Time delay for bimolecular collisions: Utility of the spectral theorem in the classical limit

Abstract

The recently proven multichannel spectral theorem for time delay is explored with the goal of providing a useful expression for ’’lifetimes’’ of bimolecular collisions. The theorem is reduced to an approximate relationship between the classical time delay and the classical excess density of states. Numerical studies on collinear atom–diatom systems confirm the accuracy of the classical relationship and a simple geometric interpretation of the excess density of states for these systems is provided. Results on the sensitivity of collisional time delays to variations in potential surface well depths and system masses are presented. the time delays being simply obtained from the excess density of states.

Microcanonical rates, gap times, and phase space dividing surfaces

“…Result ͑2.7͒ is essentially the content of the socalled classical spectral theorem. 55,56,156,157,184 If all points in the reactant region of phase space eventually react ͑that is, all points lie on crossing trajectories 97,98 ͒ then A C ͑E͒ = A ͑E͒, the full reactant phase space density of states. Apart from a set of measure zero, all phase points x M A can be classified as either trapped ͑T͒ or crossing ͑C͒.…”

“…corresponding to the Hamiltonian vector field associated with H͑x͒. 150,154,186,187 The reactant phase space volume occupied by points initiated on the dividing surface DS in with energies between E and E + dE is therefore 32,55,56,156,157,184,185 dE ͵…”

“…In the present paper we consider the problem of defining and evaluating theoretical unimolecular reaction rates in light of the penetrating analyses of Thiele, 32 Bunker and Hase, 37 Dumont and Brumer, 49 and DeLeon and co-workers, 97,98 the classical spectral theorem, 55,56 and the recent developments in phase space transition state theory 148,149,153-159 mentioned above. The particular reaction chosen for study is the isomerization HCN CNH, 155,156,[168][169][170][171][172][173][174][175][176] for which several relevant theoretical quantities have recently been computed for a ͑classical͒ model of the HCN molecule at fixed energy.…”

“…Dumont and Brumer 49,50 and Dumont and co-workers [51][52][53][54] analyzed unimolecular reaction rates in terms of the gap time distribution, while related work has been done on the so-called classical spectral theorem. 55,56 A fundamental assumption of the statistical theory of ͑classical͒ unimolecular decay is that intramolecular vibrational energy distribution occurs on a time scale much faster than that for reaction. 11,15,23 Renewed interest in the problem therefore naturally stemmed from investigations of the properties of molecular vibrational dynamics, modeled as nonlinearly coupled anharmonic oscillators, in light of the KAM theorem and the apparent existence of a threshold energy for onset of global chaos.…”

State densities and time delay in molecular collisions

“…3 we demonstrate the role of the full spectral property in providing insight into the dependence of time delays on properties of the interacting collision partners. In addition, we present an informative geometric picture of time delay for collinear atom-diatom scattering [4]. Section 4 presents a discussion of the channel projected spectral theorem, proven in detail elsewhere [3], and includes preliminary numerical studies designed to allow tests of approximate evaluations of the state densities required for the channel projected spectral property.…”