The radial proton momentum distribution of a single water molecule computed in two different fashions. The solid curve is the result when only one hydrogen path is opened, as is required in the precise methodology. The dashed curve results from a separate simulation that opens two proton paths and tabulates both end-to-end distances. Although this procedure facilitates more rapid sampling, it yields a result that is somewhat redshifted and narrowed. Also plotted is the classical momentum distribution of the system (dotted line) in order to underline the difference between the classical and quantum results.
The hydrogen-hydrogen radial distribution function in liquid water computed from a closed path integral simulation (solid line). The first peak denotes the intramolecular H–H distance, and the second peak corresponds to the distances between hydrogen atoms on other water molecules within the first solvation shell. Also plotted are two radial distribution functions that were garnered from neutron scattering data as reported by Soper et al. in 1986 (circles with dotted line) (Ref. 36) and 1999 (dashed line) (Refs. 37 and 38).
The open path end-to-end distance distribution for a bulk system of water when only one proton path is opened per configuration (solid line) and when one proton per molecule is opened for each configuration (circles with dashed line). Note the amount of noise present in the former curve, despite the fact that its simulation was carried out for many more steps.
The convergence of the radial proton momentum distribution as a function of number of system replicas is shown here. One can see that by 32 beads (long dashed line) the form of the distribution is well converged and in good agreement with the calculation at 64 beads (solid line). For 16 beads (dashed-dotted line), there is a slight blueshift introduced into the result.
The radial proton distribution for ice computed in the present work (solid line) is plotted against both a previous calculation for an ice cluster using a polarizable force field (circles with dotted line) and the experimental result (dashed line). As can be seen, the present results are in excellent agreement with the ice cluster calculation, although both fail to reproduce the correct tail behavior of the experimental distribution.
The computed radial proton momentum distribution in liquid water (solid line) is compared to the experimental result (circles with dotted line). There appears to be significant agreement between the two curves, including in the tail behavior. As one may expect, our results match the experiment best for the liquid phase.
The computed radial proton momentum distributions for the solid (circles with dotted line), liquid (solid line), and monomer at the liquid’s temperature (dashed line) are plotted against each other. As can be seen, the monomer result is redshifted with result to the liquid curve at the same temperature. The ice and liquid water distributions are very similar, including, and in contradiction to experiment, with respect to the tail behavior.
The oxygen-hydrogen radial distribution function for supercritical water in the present work (solid line) is shown alongside the results garnered from two neutron scattering experiments, that of Tassaing et al. (Ref. 56) (circles with dotted line) and Soper et al. (Ref. 37) (dashed line). The intramolecular contribution is removed from all plots. One can see that the first solvation shell is significantly more structured in simulation as compared to the experimental results.
The computed radial proton momentum distribution in supercritical water (solid line) is plotted against the experimental results and the distribution in a simple toy model (dashed line) that only includes rotation and one vibrational mode of the proton in the rigid-rotor∕harmonic oscillator approximation. The toy model is in fair agreement with the simulation, given its crudity. The simulation and the experimental curve (circles with dotted line) are in good agreement until approximately , where the path integral calculation does not produce the shoulder that is present in the latter result.
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