Franck-Condon term

Franck-Condon terms are neglected, the exciplex (charge-transfer) energy relative to the unexcited configuration energy is given approximately by... 

We conclude that the basic trend of increasing rate with increasing AG in the c/b5 system primarily reflects a Franck Condon term rather than an electronic term. However, since small rate differences may be physiologically significant, "tuning" of the electronic factor is certainly worthy of further study. 

In the above, FC is the Franck-Condon term and is defined as follows ... 

In this approach k is proportional to the square of the donor-acceptor electronic coupling matrix element (//DA) and a Franck-Condon term that contains the dependence of the ET rate on AGgy, X and factors related to the molecular structure,... 

Applying Fermiis golden rule, the rate of the electron transfer reaction is determined by the product of the probability of the nuclear transition occurring (the Franck-Condon term, FC)) and the probability of the electron tunnelling occurring ... 

In certain cases, the classical Marcus formula is not sufficient to explain the observed-dependence of the electron transfer rate on temperature or AG, which could indicate that it is necessary to use a Franck-Condon term in which the contribution of the nuclei is treated in quantum mechanical terms. In this treatment, the Franck-Condon term equals the thermally-weighted sum of the contributions from all possible vibrational states of the reactants, each multiplied by their Franck-Condon factor i.e. the square of the overlap integral of a nuclear wave function of the reactant with the nuclear wave function of the product state that has the same total energy. 

The preexponential factor v is now the Franck-Condon term which includes the free-energy change and the reorganization energy (which can also depend on separation). When the donors and acceptors lack spherical symmetry, there will also be an orientation dependence. When the donor and acceptor moieties are linked by covalent bonds, there is a possibility that the electron transfer event involves the orbitals of the linkage bonds. Evidence for this effect has been noted in certain kinds of compounds [46]. The distance dependence of photoinitiated electron transfer has recently been reviewed [21]. 

Fig. 4. Diagrams illustrating the Franck-Condon terms in the scattering tensor.

Electronic transitions are said to be vertical, meaning that absorption and emission of UV-vis radiation occurs with no movement of the nuclei, within the limits of the Bom-Oppenheimer approximation. Since most molecules are in the 0th vibrational level of the ground electronic state, the Franck-Condon term measures the probability of the transition from the 0th vibrational level of the ground state to each vibrational level of the excited state. A transition from the 0th vibrational level of So to the 0th vibrational level of Si is marked... 

In some cases the rate constant for formation of the cis isomer k ) is greater than for formation of the trans isomer (Ir,). Because the cis isomer is higher in energy than the trans, there may be better overlap of the vibrational component of the wave function of the cis ground state with the vibrational component of the excited state than is the case with the trans isomer. Thus, the Franck-Condon term for the twisted singlet to cis reaction could be more favorable than for the trans. 

Strong, the correction terms will usually be negligible compared to the Franck-Condon term, so that REPs of totally symmetric inducing modes do not show the same symmetry between the scattering cross sections of borrowing and lending states as those of nontotally symmetric modes. 

Chemical enhancement is typically explained by the CT mechanism. When a molecule is adsorbed on a metal surface, new electronic states are formed due to chemisorption. The new electronic states may serve as resonant intermediate states in Raman scattering. If the Fermi level of the metal is located between the Highest Occupied Molecular Orbital (HOMO) Lowest Unoccupied Molecular Orbital (LUMO) in energy, CT excitations may likely occur at lower energy than intrinsic intramolecular excitations of the adsorbate [55-58]. According to Albrechts notation [55], in the CT mechanism via Albrechts A term Franck-Condon term) only the totally symmetric modes are resonantly enhanced when the laser excitation is close to an allowed electronic transition, and only one excited state is involved. The resonance Raman effects for vibrational modes that are non-totally symmetric, are usually observed when these modes couple two excited states of the chromophore. The product of the symmetry of both excited states should be equal or contain the non-totally symmetry. This mechanism is known as the Herzberg-Teller mechanism or B mechanism in Albrechts notation. 

