Electron-vibration coupling strength

There are at least two ways in which detailed information about electron-vibrational coupling strengths can be obtained for mixed-valence complexes. Both are based on the fact that such coupling will be reflected in modifications of the vibrational spectrum. Thus, for example, coupling to antisymmetric modes in a symmetric ion will modify intensities and frequencies of the modes involved. 

In the MQC mean-field trajectory scheme introduced above, all nuclear DoF are treated classically while a quantum mechanical description is retained only for the electronic DoF. This separation is used in most implementations of the mean-field trajectory method for electronically nonadiabatic dynamics. Another possibility to separate classical and quantum DoF is to include (in addition to the electronic DoF) some of the nuclear degrees of freedom (e.g., high frequency modes) into the quantum part of the calculation. This way, typically, an improved approximation of the overall dynamics can be obtained—albeit at a higher numerical cost. This idea is the basis of the recently proposed self-consistent hybrid method [201, 202], where the separation between classical and quantum DoF is systematically varied to improve the result for the overall quantum dynamics. For systems in the condensed phase with many nuclear DoF and a relatively smooth distribution of the electronic-vibrational coupling strength (e.g.. Model V), the separation between classical and quanmm can, in fact, be optimized to obtain numerically converged results for the overall quantum dynamics [202, 203]. 

The shift of the emission maximum relative to the absorption maximum, the so-called Stokes shift, is determined by the value of Qq-Qo (see Fig. 1). For the equal force constant case this Stokes shift is equal to 2Shv [2], This indicates that the Stokes shift is small for the weak-coupling case and large for the strongcoupling case. It is also clear that the value of the Stokes shift, the shape of the optical bands involved, and the strength of the (electron-vibrational) coupling are related. For a more detailed account of these models the reader is referred to the literature mentioned above [1-4]. 

As a further point, we mention that the vibrational energy loss AE (Fig. 16) is also often expressed in terms of the strength of the electron-phonon coupling or of the Huang-Rhys factor S (see, for example, Stoneham, 1975,1977,1981), where... 

The vibrational reflection principle outlined in Section 6.4 provides a simple, but yet quantitative explanation of final vibrational state distributions and their variation with the coupling strength and the total energy. The central quantity is the vibrational excitation function N(ro). It comprehensively manifests the dynamical details of the fragmentation process in the upper electronic state. Usually, one needs only very few trajectories to construct N(ro) which makes the simple classical theory outlined in Section 6.4 very efficient for calculating and understanding final state distributions. This is particularly beneficial for fitting experimental data. 

Fig. 15.8. Schematic one-dimensional illustration of electronic predissociation. The photon is assumed to excite simultaneously both excited states, leading to a structureless absorption spectrum for state 1 and a discrete spectrum for state 2, provided there is no coupling between these states. The resultant is a broad spectrum with sharp superimposed spikes. However, if state 2 is coupled to the dissociative state, the discrete absorption lines turn into resonances with lineshapes that depend on the strength of the coupling between the two excited electronic states. Two examples are schematically drawn on the right-hand side (weak and strong coupling). Due to interference between the non-resonant and the resonant contributions to the spectrum the resonance lineshapes can have a more complicated appearance than shown here (Lefebvre-Brion and Field 1986 ch.6). In the first case, the autocorrelation function S(t) shows a long sequence of recurrences, while in the second case only a single recurrence with small amplitude is developed. The diffuseness of the resonances or vibrational structures is a direct measure of the electronic coupling strength.

Table 9.2. Kinetic isotope effects from exact quantum rate computations on the model of Eq. (9.12). In one case there is no protein promoting vibration, in the second case there is a promoting vibration coupled with a strength similar to that in our previous model studies. In each case there are two levels of electron coupling — essentially the rate of electron transfer between the two states. Moderate electron transfer enhances the kinetic isotope effect while strong electron coupling enhances it less. We have found that high coupling is asymptotic to sequential transfer.

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