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We employ the k-th nearest-neighbor estimator of configurational entropy inorder to decode within a parameter-free numerical approach the complexhigh-order structural correlations in fluxional molecules going beyond theusual linear, bivariate correlations. This generic entropy-based scheme fordetermining many-body correlations is applied to the complex configurationalensemble of protonated acetylene, a prototype for fluxional molecules featuringlarge-amplitude motion. After revealing the importance of high-ordercorrelations beyond the simple two-coordinate picture for this molecule, weanalyze in detail the evolution of the relevant correlations with temperatureas well as the impact of nuclear quantum effects down to the ultra-lowtemperature regime of 1 K. We find that quantum delocalization and zero-pointvibrations significantly reduce all correlations in protonated acetylene in thedeep quantum regime. At intermediate temperatures of approximately 100 to 800K, a quantum-to-classical cross-over regime is found where classical mechanicsstarts to correctly describe trends in the correlations whereas it evenqualitatively fails below 100 K. Finally, a classical description of the nucleiprovides correlations that are in quantitative agreement with the quantum onesonly at temperatures exceeding 1000 K. This data-intensive analysis has beenmade possible due to recent developments of machine learning techniques basedon neural network potentials in full dimensionality that allow us toexhaustively sample both, the classical and quantum ensemble of protonatedacetylene at essentially converged coupled cluster accuracy. The presentednon-parametric analysis of correlations is expected to complement and guide theanalysis of experimental measurements, in particular multi-dimensionalvibrational spectroscopy, by revealing the complex coupling between variousdegrees of freedom.

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