^{1}, Varadharajan Srinivasan

^{1,a)}, Daniel Sebastiani

^{2}and Roberto Car

^{3,b)}

### Abstract

Recent neutronCompton scattering experiments have detected the proton momentum distribution in water. The theoretical calculation of this property can be carried out via “open” path integral expressions. In this work, present an extension of the staging path integral molecular dynamics method, which is then employed to calculate the proton momentum distributions of water in the solid, liquid, and supercritical phases. We utilize a flexible, single point charge empirical force field to model the system’s interactions. The calculated momentum distributions depict both agreement and discrepancies with experiment. The differences may be explained by the deviation of the force field from the true interactions. These distributions provide an abundance of information about the environment and interactions surrounding the proton.

One of the authors (J.A.M.) would like to thank the Fannie and John Hertz Foundation for its support of this work. Partial support from the National Science Foundation through Grant No. CHE 021432 is also acknowledged.

I. INTRODUCTION

II. METHODOLOGY

A. The density matrix in path integral form

B. Computing the momentum distribution

C. Staging open path integral molecular dynamics

D. Proton momentum distributions in bulk

III. COMPUTATIONAL DETAILS

A. Water potential

B. Simulation parameters

IV. RESULTS

A. The solid and liquid phases

B. Supercritical water at

V. CONCLUSION

### Key Topics

- Protons
- 47.0
- Molecular dynamics
- 22.0
- Hydrogen bonding
- 14.0
- Ice
- 10.0
- Statistical mechanics models
- 8.0

## Figures

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 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 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 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 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 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 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 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 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.

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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