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Geochimica et Cosmochimica Acta 135 (2014) 203–216

Equilibrium fractionation of H and O isotopes in waterfrom path integral molecular dynamics

Carlos Pinilla a,⇑, Marc Blanchard a, Etienne Balan a, Guillaume Ferlat a,Rodolphe Vuilleumier b, Francesco Mauri a

a Institut de Mineralogie, de Physique des Materiaux, et de Cosmochimie (IMPMC), Sorbonne Universites - UPMC Univ Paris 06, UMR

CNRS 7590, Museum National d’Histoire Naturelle, IRD UMR 206, 4 Place Jussieu, F-75005 Paris, France.b Ecole Normale Superieure, Departement de Chimie, UMR CNRS-ENS-UPMC No. 8640, 24 Rue Lhomond, F-75231 Paris Cedex 05, France

Received 9 November 2013; accepted in revised form 19 March 2014; available online 29 March 2014

Abstract

The equilibrium fractionation factor between two phases is of importance for the understanding of many planetary andenvironmental processes. Although thermodynamic equilibrium can be achieved between minerals at high temperature, manynatural processes involve reactions between liquids or aqueous solutions and solids. For crystals, the fractionation factor acan be theoretically determined using a statistical thermodynamic approach based on the vibrational properties of the phases.These calculations are mostly performed in the harmonic approximation, using empirical or ab-initio force fields. In the caseof aperiodic and dynamic systems such as liquids or solutions, similar calculations can be done using finite-size molecularclusters or snapshots obtained from molecular dynamics (MD) runs. It is however difficult to assess the effect of these approx-imate models on the isotopic fractionation properties. In this work we present a systematic study of the calculation of the D/Hand 18O/16O equilibrium fractionation factors in water for the liquid/vapour and ice/vapour phases using several levels oftheory within the simulations. Namely, we use a thermodynamic integration approach based on Path Integral MD calcula-tions (PIMD) and an empirical potential model of water. Compared with standard MD, PIMD takes into account quantumeffects in the thermodynamic modeling of systems and the exact fractionation factor for a given potential can be obtained. Wecompare these exact results with those of modeling strategies usually used, which involve the mapping of the quantum systemon its harmonic counterpart. The results show the importance of including configurational disorder for the estimation of iso-tope fractionation in liquid phases. In addition, the convergence of the fractionation factor as a function of parameters such asthe size of the simulated system and multiple isotope substitution is analyzed, showing that isotope fractionation is essentiallya local effect in the investigated system.� 2014 Elsevier Ltd. All rights reserved.

1. INTRODUCTION

The fractionation of stable isotopes is of importance forgeosciences, as variations of the isotopic composition canprovide information on processes acting in atmosphere,biosphere, geosphere or hydrosphere (Hoefs, 1997).

http://dx.doi.org/10.1016/j.gca.2014.03.027

0016-7037/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Tel.: +33 1 44279822.E-mail address: pinilla@impmc.upmc.fr (C. Pinilla).

Experimental techniques have evolved continuously toincrease the accuracy and precision of isotopic compositionmeasurements. The development of reliable computationalmethods to describe the isotopic fractionation betweensolid and liquid is also a topic of continuous improvementsthat has recently opened the door to the understanding ofsome of the geochemical mechanisms responsible for theproduction of isotopic signatures. In almost all these com-putational approaches, the mass dependence of the isotope

204 C. Pinilla et al. / Geochimica et Cosmochimica Acta 135 (2014) 203–216

fractionation is dealt with through the calculation of thevibrational properties of the system of interest (Bigeleisenand Mayer, 1947; Richet et al., 1977). For crystals, the cal-culation of these vibrational properties is well establishedand many examples in the literature show that the calcu-lated infrared and Raman frequencies, vibrational densityof states, phonon dispersion curves as compared to theirexperimentally determined counterparts are reliable. Theequilibrium isotope fractionation factor can thus becomputed using methods based on an empirical descriptionof the system (Patel et al., 1991) or more accuratelythrough first-principles calculations (Schauble et al., 2006;Meheut et al., 2007; Blanchard et al., 2009; Jahn andWunder, 2009; Rustad et al., 2010a; Schauble, 2011;Sherman, 2013).

