Limits on the transition to Gaussian orthogonal ensemble behavior: Saturated radiationless transitions between strongly coupled potential surfaces.

Abstract

We examine the transition to Gaussian orthogonal ensemble (GOE) behavior in the statistical properties of the eigenspectrum and transition amplitudes of a class of model Hamiltonians, which simulates the crossing region between a bright potential surface with a low density of states and a dense set of states on a dark potential surface. The Hamiltonian contains the highly delocalized coupling of multiple bright states with the regular, Poisson-distributed, bath of dark states expected for a classically integrable system. Coupling between rovibrational states on electronic surfaces with different equilibrium molecular conformations like the ${\mathrm{NO}}_{2}^{2}$${\mathrm{A}}_{2}$${\mathrm{\ensuremath{-}}}^{2}$${\mathrm{B}}_{2}$ or acetylene-vinylidene couplings, may be of this type. Statistical properties of the eigenspectrum and of the delocalized bright state intensity are shown to follow those of the Gaussian orthogonal ensemble for increasingly longer ranges and in higher-order correlations, as the number of simultaneously interacting bright states increases, but are shown to be limited by intermediate behavior at long range, which is stable to increases in the coupling strength and depends on the number of bright states. Molecular systems that follow this model would show statistics dependent on experimental limitations on resolution and range, such as a spacing distribution function with level repulsion at short range and a nearly exponential tail, and values of ${\ensuremath{\Delta}}_{3}$ that have GOE behavior at short range, but linear at long range. Delocalization of the bright states over the sampled energy range is required for statistics stable to changes in coupling strength and is shown to be measured by the short-time behavior of the spectral Fourier transform.