The K-rotational labeling problem for eigenvectors from internal rotor calculations: Application to energy levels of acetaldehyde below the barrier

Abstract

The problem of attaching K rotational quantum number labels to computer-generated numerical eigenvectors with extensive basis set mixing is considered for the internal-rotationoverall-rotation problem in molecules with one methyl top. Quantum number labeling problems arise physically because the torsional and the rotational 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 and to acetaldehyde as a numerical example. Torsional labels vt=0, 1, and 2 for eigenfunctions generated in the first diagonalization step are assigned in order of increasing eigenvalues, and rotational energy surfaces are constructed for the A and E symmetry components of these three torsional states. Projections of rotational eigenvectors over angular momentum coherent states are used to show that classical trajectories on the appropriate rotational energy surface give faithful representations of the quantum wave functions. Irregularities in the rotational energy level structure within a given torsional state can then be understood in terms of extra separatrices in the classical phase space and unusual wave function densities in the coherent state projections. Results are used to devise an automatic K labeling scheme for numerically obtained torsion-rotation levels belonging to torsional states below the top of the internal rotation barrier.