Calculation of the vibrational properties of liquids andsolvated elements represents a bigger challenge, which mustbe overcome to produce reliable solid–liquid or solid–solu-tion isotopic fractionation factors. The aperiodic structureof a liquid phase or a solvated element is usually treatedby means of a cluster approximation (Rustad et al.,2010a,b; Sherman, 2013; Fujii et al., 2013) where the speciesof interest is surrounded by a solvation shell. The stablecluster structure is obtained at T = 0 K by minimizing theforces acting on the atoms. This method has the disadvan-tage of neglecting the constant exchange of particles withinthe solvation shells and other effects, such as the formationof chemical bonds and structural rearrangements as a func-tion of temperature and pressure considered to be impor-tant for the calculation of the isotope fractionation(Bigeleisen and Mayer, 1947; Jahn and Wunder, 2009).Another approach consists of making use of ab-initiomolecular dynamics and periodic boundary conditions.Using this approach, Rustad and Bylaska (2007) calculatedthe velocities correlation of the exchanging isotopes in theliquid phase and through its Fourier transform found thevibrational density of states to predict the fractionation fac-tor of 11B/10B in an aqueous solution. From the same typeof calculations, Kowalski and Jahn (2011) and Kowalskiet al. (2013) estimated the fractionation factor betweenaqueous solutions and solid minerals by calculating theforce constants acting on the fractionating atom only.

On the other hand, the direct relation of fractionation toquantum mechanical fluctuation effects can be used to esti-mate fractionation factors in liquid systems. Furthermore,fractionation factors can be associated to quantum kineticenergy differences of a species between two different phases(Polyakov and Mineev, 2000; Morrone and Car, 2008;Herrero and Ramirez, 2011; Ramirez and Herrero, 2011).Very recently, Markland and Berne (2012) have performedpath integral molecular dynamics simulations (PIMD) andpresented an estimation of the fractionation factors of D/Hbetween liquid and vapour phases of water. In this workthey have used the kinetic energy of the substituted atomas a way to estimate the difference of free energy betweenthe different phases. The advantage of this method lies onthe inclusion of quantum effects of relevance in water,and on taking into account anharmonic effects and systemsize when calculating such quantities. In addition, Ceriottiand Markland (2013) have performed a study on the

efficiency of methods for the simulations of isotope effects.In their work these authors discuss the importance on theconvergence of results as a function of the number ofparameters used in PIMD calculations. They alsodeveloped an accurate perturbative method to calculateisotope effects in homogeneous systems using onlyproperties of the system containing the most common typeof isotope.

In this contribution we present a systematic computa-tional study of D/H and 18O/16O fractionation in the liquid,solid and vapour phases of water using PIMD methods andcompare their performance with more traditional methodsbased on the computation of vibrational properties. We willdiscuss the convergence of the fractionation factor as afunction of parameters such as temperature and size ofthe simulated system and will discuss how the variousapproximations, inherent to the methods used, affect thecalculation of the fractionation factor. This study will con-centrate on an assessment of the minimum number of ingre-dients necessary to determine accurate estimates of theequilibrium fractionation factor involving a liquid phase.We have chosen water given its importance in many naturalprocesses. The structure of this paper is as follows: In Sec-tion 2 a summary of the methods and computational tech-niques will be given. This will be followed by a discussionon the equilibrium fractionation factors found from theseapproaches and compared with experimental data. Finally,we will summarise our conclusions.

2. METHODS

2.1. The equilibrium isotope fractionation factor from

vibrational properties

The equilibrium isotope fractionation factor (a factor)of an element displaying two isotopic forms Y and Y*

between two phases a and b (with composition AYn andBYn respectively) is related to isotopic abundances as:

aða; b; Y Þ ¼ ðnY �=nY ÞaðnY �=nY Þb

; ð1Þ

where nY,a is the mole fraction of isotopes Y in phase a.This temperature-dependent thermodynamic quantity isrelated to the partition function of phase a and b by:

aða; b; Y Þ ¼ QðAY n�1Y �i ÞQðAY nÞ

� �a

QðBY nÞQðBY n�1Y �i Þ

� �b

ð2Þ

where QðAY n�1Y �i Þ represents the partition function of thephase a having the atom Y on the site i substituted withY*, Q(AYn) is the partition function of the phase a withoutany substitutions.

