^{1,a)}and C. P. Herrero

^{1}

### Abstract

The isotope effect in the melting temperature of ice Ih has been studied by free energy calculations within the path integral formulation of statistical mechanics. Free energy differences between isotopes are related to the dependence of their kinetic energy on the isotope mass. The water simulations were performed by using the q-TIP4P/F model, a point charge empirical potential that includes molecular flexibility and anharmonicity in the OH stretch of the water molecule. The reported melting temperature at ambient pressure of this model increases by and upon isotopic substitution of hydrogen by deuterium and tritium, respectively. These temperature shifts are larger than the experimental ones (3.8 and 4.5 K, respectively). In the classical limit, the melting temperature is nearly the same as that for tritiated ice. This unexpected behavior is rationalized by the coupling between intermolecular interactions and molecular flexibility. This coupling makes the kinetic energy of the OH stretching modes larger in the liquid than in the solid phase. However, the opposite behavior is found for intramolecular modes, which display larger kinetic energy in ice than in liquid water.

This work was supported by Ministerio de Ciencia e Innovación (Spain) through Grant No. FIS2009-12721-C04-04 and by Comunidad Autónoma de Madrid through Project No. MODELICO-CM/S2009ESP-1691. The authors benefited from the discussions with L. M. Sesé, C. Vega, E. G. Noya, and E. R. Hernández.

I. INTRODUCTION

II. METHODOLOGY

A. Computational conditions

B. Relative free energy

C. Isotope effect on the free energy

D. Quantum-classical free energy difference

E. Temperature dependence of the free energy

III. TEST SIMULATIONS

A. Radial distribution function

B. Temperature of maximum density

IV. ISOTOPE EFFECTS IN THE MELTING TEMPERATURE

A. and

B. Classical limit

C. Kinetic energy and molecular mass

V. CONCLUSIONS

## Figures

OO RDFs derived from quantum and classical simulations of water at 298 K and density of . For comparison, the PI MD results of Ref. 12 are shown as open circles.

OO RDFs derived from quantum and classical simulations of water at 298 K and density of . For comparison, the PI MD results of Ref. 12 are shown as open circles.

OH RDFs derived from quantum and classical simulations of water at 298 K and density of . For comparison, the PI MD results of Ref. 12 are shown as open circles.

OH RDFs derived from quantum and classical simulations of water at 298 K and density of . For comparison, the PI MD results of Ref. 12 are shown as open circles.

HH RDFs derived from quantum and classical simulations of water at 298 K and density of . For comparison, the PI MD results of Ref. 12 are shown as open circles.

HH RDFs derived from quantum and classical simulations of water at 298 K and density of . For comparison, the PI MD results of Ref. 12 are shown as open circles.

Density of water at 1 atm pressure obtained from classical and PI MD simulations. Lines are cubic polynomial fits to the simulation data.

Density of water at 1 atm pressure obtained from classical and PI MD simulations. Lines are cubic polynomial fits to the simulation data.

The relative free energy of liquid water as a function of the hydrogen isotope mass. The open circles correspond to the masses of H, D, and T, respectively. The result was obtained by nonequilibrium simulations with the AS method at the reference state point .

The relative free energy of liquid water as a function of the hydrogen isotope mass. The open circles correspond to the masses of H, D, and T, respectively. The result was obtained by nonequilibrium simulations with the AS method at the reference state point .

The relative free energy of normal water and ice at pressure of 1 atm as determined by our RS simulations. The melting point is .

The relative free energy of normal water and ice at pressure of 1 atm as determined by our RS simulations. The melting point is .

The relative free energy of deuterated water and ice at pressure of 1 atm as determined by our RS simulations. The melting point is .

The relative free energy of deuterated water and ice at pressure of 1 atm as determined by our RS simulations. The melting point is .

The relative free energy of tritiated water and ice at pressure of 1 atm as determined by our RS simulations. The melting point is .

The relative free energy of tritiated water and ice at pressure of 1 atm as determined by our RS simulations. The melting point is .

The function obtained for liquid water by nonequilibrium AS simulations at the reference state point . The extrapolation is shown for the solid and liquid phases in the inset of the figure.

