic coupling model can be improved by adding diagonal quadratic terms 7." Q for the non totaUy-symmetric modes for which the diagonal linear terms vanish [63], In this case, the 7/" constants can be conveniently... 

We have constructed several linear vibronic coupling model Hamiltonians augmented with diagonal quadratic terms for the non-totally symmetric modes. The total Hamiltonian of the molecule in the diabatic representation reads... 

A linear vibronic coupling model Hamiltonian [33], augmented with a diagonal quadratic term along the vioa mode [31] is adopted for the molecular Hamiltonian. Its matrix representation in the basis of the diabatic electronic states reads... 

As the three effective mode model starts from the linear vibronic coupling Hamiltonian (LVC) [9] it may also have some relevance to generalize it and start from the quadratic vibronic coupling Hamiltonian (QVC) to obtain the appropriate quadratically extended (three)-effective mode equations. The motivation for this work has arisen that, in addition to the numerous applications of the LVC model, some other works in which the QVC model is used are also available [32,35], Our aim is to proceed along this direction. Following [21], we set up the QVC three-effective mode Hamiltonian and, using it for the pyrazine molecule we can calculate the autocorrelation function, the spectrum and the diabatic populations. The obtained results can be compared to those calculated by the LVC three-effective mode method. 

Fig. 16.3 The diabatic-state populations for the pyrazine as a function of time. Panei A the population for the lower state (solid line) and for the upper state (dotted line) using the exact 24-mode model. Panel B the population for the lower state (solid line) and for the upper state (dotted line) using the linear vibronic coupling model (6-mode). Panel C the population for the lower state (solid line) and for the upper state (dotted line) using the three effective mode approach. Panel D the population for the lower state (solid line) and for the upper state (dotted line) using our quadratically extended three effective mode approach...

Table 1 Linear (V and V, , in cm /A) and quadratic (Lj and in cm / ) vibronic coupling parameters, Jahn-TeUer stabilization energies [Ejr(D4h) and Ejr(D3d) in cm ] and vibronic coupling strengths = En-(D4i,)/hoOj, = Ejp(D3d)/ha)i ] of the T2g <8> Sg and T2g <8> t2g Jahn-TeUer problems in [Ee(CN)5] as deduced from DFT calculations on a charge-compensated model complex, using water as a solvent and a LDA(VWN) functional as well as a triple zeta basis set) ...

By using the LVC model, augmented by purely quadratic couplings only for totally symmetric modes (thus adding QVC contributions) the symmetry-selection rule, (6), can be directly applied to deduce the vibronic Hamiltonian matrices for the description of the five lowest X — D doublet states of these fluorobenzene cations. We shall not write down all five matrices here, but rather provide the basic features regarding their QVC Hamiltonian. The general form of the QVC potential energy matrix, W// oro, for the above mentioned fluorobenzene cations is depicted below ... 

This question provided the motivation for a simple 2D model description of the B-band, focussing exclusively on the PJT interaction for comparison, also the earher JT-type approach was reconsidered. The vibronic Hamiltonians are given by Eqs. (3) and (29), respectively, augmented by the appropriate quadratic couphng constants. The two coupling constants... 

---