Extended quantization condition for constructive and destructive interferences and trajectories dominating molecular vibrational eigenstates.

The role of destructive quantum interference in semiclassical quantization of molecular vibrational states is studied. This aspect is crucial for correct quantization, since failure in the appropriate treatment of destructive interference quite often results in many spurious peaks and broad background to hide the true peaks. We first study the time-Fourier transform of the autocorrelation function without performing summation over the trajectories. The resultant quantity, the prespectrum which is a function of individual classical trajectories, provides a clear view about how destructive interference among the trajectories should function. It turns out that the prespectrum is oscillatory but never a random noise. On the contrary, it bears a systematic and regular structure, which is sometimes characterized in terms of very sharp and high peaks in the energy space of the sampled classical trajectories. We have found an extended quantization condition that is responsible for generating these peaks in the prespectrum, which we call the prior quantization condition. Integration of the prespectrum over the trajectory space is supposed to give "zero" (practically a small value of the order of the Planck constant) at a noneigenvalue energy, which is actually a materialization of the destructive interference. Besides, certain finite peaks in the prespectrum survive after the integration to form the true spikes (eigenvalues) in the final spectrum, if they satisfy an additional resonance condition. For these resonance components, the prior quantization condition is reduced to the Einstein-Brillouin-Keller quantization condition. Based on these analyses, we propose a rather conventional filtering technique to efficiently handle tedious computation for destructive interference, and numerically verify that it works well even for multidimensional chaotic systems. This filtering technique is further utilized to extract a few trajectories that dominate an eigenstate of molecular vibration.