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The K-Rotational Labeling Problem for Eigenvectors From Internal Rotor Calculations: Application to Energy Levels of Acetaldehyde Below the Barrier



J Ortigoso, I Kleiner, Jon T. Hougen


The problem of attaching torsional and rotational quantum number labels to computer-generated-numerical eigenvectors with extensive basis set mixing is considered for the internal-rotation-overall-rotation problem in molecules with one methyl top. Quantum number labeling problems arise physically because the torsion and the rotation degrees of freedom both pass from one limiting case to another as the torsional energy moves from below the top of the internal rotation barrier to above it, i.e., the torsional degree of freedom changes from a vibration to an internal rotation, while the rotational degree of freedom moves its direction of quantization from a principal axis to an axis depending also on angular momentum generated by the methyl top rotation. Since the choice of axis system, basis set, and computational scheme all influence the eigenfunction labeling procedure, consideration is limited to a commonly used two-step matrix-diagonalization scheme. Torsional labels vt = 0, 1, and 2 for eigenfunctions generated in the first diagonalization step are chosen in order of increasing eigenvalue. Rotational energy surfaces are then constructed for the different torsional states. Projections of rotational eigenvectors over angular momentum coherent states are used to show that classical trajectories in the appropriate rotational energy surface give faithful representations of the quantum wavefunctions. We therefore conclude that irregularities in the rotational energy level structure within a given torsional state are related to the existence of extra separatrices on the classical phase space. We have applied our results to rotational levels of acetaldehyde arising from torsional states below the barrier and have devised a useful labeling scheme for rotational energy levels over a wide range of rotational quantum numbers in the presence of strong coupling with the torsion.
Journal of Chemical Physics
No. 24


acetaldehyde, internal rotation, k-rotation


Ortigoso, J. , Kleiner, I. and Hougen, J. (1999), The K-Rotational Labeling Problem for Eigenvectors From Internal Rotor Calculations: Application to Energy Levels of Acetaldehyde Below the Barrier, Journal of Chemical Physics (Accessed April 17, 2024)
Created June 1, 1999, Updated February 17, 2017