In the classical limit, the ratio of the classical partitionfunctions determined for two different isotopes does notdepend on the potential describing the interaction betweenthe atomic constituents of the system. The isotopic fraction-ation thus only depends on the quantum behavior of thesystem and it is useful to define the reduced partition func-tion ratio b(a,Y) as the ratio of the quantum and classicalpartition function ratios. At variance with the isotopic frac-tionation factor a, the b factor is a quantity defined for a

C. Pinilla et al. / Geochimica et Cosmochimica Acta 135 (2014) 203–216 205

single phase. It can be envisioned as the fractionation factorbetween the considered system and a reference one,corresponding to an ideal gas of atoms Y (Richet et al.,1977). Isotopic fractionation factors are usually expressedin permil (&) and using the common notation:103ln(a(a,b,Y)) = 103ln(b(a,Y)) � 103ln(b(b,Y)).

Assuming that the Y* species is in the dilute limitand that the free energy on one site does not dependon the occupancy of the same site in a neighbouringcell, the b(a,Y) function of a crystalline phase a canbe represented as the average over the occupancy ofall equivalent sites within the crystal lattice. Within theharmonic approximation and using the Teller-Redlichrule (Wilson et al., 1955) to improve the numerical con-vergence, the b(a,Y) factor of a crystal can be written as(Meheut et al., 2007):

bða;Y Þ ¼Y3Nat

i¼1

Yfqg

m�q;imq;i

e�hm�q;i=ð2kBT Þ

1� e�hm�q;i=ðkBT Þ �1� e�hmq;i=ð2kBT Þ

e�hmq;i=ð2kBT Þ

" #ð1=N�N qÞ

ð3Þ

where mq,i are the frequencies of the phonons of the ithbranch with wave vector q. Nat is the number of atoms inthe unit-cell, and N is the number of sites occupied by theY atom in the unit-cell. T is the temperature and kB isthe Boltzmann constant. m*

q,i correspond to frequencies ofthe phase containing the rare isotope. The product ofEq. (3) is usually performed on a grid containing Nq qvectors to ensure full convergence.

It can also be shown (Meheut et al., 2007; Liu et al.,2010) that a relationship similar to Eq. (3) holds in the caseof the gas phase, where rotations and translations contribu-tion are also present (Chacko et al., 2001). Usually, thesecontributions can be safely accounted for using the Teller-Redlich rule (also referred to as the high-temperature prod-uct rule) (Stripp and Kirkwood, 1951; Wilson et al., 1955).In the present work we have verified that b(a,Y) factors forthe gas phases using the Teller-Redlich rule differ by lessthan 0.01& with respect to those obtained using a full rep-resentation of the translational and rotational degrees offreedom in the partition functions.

2.2. The equilibrium isotope fractionation factor and the

kinetic energy

The b factor is exactly related to the free energy of theisotopologues by (Urey, 1947; Bigeleisen and Mayer,1947; Polyakov and Mineev, 2000):

lnðbða; Y ÞÞ ¼ � F � � FkBT

þ F � � FkBT

� �classic

ð4Þ

where F is the free energy and the subscript classic refers toquantities calculated using classical mechanics. Unfortu-nately, the absolute value of the free energy is not a quan-tity that can be directly obtained for any arbitrary system.Relating the free energy to another physical property, suchas the kinetic energy, can circumvent this problem. On thisline, it can be shown (Landau and Lifshitz, 1980, pp. 50–51)that the free energy of an isotopic species depends on itskinetic energy and mass by:

@F@m¼ �hKi

mð5Þ

where h i represents a thermodynamic average in the canon-ical ensemble. Inserting Eq. (5) into Eq. (4) and taking intoaccount that in the ideal limit the kinetic energy of an atomis hKi ¼ 3kBT

2, the b-factor is then given by:

lnðbða; Y ÞÞ ¼ 1

kBT

Z m�

mdm0hKaðm0Þi

m0� 3

2ln

m�

m

� �ð6Þ

where hKaðm0Þi is the average kinetic energy of an atom ofmass m0 in phase a. In this expression, the b factor is thusobtained by thermodynamic integration from mass m tomass m*.

