GROUP PARAMETRIZED TUNNELING AND LOCAL SYMMETRY CONDITIONS

\underline{WILLIAM HARTER}, JUSTIN MITCHELL, Department of Physics, University of Arkansas, Fayetteville, AR 72701.

Recently, Hougen^{a} 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 SF_{6}^{b} or NH_{3} maser inversion dynamics by Feynman et.al.^{c} 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 γ̄^{i}ḡ_{i} of operators in dual symmetry group Ḡ. The coefficients γ̄^{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 Gvs.Ḡ duality generalizes lab-vs.-body and state-vs.-particle. The number of relevant γ̄^{i}-parameters is reduced if a system tends to stick in states of a local symmetry subgroup L⊂G so the H spectrum forms level clusters labeled by induced representations d^{(ℓ)}(L)↑G. A cluster-(ℓ) has one E^{(ϵ)}-level labeled by G species (ϵ) for each L species (ℓ) in D^{ϵ}(G)↓L by Frobenius reciprocity^{d}. Then we apply local symmetry conditions to each irrep D^{ϵ}(γ̄^{i}ḡ_{i}) that has already been reduced with respect to local symmetry L. This amounts to setting each off-diagonal component D_{j,k}^{ϵ}(H) to zero. Local symmetry conditions may tell which γ̄^{i}-parameters are redundant or zero and directly determine d^{(ℓ)}↑G tunneling matrix eigenvalues that give E^{(ϵ)}-levels as well as eigenvectors. Otherwise one may need to choose a particular localizing subgroup chain L⊂L_{1}⊂L_{2}...G and further reduce the number of path parameters to facilitate spectral fitting.

aJ.T. Hougen, 2009 MSS RJ01, J Mol Spect 123, 197 (1987)

bW.G. Harter and J. C. Mitchell, 2009 MSS RJ05 (See also following talk.)

cR.P. Feynman, R. B. Leighton, M. Sands, Lectures on Physics Vol.3 (Addison Wesley 1964) p.9-1

dW.G. Harter, Principles of Symmetry, Dynamics, and Spectroscopy, (Wiley Interscience, 1993)p.265