Golden-rule transition rates

The contribution to the X-ray absorption coefficient due to the excitation of a deep core level may be expressed as /rc = nco-c, where nc is the density of atoms with the core level of concern and absorption cross section for this level on a single atom. Assuming the X-ray field to be a small perturbation, the latter can be evaluated from the golden rule transition rate per unit photon flux. The general X-ray absorption cross section is given by... 

Equations (12.55), sometime referred to as multiphonon transition rates for reasons that become clear below, are explicit expressions for the golden-rule transitions rates between two levels coupled to a boson field in the shifted parallel harmonic potential surfaces model. The rates are seen to depend on the level spacing 21, the normal mode spectrum mo,, the normal mode shift parameters Ao-, the temperature (through the boson populations ) and the nonadiabatic coupling... 

Golden-rule transition rates 12.4.1 The decay of an initially prepared level... 

Field ionization can also be treated as a direct electronic transition from the atomic state to a vacant state at the surface. Following the Fermi golden rule, the rate of field ionization is given by31... 

Applying Fermiis golden rule, the rate of the electron transfer reaction is determined by the product of the probability of the nuclear transition occurring (the Franck-Condon term, FC)) and the probability of the electron tunnelling occurring ... 

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

Note that if we identify the sum over 8-fimctions with the density of states, then equation (A1.6.88) is just Femii s Golden Rule, which we employed in section A 1.6.1. This is consistent with the interpretation of the absorption spectmm as the transition rate from state to state n. 

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... 

A simple method for predicting electronic state crossing transitions is Fermi s golden rule. It is based on the electromagnetic interaction between states and is derived from perturbation theory. Fermi s golden rule states that the reaction rate can be computed from the first-order transition matrix and the density of states at the transition frequency p as follows ... 

The golden rule is a reasonable prediction of state-crossing transition rates when those rates are slow. Crossings with fast rates are predicted poorly due to the breakdown of the perturbation theory assumption of a small interaction. 

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 ... 

The microscopic rate constant is derived from the quantum mechanical transition probability by considering the system to be initially present in one of the vibronic levels on the initial potential surface. The initial level is coupled by spin-orbit interaction to the manifold of vibronic levels belonging to the final potential surface. The microscopic rate constant is then obtained, following the Fermi-Golden rule, as ... 

Energy transfer in solution occurs through a dipole-dipole interaction of the emission dipole of an excited molecule (donor) and the absorptive moment of a unexcited molecule (acceptor). Forster<40) treated the interaction quantum mechanically and derived and expression for the rate of transfer between isolated stationary, homogeneously broadened donors and acceptors. Dexter(41) formulated the transfer rate using the Fermi golden rule and extended it to include quadrupole and higher transition moments in either the donor or the acceptor. Following the scheme of Dexter, the transfer rate for a specific transition is... 

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 ... 

For a quantitative treatment of establishing connections between vibronic coupling and vibrational progressions in electronic spectra, band profiles from vibronic wavefunctions must be calculated using common procedures of time-dependent perturbation theory and Fermi s golden rule [57], For emission, e.g., the transition rate which is the transition probability per unit time summed over... 

Dick [1977] explained this behavior within the framework of a phonon-assisted tunneling mechanisms using the TLS approximation and golden rule formalism (see Sections 2.3 and 6.4). One-phonon transitions dominate the mechanism at low temperatures, resulting in a linear dependence of k with 7 this follows directly from relation (6.27) when j3/Wl. At higher temperatures, the main contribution comes from Raman processes, leading to a T4 dependence of the rate constant. This predicted T4 temperature dependence for RbBr OH- is analogous to results obtained by Silbey and Trommsdorf [1990] for two-proton transfer in benzoic acid crystals (see Section 6.4). 

Let us consider first the T = 0 case. If the non-diagonal interaction Hint is weak then the rate of the non-radiative transition is determined by the Fermi golden rule... 

Fig. 4. The temperature dependence of the rate of non-radiative transitions y for some (given) values of the interaction parameter w parameters co0 and cr0 are the same as in Fig. 1(a). The sharp peaks result from the divergence of the resolvent in equation (31) for w > 10 at some positive co > o>m- F°r comparison, the golden rule result is also presented (the thick line below).

The next step is to treat tunneling as a perturbation. Following this idea, the transition rates TAA from the state A to the state A are calculated using the Fermi golden rule... 

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