Optical Franck-Condon factors

Absorption of light by molecules, resulting in electronic excitations, is caused by the interaction of the bound molecular electrons with the electric field of the radiation. In the dipolar approximation, the interaction of the dipole operator of the solute mo with the time-dependent electric field E(t) 

The extinction coefficient and emission rate are defined through the spectral density function G (v) that combines the effects of solvent-induced inhomogeneous broadening and vibrational excitations of the donor-acceptor complex. A substantial simplification of the description can be achieved if the two types of nuclear motions are not coupled to each other. The spectral density G (v) is then given by the convolution  

The vibronic envelope ECWD (v) in Eq. [129] can be an arbitrary gas-phase spectral profile. In condensed-phase spectral modeling, one often simplifies the analysis by adopting the approximation of a single effective vibrational mode (Einstein model) with the frequency Vv and the vibrational reorganization energy Xy. The vibronic envelope is then a Poisson distribution of 

Equation [134], given in the form of a weighted sum of individual solvent-induced line shapes, provides an important connection between optical band shapes and CT free energy surfaces. Before turning to specific models for the Franck-Condon factor in Eq. [134], we present some useful relations, following from integrated spectral intensities, that do not depend on specific features of a particular optical line shape. 

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A more general definition of the FCWD includes overlap integrals of quantum nuclear modes. The definition given by Eq. [19] includes only classical solvent modes (superscript s ) for which these overlap integrals are identically equal to unity. An extension of Eq. [19] to the case of quantum intramolecular excitations of the donor-acceptor complex is given below in the section discussing optical Franck-Condon factors. 

If the equilibrium position of the excited state C is located outside the configurational coordinate curve of the ground state, the excited state intersects the ground state in relaxing from B to C, leading to a nonradiative process. As described above, the shape of an optical absorption or emission spectrum is decided by the Franck-Condon factor and also by the electronic population in the vibrational levels at thermal equilibrium. For the special case where both ground and excited states have the same angular frequency, the absorption probability can by calculated with harmonic oscillator wavefunctions in a relatively simple form ... 

Fig. 19. Predicted dependence of the photoionization spectral dependence on the Franck-Condon factor [dF c—see Eq. (53)]. The parameter values are appropriate for the electron cross section ( ) for in GaP. The level depth is E, = 0.9 eV, the band gap is Et = 2.2 eV, the average optical gap (the Penn gap) is Ep = 5.8 eV, and the temperature is 400°K. [After Jaros (1977, Fig. 5e).]...

The free energy gap, equal to the energy of the incident light, is basis independent. It defines the Franck-Condon factor entering the optical band shapes. The analysis below follows this general scheme (Figure 4). 

Equation [48] gives the Franck-Condon factor that defines the probability of finding a system configuration with a given magnitude of the energy gap between the upper and lower CT free energy surfaces. It can be directly used to define the solvent band shape function of a CT optical transition in Eq. [134]... 

The Franck-Condon factors of polarizable chromophores in Eq. [153] can be used to generate the complete vibrational/solvent optical envelopes according to Eqs. [132] and [134]. The solvent-induced line shapes as given by Eq. [153] are close to Gaussian functions in the vicinity of the band maximum and switch to a Lorentzian form on their wings. A finite parameter ai leads to asymmetric bands with differing absorption and emission widths. The functions in Eq. [153] can thus be used either for a band shape analysis of polarizable optical chromophores or as probe functions for a general band shape analysis of asymmetric optical lines. 

The rate constant for ET can mathematically be regarded as the optical spectrum of a localized electron in the limit where the photon energy to be absorbed or emitted approaches zero. Erom the theory of radiative transitions [10, 12] and r / -b 1) = / for a positive integer /, we see that the factor multiplied to on the right-hand side of Eq. 27 represents the thermally renormalized value of the Franck-Condon factor [i.e., the squared overlap integral between the lowest phonon state in Vy(Q) and the ( AG /te)-th one in piQ)] for ET. The renormalization manifests itself in the Debye-Waller factor exp[—,vcoth( / (y/2)], smaller than e which appears also in neutron or X-ray scattering 12a]. Therefore, yen in Eq- 27 represents the effective matrix element for electron tunneling from the lowest phonon state in the reactant well with simultaneous emission of i AG /liw) phonons. 

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