Contrary to what happens with the related expression (1.5) for radiative transitions, the Franck-Condon term in (1.7) is made up of a single overlap integral for each vibrational mode, corresponding to the unique isoenergetic transition. In order to discuss the role of the Franck-Condon factor in radiationless transitions, it is worthwhile considering the two archetypal simations shown in Fig. 1.10. 

Section 6.13.2 and illustrated in Figure 6.5. The possible inaccuracies of the method were made clear and it was stressed that these are reduced by obtaining term values near to the dissociation limit. Whether this can be done depends very much on the relative dispositions of the various potential curves in a particular molecule and whether electronic transitions between them are allowed. How many ground state vibrational term values can be obtained from an emission spectrum is determined by the Franck-Condon principle. If r c r" then progressions in emission are very short and few term values result but if r is very different from r", as in the A U — system of carbon monoxide discussed in Section 7.2.5.4, long progressions are observed in emission and a more accurate value of Dq can be obtained. 

Excited-State Relaxation. A further photophysical topic of intense interest is pathways for thermal relaxation of excited states in condensed phases. According to the Franck-Condon principle, photoexcitation occurs with no concurrent relaxation of atomic positions in space, either of the photoexcited chromophore or of the solvating medium. Subsequent to excitation, but typically on the picosecond time scale, atomic positions change to a new equihbrium position, sometimes termed the (28)- Relaxation of the solvating medium is often more dramatic than that of the chromophore... 

Sethna [1981] considered two limiting cases. The calculation of action in the fast flip approximation (a>j CO ) proceeds by utilizing the expansion exp ( — cu,-1t ) 1 — cu t. After substituting the first term, i.e. the unity, in (5.72) we get precisely the quantity which yields the Franck-Condon factor in the rate constant. The next term cancels the adiabatic renormalization and changes KM)... 

The Franck-Condon model, which decomposes the initial quasistable state of the wave function in terms of free-rotor states and gives the product rotational distribution if no torques are present, was apphed to Ne CI2 and He CI2 and correctly predicted the low / behavior of the distribution, as well as the invariance of the distributions in the Av = 1 and —2 channels [99, 100]. 

Developed into a power series in R 1, where R is the intermolecular separation, H exhibits the dipole-dipole, dipole-quadrupole terms in increasing order. When nonvanishing, the dipole-dipole term is the most important, leading to the Forster process. When the dipole transition is forbidden, higher-order transitions come into play (Dexter, 1953). For the Forster process, H is well known, but 0. and 0, are still not known accurately enough to make an a priori calculation with Eq. (4.2). Instead, Forster (1947) makes a simplification based on the relative slowness of the transfer process. Under this condition, energy is transferred between molecules that are thermally equilibriated. The transfer rate then contains the same combination of Franck-Condon factors and vibrational distribution as are involved in the vibrionic transitions for the emission of the donor and the adsorptions of the acceptor. Forster (1947) thus obtains... 

Certain features of light emission processes have been alluded to in Sect. 4.4.1. Fluorescence is light emission between states of the same multiplicity, whereas phosphorescence refers to emission between states of different multiplicities. The Franck-Condon principle governs the emission processes, as it does the absorption process. Vibrational overlap determines the relative intensities of different subbands. In the upper electronic state, one expects a quick relaxation and, therefore, a thermal population distribution, in the liquid phase and in gases at not too low a pressure. Because of the combination of the Franck-Condon principle and fast vibrational relaxation, the emission spectrum is always red-shifted. Therefore, oscillator strengths obtained from absorption are not too useful in determining the emission intensity. The theoretical radiative lifetime in terms of the Einstein coefficient, r = A-1, or (EA,)-1 if several lower states are involved,... 

Evaluation of the Work Term from Charge Transfer Spectral Data. The intermolecular interaction leading to the precursor complex in Scheme IV is reminiscent of the electron donor-acceptor or EDA complexes formed between electron donors and acceptors (21). The latter is characterized by the presence of a new absorption band in the electronic spectrum. According to the Mulliken charge transfer (CT) theory for weak EDA complexes, the absorption maximum hv rp corresponds to the vertical (Franck-Condon) transition from the neutral ground state to the polar excited state (22). 

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