Path-integral studies of quantum statistical effects in vibrational spectroscopy
We examine the role of nuclear quantum statistical effects in the vibrational spectroscopy of molecular systems at finite temperatures.
Our starting point is Matsubara dynamics, which rigorously combines quantum Boltzmann statistics with classical real-time trajectories, by filtering out the high-frequency components of the imaginary-time Feynman paths that are responsible for quantum coherence. Using a simple model of perturbed harmonic oscillators, it is shown that most of the intensity differences between quantum and classical non-fundamental (overtone, combination, and difference) bands can be accounted for by the anharmonic coupling between the imaginary-time path centroid and the fluctuations around it. This coupling causes the amplitudes of the relevant centroid vibrations to be ‘Matsubara heated’ to an effective temperature that is consistent with the underlying quantum distribution. Quantum coherence thus appears to play no major role in describing such features, contrary to what has sometimes been assumed in the literature.
Practical path-integral methods for calculating vibrational spectra, such as centroid and thermostatted ring-polymer molecular dynamics, faithfully capture the harmonic behaviour of the centroid but make drastic approximations to the fluctuation dynamics. We explain how these approximations render such methods incapable of predicting non-fundamental bands with significantly more accuracy than classical mechanics. This is borne out by illustrative simulations of water in its gas, liquid, and ice phases; the linearised semiclassical initial value representation is shown to be the only established trajectory-based approach that reproduces most non-fundamental bands with qualitative accuracy, even though it suffers severe zero-point energy leakage on a sub-picosecond timescale. However, a simple quantum–classical correction formula (derived from first-order perturbation theory) is shown to bring the predictions of path-integral methods into much closer agreement with exact quantum results.
Finally, a simple model of the carbon dioxide molecule is studied numerically to establish that ‘Matsubara heating’ accounts for the quantum (temperature quasi-independent) behaviour of the Fermi resonance splitting, which is underestimated by path-integral methods analogously to overtone band intensities.