Energy randomisation: how much of rotational phase space is explored and how long does it take?

Abstract

The assumption that the internal energy of a molecule is randomised on a timescale that is short compared with the reaction time is at the heart of modern theories of unimolecular reaction. In applying such theories it is necessary to decide the volume of phase space in which the energy is assumed to be randomised. The question of whether the K rotational quantum number is conserved has an impact on that choice. The conceptual sequence from experimental spectra, through analysis, and interpretation in terms of K relaxation is described below.

At low resolution, intramolecular vibrational energy randomisation results in the broadening of the features of IR absorption spectra. At high resolution in bound systems, such broadened features are revealed to be clumps of discrete lines, each of which is a transition to a molecular eigenstate. Since the discrete lines can be assigned by spectroscopic means, the erroneous assignment of inhomogeneous broadening to rate processes can be avoided. Each clump of eigenstates is characterised by its dilution factor, interaction width and effective level density. Examples include the IR spectra of ethanol and but-1-yne in the 3 µm region.

The interpretation of molecular eigenstate spectra involves several conceptual stages: (1) identification of the bright state which would be prepared by the coherent excitation of a certain section of spectrum, (2) evaluation of the rate of energy transfer out of the bright state, (3) use of the rotational quantum number dependence of the spectra and the trends among related systems to deduce the mechanisms by which the bright state is coupled to the bath, and (4) modelling the spectra with random matrix calculations in order to determine the average coupling parameters for anharmonic coupling and x, y and z-type Coriolis interactions.

Random matrix simulations provide the opportunity to address the title questions. The simulations focused particularly on rotationally mediated vibrational relaxation and were constrained to obey the rotational quantum number dependence of the Coriolis interaction. For ethanol, when the system is prepared with a specific K quantum number, one finds that K is not conserved but neither is the population completely randomised among the 2J+ 1 available K states even at long times. The time needed for the final (non-random) distribution among K-states to be achieved is typically of the order of 1 ns, even though the energy leaves the bright state an order of magnitude more quickly.