^{1}, Carlos Vega

^{1,a)}, Eva G. Noya

^{1}, Rafael Ramírez

^{2}and Luis M. Sesé

^{3}

### Abstract

With a view to a better understanding of the influence of atomic quantum delocalization effects on the phase behavior of water, path integral simulations have been undertaken for almost all of the known ice phases using the TIP4P/2005 model in conjunction with the rigid rotor propagator proposed by Müser and Berne [Phys. Rev. Lett.77, 2638 (1996)]. The quantum contributions then being known, a new empirical model of water is developed (TIP4PQ/2005) which reproduces, to a good degree, a number of the physical properties of the ice phases, for example, densities, structure, and relative stabilities.

The authors would like to thank J. L. Abascal for insightful conversations and M. I. J. Probert for his hospitality and illuminating discussions while one of the authors, C.V., was in York. We would also like to thank the reviewers for their useful comments regarding the manuscript. This work has been funded by DGI (Spain) (Grant Nos. FIS2007-66079-C02-01 and FIS2006-12117-C04-03), by the Comunidad Autonoma de Madrid (Grant No. S-0505/ESP/0299) (MOSSNOHO), and by the Universidad Complutense de Madrid (Grant No. 910570). E.G.N. would like to thank the MEC for a Juan de la Cierva fellowship.

I. INTRODUCTION

II. METHODOLOGY

A. Path integrals for a rigid molecule

B. Path integrals for an ensemble of rigid molecules

C. Simulation details

III. RESULTS

IV. CONCLUSIONS

### Key Topics

- Ice
- 94.0
- Quantum effects
- 40.0
- Numerical modeling
- 15.0
- Phase diagrams
- 9.0
- Particle distribution functions
- 8.0

## Figures

Kinetic rotational energy from PIMC simulations of the isolated molecule (filled circles) as a function of temperature. Between 10 and 50 replicas have been used depending on the temperature. There is a good agreement between the simulation data and the rotational energy obtained from the theoretical partition function of an asymmetric top having the geometry (solid line). The magnitude of the error is less than the size of the symbols shown.

Kinetic rotational energy from PIMC simulations of the isolated molecule (filled circles) as a function of temperature. Between 10 and 50 replicas have been used depending on the temperature. There is a good agreement between the simulation data and the rotational energy obtained from the theoretical partition function of an asymmetric top having the geometry (solid line). The magnitude of the error is less than the size of the symbols shown.

Radial distribution function of ice for TIP4P/2005 (dashed green line) and (solid red line) at 250 K and . The blue dotted line corresponds to the experimental data of Soper at 220 K (Ref. 73).

Radial distribution function of ice for TIP4P/2005 (dashed green line) and (solid red line) at 250 K and . The blue dotted line corresponds to the experimental data of Soper at 220 K (Ref. 73).

Radial distribution function of ice II for TIP4P/2005 (dashed green line) and (solid red line) at 123 K and .

Radial distribution function of ice II for TIP4P/2005 (dashed green line) and (solid red line) at 123 K and .

Radial distribution function of ice VI for TIP4P/2005 (dashed green line) and (solid red line) at 225 K and .

Radial distribution function of ice VI for TIP4P/2005 (dashed green line) and (solid red line) at 225 K and .

Equations of state for ice at . Classical TIP4P/2005 model (gray dot-dashed line/filled triangles) (Ref. 74), experimental data (red solid line) (Ref. 75), (blue dotted line/filled squares), and the new TIP4PQ/2005 model (black double-dotted line/filled circles). The error in the density is of order .

Equations of state for ice at . Classical TIP4P/2005 model (gray dot-dashed line/filled triangles) (Ref. 74), experimental data (red solid line) (Ref. 75), (blue dotted line/filled squares), and the new TIP4PQ/2005 model (black double-dotted line/filled circles). The error in the density is of order .

Plot of the total energy of ices , II, III, V, and VI at low temperatures for for . Lines correspond to the fit . The error in the total energy is of order .

Plot of the total energy of ices , II, III, V, and VI at low temperatures for for . Lines correspond to the fit . The error in the total energy is of order .

Radial distribution function of ice for the TIP4PQ/2005 model using PIMC (dashed blue line) compared to the classical TIP4P/2005 model (dotted red line) and with experimental data (solid red line) (Ref. 96) at 77 K and .

Radial distribution function of ice for the TIP4PQ/2005 model using PIMC (dashed blue line) compared to the classical TIP4P/2005 model (dotted red line) and with experimental data (solid red line) (Ref. 96) at 77 K and .

