Can the Bloch equations describe the vibrational spectra of a reacting molecule?
The vibrational spectra of molecules that are rapidly interconverted among distinct species by very fast motions, for example, during ordinary chemical reactions or during the rotation of a molecule between different sites in a solid, are considered. The question of the title is addressed in a number of distinct stages. First, the spectra predicted by Bloch equations with the inclusion of exchange terms are derived. The results differ from those familiar from magnetic resonance spectroscopy, since the vibrational transition moment can have a different magnitude and orientation in each site. Next, the question of whether a reaction can be fast enough on the time scale required for the simple vibrational Bloch equations to be valid is addressed, and it is concluded that this is unlikely. The observed spectrum may be fit with the result of the Bloch equation analysis (as has been done often in the past), but we conclude that the rate of the reaction cannot be simply extracted from the parameters used in this analysis. Instead, a more useful and general analysis of the spectra proceeds from a correlation function approach. We briefly discuss the results of a Redfield analysis. Next, we use the Mori-Zwanzig formalism to derive equations for vibrational spectra of reacting molecules. We outline the assumptions that are necessary to simplify the Mori equations sufficiently to reproduce the Bloch equations. The most important assumption is that the reaction goes over a high barrier. However, this results in a reaction too slow to have an observable effect. For lower barriers, the effect of the motion along the reaction coordinate cannot be separated into a reactive and a nonreactive part. This analysis demonstrates, in detail, the failure of the simple Bloch equations with exchange. Observed spectra can be interpreted using the equations derived from a Redfield or a Mori analysis. © 1985 American Institute of Physics.
MacPhail, RA; Strauss, HL
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