Radiationless electronic transitions

Thus, in the classical limit, the consequence of the energy-conserving golden-rule expression (Eq. 31) is simply to invoke the classical density of states at the transition state (i.e., the crossing, where g — g = 0, the point to which a radiationless electronic transition is confined due to the constraints of the Franck-Condon principle). 

The quantity of primary interest for the description of radiationless electronic transitions is the time-dependent population probability of excited electronic states. The population Pn t) of the nth diabatic electronic state is defined as the expectation value of the projection operator... 

Figure 6.11. Electronic transitions to the first excited singlet (s) and lowest triplet (0 states from the ground states (g) of benzophenone (B) and naphthalene (N) moieties in compounds (4), n = 1-3. Possible radiative transitions are represented by straight arrows, radiationless transitions by wavy arrows.(80> Reprinted by permission of the American Chemical Society.

Instead of the quantity given by Eq. (15), the quantity given by Eq. (10) was treated as the activation energy of the process in the earlier papers on the quantum mechanical theory of electron transfer reactions. This difference between the results of the quantum mechanical theory of radiationless transitions and those obtained by the methods of nonequilibrium thermodynamics has also been noted in Ref. 9. The results of the quantum mechanical theory were obtained in the harmonic oscillator model, and Eqs. (9) and (10) are valid only if the vibrations of the oscillators are classical and their frequencies are unchanged in the course of the electron transition (i.e., (o k = w[). It might seem that, in this case, the energy of the transition and the free energy of the transition are equal to each other. However, we have to remember that for the solvent, the oscillators are the effective ones and the parameters of the system Hamiltonian related to the dielectric properties of the medium depend on the temperature. Therefore, the problem of the relationship between the results obtained by the two methods mentioned above deserves to be discussed. 

In Chapters 2 and 4, the Franck-Condon factor was used to account for the efficiency of electronic transitions resulting in absorption and radiative transitions. The efficiency of the transitions was envisaged as being related to the extent of overlap between the squares of the vibrational wave functions, /2, of the initial and final states. In a horizontal radiationless transition, the extent of overlap of the /2 functions of the initial and final states is the primary factor controlling the rate of internal conversion and intersystem crossing. 

Dissociation at a surface appears to be analogous to dissociation in the gas phase. The impinging electron causes a Franck-Condon transition to an electronic state which subsequently dissociates. This one-dimensional Franck-Condon excitation model is illustrated schematically in Fig. 31. The cross section for the electronic transition is probably comparable to gas phase excitation processes. After excitation the particle, which is now in a repulsive state, begins to move away from the surface. If it has sufficient energy it may escape from the surface. If not the fragments remain adsorbed. Moreover, radiationless de-excitation may occur... 

Charge transfer between electronic states in the electrode and solution are radiationless processes whose standard rate coefficients span over a considerable range of magnitude. The electronic transition occurs between levels of the same energy for, otherwise, radiation would be emitted. 

The finite lifetime of each excited state is the reflection of a fundamental law of nature - tendency towards minimum total energy of a system. The quantum mechanical system tends to occupy the state in which its total energy would be minimal. However, the transition of an atom to the lowest (ground) state depends on many circumstances (first of all, on the sort of excited configuration, on the presence of external fields, on the character of the matter itself - density of gas, vapours or plasma, etc.). There are two main channels of decay of the excited states radiative and radiationless. In the first case the electronic transition from the higher to the lower state is connected with the radiation of one or several quanta of... 

The energy stored in an excited state is dissipated by the unimolecular radiative and radiationless relaxations. For strongly allowed electronic transitions, Strickler and Berg have obtained an expression for the rate constant of the excited-state radiative decay (Equation 6.67).31... 

The theory of the multi-vibrational electron transitions based on the adiabatic representation for the wave functions of the initial and final states is the subject of this chapter. Then, the matrix element for radiationless multi-vibrational electron transition is the product of the electron matrix element and the nuclear element. The presented theory is devoted to the calculation of the nuclear component of the transition probability. The different calculation methods developed in the pioneer works of S.I. Pekar, Huang Kun and A. Rhys, M. Lax, R. Kubo and Y. Toyozawa will be described including the operator method, the method of the moments, and density matrix method. In the description of the high-temperature limit of the general formula for the rate constant, specifically Marcus s formula, the concept of reorganization energy is introduced. The application of the theory to electron transfer reactions in polar media is described. Finally, the adiabatic transitions are discussed. 

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