General Case of N-Coupled Modes

With this machinery in hand, we look at the general Nd problem, vhere N may be identified vith some or all molecular modes. Since the molecule in the sth electronic state generally does not share the symmetries of the Ith state, we must consider the largest subgroup common to both the s and I state conformations to classify the molecular normal modes. As we have already noted the mass-weighted normal coordinates of the I electronic state manifold and of an arbitrary secondary manifold s are linearly related 

In the notation of Section 3.2.1, the multidimensional GF associated with the s l transition is given by 

The overlap integral. Equation 4.58, may be evaluated according to the procedure used in Section 3.2. The only difference in the treatment in this section is that the generating function has a higher dimension and depends on the above 2N complex variables and z,((i = 1,2. N). As before, to facilitate integration we define a new set of coordinates q, and qj, 

Equations 4.59 and 4.60 guarantee that the transformation from to qj, and from to are of the same form as the transformation (4.57) from q[, to The only distinction is that the displacement vector in this transformation is. .. n and 0, respectively (see, for example, Equation 3.23). 

On substitution of Equations 4.59 and 4.60 into Equation 4.58 and considering the transformation between and q (and also between and q ) discussed above, we obtain 

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Fig. 17. A comparision of the temperature dependence of the line-shape function (G) of the transition probability for the multimode case (solid line) as against a single mode approximation (dashed line). Here the phonon frequency spectrum (A) is assumed to be of Gaussian form, A a>) = 2 2) 1,2exp [—(to — cu0)2/2

The addition of the extra degree of freedom leads to a correction to the one-dimensional result in the form of a prefactor which is just the reduced barrier frequency at the saddle point. This result is very general. The KGH expression for the rate in the spatial diffusion limit (Eq. (78)) is just a special case in which the bath modes are harmonic and the coupling between the system and the bath is bilinear. However, Eq. (118) is much more general, in fact it is not yet completely defined since we have not yet shown how to determine the transformation coefficients a0, j = 1,. . ., N. 

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