Here, we stress that the kinetic energy used in the ther-modynamic integration is that of the quantum system. Itdiffers from the kinetic energy determined using standardmolecular dynamic methods. These latter methods solvethe classical equation of atomic motions in a force field,which can be defined either empirically or using ab initioelectronic structure calculations (e.g. in Car-Parinellomolecular dynamics). In the present case, the determinationof the kinetic energy has to take into account the fact that,in a quantum system, the atomic trajectories are notdefined. The atoms display some degree of delocalization(i.e. some uncertainty on their position); which is inverselyrelated to their mass. Path integral methods (Feynman andHibbs, 1965; Tuckerman, 2010) enable the treatment ofsuch effect by replacing the standard classical system by alarger number of replicated classical systems. The replicatedsystems interact through harmonic springs connecting agiven atom to its counterpart in the next replica. The quan-tum thermodynamic averages can be calculated exactly forany force field using path integral methods.

More specifically, in path integral methods the quantumpartition function is calculated through a discretization ofthe integral representing the density matrix (see AppendixA). This discretization leads to a representation where thequantum particle is isomorphic to a classical one definedby a ring polymer of P classical replicas (called beads) con-nected by harmonic springs (Feynman and Hibbs, 1965;Ceperley, 1995; Tuckerman, 2010). This isomorphism onlyapplies to equilibrium thermodynamic properties. The leftside of Fig. 1 shows a representation of O and H in a watermolecule where each quantum particle has been substitutedby a ring polymer containing P = 4 beads. In practice, eachbead represents a classical system containing a replica foreach atom of the original quantum system as shown onthe right side of Fig. 1 for a water molecule.

Here we have used the centroid virial estimator for thecalculation of the kinetic energy (see Appendix A for moreinformation):

KaðmÞ ¼3

2kBT þ 1

2P

XP

k¼1

ðrðkÞi � rðCÞi Þ �@UðrðkÞi Þ@rðkÞi

* +a

ð7Þ

where hia represents an ensemble average on phase a and

rðkÞi � rðCÞi is a vector from the centre of the polymer to theposition of the kth bead as depicted in Fig. 1. The term

� @UðrðkÞi Þ@rðkÞi

is the total force on the ith atom of the kth bead

O

H H

1

2

3

4

3

4

1

2

4

3

2

1

O

H H

1

2

3

4

3

4

1

2

4

3

2

1

O

H H

1

2

3

4

3

4

1

2

4

3

2

1

O

H H

1

2

3

4

3

4

1

2

4

3

2

1Δ rO

H H

2

3

4

3

4

1

2

4

3

2

1

c

1

c c

(a) (b)

Fig. 1. (a) Schematic representation of a water molecule using the path integral formalism with P = 4 beads. The vector DrðkÞi � r

ðCÞi is the

difference of positions between the kth bead and the centroid for a particle i. The grey dashed circles represent the centroid position as definedby Eq. (A.8). (b) The bead configurations of the water molecule obtained from this representation. In colour the water molecule representationfrom classical replicas of each atom that correspond to the same bead. (For interpretation of the references to colour in this figure legend, thereader is referred to the web version of this article.)

206 C. Pinilla et al. / Geochimica et Cosmochimica Acta 135 (2014) 203–216

arising from the system’s force field. The first part of Eq. (7)can be seen as the kinetic energy of a free particle whilst thesecond one represents the quantum confinement of a parti-cle due to the neighbouring atoms. It is clear that if theatom behaves classically the total fractionation will amountto zero. Therefore, the average in Eq. (7) should be per-formed taking into account quantum effects. This systemcan be readily explored and forces easily calculated fromPIMD methods (see Tuckerman, 2010, for a description).