The function obtained for liquid water by nonequilibrium AS simulations at the reference state point . The extrapolation is shown for the solid and liquid phases in the inset of the figure.

The relative free energy of water and ice at pressure of 1 atm as determined by our RS simulations in the classical limit. The melting point is .

The relative free energy of water and ice at pressure of 1 atm as determined by our RS simulations in the classical limit. The melting point is .

Gibbs free energy difference between ice and water as a function of the molecular mass at the reference state point . Depending on the molecular mass, the KE of the liquid may be larger than that of the solid .

Gibbs free energy difference between ice and water as a function of the molecular mass at the reference state point . Depending on the molecular mass, the KE of the liquid may be larger than that of the solid .

Kinetic energy difference between ice and water as a function of the molecular mass at the reference state point . The line is a guide to the eye.

Kinetic energy difference between ice and water as a function of the molecular mass at the reference state point . The line is a guide to the eye.

Intramolecular OH distance of ice and water as a function of the molecular mass at the reference state point . Lines are guides to the eye. The inset shows the quasiharmonic stretch frequency (in ) as a function of the OH distance for the q-TIP4P/F potential. was derived from the second derivative of the potential energy with respect the OH bond distance and by considering the actual O and H masses.

Intramolecular OH distance of ice and water as a function of the molecular mass at the reference state point . Lines are guides to the eye. The inset shows the quasiharmonic stretch frequency (in ) as a function of the OH distance for the q-TIP4P/F potential. was derived from the second derivative of the potential energy with respect the OH bond distance and by considering the actual O and H masses.

## Tables

Molecular properties (bond distance, bond angle, and dipole moment) as well as kinetic and potential energy of liquid water at 298 K and density . is the distance of the RDF maximum associated with the H bond. The KE is partitioned into H-isotope and O-atom contributions ( and , respectively). is the maximum density of water at TMD, as derived from simulations at . Both classical and quantum results are given. The quantum results correspond to normal , heavy , and tritiated water.

Molecular properties (bond distance, bond angle, and dipole moment) as well as kinetic and potential energy of liquid water at 298 K and density . is the distance of the RDF maximum associated with the H bond. The KE is partitioned into H-isotope and O-atom contributions ( and , respectively). is the maximum density of water at TMD, as derived from simulations at . Both classical and quantum results are given. The quantum results correspond to normal , heavy , and tritiated water.

Relative free energies of tritiated water at the reference point as derived by independent AS simulations of different lengths. For a given simulation length, two results are presented, corresponding to simulations where the initial and final integration limits are interchanged. The last column shows the average of both independent runs.

Relative free energies of tritiated water at the reference point as derived by independent AS simulations of different lengths. For a given simulation length, two results are presented, corresponding to simulations where the initial and final integration limits are interchanged. The last column shows the average of both independent runs.

Relative free energy at the reference point (, ) of solid and liquid phases of normal, heavy, and tritiated water. is given in , and its estimated error is . The last column summarizes the results obtained in the classical limit .

Relative free energy at the reference point (, ) of solid and liquid phases of normal, heavy, and tritiated water. is given in , and its estimated error is . The last column summarizes the results obtained in the classical limit .

Computational conditions used in the nonequilibrium RS simulations to determine the temperature dependence of the relative Gibbs free energy of the studied isotope compositions of water and ice.

Computational conditions used in the nonequilibrium RS simulations to determine the temperature dependence of the relative Gibbs free energy of the studied isotope compositions of water and ice.

Melting temperature, entropy, and enthalpy for normal, heavy, and tritiated water as well as classical limit results at ambient pressure. The melting enthalpy was estimated by two independent methods. The change in the kinetic and potential energies upon melting (liquid minus solid values) and the molar volume of the solid and liquid phases are also given. The standard error in the final digits is given in parenthesis.

Melting temperature, entropy, and enthalpy for normal, heavy, and tritiated water as well as classical limit results at ambient pressure. The melting enthalpy was estimated by two independent methods. The change in the kinetic and potential energies upon melting (liquid minus solid values) and the molar volume of the solid and liquid phases are also given. The standard error in the final digits is given in parenthesis.

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