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Work under this heading consists of developing and/or applying Hamiltonians containing interactions between electronic, vibrational, rotational, and nuclear spin motions. The present focus is on one-top torsion-rotation problems and multi-dimensional tunneling-rotation problems. Analysis and fitting of small one-top internal-rotor prototypes Using a program first developed in Brussels, based on a theoretical Hamiltonian formalism from the literature, high quality least-squares fits of several thousand transitions in two or more torsional states have been carried out for CH3CHO, CH3OH, and CH3COOH. These fits give residuals essentially equal to the experimental error in the measurements, and permit (i) the generation of atlases for radioastronomy searches, (ii) the generation of ground-state energy levels for assignments in local mode and other vibrational studies, and (iii) comparison of the higher-order torsion-rotation interaction parameters and the internal-rotation potential with state-of-the-art ab inito electronic structure calculations. These fits also pointed out the need for further theoretical understanding (i) of the quantum number labeling of the eigenvectors obtained by diagonalizing the large torsion-rotation matrices, and (ii) of the use of contact transformations to reduce the number of determinable parameters in a fit. The energy level diagram for CH3CHO is shown below, where the torsional plus K-rotational energy is plotted against the K quantum number, illustrates some of the theoretical problems mentioned above.  The three rather clear parabolas in the lower part of the figure correspond to levels below the top of the internal rotation barrier. These levels are relatively well understood at present. The irregularly arranged levels in the upper part of the figure correspond to levels above the barrier, which are not well understood. The figure below gives a slightly different display of the levels above the barrier, where A and E labels from the permutation-inversion molecular symmetry group have been added, as well as circles indicating a systematic pattern of large torsion-rotation interactions.  One of the theoretical tools, namely coherent state projections obtained from a combined classical mechanics and quantum mechanics treatment of the labeling problem, is illustrated in the figure below.  Such diagrams, along with rotational energy surfaces, alternative torsional quantum numbers, etc. have been used to rationalize the somewhat bizarre behavior of level sequences under the influence of strong torsion-rotation interactions. Effective Hamiltonians for multidimensional tunneling-rotation problems When many large amplitude vibrational motions are present in a molecule with sufficient symmetry, the molecule will tunnel between the various structurally and energetically equivalent equilibrium configurations. These tunneling motions frequently alter the moments of inertia of the molecule and/or generate angular momentum, both of which affect the overall rotation of the molecule and therefore affect the positions of the rotational energy levels observed in any high resolution spectroscopic study. The most recent examples of such studies at NIST, which (i) involve the extensive use of group theory to derive the correct form for the tunneling-rotation Hamiltonian matrices, (ii) do not require precise knowledge of the multi-dimensional potential energy surface, and (iii) lead generally to least squares fits with residuals approaching the experimental uncertainties, include Na3 and (CH3OH)2. The figure below illustrates a recently derived energy level diagram for (CH3OH)2 showing the hierarchy of tunneling splittings induced by (i) the less hindered methyl-top torsional tunneling, (ii) the more hindered methyl-top torsional tunneling, (iii) the lone-pair exchange tunneling, and (iv) the proton-donor - proton-acceptor exchange tunneling.  Peptide mimetics and mimetic precursors The tools described in the two sections above can be applied to the spectra of molecules containing functional groups of importance in proteins. A very common benchmark molecule for testing ab inito and other protein modeling algorithms is alanine dipeptide, which is formed by condensing the amino acid alanine H2N-CH(-CH3)-COOH with a CH3-COOH at its amine end and a H2N-CH3at its carboxyl end, in order to mimic the two-peptide-linkage environment in which alanine is found in proteins. In principle (and with some effort) high resolution spectroscopic studies can determine precise and unambiguous (albeit gas-phase rather than solvated) values for (i) the number of conformers that exist for this molecule, (ii) the shape of the heavy atom skeleton in each conformer, and (iii) the forces resisting rotation about the N-C and C-C single bonds between the two peptide linkages. The latter are in fact the forces resisting the protein folding motions of longer polypeptides. To carry out this plan, however, it is first necessary to be able to fit and predict the spectra of each conformer, taking into account any splittings caused by internal rotation of any or all of the methyl groups. Work on alanine dipeptide and some of its simpler fragments and/or related species is just beginning in our group.
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