2.3. Computational details

We have carried out PIMD simulations within the stag-ing approach (Tuckerman and Berne, 1993) as implementedin the PINT module of the CP2K suite of codes (Krack andParrinello, 2004; Yujie et al., 2006) for water in the liquid,vapour and solid phases at different temperatures and den-sities. We have used the point charge, flexible q-SPC/FWforce field for water (Paesani et al., 2006). This model hasbeen parameterized to reproduce many of the experimentalstructural and dynamical properties of water including iso-tope effects. However, the q-SPC/FW force field treats OHstretch through a harmonic potential and so lacks theeffects to describe correctly OH anharmonicity. Here wehave used the approach of Markland and Berne (2012)and replaced the harmonic OH stretch potential by thequartic potential:

V OHðrÞ ¼ Dr a2r ðr � reqÞ2 � a3

r ðr � reqÞ3 þ7

12a4

r ðr � reqÞ4� �

;

ð8Þ

where Dr, ar and req have been taken from the q-TIP4P/Fwater potential of Habershon et al. (2009). Structural prop-erties for liquid water using this combined force fieldtogether with PIMD methods can be found in the Supple-mentary information. The O–H radial distribution

function, which concerns directly with our choice of usingan anharmonic description of the O–H bond agrees wellwith experimental results. Unless specified, all results pre-sented in this work have been obtained using this modifiedpotential and that from here on we will refer as the q-SPC/FW+anh force field.

Liquid water simulations were performed using simula-tion boxes containing up to 256 molecules in the case ofwater. For ice Ih we have started from the structures pro-posed by Hirsch and Ojamae (2004). In particular, we havefound configuration No. 6 from that work to have the low-est formation energy with the q-SPC/Fw+anh force fieldand we used it as the basis for ice modeling in this work.Calculations have been carried out within the NVT ensem-ble using P = 32 beads for the discretisation of the quantumsystem during tests of convergence for the fractionation fac-tor as will be shown later. All calculations have been per-formed at a constant volume, corresponding in each caseto the experimental coexistence density for the liquid andvapour phases at a given temperature (Wagner and Prub,2002) and to the experimental density of ice (Rottgeret al., 1994). The temperature was controlled via a Nose-Hoover chain of three thermostats (Martyna et al., 1992).Expected averages were derived in runs of 3 � 105 molecu-lar dynamics steps for water and 2 � 106 for the water mol-ecule in the gas phase, in both cases a time step of 0.5 fs wasused. Finally, the integrals (Eq. (6)) have been estimatedusing the trapezoidal rule with 11 masses (6 masses in thecase of O) equally spaced between MH and MD

(M16Oand M18O

).In addition to PIMD calculations, fractionation factors

have been estimated using phonon frequencies computedfrom snapshots taken from equilibrated classical moleculardynamics runs performed at several temperatures for thesolid, liquid and vapour phases. We have used the sameforce field as described earlier and carried out runs of

C. Pinilla et al. / Geochimica et Cosmochimica Acta 135 (2014) 203–216 207

5 � 105 ps containing 256 and 16 water molecules for theliquid and ice phases respectively. Reported fractionationfactors have been calculated using an average of up to3000 different snapshots for each one of the studied temper-atures. In a first step, the atomic positions have not beenrelaxed and the subsequent imaginary frequencies havebeen neglected in the calculation of the fractionation factor.Fractionation factors reported using this method arereferred to as the MD results. In a second step, the atomicpositions of the MD snapshots have been optimized usingthe same force field, leading to stable configurations atT = 0 K. The vibrational properties of these relaxedconfigurations were then used to calculate the isotopefractionation factors, which will be referred to as theRELAX results.

