Golden rule formula

The first type of interaction, associated with the overlap of wavefunctions localized at different centers in the initial and final states, determines the electron-transfer rate constant. The other two are crucial for vibronic relaxation of excited electronic states. The rate constant in the first order of the perturbation theory in the unaccounted interaction is described by the statistically averaged Fermi golden-rule formula... 

Comparison of (1.14), (2.47a) and (2.60a) reveals the universality of the golden rule in the description of both the nonadiabatic and adiabatic chemical reactions. However, the matrix elements entering into the golden-rule formula have quite a different nature. In the case of an adiabatic reaction it comes from tunneling along the reaction coordinate, while for a nonadiabatic... 

The expression for the rate R (sec ) of photon absorption due to coupling V beriveen a molecule s electronic and nuclear charges and an electromagnetic field is given through first order in perturbation theory by the well known Wentzel Fermi golden rule formula (7,8) ... 

Spectral densities are positive, or at least nonnegative, functions of frequency. This follows from their physical interpretations as transition probabilities, and is clear analytically from the golden rule formula, (Eq. (3)). The positive nature of spectral densities is essential to the methods of analysis we will use in the next two sections. 

Within the framework of first-order perturbation theory, the rate constant is given by the statistically averaged Fermi golden rule formula ... 

For interacting electrons the calculation is a little bit more complicated. One should establish the relation between many-particle eigenstates of the system and single-particle tunneling. To do this, let us note, that the states /) and i) in the golden rule formula (83) are actually the states of the whole system, including the leads. We denote the initial and final states as... 

The V-B coupling Hamiltonian to first order in the three HOD dimensionless normal coordinates is Hv b = —2, c], l , where F, is the inter-molecular force due to the solvent exerted on the harmonic normal coordinate, evaluated at the equilibrium position of the latter. This force obviously depends on the relative separations of all molecules, and on their relative orientations. In the most rigorous quantum description of rotations, this term would depend on the excited molecule rotational eigenstates and of the solvent molecules. Instead rotation was treated classically, a reasonable approximation for water at room temperature. With this form for the coupling, the formal conversion of the Golden Rule formula into a rate expression follows along the lines developed by Oxtoby (2,53), with a slight variation to maintain the explicit time dependence of the vibrational coordinates (57),... 

Note that the results derived from this definition are equivalent to those derived from Fermi s golden rule [23]. Hence we refer to them as a Fermi s golden rule formula. 

Second, several observables are obtained as products of matrix elements that scale like IxY (therefore like and the density of states that scales like Q (Eq. (2.95) or (2.97) below). A well-known example is the golden rule formula (9.25) for the inverse lifetime of a local state interacting with a continuum. Such products remain finite and physically meaningful even when Q oo. 

Ubrium nuclear configuration is smaU. The interaction V(Xeq + q, B) between the oscillator and the surrounding bath B can then be expanded in powers of q, keeping terms up to first order. This yields V = C — Fq where C = V(xeq, B) is a constant and F = —(9 E/9 ) =o- When the effective interaction —Fq is used in the golden rule formula (9.25) for quantum transition rates, we find that the rate between states i and j is proportional to qy This is true also for radiative transition probabilities, therefore the same formalism can be apphed to model the interaction of the oscillator with the radiation field. 

Our model assumptions lead to exponential decay of the probability that the system remains in the initial state, where the decay rate ki is given by the so-called Fenni golden rule formula. 

The rate can be calculated by the Golden-rule formula. As discussed at length in Chapter 9, this assumes that the quasi-continuum of final states is broad and relatively unstmctured. 

The Golden Rule formula Eq. (7.5.16) for the FWHM and Eq. (7.5.9) for the level shift are expressed in terms of the unperturbed vibrational wavefunc-tions. For strong predissociations, this approximation becomes untenable. Exant methods exist that can determine both the linewidth and the level shift. One method consists of numerically solving the following coupled equations (Lefebvre-Brion and Colin, 1977 Child and Lefebvre, 1978) ... 

Figure 7.37. Variation of r versus (He)2. The full line corresponds to the v = 21 diabatic level of a model molecule. The arrow indicates the value of He for which the adiabatic v = 0 level coincides with the diabatic v = 21 level. The dotted line gives the F-values expected from the Golden Rule formula. [Adapted from data of the model of Child and Lefebvre (1978) and unpublished results.]...

The observed autoionization width is the result of decay into several different continua. Therefore, it is useful for calculations to define a quantity, the partial autoionization width, in which the initial and final states of the autoionization process are defined perfectly. One can then describe the total width, the only experimentally observable autoionization width, by summing over all possible final states. The initial state is specified by 1, n, and v, the final state by 2, v+ of the resultant ion, and e of the ejected electron. The partial autoionization width can be expressed by the Golden Rule formula ... 

As the bright state mixes with more and more dark states, the resultant lineshape evolves toward a smooth Lorentzian shape (Bixon and Jortner, 1968). The width of this composite line approaches that predicted by Fermi s Golden rule formula... 

The Golden Rule formula (9.5) for the mean rate constant assumes the Unear response regime of solvent polarization and is completely equivalent in this sense to the result predicted by the spin-boson model, where a two-state electronic system is coupled to a thermal bath of harmonic oscillators with the spectral density of relaxation J(o)) [38,71]. One should keep in mind that the actual coordinates of the solvent are not necessarily harmonic, but if the collective solvent polarization foUows the Unear response, the system can be effectively represented by a set of harmonic oscillators with the spectral density derived from the linear response function [39,182]. Another important point we would like to mention is that the Golden Rule expression is in fact equivalent [183] to the so-called noninteracting blip approximation [71] often used in the context of the spin-boson model. The perturbation theory can be readily applied to... 

Figure 9.2. Renormalized quantum energy gap law, that is, the activation energy —AP versus the free energy change of reaction Ap° — E IE vs. AG/ , in our notation), for H O with immersed donor and acceptor molecules of radii 3.5 A (point chaiges in spherical cavities) at different separation, as compared against the classical Marcus law. The parameter AP was evaluated from the Golden Rule formula by the method of steepest descents. The corresponding simulated relaxation spectrum is shown in Figure 9.3. (Reproduced from [41c] with permission. Copyright (1997) by the American Institute of Physics.)...

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