Frank-Condon factors

The rate constants for these relatively short range hole transfer reactions generally decrease exponentially with distance. Yet, characterizing these DNA-mediated reactions with the parameter (3 is a simplification and is certainly inappropriate in cases where the Frank-Condon factor varies with distance (such as has been observed for the acridine photooxidant). Keeping these limitations in mind, however, /i-values for DNA-mediated hole transfer of -0.6-0.7 A-1 have been suggested using several different oxidant-DNA assemblies (Ap, St, Ap radical cation). 

In the quantum mechanical formulation of electron transfer (Atkins, 1984 Closs et al, 1986) as a radiationless transition, the rate of ET is described as the product of the electronic coupling term J2 and the Frank-Condon factor FC, which is weighted with the Boltzmann population of the vibrational energy levels. But Marcus and Sutin (1985) have pointed out that, in the high-temperature limit, this treatment yields the semiclassical expression (9). 

SF-OD level with the basis set composed of a cc-pVTZ basis on carbons and a cc-pVDZ basis on hydrogens). These energies are very close to the MRPT values (26) of 0.72 and 0.83 eV (for the 1 fi and 1 Ai states, respectively). With regard to experiment, the lowest adiabatic state, 1 B, has not been observed in the photoelectron spectrum (40) because of unfavorable Frank-Condon factors. The experimental adiabatic energy gap (including ZPE) between the ground triplet state and the VA state is 0.70 eV. The estimated experimental >s 0.79 eV, which is 0.15 eV lower than the SF-OD estimate. 

Corresponding to the Frank-Condon principle is an associated concept called the Frank-Condon factors. Thus, when an electronic transition occurs from the vibrational levels of a lower vibrational state to the corresponding vibrational levels of a higher electronic state, there are various intensities of transition, depending on the vibrational states to which a transition is made. 

Development of the Frank-Condon principle in quantum mechanical terms (involving a transition dipole moment14) allows a calculation of the intensities referred to in terms of a series of Frank-Condon factors by which expressions for the transition probabilities are multiplied to obtain a net transition probability from one level to another for an electron-transfer process. 

The excitation of a molecule may result in a change of its electron and rotational-vibrational quantum numbers. In the adiabatic approximation," the total wavefunction of a molecule can be presented as a product of the electron wave and the rovibrational wavefunction. In those cases where the former is weakly affected by the changes in the relative position of the nuclei (this is usually the case with lower vibrational levels), we can use the Condon approximation considering the electron wavefunction only at equilibrium configuration of the nuclei. In this case the oscillator strength factorizes into an electron oscillator strength and the so-called Frank-Condon factor, which is the overlap integral of the vibrational wavefunctions of the initial and the final states of the molecule.115,116... 

There are two different temperature regimes of diffusive behavior they are analogous to those described by Holstein [1959] for polaron motion. At the lowest temperatures, coherent motion takes place in which the lattice oscillations are not excited transitions in which the phonon occupation numbers are not changed are dominant. The Frank-Condon factor is described by (2.51), and for the resonant case one has in the Debye model ... 

Rate Constants, Tunneling Matrix Elements, and Frank-Condon Factors for Orientational Relaxation of OH Dipoles in Alkali Halide Crystals... 

Finally, for Wbu,av we need to calculate the Frank-Condon factor av bu) 2 Which is given by... 

Radiationless transitions often show a pronounced deuteration effect. This effect should be discussed. Notice that coh — /2cod, and Sh — Sd/V2 and that the deuteration effect can be estimated by using the Frank-Condon factor for a displaced oscillator by using the energy law expression. In this case, it is found that... 

The relative contributions of A-and 6-term scattering under resonance conditions is a subject of considerable interest and different conclusions have been found for different systems. For example the A-term predominates for n-electron systems, e. g. polyenes, especially for their main intense absorption band (Warshel 1977). Vibrational wavefunc-tions of non-totally symmetric modes are more nearly orthogonal. Thus, vibrations may only derive their intensities from the 6-teim. A-term and 6-term enhancement can be distinguished experimentally by their excitation profiles. For A-term scattering a peak in the excitation profile is expected at the origin of the resonant electronic transition and subsidiary peaks at successive excited state vibrational levels. The amplitudes of the peaks depend on the successive Frank-Condon factors. For 6-term scattering excitation profile maxima are expected at the 0 0 and 1 0 positions for each of the mixing... 

The interchain hopping transport can be described by the probability Ql(x) -where x is the effective conjugation or delocalization length-of finding a comparable mean free path on another chain weighted by the Frank-Condon factor FC. The conductivity a can then be described as a function of the concentration C of polarons, multiplied by the integral over the interchain hopping probability Pp(x) [96, 97]. 

Where po is the dipole moment, Eo the incident electric field, and M(cOl) the field enhancement at the fi-equency of the laser line. f[n,m) is the Frank-Condon factor, defined as ... 

Q.25.4 Considering only AG° and X, when is the Frank-Condon factor maximized What is X and what does it represent ... 

A.25.4 The Frank-Condon factor has a maximum when AG = X. X is the reorganization energy and it represents the energy needed to distort the product state into the same shape as the initial state without electron transfer occurring. 

D14.2 The Franck-Condon principle states that because electrons are so much lighter than nuclei, an electronic transition occurs so rapidly compared to vibrational motions that the internuclear distance is relatively unchanged as a resu It of the transition. This implies that the most probable transitions vf <— vj are vertical. This vertical line will, however, intersect any number of vibrational levels Vf in the upper electronic state. Hence transitions to many vibrational states of the excited state will occur with transition probabilities proportional to the Frank-Condon factors which are in turn proportional to the overlap integral of the wavefunctions of the initial and final vibrational states. A vibrational progression is observed, the shape of which is determined by the relative horizontal positions of the two electronic potential energy curves. The most probable transitions are those to excited vibrational states with wavefunctions having a large amplitude at the internuclear position Re. 

P24.28 The theoretical treatment of section 24.11 applies only at relatively high temperatures. At temperatures above 130 K, the reaction in question is observed to follow a temperature dependence consistent with eqn 24.81, namely increasing rate with increasing temperature. Below 130 K, the temperature dependent terms in eqn 24.81 are replaced by Frank-Condon factors that is, temperature-dependent terms are replaced by temperature-independent wavefunction overlap integrals. 

Fig. 4. Low resolution emission spectra of some vibronic states of NO, obtained by two-photon excitation. Relative intensities of the bands of a given progression agree with those calculated from known Frank-Condon factors, except for the 0-0 bands which are attenuated by self-absorption. Intensities of bands due to different upper states were recorded at arbitrary sensitivity levels. The spectra shown are direct box car output recordings. Comparison with calculated spectra was made by taking into account the spectral response of the detection optics and the calibration curve of the monochromator.

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