Plot of the total energy of ices , II, III, V, and VI at low temperatures for for the TIP4PQ/2005 model. Lines correspond to the fit . The error in the total energy is of order .

Plot of the total energy of ices , II, III, V, and VI at low temperatures for for the TIP4PQ/2005 model. Lines correspond to the fit . The error in the total energy is of order .

## Tables

Parameters for both the TIP4P/2005 and the TIP4PQ/2005 models. The distance between the oxygen and hydrogen sites is . The angle, in degrees, formed by hydrogen, oxygen, and the other hydrogen atom is denoted by . The Lennard-Jones site is located on the oxygen with parameters and . The charge on the proton is . The negative charge is placed in a point at a distance from the oxygen along the H–O–H bisector.

Parameters for both the TIP4P/2005 and the TIP4PQ/2005 models. The distance between the oxygen and hydrogen sites is . The angle, in degrees, formed by hydrogen, oxygen, and the other hydrogen atom is denoted by . The Lennard-Jones site is located on the oxygen with parameters and . The charge on the proton is . The negative charge is placed in a point at a distance from the oxygen along the H–O–H bisector.

Results for the model for the systems studied along with a comparison with classical results for the same model. All energies are in units of kcal/mol and the densities are in . The errors (in kcal/mol) are in , in , in , in , and in .

Results for the model for the systems studied along with a comparison with classical results for the same model. All energies are in units of kcal/mol and the densities are in . The errors (in kcal/mol) are in , in , in , in , and in .

Oxygen-oxygen radial distribution function of ice for various water models at 250 K and .

Oxygen-oxygen radial distribution function of ice for various water models at 250 K and .

Results for the model for the low temperature ice phases at a pressure of 1 bar. The energies are in units of kcal/mol and the densities are in . The errors (in kcal/mol) are in , in , in , in , and in .

Results for the model for the low temperature ice phases at a pressure of 1 bar. The energies are in units of kcal/mol and the densities are in . The errors (in kcal/mol) are in , in , in , in , and in .

Comparison of the energies at 0 K for a selection of phases for both the and the TIP4PQ/2005 models as well as results for the classical TIP4P/2005 model (Ref. 78). The energies are in units of kcal/mol. The lowest energy (most stable phase) is shown in bold font. The lower section provides the relative energies with respect to ice II.

Comparison of the energies at 0 K for a selection of phases for both the and the TIP4PQ/2005 models as well as results for the classical TIP4P/2005 model (Ref. 78). The energies are in units of kcal/mol. The lowest energy (most stable phase) is shown in bold font. The lower section provides the relative energies with respect to ice II.

PIMC results for the TIP4PQ/2005 model for the systems studied and their relation to the experimental densities. All energies are in units of kcal/mol and the densities are in . The errors (in kcal/mol) are in , in , in , in , and in .

PIMC results for the TIP4PQ/2005 model for the systems studied and their relation to the experimental densities. All energies are in units of kcal/mol and the densities are in . The errors (in kcal/mol) are in , in , in , in , and in .

Unit cell parameters for the TIP4PQ/2005 model for a selection of ice phases. Experimental values are from Table 11.2 of Ref. 2. Note that for ice II the hexagonal unit cell rather than the rhombohedral unit cell is given. All distances are in angstrom.

Unit cell parameters for the TIP4PQ/2005 model for a selection of ice phases. Experimental values are from Table 11.2 of Ref. 2. Note that for ice II the hexagonal unit cell rather than the rhombohedral unit cell is given. All distances are in angstrom.

PIMC results for the TIP4PQ/2005 model for the low temperature ice phases at a pressure of 1 bar. All energies are in units of kcal/mol and the densities are in . The errors (in kcal/mol) are in , in , in , in , and in .

PIMC results for the TIP4PQ/2005 model for the low temperature ice phases at a pressure of 1 bar. All energies are in units of kcal/mol and the densities are in . The errors (in kcal/mol) are in , in , in , in , and in .

Estimates of the coexistence pressures (in bar) for the TIP4PQ/2005 model extrapolated to 0 K. Experimental values are taken from the work of Whalley (Ref. 36) and the values for the classical TIP4P/2005 model are from Ref. 78.

Estimates of the coexistence pressures (in bar) for the TIP4PQ/2005 model extrapolated to 0 K. Experimental values are taken from the work of Whalley (Ref. 36) and the values for the classical TIP4P/2005 model are from Ref. 78.

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