Golden rule rates

The golden rule rate expression is a standard quantum-mechanical result for the relaxation rate of a prepared state z) interacting with a continuous manifold of states 1/ The result, derived in Section 9.1, for this rate is the Fermi golden rule 

The equality (i E i ) = 0 for all i. i is a model assumption The picture is of two manifolds of states, / and / and an mteraction that couples between, but not within, them. 

)t is the quantum thermal average, (... )t = Tr[e ... ]/Tr[e ], and Tr denotes a trace over the initial manifold z. We have thus identified the golden rule rate as an integral over time of a quantum time correlation function associated with the interaction representation of the coupling operator. 

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The technique of complex-valued dielectric functions was originally applied to solvation problems by Ovchinnikov and Ovchinnikova [2] in the context of the electron transfer theory. They reformulated in terms of s(k, to) the familiar golden rule rate expression for electron transfer [3], This idea, thoroughly elaborated and extended by Dogonadze, Kuznetsov and their associates [4-7], constitutes a background for subsequent nonlocal solvation theories. 

The two-dimensional electron transfer diabatic free energy surfaces in Figure 7 have been analyzed with the Golden Rule rate expression given in Eq. 46. This analysis suggests that FT and EPT are possible for both systems, but FT is the dominant path due to significant overlap between the proton vibrational wave... 

In practive, however, only the lowest order nonvanshing contributions in (28) are considered. If these terms are then taken as the effective interactions vri, the approximate golden-rule rate is written as... 

This simple steady-state argument thus leads to the same golden rule rate expression, Eq. (9.25), obtained before for this model. 

The golden-rule rate expressions obtained and discussed above are very useful for many processes that involve transitions between individual levels coupled to boson fields, however there are important problems whose proper description requires going beyond this simple but powerful treatment. For example, an important attribute of this formalism is that it focuses on the rate of a given process rather than on its full time evolution. Consequently, a prerequisite for the success of this approach is that the process will indeed be dominated by a single rate. In the model of Figure 12.3, after the molecule is excited to a higher vibrational level of the electronic state 2 the relaxation back into electronic state 1 is characterized by the single rate (12.34) only provided that thermal relaxation within the vibrational subspace in electronic state 2 is faster than the 2 1 electronic transition. This is... 

Equations (13.26) and (13.29b) now provide an exact result, within the bilinear coupling model and the weak coupling theory that leads to the golden rule rate expression, for the vibrational energy relaxation rate. This result is expressed in terms of the oscillator mass m and frequency ca and in tenns of properties of the bath and the molecule-bath coupling expressed by the coupling density A ((a)g (a) at the oscillator frequency... 

Equation (13,35) is the exact golden-rule rate expression for the bilinear coupling model. For more realistic interaction models such analytical results cannot be obtained and we often resort to numerical simulations (see Section 13.6). Because classical correlation functions are much easier to calculate than their quantum counterparts, it is of interest to compare the approximate rate ks sc, Eq. (13.27), with the exact result kg. To this end it is useful to define the quantum correction factor... 

With this understanding, we can continue in two ways. First we can use the interaction (13.13) in the golden-rule rate expression—approach we take in Section 13.4.4. Alternatively, we may use the arguments that (1) transitions between states of the high-frequency impurity oscillator can occur with appreciable probability only during close encounters with a bath atom (see footnote 3), and (2) during such encounters, the interactions of the oscillators with other bath atoms is relatively small and can be disregarded, in order to view such encounters as binary collision events. This approach is explored in the next section. 

Problem 18.1. Show that if the 8 functions in golden-rule rate expressions like (18.5) are replaced by Lorentzians with constant width P independent of the individual transitions, the corresponding correlation function expressions for these rates, for example, Eqs (12.41) and (12.44) become... 

Evaluation of the golden-rule rate involves the absolute square of matrix elements ofthe form (2Dzj ° lAXa l lDxj ° 2Axj b- Equations (18.27) and (18.28) imply that such matrix elements can be constructed from dipole matrix elements of the individual molecules that take forms like (2d, //d I Id, ZrfJuD,... 

The overdamped limit gives the golden rule rate, with k proportional to the square of the coupling, J, and inversely proportional to the damping strength, /. Note that the population relaxation rate in the eigenstate representation is simply f M in both regimes. Thus,... 

Skinner, J. L. 1997, Semiclassical approximations to golden rule rate constants . 

Within the framework of the effective, time independent Hamiltonian of the QRA in equation (56), equation (117) is a golden rule rate coefficient for exponential decay. An obvious condition for the applicability of the QRA is given by the inequality (118) ... 

To conclude this subsection we observe that the utilization of the standard Fermi-Golden rule rate expression, which leads to Eq. 1, is subject to a number of constraints. In particular, when the adiabatic electronic energy gap between P BH and P B "H is too small ( kT at room T) its use would be questionable. Nevertheless, even in this case an exact theoretical treatment of the problem is unlikely to reveal dispersive kinetics at room T from the Q distribution. 

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