Multi-phonon theory of vibrational relaxation

A different approach to the calculation ofthe rates kf i, Eq. (13.40), and the related rates k and ks is to evaluate the force autocorrelation function associated with the interaction (13.13) and (13.14) and the coiTesponding force 

In the second equality we have expanded the coordinate deviation 5x in normal modes coordinates, and expressed the latter using raising and lowering operators. The coefficients are defined accordingly and are assumed known. They contain the parameter a, the coefficients of the normal mode expansion and the transformation to raising/lowering operator representation. Note that the inverse square root of the volume Q of the overall system enters in the expansion of a local position coordinate in normal modes scales, hence the coefficients scale like 

Recall that the interaction form (13.52) was chosen to express the close encounter nature of a molecule-bath interaction needed to affect a transition in which the molecular energy change is much larger than Awd where cfD is the Debye cutoff frequency of the thermal environment. This energy mismatch implies that many bath phonons are generated in such transition, as will be indeed seen below. 

This will be accomplished in two stages described in Problem 12-5 and Eq. (12.53). First we bring all operators onto a single exponent. This leads to 

Second we use the Bloch theorem (cf. Eq. (10.38)) that states that for a harmonic system, if A is linear in the coordinate and momentum operators, then (e )T = exp[(l/2)(242)y]. In our case 

The structure of Eqs. (13.53)-( 13.55) is similar to that encountered in the evaluation of rates in the spin-boson problem, Sections 12-4.2 and 12-4.3 (see Eqs (12.44), (12.49)-(12.50)) and our evaluation proceeds in the same way. We need to calculate the thermal average 

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