Intramolecular vibrational relaxation seen as expansion in phase space. II. Reference ergodic systems
Authors: Pavlov-Verevkin V. B. , Lorquet J. C. 
Published: July 2002

The aim of the paper is to estimate the volume of phase space that is, in principle, available to a nonstationary wave packet during its intramolecular vibrational relaxation. For that purpose, use is made of the maximum entropy method, together with the concept of constrained ergodicity to construct two so-called reference ergodic systems. The first one concerns thermal excitation processes. In that case, the only two constraints that are imposed on the intramolecular dynamics arise from the normalization of the wave function and from the conservation of energy. These constraints affect the zeroth and first moments of the spectrum. The second reference system concerns a situation where, as an additional constraint, use is made of the information that the system has been prepared spectroscopically, i.e., by a specific excitation process, consisting in the coherent excitation of an initial pure state. Then, the second moment of the spectrum, denoted σ, is shown to provide the appropriate additional constraint. Translated into the time domain, the prior knowledge of the dynamics used as a constraint is limited to an infinitesimally brief period of time [0,dt] with the remaining evolution determined by the maximum entropy method. The spectroscopic reference system constructed in that way can be understood as the one that samples the maximal volume of phase space available to a wave packet having a specified average energy and being put in motion by a specified initial force.
Closed-form expressions are obtained for the phase space volumes occupied by these two reference systems for various simple parametrizations of the function D(E) that expresses the density of states as a function of the internal energy (power laws or exponential increase). Thermal reference systems are found to sample a larger volume of phase space than their spectroscopic counterparts. The difference between these two cases depends critically on the value of σ, and also on the symmetry characteristics of the excitation process. In general, the volumes occupied by the reference systems, thermal as well as spectroscopic, can be expressed as ηEavD(Eav), where Eav is the (conserved) average energy of the wave packet and η is a correcting factor that depends on the functional form of D(E) and on the nature of the imposed constraints. In all cases studied, the value of η was found not to greatly differ from 1. The method has been applied to the analysis of three experimental photoelectron spectra presenting different spectral characteristics (X̃ 2A1 state of NH+3, X̃ 2B3 state of C2H+4, and the X̃ 2A″ state of C2H3F+). The fractional occupancy index F defined by Heller as the fraction of the available phase space eventually explored up to the break time TB could be determined. After a time of the order of 100 fs, F was found to be of the order of a few percent for thermal excitation. When the molecule presents some symmetry, the expansion of the wave packet is restricted to that part of phase space spanned by the totally symmetric wave functions. The use of this additional a priori knowledge increases the fractional index F.