As mentioned before each bead contains a full classicalrepresentation of the quantum system and thus after run-ning a PIMD simulation a “pseudo-quantum” trajectorycorresponding for each bead is produced. We have takenadvantage of these “pseudo-quantum” trajectories createdfor centroid and beads to calculate b factors. We haveattempted to use the dynamics of the centroid where theposition rc for each atom is defined by Eq. (A.8), as away to calculate the fractionation factor but the obtainedvalues were much larger than the ones obtained from othermethods. The failure on using the centroid trajectory ismost likely related to the inability of the centroid dynamicsto describe correctly the vibrational properties of the watermolecule. Witt et al. (2009) used the centroid positions toobtain the infrared spectra of the OH molecule and theyfound it to be red shifted. They concluded that this errorwas possibly due to the presence of many configurationsin the centroid trajectory where the ”real” OH bond repre-sented by the O and H centroids that depends on the spreadof the ring polymer was wrongly described. This situationcould generate wrong OH bond lengths. In fact, Wittet al. (2009) observed even negligible centroid bond lengthswhen T! 0 K. Instead, we have used the trajectory fromone of the beads (referred here as the BEAD results) repre-senting one of the classical replicas of the quantum system.We believe this approach to be more consistent, as inter-atomic interactions are unequivocally defined by the forcefield, which is not the case for the interaction betweencentroids. As a reading help, a summary of the variousmodeling approaches is reported in Table 1.

Table 1Summary of all the methods used in this work for the calculation of the

Method Configurations

PIMD Full trajectory from a 32 bead PIMD calculation perfotemperature

MD Snapshots taken from a classical MD trajectory – No ratomic positions to the equilibrium point performed

RELAX Snapshots taken from a classical MD trajectory – Relaxpositions to the equilibrium point performed

BEAD Snapshot taken from one of the beads trajectory of a PI

CENTROID Snapshots taken from the centroid trajectory of a PIM

Additionally, we have performed density functional the-ory calculations (DFT) of the relaxed structure of ice Ihusing the generalized gradient approximation as proposedby Perdew et al. (1996). In the case of liquid water we havetaken snapshots from the classical molecular dynamics tra-jectory and relaxed atomic position using DFT. Phononfrequencies of the relaxed structures were computed fromfirst-principles using the linear response method (Baroniet al., 2001). Calculations were done with the PWscf andPHonon modules of the QUANTUM-ESPRESSO code(Giannozzi et al., 2009). We have used the same calculationparameters as reported in an earlier work by Blanchardet al. (2010).

3. RESULTS AND DISCUSSION

3.1. Kinetic energy of H and O nuclei

A typical quantum effect related to the atomic motion isthat the kinetic energy converges to a finite value at lowtemperatures rather than to zero as in the classical model.Here we have used Eq. (7) to evaluate KH and KO, the con-tribution to the kinetic energy from an H and O atom,respectively, belonging to a water molecule in the liquidphase.

The kinetic energy for O and H as a function oftemperature are reported in Figs. 2 and 3, respectively.For comparison we have also included the classical limit(P = 1) of the kinetic energy, which is directly proportionalto temperature (equipartition theorem). In all cases, thekinetic energy increases with temperature and quantumeffects become important at low temperatures (see Fig. 4).

The kinetic energy of H and O is slightly larger in ice Ihthan in water. The KH contribution to the total kineticenergy of the water molecule in a condensed phase is largerthan the KO and also much larger than its classical counter-part. For both H and O, the quantum contribution to thekinetic energy related to the mass is important. Kineticenergy differences due to isotope mass differences are alsonoticeable in both cases. For oxygen there is a decrease of5% in the kinetic energy when changing from 16O to 18O.For hydrogen, a 30% decrease in kinetic energy is observedfrom H to D. This indicates that the quantum contributionto the kinetic energy is very important for these species atthe studied temperatures. For very large temperatures,

isotope fractionation factors.

103ln(b)

rmed at a given Calculated using the isotope’s kinetic energy(Eq. (6)). Exact reference method.

elaxation of Calculated through vibrational properties(Eq. (3))

ation of atomic Calculated through vibrational properties(Eq. (3))

MD calculation Calculated through vibrational properties(Eq. (3))

D calculation. Calculated through vibrational properties(Eq. (3))

Fig. 2. Calculated oxygen contribution (KO) to the kinetic energyin ice Ih (right) and liquid water (left) as a function of temperature.Circles and squares correspond to estimates using the masses of 16Oand 18O respectively. Dash-dotted line represents the kinetic energyof 16O in the vapour phase. Error bars are of the order 1 � 10�3 kJ/mol and have been omitted. The solid line represents the classiclimit for the kinetic energy that for ice and water amounts to (3/2)kBT per atom. Inset shows the region where ice and water curvesoverlap.

