Vibronic Hamiltonian coupling

The effective-mode transformation described here is closely related to earlier works which led to the construction of so-called interaction modes [75, 76] or cluster modes [77, 78] in Jahn-Teller systems. The approach of Refs. [54,55,72] generalizes these earlier analyses to the generic form - independent of particular symmetries - of the linear vibronic coupling Hamiltonian Eq. (8). 

Diabatic electronic states (previously termed crude adiabatic states ) are defined as slowly varying functions of the nuclear geometry in the vicinity of the reference geometry [9-11]. The final vibronic-coupling Hamiltonian is obtained by adding the nuclear kinetic-energy operator which is assumed to be diagonal in the diabatic representation. 

Modern electronic-structure theory provides all tools which are required to determine the parameters entering vibronic-coupling Hamiltonians from first principles. 

The computation of the inter-state coupling constants being defined as first derivatives of off-diagonal elements of the electronic Hamiltonian in the diabatic representation (see Eq. (11)), appears at first sight to be more difficult. It can be shown, however, by analj ng the adiabatic PE functions associated with the vibronic-coupling Hamiltonian (13) that the A -"" can be determined from second derivatives of the adiabatic energies with respect to the nontotally symmetric coordinate Qj. For an electronic two-state system, the following simple formula results ... 

As well as providing an overview of the MCTDH method and the vibronic coupling Hamiltonian, results will be presented in this chapter that highlight the features of three photo-physical systems in which vibronic coupling plays a crucial role. Here it will become clear that in some cases many degrees of freedom are required for a faithful description of the system dynamics through a conical intersection, and for these the MCTDH method is able to provide accurate information. 

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. 

Here k, A, 7, /r stand for all the coupling constants and denote the adiabatic surfaces obtained by diagonalising the potential part of the vibronic coupling Hamiltonian, Eq. (4). There is some unavoidable arbitrariness in this approach, as the points Q and their weights Wn have to be chosen. However, the least-squares approach is to be preferred when more than linear coupling terms are included. 

The vibronic coupling Hamiltonian provides a realistic model for accurately describing the short-time multi-mode dynamics in nonadiabatic systems. 

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. 

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. 

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