Assignment and extracting dynamics from experimentally and theoretically obtained spectroscopic hamiltonians in the...

An analysis of existing algebraic multiresonance spectroscopic Hamiltonians, derived by fitting to experimental data or from classical canonical or quantum Van Vleck perturbation theory, allows without any significant further classical or quantum calculation the assignment of quantum numbers and motions to states observed in spectra that were previously thought to be irregular or just unassignable. In such cases, inspection of the amplitude and phase of eigenfunctions previously calculated in the Hamiltonians derivation process but now transformed to a reduced dimension semiclassical action-angle representation reveals extremely simple albeit unfamiliar topologies that give quantum numbers by simply counting nodes and phase advances. The topology allows these simple wave functions to be sorted into dynamically different excitation ladders or classes of states which are associated with different regions of phase space. The rungs of these ladders or the states in the classes intersperse in energy causing the spectral complexity. No experimental procedure allows such sorting. The success of the work stems from (1) the qualitative insights of nonlinear dynamics, (2) the conversion of the quantum problem in full dimension to a semiclassical one in reduced dimension by use of a canonical transform that takes advantage of the polyad and other constants of the motion, and (3) the judicious choice of the reduced angle variables to reflect rational ratio resonance frequency conditions. This leads to localization of those semiclassical wave functions, which are affected by the particular resonance. In reverse, the localized appearance of the reduced dimension wave function reveals which resonances govern it and makes sorting visually simple. The success of the work also stems from (4) the revealing use of plots of phase advances as well as the usual densities of the eigenstates for sorting and assignment purposes. Even in classically chaotic regions, organizing trajectories, which correspond to averages over regional phase space structures that need not be computed, can easily be drawn as the structure about which eigenfunction localization takes place. The organizing trajectories when transformed back to the full dimensional configuration space reveal the internal molecular motions. The complexity of the usual quantum stationary and propagated wave functions and associated classical trajectories forbids most often such assignments and sorting. This procedure brings the ability to interpret complex vibrational spectra to a degree previously thought possible only for lower excitations. The new methodology replaces and extends the computationally more difficult parts of a procedure used by the authors that was applied successfully to several molecules during the past few years. The new methodology is applied to DCO, CHBrClF, and the bending of acetylene.