The generation of free energy landscapes corresponding to conformational equilibria in complex molecular systems remains a significant computational challenge. Adding to this challenge is the need to represent, store, and manipulate the often high-dimensional surfaces that result from rare-event sampling approaches employed to compute them. In this Letter, we propose the use of artificial neural networks as a solution to these issues. Using specific examples, we discuss network training using enhanced-sampling methods and the use of the networks in the calculation of ensemble averages.

The effect of proton tunneling on many-body correlated proton transfer in hexagonal ice is investigated by quantum simulation. Classical single-particle hopping along individual hydrogen bonds leads to charge defects at high temperature, whereas six protons in ringlike topologies can move concertedly as a delocalized quasiparticle via collective tunneling at low temperature, thus preventing the creation of high-energy topological defects. Our findings rationalize many-body quantum tunneling in hydrogen-bonded networks and suggest that this phenomenon might be more widespread than previously thought.

The transfer of multiple protons in hydrogen-bonded networks usually occurs one proton at a time. At sufficiently high temperatures, each proton transfers via thermally activated hopping along its hydrogen bond, thereby moving a charge defect through the network. At low temperatures, quantum-mechanical tunnelling might set in instead, thus avoiding hopping over the energy barriers. In the case of several transferring protons, independent thermal hopping or quantum tunnelling of the individual protons becomes less favourable because of a significant creation of charge defects. In individual molecules or hydrogen-bonded molecular complexes, for instance, double proton transfer is often found to be concerted. Multiple proton transfer that avoids charge defects can occur in cyclic topologies built from several hydrogen bonds that allow for directional chains of proton transfer. This requires perfect proton order within these rings, which imposes handedness and thus chirality, and changes parity upon transfer of all protons. Ordinary ice, which is hexagonal ice I, is the most stable form of crystalline ice obtained upon freezing liquid water at ambient pressure and consists of interconnected six rings of oxygen atoms that host six protons each. These hexagonal rings remain proton disordered even down to low temperatures, as heralded by the residual entropy of ice I. However, owing to combinatorics, a certain number of these six rings is proton ordered in macroscopic crystals. These chiral hexameric rings might support coherent tunnelling of the hosted protons. Indeed, there is some evidence in the recent literature, both experimental and simulational, that correlated tunnelling of all six protons might be possible in proton-ordered six rings in ice I if temperatures are low enough. In this Perspective, the key ideas and previous findings will be reviewed in the light of relevant experiments with a focus on available ab initio path integral simulation work supplemented with additional data provided herein.

Multiple proton transfer controls many chemical reactions in hydrogen-bonded networks. However, in contrast to well-understood single proton transfer, the mechanisms of correlated proton transfer and of correlated proton tunneling in particular have remained largely elusive. Herein, fully quantized ab initio simulations are used to investigate H/D isotopic-substitution effects on the mechanism of the collective tunneling of six protons within proton-ordered cyclic water hexamers that are contained in proton-disordered ice, a prototypical hydrogen-bonded network. At the transition state, isotopic substitution leads to a Zundel-like complex, [HO⋅⋅⋅D⋅⋅⋅OH], which localizes ionic defects and thus inhibits perfectly correlated proton tunneling. These insights into fundamental aspects of collective proton tunneling not only rationalize recent neutron-scattering experiments, but also stimulate investigations into multiple proton transfer in hydrogen-bonded networks much beyond ice.

Activation parameters for the model oxidation half reaction of the classical aqueous ferrous ion are compared for different molecular simulation techniques. In particular, activation free energies are obtained from umbrella integration and Marcus theory based thermodynamic integration, which rely on the diabatic gap as the reaction coordinate. The latter method also assumes linear response, and both methods obtain the activation entropy and the activation energy from the temperature dependence of the activation free energy. In contrast, transition path sampling does not require knowledge of the reaction coordinate and directly yields the activation energy [C. Dellago and P. G. Bolhuis, Mol. Simul. 30, 795 (2004)]. Benchmark activation energies from transition path sampling agree within statistical uncertainty with activation energies obtained from standard techniques requiring knowledge of the reaction coordinate. In addition, it is found that the activation energy for this model system is significantly smaller than the activation free energy for the Marcus model, approximately half the value, implying an equally large entropy contribution.