Vibronic coupled system

To apply the mapping formalism to vibronically coupled systems, we identify the / ) with electronic states and the h m with operators of the nuclear dynamics. Hereby, the adiabatic as well as a diabatic electronic representation may be employed. In a diabatic representation, we have [cf. Eq. (1)]... 

To represent the POs for a vibronically coupled system in a intuitively clear way, we consider the classical electronic population variable A dia =... 

To perform a PO analysis of nonadiabatic quantum dynamics, we employ a quasi-classical approximation that expresses time-dependent quantities of a vibronically coupled system in terms of the vibronic POs of the system [123]. Considering the quasi-classical expression (16) for the time-dependent expectation value of an observable A, this approximation assumes that the integrable islands in phase space represent the most significant contributions to the dynamics of the observables considered [236]. As a consequence, the short-time dynamics of the system is determined by its shortest POs and can be approximated by a time average over these orbits. Denoting the A th PO with period 7 by qk t),Pk t) we obtain [123]... 

Schneider, R., Domcke, W., and Koppel, H. (1990). Aspects of dissipative electronic and vibrational dynamics of strongly vibronically coupled systems, J. Chem. Phys. 92, 1045-1061. 

It quickly becomes clear that there is a rich variety of vibronically coupled systems that may be present in compounds containing doped Ceo. There are other complications that arise that further complicate the theoretical description of these problems. One such complication is that each of these coupling problems is actually a multimode problem. That is, there are several modes of vibration of Ceo that can simultaneously couple to the aforementioned electronic states (ygg and ihg, to be precise). Although this complication can be dealt with (see, e.g. [48]), it is often easier to work in terms of a single, effective mode as this is far simpler and reproduces most of the important aspects of the problem. In fact, in the current context, even the details of some effective mode of vibration are effectively irrelevant as we are ignoring the distortion of the Ceo cage. 

This holds true especially for vibronically coupled systems. In absorption spectra, vibronic coupling effects are often hidden by Franck-Condon effects. Resonance Raman spectroscopy makes it possible to selectively enhance such hidden vibronic coupling effects by measuring the REPs of inducing modes, in particular those that are nontotally symmetric. A proper experimental study should include both the intensity and the depolarization ratio across the absorption band and should cover not only the fundamental but also overtones. 

In this review we have considered the question how resonance Raman spectroscopy can help us to gather information on vibronic coupling. We now briefly consider the kind of information we may expect to obtain. To describe a vibronically coupled system, we first have to identify the states and modes involved in the coupling. If this standard assignment problem is solved, we have to determine the strength of the coupling and the analytical form of the coupling operator. 

The concept of diabatic electronic states plays an important role in the theoretical description of strongly vibronically coupled systems. Contrary to the usual adiabatic electronic states, they are not eigenstates of the electronic Hamiltonian which thus acquires off-diagonal matrix elements in... 

Diabatic electronic states represent a useful concept in theoretical studies of strongly vibronically coupled systems. They are slowly varying functions... 

It was implicitly assumed in the discussion so far that the motion of the molecular nuclei is determined by a single Born-Oppenheimer potential energy surface. However, in vibronic coupling systems, there are several coupled electronic states of importance. The MCTDH algorithm hence has to be extended to deal with more than one electronic state. 

The single-set formulation is to be preferred when the different electronic states have a similar form. The well known spin-boson model is in this category. When the motion on the different electronic states is very different, the multi-set formulation becomes the appropriate choice. The smaller number of SPFs needed for convergence then over-compensates the overhead of the multi-set formulation. For vibronically coupled systems, we have always chosen the multi-set formulation. 

The first numerically exact simulation of a time-resolved photoelectron spectrum for a multidimensional vibronic-coupling system was reported... 

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]. 

Special realizations of vibronic coupling systems, such as occur for well-localized, weakly coupled molecular subsystems have been touched upon only briefly, in Sect. 6.1.2.5. Another special case, that of light-induced Coins [99,100] has not been discussed above, but is expected to emerge as highly relevant in strong laser fields in the future. This may, of course, also apply to other realizations of specific sysfems, and likewise regarding their experimental detection. 

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