Golden rule approach

The Golden Rule approach has been used for many years by Levich, Dogonadze, and co-workers (39, 40), who have stressed the difference between the roles of "quantum" (high-frequency) and "classical" coupling modes in discussing the theory of electron transfer, and by a number of subsequent workers (16, 41). 

For typical outer-sphere exchanges at ordinary temperatures, it seems probable that the original assumption of Hush and of Marcus that barrier penetration is a comparatively minor effect is correct. Moreover> it is, in a particular case, quite simple to calculate. The more general questions to which we do not yet have an answer are how adequate is the Golden Rule approach in discussing tunnelling, and, in particular, what would be expected for systems strictly remaining on one surface (electronically adiabatic) A number of fundamental issues involved here have been discussed in a recent series of papers (42-45). 

Finally, I refer back to the beginning of this paper, where the assumption of near-adiabaticity for electron transfers between ions of normal size in solution was mentioned. Almost all theoretical approaches which discuss the electron-phonon coupling in detail are, in fact, non-adiabatic, in which the perturbation Golden Rule approach to non-radiative transition is involved. What major differences will we expect from detailed calculations based on a truly adiabatic model—i.e., one in which only one potential surface is considered [Such an approach is, for example, essential for inner-sphere processes.] In work in my laboratory we have, as I have mentioned above,... 

In this work the electronic predissociation from the A,B and B states has been studied using a time dependent Golden rule approach in an adiabatic representation. The PES s previously reported[31 ] to simulate the experimental spectrum[22] were used. Non-adiabatic couplings between A-X and B-X were computed using highly correlated electroiric wavefunctions using a finite difference method, with the MOLPRO package[42]. 

Other methods of calculating the O N separation dependent proton transfer rates, such as a Fermi Golden Rule approach (Siebrand et al. 1984), can also be employed. In this approach, two harmonic potential wells (e.g., O-H N and, O H-N) are considered to be coupled by an intermolecular term in the Hamiltonian. Inclusion of the van der Waals modes into this approximation involves integration of the coupling term over the proton and van der Waals mode wavefunctions for all initial and final states populated at a given temperature of the system. Such a procedure requires the reaction exothermicity and a functional form for the variation of the coupling as a function of well separation. In the present study, we employ the barrier penetration approach this approach is calculationally straightforward and leads to a clear qualitative physical picture of the proton transfer process. 

Using a Fermi s Golden Rule approach, if the coupling between the oscillator and the bath modes is weak, then, to first order, the transition rate from the first excited vibrational level to the ground state is given by (3)... 

Royea W. J., Fajardo A. M. and Lewis N. S. (1997), Fermi golden rule approach to evaluating outer-sphere electron-transfer rate constants at semiconductor/liquid interfaces , J. Phys. Chem. B 101, 11152-11159. 

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... 

If the predissociation line is sharp, indicating only a small probability of the resonance state breaking up, then a perturbation type approach may be used. This approach is very clearly described in Shapiro s paper of 1972 where he introduces and tests out numerical procedures for evaluating the bound-continuum integrals needed in both this approximate perturbation (or so called golden rule) approach and also in the exact theory of photodissociation processes. The basic theory of the golden rule expressions has been presented by Levine.It has also been carefully derived by Beswick and Jortner who have used it in a pioneering study of vibrational predissociation. 

These partial widths give the "final quantum state distributions in that they are proportional to the probability of the resonant state decomposing to yield a particular final state f . The total width (eq. 8) is the sum of all the partial widths (eq. 10). Recently, in an extremely thorough and elegant study, Halberstadt, swick and Janda have applied both the complete photodisocciation theory (eq. 4b) and the golden rule approach to the study of vibrational predissociation in the Ne-C 2 system. 

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