Deciphering High-order Structural Correlations within Fluxional Molecules from Classical and Quantum Configurational Entropy.

We employ the k-th nearest-neighbor estimator of configurational entropy in order to decode within a parameter-free numerical approach the complex high-order structural correlations in fluxional molecules going much beyond the usual linear, bivariate correlations. This generic entropy-based scheme for determining many-body correlations is applied to the complex configurational ensemble of protonated acetylene, a prototype for fluxional molecules featuring large-amplitude motion. After revealing the importance of high-order correlations beyond the simple two-coordinate picture for this molecule, we analyze in detail the evolution of the relevant correlations with temperature as well as the impact of nuclear quantum effects down to the ultra-low temperature regime of 1 K. We find that quantum delocalization and zero-point vibrations significantly reduce all correlations in protonated acetylene in the deep quantum regime. Even at low temperatures up to about 100 K, most correlations are essentially absent in the quantum case and only gain importance at higher temperatures. In the high temperature regime, beyond roughly 800 K, the increasing thermal fluctuations are found to exert a destructive effect on the presence of correlations. dm At intermediate temperatures of approximately 100 to 800 K, a quantum-to-classical cross-over regime is found where classical mechanics starts to correctly describe trends in the correlations whereas it even qualitatively fails below 100 K. Finally, a classical description of the nuclei provides correlations that are in quantitative agreement with the quantum ones only at temperatures exceeding 1000 K. This data-intensive analysis has been made possible due to recent developments of machine learning techniques based on high-dimensional neural network potential energy surfaces in full dimensionality that allow us to exhaustively sample both, the classical and quantum ensemble of protonated acetylene at essentially converged coupled cluster accuracy from 1 to more than 1000 K. The presented non-parametric analysis of correlations beyond usual linear two-coordinate terms is transferable to other system classes. The technique is also expected to complement and guide the analysis of experimental measurements, in particular multi-dimensional vibrational spectroscopy, by revealing the complex coupling between various degrees of freedom.