Quantum ergodicity and vibrational relaxation in isolated molecules

Abstract

Recent advances in classical ergodic theory have produced qualitative information about the dynamics of anharmonic systems that is relevant to internal molecular dynamics and to unimolecular reaction rate theories. We describe here the results of an attempt to translate this progress into quantum molecular dynamics. Existing theories of quantum ergodicity are found to be ill suited to our task, so an extended concept of quantum ergodicity closely adhering to the well established classical concept is suggested. The new concept has the desired relevance to relaxation processes in isolated molecules, and it allows straightforward numerical investigation of molecular models. We report results for several triatomic models and one model of a linear molecule with four atoms. A prevalence of nonergodic behavior is observed for energies below the minimum required for dissociation.

Ergodicity and rapid electron delocalization – The dynamical mechanism of atomic reactivity and covalent bonding

“…From our point of view, the spectral energy gap distributions certainly do relate to the ergodic properties of the system in much the way proposed in the Wigner analysis but there are exceptions when the analysis does not apply and when it applies it is, in our view, the consequence of a more fundamental analysis of quantum ergodicity in terms of the eigenfunctions rather than the corresponding eigenvalues alone [49,50]. Let us recall the spectral formulation of quantum dynamics where the time development of a wavepacket ψ(t) is obtained by expansion in terms of the energy eigenvalues {φ k } as…”

“…In an earlier work [49,50], we have studied the apparent degree of ergodicity of coupled systems of two and three harmonic oscillators within the basis set of vibrational energy eigenfunctions in the absence of anharmonic coupling. In molecular systems, the atomic orbitals are the natural basis functions within which to study apparent ergodicity.…”

The Role of Quantum Dynamics in Covalent Bonding – A Comparison of the Thomas-Fermi and Hückel Models

“…While there is no doubt about the validity of such a correlation between ergodic properties of a system and its spectral properties, we believe that quantum ergodicity is more directly related to the character of the energy eigenfunctions. This suggestion has been made and explored in earlier work on anharmonically coupled harmonic oscillators [52]. Consequently, the definition we adopt here is that a quantum system is ergodic if the energy eigenfunctions are maximally delocalized in any local basis set.…”

Quantization with operators appropriate to shapes of trajectories and classical perturbation theory

“…Variational quantum mechanical results for the energy levels were given there and by N ordholm and Rice. 15 The Birkhoff-Gustavson perturbation theory 5 (BGPT), which consists of a series of canonical transformations that converts the classical Hamiltonian to a "normal" form, was first applied to the same system by Swimm and Delos. 10 Their treatment gave most of the energy levels accurately.…”