The Thermally Averaged Franck-Condon Factor

In the following discussion we first consider the form of //ab and then expressions for the thermally averaged Franck-Condon factor, FC. 

Three broad classes of vibrational modes may contribute to the thermally averaged Franck-Condon factor the high-frequency (fast) modes hv > 1000 cm ) which are mainly intraligand vibrations, intermediate modes (1000 cm > hv > 100 cm ) that typically include the metal-ligand stretching vibrations and higher frequency solvent orientational-vibrational modes, and the low-frequency (slow) modes hv < 100 cm ) which are primary solvent modes but can include low-frequency intramolecular modes. At ordinary temperatures hv kT k hv hv and the low-frequency modes can be treated using classical (continuum) expressions. 

The thermally averaged Franck-Condon Factor at high temperatures is given by... 

The FCWD term (often referred to as the "nuclear factor of the rate constant) is a thermally averaged Franck-Condon factor connecting the initial and final states. It contains a sum of overlap integrals between the nuclear wavefunctions of reactant and product of the same energy. The nuclear wavefunctions include both (inner) vibrational modes and (outer) solvent reorganizational modes. The summation is made over the initial levels of the... 

The function G in eq 1 is the Franck-Condon factor which accounts for the contribution of nuclear degrees of freedom and represents the thermal average of the overlap integrals between nuclear wavefunctions with respect to conservation of energy, and is given by (2, 3, 8, 9)... 

The branching ratios are proportional to the Franck-Condon factors for the molecular ionization transition, with the projectile at distance R from the molecule. The bl,(R) depend parametrically on R. For the bj,(R) to be well defined in an actual collision with velocity R, it must be true that R is small compared to the vibrational intramolecular motion. This is a severe limitation, even at thermal velocities. Therefore, only when the interaction between the projectile and the molecule is weak, leading to a correspondingly weak variation of the b (R) in the course of the collision, approximation (III.2) may be used. We may then further approximate by introducing an average branching ration defined by... 

Where V is the effective electronic coupling and O is the thermally averaged nuclear vibrational Franck-Condon factor. Also the constant h is equal to h/27i, where h is the Planck s constant. 

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