# LOW-BARRIER ROTATION-PSEUDOROTATION HAMILTONIAN AND APPLICATION TO THE B STATE OF $Na_{3}$

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 Title: LOW-BARRIER ROTATION-PSEUDOROTATION HAMILTONIAN AND APPLICATION TO THE B STATE OF $Na_{3}$ Creators: Ohashi, N.; Tsuura, Makiko; Hougen, Jon T.; Ernst, Wolfgang E.; Rakowsky, Stefan Issue Date: 1996 Abstract: A formalism has been developed for treating the rotation-pseudorotation problem in $Na_{3}$ molecules which has many analogies with, but is not identical to, the traditional principal axis formalism for methyl-top internal rotor molecules. The pseudorotational coordinate (angle) $X_{p}$ and $D_{3}h$ group theory are taken from our previously developed high barrier $formalism^{1}$, and lead to a molecular Hamiltonian containing the following terms (up to second order) $$H = FP_{X_{p}}^{2}+\frac{1}{2}V_{3}(1-\cos3x_{p})+BJ^{2}+(C-B)J_{z}^{2}+QP_{X_{p}}J_{z}+f_+J_+^{2}+f_-j_-^{2}$$ The first two terms ($F, V_{3}$) represent the three-indentical-well particle-on-a-ring pseudorotational problem. The next two terms (B,C) represent an oblate symmetric top rotational Hamiltonian (z axis perpendicular to the $Na_{3}$ plane). The fifth term (Q) represents the coriolis interaction between the pseudo-rotational and overall-rotational angular momentum. Up to this point the Hamiltonian can be mapped onto that for methyl-group internal rotation in an oblate symmetric top. The final two terms are peculiar to the present problem. They contain products of the rotational angular momentum ladder operators $J_{\pm} = J_{x}iiJ_{y}$ and coefficients f$_+$, which unlike the coefficients $F, v_{3}$, B, C, Q above cannot be constants, but must be expressed as a Fourier series in the pseudorotation functions $\exp(\pm imx_{p}$), where $m = +1$ mod 3. These final two terms allow for ""rotation"" of the asymmetric rotor A, B, C axes when pseudorotation occurs and each of the Na atoms in turn occupies the apex position in the distorted triangle equilibrium structure. As might be expected, they are responsible for a number of unusual features in the rotational energy level pattern. Various aspects of the energy levels arising from this Hamiltonian when the barrier $\nu_{3}$ = 0 will be discussed, as well as a relatively successful fit of $published^{2}$ and unpublished measurements from earlier B-X pump-probe experiments. URI: http://hdl.handle.net/1811/13761 Other Identifiers: 1996-WH-05