Approximation Condon

At this stage we may distinguish between excitation involving different electronic states and excitation occurring within the same electronic (ground) state. Wlien the spectroscopic states are located in different electronic states, say the ground (g) and excited (e) states, one frequently assumes the Franck-Condon approximation to be applicable ... 

Albrecht A C, Clark R J H, Oprescu D, Owens S J R and Svensen C 1994 Overtone resonance Raman scattering beyond the Condon approximation transform theory and vibronic properties J. Chem. Phys. 101 1890-903... 

Using the Condon approximation, the transition dipole moment is taken to be a constant with respect to the nuclear coordinates. Equation (26) then reduces to the familiar expression... 

In an ambitious study, the AIMS method was used to calculate the absorption and resonance Raman spectra of ethylene [221]. In this, sets starting with 10 functions were calculated. To cope with the huge resources required for these calculations the code was parallelized. The spectra, obtained from the autocorrelation function, compare well with the experimental ones. It was also found that the non-adiabatic processes described above do not influence the spectra, as their profiles are formed in the time before the packet reaches the intersection, that is, the observed dynamic is dominated by the torsional motion. Calculations using the Condon approximation were also compared to calculations implicitly including the transition dipole, and little difference was seen. 

The Franck-Condon approximation (see Section 7.2.5.3) assumes that an electronic transition is very rapid compared with the motion of the nuclei. One important result is that the transition moment for a vibronic transition is given by... 

Owing to the separation of the active and inactive modes, in the Condon approximation the matrix element (2.56) breaks up into the product of overlap integrals for inactive modes and a constant factor V responsible for interaction of the potential energy terms due to the active modes. In this approximation the survival probability of Ai develops in time as... 

EPR studies on electron transfer systems where neighboring centers are coupled by spin-spin interactions can yield useful data for analyzing the electron transfer kinetics. In the framework of the Condon approximation, the electron transfer rate constant predicted by electron transfer theories can be expressed as the product of an electronic factor Tab by a nuclear factor that depends explicitly on temperature (258). On the one hand, since iron-sulfur clusters are spatially extended redox centers, the electronic factor strongly depends on how the various sites of the cluster are affected by the variation in the electronic structure between the oxidized and reduced forms. Theoret-... 

Following Ref. 14, in the classical limit in the Condon approximation we transform Eq. (16) to the form... 

Equation (47) shows that in the Condon approximation the probabilities of forward and reverse transitions satisfy the detailed balance principle since the point q corresponds to the intersection of the potential energy surfaces (and free energy surfaces) where Haa = Hbb. Therefore, at the point q we have... 

If the dependence of nA and nB on q is taken into account in the calculation of the statistical operators for heavy particles, we obtain the improved Condon approximation (ICA) which differs from Eq. (17) only by the change of p and p°f to p, and pf, respectively. In the classical limit for p, and p/ the expression for the transition probability takes the form... 

Using Eq. (49) one may consider all the cases of the Condon approximation (CA, ICA), and in some models go beyond the Condon approximation (BCA). 

It has been shown16 that in the Condon approximation the value of the polarization in the transitional configuration P is equal to... 

To take into account the additional effect of diagonal dynamic disorder in the improved Condon approximation it was suggested in Ref. 16 that fluctuations of the polarization of the type... 

Kinetic Parameters" for the Symmetric Transition in the Improved Condon Approximation... 

Kinetic Parameters for the Symmetric Transition Taking Account of the Modulation of the Zeroth-Order Electron Densities (Beyond the Condon Approximation)... 

Figure 4. Path of the transition on the potential energy surfaces in the Condon approximation and beyond the Condon approximation.

The calculation of the integrals in Eq. (55) in the classical limit in the improved Condon approximation (for the nuclear subsystem) using the saddle point method leads to two coupled equations for the electron wave functions of the donor and the acceptor in the transitional configuration ... 

A direct variational method was used in Refs. 23 and 24 to go beyond the Condon approximation. Functions of the type... 

In the classical limit, the transitional configuration q, Q in the Condon approximation is determined by the equations... 

Next we consider the triplet-triplet transition. In the Condon approximation we have... 

Unfortunately, the above analysis can never be widely applicable to the determination of excited-state geometries since so few molecules and ions exhibit vibronically structured absorption bands and excitation profiles, even at low temperatures. Moreover, some questions arise as to the possible breakdown of the Condon approximation. Other types of molecule for which similar analyses have been carried out include 3-carotene, carotenoids (9) and certain carotenoproteins such as ovorubin (10). In these cases the excitation profiles of three skeletal a bands are monitored, and estimates for the change in C-C and C=C bonds lengths ( 0.02 A) have been made. 

The integral in large parentheses is over the electronic coordinates r only, and still depends on the nuclear coordinates R. At this stage we invoke the Condon approximation, which is familial from the theory of electronic spectroscopy. Because of the large nuclear mass the wave-functions Xi and Xf are much more strongly localized than the electronic wavefunctions 4>i and some value R, and it suffices to replace the electronic matrix element by its value M at R. So we write ... 

Charge Localized vs. Delocalized Wavefunctions. In the spirit of the Condon approximation discussed above, we do not include the full dependence of the purely electronic matrix elements on q n, but rather evaluate them where the diabatic curves... 

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