Partition function adiabatic approximation

These equations lead to fomis for the thermal rate constants that are perfectly similar to transition state theory, although the computations of the partition functions are different in detail. As described in figrne A3.4.7 various levels of the theory can be derived by successive approximations in this general state-selected fomr of the transition state theory in the framework of the statistical adiabatic chaimel model. We refer to the literature cited in the diagram for details. 

Now we make the usual assumption in nonadiabatic transition theory that non-adiabaticity is essential only in the vicinity of the crossing point where e(Qc) = 0- Therefore, if the trajectory does not cross the dividing surface Q = Qc, its contribution to the path integral is to a good accuracy described by adiabatic approximation, i.e., e = ad Hence the real part of partition function, Zq is the same as in the adiabatic approximation. Then the rate constant may be written as... 

The proper evaluation of the quantized energy levels within the SACM requires a separable reaction coordinate and thus numerical implementations have implicitly assumed a center-of-mass separation distance for the reaction coordinate, as in flexible RRKM theory. Under certain reasonable limits the underlying adiabatic channel approximation can be shown to be equivalent to the variational RRKM approximations. Thus, the key difference between flexible RRKM theory and the SACM is in the focus on the underlying potential energy surface in flexible RRKM theory as opposed to empirical interpolation schemes in the SACM. Forst s recent implementation of micro-variational RRKM theory [210], which is based on interpolations of product and reactant canonical partition functions, provides what might be considered as an intermediate between these two theories. 

At higher temperatures there can be some contribution from low lying excited electronic states (e.g., a second spin-orbit state). In this case, the multiple surface effects are generally treated via the consideration of the number of surfaces on which the potential is attractive. The single surface rate coefficient is then multiplied by the ratio fe given by the electronic partition function for the reactive surfaces divided by that for the reactants. This result corresponds to an adiabatic approximation for the electronic d3mamics. 

The full computation of the coupled partition function using the adiabatic approximation can be problematic. For each set of quantum numbers describing excitations of the fast vibrational modes, a distinct vibrationally adiabatic potential curve is generated for the torsion. In prineiple, this may... 

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