Knowledge Bank

Recently, Hougen \textbf{RJ01}, {J Mol Spect {\bf{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 \textit{S}-parameter analysis of superfine structure in $SF6$ \textbf{RJ05} (See also following talk.)} or $NH3$ maser inversion dynamics by Feynman \textit{et.al.} (Addison Wesley 1964)\textit{ p.9-1}} 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 $\gamma ¯ig¯i$ of operators in dual symmetry group $G¯$. The coefficients $\gamma ¯i$ 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$$vs.$$G¯$ duality generalizes $\textit lab $$\textit -vs.$$\textit -body$ and $\textit state $$\textit -vs.$$\textit -particle$. The number of relevant $\gamma ¯i$-parameters is reduced if a system tends to stick in states of a local symmetry subgroup $L$$\subset$$G$ so the $H$ spectrum forms level clusters labeled by induced representations $d(\ell )(L)$$\uparrow$$G$. A cluster-$(\ell )$ has one $E(ϵ)$-level labeled by $G$ species $(ϵ)$ for each $L$ species $(\ell )$ in $Dϵ(G)$$\downarrow$$L$ by Frobenius reciprocity}, {(Wiley Interscience, 1993)}\textit p.265}. Then we apply $local\; symmetry\; conditions$ to each irrep $Dϵ(\gamma ¯ig¯i)$ that has already been reduced with respect to local symmetry $L$. This amounts to setting each off-diagonal component $Dj,kϵ(H)$ to zero. Local symmetry conditions may tell which $\gamma ¯i$-parameters are redundant or zero and directly determine $d(\ell )$$\uparrow$$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$$\subset$$L1$$\subset$$L2$$...$$G$ and further reduce the number of path parameters to facilitate spectral fitting.