15min:

WILLIAM HARTER, JUSTIN MITCHELL, Department of Physics, University of Arkansas, Fayetteville, AR 72701.

Recently, Hougen\footnoteJ.T. Hougen, 2009 MSS \textbfRJ01, J Mol Spect 123, 197 (1987) showed an ad hoc symmetry-based parameterization scheme for analyzing tunneling dynamics and high resolution spectra of fluxional molecular structure similar to S -parameter analysis of superfine structure in SF₆ or NH₃ maser inversion dynamics by Feynman et.al. The problem is that ad hoc parametrization, like path integration in general, can lead to logjams of parameters or ``paths'' with no way to pick out the relevant ones.

We show a way to identify and use relevant parameters for a tunneling Hamiltonian H having global G-symmetry-defined bases by first expressing H as a linear combination \bar ⁱ \bar g_i of operators in dual symmetry group \bar G. The coefficients \bar ⁱ are parameters that define a complete set of allowed paths for any H with G-symmetry and are related thru spectral decomposition of G to eigensolutions of H. Quantum G vs.\bar G duality generalizes \textit lab \textit -vs.\textit -body and \textit state \textit -vs.\textit -particle.

The number of relevant \bar ⁱ-parameters is reduced if a system tends to stick in states of a local symmetry subgroup L\subsetG so the H spectrum forms level clusters labeled by induced representations d^(\ell) (L) G. A cluster-(\ell) has one E^( )-level labeled by G species ( ) for each L species (\ell) in D^ (G) L by Frobenius reciprocity\footnoteW.G. Harter, Principles of Symmetry, Dynamics, and Spectroscopy , (Wiley Interscience, 1993) p.265 . Then we apply local symmetry conditions to each irrep D^ (\bar ⁱ \bar g_i) that has already been reduced with respect to local symmetry L. This amounts to setting each off-diagonal component D_j,k^ (H) to zero.

Local symmetry conditions may tell which \bar ⁱ-parameters are redundant or zero and directly determine d^(\ell) G tunneling matrix eigenvalues that give E^( )-levels as well as eigenvectors. Otherwise one may need to choose a particular localizing subgroup chain L\subsetL₁\subsetL₂...G and further reduce the number of path parameters to facilitate spectral fitting.