Golden Rule expression

Femii s Golden Rule expresses the rate of transitions between b and a as... 

The first-order El "golden-rule" expression for the rates of photon-induced transitions can be recast into a form in which certain specific physical models are easily introduced and insights are easily gained. Moreover, by using so-called equilibrium averaged time correlation functions, it is possible to obtain rate expressions appropriate to a... 

So, the final result is the order golden rule expression ... 

Substitution of this for the golden-rule expression (1.14) together with the renormalized tunneling matrix element from (5.60) gives (5.64), after thermally averaging over the initial energies E-,. In the biased case the expression for the forward rate constant is... 

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 ... 

Thus, having prepared the system at the initial time t=0 in the state i f >, the probability of finding the system in the state f f > at time t is given, as usual, by Cf(t) 2. The Fermi Golden-rule expression (to first order in TDPT) has the form [47]... 

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... 

Recently, the electron-transfer kinetics in the DSSC, shown as a schematic diagram in Fig. 10, have been under intensive investigation. Time-resolved laser spectroscopy measurements are used to study one of the most important primary processes—electron injection from dye photosensitizers into the conduction band of semiconductors [30-47]. The electron-transfer rate from the dye photosensitizer into the semiconductor depends on the configuration of the adsorbed dye photosensitizers on the semiconductor surface and the energy gap between the LUMO level of the dye photosensitizers and the conduction-band level of the semiconductor. For example, the rate constant for electron injection, kini, is given by Fermi s golden rule expression ... 

The great success of Forster theory lies on the simplicity of these expressions, which can be applied from purely spectroscopic data. However, the approximations underlying these equations are not evident at first sight. It is better to turn to the Golden Rule expression of the rate ... 

The transitions rates are defined by the same golden rule expressions, as before, but with explicitly shown single-particle state a... 

Both approaches rest upon the Golden Rule expression for the photodissociation cross section and comprise the same basic assumption, namely the weak interaction between the light pulse on one hand and the molecule on the other hand (Henriksen 1988). 

Fig. 12.2. Schematic illustration of the absorption of an IR photon by the van der Waals complex Ar H2 and the subsequent dissociation into Ar+H2(n = 0). Voo(R) and Vn(i ) are the diagonal elements of the potential coupling matrix defined in Equation (3.6) which serve to define the zero-order radial wavefunc-tions employed in the Golden Rule expression for the dissociation rate. The assignment of the bound levels is (m,n), where m and n denote the number of quanta of excitation in the dissociation mode R and the vibrational mode of H2, respectively.

The manifestation of the dipole-dipole approximation can be seen explicitly in Equation (3.134) as the R 6 dependence of the energy transfer rate. In Equation (3.134) the electronic and nuclear factors are entangled because the dipole-dipole electronic coupling is partitioned between k24>d/(td R6) and the Forster spectral overlap integral, which contains the acceptor dipole strength. Therefore, for the purposes of examining the theory it is useful to write the Fermi Golden Rule expression explicitly,... 

The nature of the molecular system implements a change in the physical mechanism of the photoabsorption process. Once again we may, however, employ the golden rule expression. We use it in a general sense to express the absorption rate in the form of ... 

In eq. (5-2), vt, refers to the frequency of the ith chromophore vibration (populated by the IVR transition), and the proportionality to l/v( is purely phenomenological. This proportionality reflects the expectation that low frequency chromophore modes will couple most efficiently to the (low frequency) vdW modes. Fermi s Golden Rule expression has two important consequences. First, it predicts that... 

The second theoretical approach is quantum mechanical in nature and is based on the Fermi Golden Rule expression for nonradiative decay processes [45,46]... 

Equation (34) is the Fermi Golden Rule expression, where Hc, is the electronic interaction, and FC is the F ranck-Condon factor. The analytical version of Eq. (34), applicable to high temperature, is given by Eq. (35) ... 

The matrix element for inelastic electron-electron scattering is given by the following golden-rule expression (spin-averaged) /85/... 

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). 

Solution of the golden-rule expression for the non-adiabatic ET rate constant by applying the saddle-point (or stationary phase) approximation yields [36],... 

The rate constant for the energy transfer process is thus given by a golden rule expression similar to that seen above (Eq. 5) for electron transfer. 

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