Electron-vibrational coupling

Equation 1 describes the radiationless decay rate for a single-frequency model with weak electron-vibration coupling in the low temperature limit as derived by Englman and Jortner. 

The reorganization free energy /.R represents the electronic-vibrational coupling, ( and y are fractions of the overpotential r] and of the bias voltage bias at the site of the redox center, e is the elementary charge, kB the Boltzmann constant, and coeff a characteristic nuclear vibration frequency, k and p represent, respectively, the microscopic transmission coefficient and the density of electronic levels in the metal leads, which are assumed to be identical for both the reduction and the oxidation of the intermediate redox group. Tmax and r max are the current and the overvoltage at the maximum. 

Figure 1. Relationship between activation free energy and overall free energy (values in units of electron-vibrational coupling energy x) n high-temperature...

In this expression A and Q are distance dispersion resulting from electron-vibrational coupling, and frequency tensor (assumed identical in reactant and product states), respectively (work of formation of precursor and successor states is omitted). If we assume that the frequency tensor is diagonal, then we have simply a sum of independent terms for all inner and outer contributing modes. At sufficiently high temperature, the hyperbolic tangents become unity and we obtain the usual (in this approximation) high-temperature expression ... 

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]. 

Gas-phase photoelectron spectroscopy (PES) has been used in conjunction with theoretical calculations to investigate the hole-vibrational and electron-vibrational couplings in fused benzodithiophenes. The first ionization energies of benzojl,2- 5,4- ]dithiophene 21 and benzo[l,2-A4,5- ]dithiophene 22 were found to be to be 7.585 and 7.573eV, respectively <2006CEJ2073>. 

There are several nonadiabatic interactions (see the appendix), for example, electron-vibration coupling and spin-orbit interaction. The electron-vibration interaction is described by the operator ... 

The previously described electron-vibration coupling describes radiationless transitions between different potential energy surfaces. This channel makes a noticeable contribution, if the potential energy surfaces cross or their separation is small. 

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]. 

Figure 6 A scheme of the three possible resonances in OOTF. i) Global resonance (A). Very weak electron-vibration interaction is expected ii) Localized resonances or traps (B).Usually the LEPS experiments are not detecting electrons trapped in these resonances and they appear as a reduction in the transmission probability, iii) Quantum well structure (C). Here the electron is localized in one dimension, while it is delocalized in the other two dimensions. There is a significant electron-vibration coupling.

A third type of resonance may result from the two dimensional structure of the organized organic films. In this case the electron is delocalized in two dimensions but localized in the third one (see Fig. 6C).40 Hence, the structure in the transmission probability is not a result of complete delocalization of the electron (as in the case of band structure) nor to complete localization of the electron on a single molecule (as in a typical trap ). As expected, also in this process strong electron-vibration coupling occurs and evidence for inelastic scattering is observed. 

The observation that the fine structure in the spectra relate to the vibrational states of the anions of the aromatic parts of the molecule, indicates that the extra electron is delayed near the aromatic groups and therefore electron-vibrational coupling occurs. 

Here V is the matrix element which describes the coupling of the electronic states of reactants and products, S is known as the electron-vibration coupling constant which is equal to the inner-shell reorganization energy X- expressed in units of vibrational quanta,... 

A theory should just consider electron propagation with a weak probability of exciting a vibration. Perturbation theory seems to be justified due to the smallness of the electron-vibration coupling (Migdal s... 

The electron-vibration coupling V has the same symmetry of the vibration. This is because the Hamiltonian is totally symmetric under transformations of the point group of the ensemble molecule plus substrate. In Appendix we give further details in particular Eq. (A4) shows that in order to preserve the invariance of the Hamiltonian under transformation of the nuclear coordinates, the electronic coordinates must transform in the same way [37]. Hence, if a symmetric mode is excited, the electron-vibration coupling will also be symmetric in the electronic-coordinate transformations. Thus only electronic states of the same symmetry will give non-zero matrix elements for a symmetric vibration. This kind of reasoning can be used over the different vibrations of the molecule. 

Hamiltonian (2) is a very useful simplification to treat the interplay between electrons and vibrations in condensed matter systems. More complete approaches use some electronic structure calculation to go beyond the approximation of Hamiltonian (2). Nevertheless they remain at the same level of approximation concerning the electron-vibration coupling term ... 

All of the approaches reviewed in this article rely on two approximations the adiabatic one for the electronic structure calculation, and the linear one in the electron-vibration coupling Eq. (A.l). 

Yet, the information about the electron-vibration coupling is known because the adiabatic approximation has parametrized the electronic structure for each nuclear conformation. 

Figure 2.4. The energy-gap dependence of the nuclear Franck-Condon factor, which incorporates the role of the high-frequency intramolecular modes. Sc = A/2 is the dimensionless electron-vibration coupling, given in terms which reduce replacement (A) between the minimum of the nuclear potential surfaces of the initial and final electronic states. (Bixon and Jortner, 1999) Reproduced with permission.

It was quantitatively interpreted (Rice et al., 1977) as originating from bond alternation phase oscillations (in contrast to the bond alternation amplitude oscillations mentioned in subsection 4.8.2D). The vibrational absorption lines labeled 2 to 10 are directly related to the Ag Raman lines of TCNQ. The broad peak above 1600 cm originates from the single electron transition across the gap, and the indented line shape of mode 2 is a consequence of Fano interference between the single electron continuum and the phonon mode. The line intensities are determined by the respective electron - vibration coupling constants. 

The numbers of IR- and RS-active modes are 4 (4tiu) and 10 (2ag -i- 8hg), respectively. All other modes are IR- and RS-forbidden silent modes (au, tig, t2g, t2 , gg, gu. hu)- Conversely, the number of HRS-active modes are 22 (4tiu -i- 5t2u + 6g -i-7hu) including 18 silent modes. Since these HRS modes are mutually exclusive with RS-active modes, combination of resonance RS and resonance HRS should be useful for understanding the electron-vibration coupled system. 

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