The FFT method for determining semiclassical eigenvalues: Application to asymmetric top rigid rotors

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Abstract
A Fourier transform method for calculating action variables and semiclassical eigenvalues in molecules starting from Cartesian coordinates and momenta is developed and applied to the determination of rotational semiclassical eigenvalues for rigid asymmetric top molecules. The method involves calculating rotational actions by using Fourier representations of the symmetric top action‐angle variables to express the integrals. The question of what is the optimum quantization axis is considered, and it is found that the same primitive semiclassical eigenvalues are obtained independent of which principal axis is used for quantization provided that the replacement J → J+ 1/2 is made. The replacement J → [J(J+1)]1/2 leads to eigenvalues that are slightly dependent on quantization axis, but more accurate. For either choice of J replacement, the primitive eigenvalues are found to be double valued when trajectory motion is librational relative to both the prolate and oblate top quantization axes, and also when motion is rotational relative to both axes. This problem can be resolved by implementing uniform semiclassical theory, but this is hard to generalize to nonrigid molecule eigenvalue calculations. Here we propose several simple modifications to the primitive semiclassical method which determine uniquely defined eigenvalues for both the J(J+1) or (J+ 1/2 )2 replacements. These modified primitive semiclassical methods are also applicable to nonrigid molecule calculations, and one of these (the linear interpolation method) is found to give eigenvalues that are quite accurate without requiring information beyond that which is needed in a primitive semiclassical calculation.