The Vibronic Coupling Model

To obtain an expression of the A coupling constant as a function of the adiabatic potential energies, one first writes the diabatic potential energy matrix along the coordinate Qi 

From this last two equations, the a coupling constant can be written as 

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 

The vibronic coupling model Hamiltonian is well suited for a combination with the MCTDH method as it has the required product form (see Sect. 4.2.3). Usually, molecular systems affected by strong vibronic couplings have complicated spectra with very dense bands. Therefore, a detailed analysis of the spectrum in terms of individual vibronic states is in general impossible and one is more interested in the overall electronic band profiles. In this case, the use of a time-dependent fl-amework can be advantageous since the absorption profile can be obtained from the time-dependent wavepacket propagated over relatively short times, as exposed below. 

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D. The Vibronic-Coupling Model Hamiltonian IV, Non-Adiabatic Molecular Dynamics... 

A B2u- The vibronic coupling model Hamiltonian is set up using the ground-state... 

The vibronic coupling model has been applied to a number of molecular systems, and used to evaluate the behavior of wavepackets over coupled surfaces [191]. Recent examples are the radical cation of allene [192,193], and benzene [194] (for further examples see references cited therein). It has also been used to explain the lack of structure in the S2 band of the pyrazine absoiption spectrum [109,173,174,195], and recently to study the photoisomerization of retina] [196],... 

The Hamiltonian provides a suitable analytic form that can be fitted to the adiabatic surfaces obtained from quantum chemical calculations. As a simple example we take the butatriene molecule. In its neutral ground state it is a planar molecule with Dy, symmetry. The lowest two states of the radical cation, responsible for the first two bands in the photoelectron spectrum, are X2B2g and A1 By The vibronic coupling model Hamiltonian is set up using the ground-state... 

Figure 7. The PES of the X2B2S and A -Bi, states of the butatriene radical cation, (a) Diabatic surfaces, (b) Adiabatic surfaces. The surfaces are obtained as eigenfucations of the vibronic coupling model Hamiltonain that fitted to reproduce quantum chemical calculations. The coordinates are shown in Figure lc. See Section III. D for further details.

Muller and Stock [227] used the vibronic coupling model Hamiltonian, Section III.D, to compare surface hopping and Ehrenfest dynamics with exact calculations for a number of model cases. The results again show that the semiclassical methods are able to provide a qualitative, if not quantitative, description of the dynamics. A large-scale comparison of mixed method and quantum dynamics has been made in a study of the pyrazine absorption spectrum, including all 24 degrees of freedom [228]. Here a method related to Ehrenfest dynamics was used with reasonable success, showing that these methods are indeed able to reproduce the main features of the dynamics of non-adiabatic molecular systems. 

In the following, we summarize the pertinent results of our analysis of Refs. [50-53] where we applied the LVC Hamiltonian Eq. (1) in conjunction with a 20-30 mode phonon distribution composed of a high-frequency branch corresponding to C=C stretch modes and a low-frequency branch corresponding to ring-torsional modes. In all cases, the parametrization of the vibronic coupling models is based on the lattice model of Sec. 3.1 and the complementary diabatic representation of Sec. 3.2. 

A closer analysis of the DFT results within the vibronic coupling model shows5 that the lone-pair stabilization—as has been emphasized before7,10—is a purely orbital effect. It is caused by the changes in overlap due to the electronic rearrangements, which accompany the nuclear Td—>C2v... 

The model which has been most widely applied to the calculation of vibronic intensities of the Cs2NaLnCl6 systems is the vibronic coupling model of Faulkner and Richardson [67]. Prior to the introduction of this model, it was customary to analyse one-phonon vibronic transitions using Judd closure theory, Fig. 7d, [117] (see, for example, [156]) with the replacement of the Tfectromc (which is proportional to the above Q2) parameters by T bromc, which include the vibrational integral and the derivative of the CF with respect to the relevant normal coordinate. The selection rules for vibronic transitions under this scheme therefore parallel those for forced electric dipole transitions (e.g. A/ <6 and in particular when the initial or final state is /=0, then A/ =2, 4, 6). 

Importantly, the vibronic coupling model Hamiltonian described in detail in Sec. 2 is of a form that allows the full efficiency of the method to be realized. This combination of model and propagation technique thus allows us to study completely and in detail the multidimensional molecular dynamics through a conical intersection. 

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. 

A very useful starting point for the study of non-adiabatic processes, which are common in photochemistry and photophysics, is the vibronic coupling model Hamiltonian. The model is based on a Taylor expansion of the potential surfaces in a diabatic electronic basis, and it is able to correctly describe the dominant feature resulting from vibronic coupling in polyatomic molecules a conical intersection. The importance of such intersections is that they provide efficient non-radiative pathways for electronic transitions. Not only is the position and shape of the intersection described by the model, but it also predicts which nuclear modes of motion are coupled to the electronic transition which takes place as the system evolves through the intersection. 

The advent of MCTDH has made it possible to solve the (second-order) vibronic coupling Hamiltonian of small to medium sized molecules (5-12 atoms, say), including all internal degrees of freedom. In fact, it is the combination of the vibronic coupling model with MCTDH which is numerically so successful. The vibronic coupling model provides a realistic multi-mode Hamiltonian, and this Hamiltonian is, from its ansatz, in the product form advantageous for MCTDH. MCTDH then solves the dynamics problem accurately and efficiently. 

The methodology of molecular quantum dynamics applied to non-adiabatic systems is presented from a time-dependent perspective in Chap. 4. The representation of the molecular Hamiltonian is first discussed, with a focus on the choice of the coordinates to parametrize the nuclear motion and on the discrete variable representation. The multi-configuration time-dependent Hartree (MCTDH) method for the solution of the time-dependent Schrddinger equation is then presented. The chapter ends with a presentation of the vibronic coupling model of Kdppel, Domcke and Cederbaum and the methodology used in the calculation of absorption spectra. 

These last equations show that the MCTDH method is capable of treating the nuclear dynamics of molecular systems on several coupled electronic states. This formalism has been used, in combination with the vibronic coupling model of Kdppel et al. 

We note that a very efficient version of this method based on the Lanczos iterative eigensolver has been implemented for the specific case of vibronically coupled systems described by the vibronic coupling model Hamiltonian, allowing for the computation of absorption or photoelectron spectra for systems with bases containing up to 10 basis functions. This method is described in details in the Chap. 7 of Ref. [64]. 

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