Quantumness of classical-trajectory-based methods for vibrational spectroscopy

Classical-trajectory-based methods calculate the vibrational spectrum of a molecular system as the Fourier transform of an appropriate time correlation function. In this paper, we assess the quantumness of different approaches derived from the path-integral representation of quantum mechanics. We focus on power spectra obtained by means of semiclassical (SC) dynamics, centroid molecular dynamics (CMD), ring polymer molecular dynamics (RPMD), and its thermostatted version (TRPMD). Our calculations also include classical and quasi-classical trajectory (QCT) simulations as examples of results based on a purely classical propagator. Calculations are performed for a three-dimensional anharmonic model system and the non-rotating gas-phase water molecule. We show that typical features of classical calculations, such as sum-of-frequency combination bands and overtones, difference bands, and spectroscopic signals at negative frequencies, are found for classical, QCT, CMD, and (T)RPMD spectra. Conversely, these features are basically absent in semiclassical calculations, which show just a reminiscence of the underlying classical trajectory. The overall accuracy of the results compared to quantum mechanical values is always better for SC methods. Classical results depend on the initial sampling distributions, and their accuracy is of the same order as CMD, RPMD, and TRPMD simulations, i.e., an order of magnitude lower than for semiclassical approaches. Our main conclusion is that when it comes to molecular vibrational spectroscopy calculations, semiclassical methods have a predominant quantum character, being able to include also real-time coherence effects, while CMD, RPMD, and TRPMD are prevalently classical, reproducing just the anharmonicity related to the zero point energy or quantum statistical distribution.