Radiationless golden rule

Let us now consider how similar the expression for rates of radiationless transitions induced by non Bom-Oppenheimer couplings can be made to the expressions given above for photon absorption rates. We begin with the corresponding (6,4g) Wentzel-Fermi golden rule expression given in Eq. (10) for the transition rate between electronic states Ti,f and corresponding vibration-rotation states Xi,f appropriate to the non BO case ... 

Electron transfer reactions have also been treated from the quantum mechanical point of view in formal analogy to radiationless transitions, considering the weakly interacting states of a supermolecule AB the probability (rate constant) of the electron transfer is given by a golden rule expression of the type17... 

This form of the rate constant of the radiationless transition is known as the golden rule . It depends on two limiting conditions, which are ... 

The rate constant for the radiationless relaxation of the excited states of inorganic compounds has been derived on the basis of the Fermi s golden rule (Equation 6.69). 

The transition probability per unit time given by the time-dependent perturbation theory, that Fermi named Golden Rule in view of its prevalence in radiationless transitions, has the form... 

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 quantum mechanical treatment of non-adiabatic electron transfers are normally considered in terms of the formalism developed for multiphonon radiationless transitions. This formalism starts from Fermi s golden rule for the probability of a transition from a vibronic state Ay of the reactant (electronic state A with vibrational level v) to a set of vibronic levels B of the product... 

This is the most direct experimental manifestation of the existence of an electronic interaction. It can occur spontaneously in mixed-valence complexes, but also in bimetallic systems after a photochemical excitation (photoinduced electron transfer). The general theory considers electron transfer as a special case of radiationless transition, with a perturbative treatment based on Fermi s Golden Rule [42]. In the nonadiabatic case, the rate constant can be written as [43] ... 

In the golden rule approach, developed by Jortner, Bixon and others (Kestner et al., 1974 Ulstrap and Jortner, 1975 Jortner, 1976 Siders and Marcus, 1981a and 1981b Bixon and Jortner, 1982), D and A are treated as weakly coupled but distinct entities and ET as a nonadiabatic radiationless transition between them governed by Fermi s golden rule, which may be written in the form... 

In the large molecule statistical limit, the rate constant knr for the nonradiative (radiationless) transition from the initial state tJ/s of energy Es to the final state ijjt of energy Ef can be expressed by the Golden rule... 

As seen in Sec. 2.3, conical intersections that mediate unsuccessful chemical reactions have been shown to provide the decay channel associated with processes that are usually thought to occur through a photophysical mechanism (e.g. controlled by the Fermi Golden Rule ) such as the radiationless deactivation and/or quenching processes. Furthermore, important organic chromophores have been demonstrated to undergo either photochemical reactions or internal conversion processes on an extremely fast (usually sub-picosecond) time scale (i.e. emission is not observed from the excited state, since the time scale of the reaction is faster than the radiative lifetime). 

The simplest approach to estimate the probability of predissociation due to spin-orbit coupling is the use of the Fermi-Wentzel-Rice golden rule. The line width for radiationless transition from the bound state level Xv to a level of the continuum at energy e can be approximated by... 

For radiationless transitions due to nonadiabatic coupling the golden rule formula can be used in an equivalent manner as a first approximation the spin-orbit matrix element in equation (23) must then to be replaced by the nonadiabatic coupling term. 

Equation (1.5) is the simplest of all Golden Rule expressions because both the electronic matrix element and the Franck-Condon factor are taken as averages over all interacting vibronic states. A better model employs a Franck-Condon weighted density of states, Pf(F), in order to account for the fact that not all states in the dense manifold couple with i> with the same probability. In any case, the variation in the Franck-Condon factors with electronic energy gap, AE = Ej - Ef, determines the relative rates of radiationless transitions in compounds that contain the same chromophores and hence exhibit similar values of Ujf. The relative magnitudes of the Franck-Condon factors for different vibrational modes also determines the nature of the accepting modes populated preferentially by the radiationless transition. 

From a quanmm mechanical viewpoint, both the photoinduced and back-electron transfer processes can be viewed as radiationless transitions between different, weakly interacting electronic states of the A-L-B supermolecule (Fig. 2.6). The rate constant of such processes is given by an appropriate Fermi golden rule expression ... 

Energy transfer, particularly in supramolecular systems, can be viewed as a radiationless transition between two locahzed , electronically excites states (2.15). Therefore, the rate constant can be again obtained by an appropriate golden rule expression ... 

Fermi called eq. (15.36) the Golden Rule of time-dependent perturbation theory because of its prevalence in radiationless transitions. Sometimes it is referred to as Fermi s Golden Rule. 

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