Fig. 3. Calculated hydrogen contribution (KH) to the kineticenergy of ice Ih (right) and water (left) as a function oftemperature. Circles and squares correspond to estimates usingthe mass of H and D respectively. Dash-dotted line represents thekinetic energy of H in the vapour phase. Error bars are of the order3 � 10�3 kJ/mol and have been omitted. The solid line representsthe classic limit for the kinetic energy that for ice and wateramounts to (3/2)kBT per atom. Inset shows the region where iceand water curves overlap.

Fig. 4. The D/H b factor in & for the condensed (top) and vapour(bottom) phases of water from several of the outlined methods. Seetext for a description on the calculation methods.

208 C. Pinilla et al. / Geochimica et Cosmochimica Acta 135 (2014) 203–216

one should expect the kinetic energy to converge to the clas-sical limit. In Figs. 2 and 3 we have also included the kineticenergy for O and H in the isolated molecule (i.e. in the gasphase). It can be seen that quantum effects are slightlystronger in the liquid or solid phase due to the moresignificant quantum confinement, which as seen in Eq. (7)

depends on the forces exerted on the particle by the othersurrounding atoms.

3.2. H and O reduced partition function ratios (b-factors)

We have calculated the reduced partition function ratios(b-factors) from harmonic vibrational properties using Eq.(3) and from PIMD using Eq. (6). The calculated b factor inthe liquid and ice Ih phases for D/H and 18O/16O can beseen in Figs. 4 and 5. The RELAX and MD methods leadto different reduced partition ratios. The difference is largerin the case of H/D (50& in average) than for 18O/16O(1.5&). In both cases, it is of importance when evaluatingthe equilibrium fractionation factor a. The difference sug-gests a larger rearrangement involving H positions afterthe relaxation process. The comparison between the PIMDand the BEAD follows a similar trend with differencesbetween b-factors smaller for the liquid water than for theice. In general, these differences become smaller when tem-perature increases. The similarity of PIMD and BEAD onone hand and of MD and RELAX on the other, comesfrom the fact that equivalent configurations are used. Inall cases, the PIMD and the BEAD give in general b-factorsthat are lower than the MD and RELAX cases. This is most

C. Pinilla et al. / Geochimica et Cosmochimica Acta 135 (2014) 203–216 209

likely due to noticeable differences between the pseudoquantum and classical configurations generated by PIMDand classical MD calculation, respectively, as alreadyknown for the structural properties of pure water(Paesani et al., 2006). Calculations in the gas phase followa similar trend to the one presented above for the con-densed phases with b-factors that differ by an average of12& for D/H and 0.5& for 18O/16O.

Frequently, calculations to estimate b and a factors arecarried out using only the harmonic vibrational propertiesof atomic configurations pertaining to a single MD run per-formed at a fixed temperature. These vibrational propertiesare subsequently used to extrapolate the b or a factor over alarge temperature range by considering the related changesin the occupation of vibrational states (e.g. Eq. (3)). Wefound this to be an acceptable approximation in the caseof the RELAX method where differences of �0.5& areobserved on the reduced partition function ratio of hydro-gen obtained from MD runs at each temperature of interestand an extrapolation from a single MD run at 300 K. In thecase of the MD method (i.e. clamped configurations fromMD runs), differences as large as 50& are observed (seeFig. 5).

Fig. 5. The 18O/16O b factor in & for the condensed (top) andvapour (bottom) phases of water from several of the outlinedmethods. See text for a description on the calculation methods.

3.3. Convergence of b and a factors

As mentioned in the methods section, PIMD relies onproviding a number of beads (P) large enough to ensurethe convergence of the results. In the case of the structureof liquid water, this number has been established to a min-imum of P = 32 beads (Morrone et al., 2007). Using a smal-ler number of P slices causes a blue shift and slightbroadening of the proton momentum distribution withrespect to converged results (Morrone et al., 2007). In thecase of H/D isotope fractionation, Markland and Berne(2012) show that P = 32 is again a good choice to get agood convergence on the final value of a.

In the present study, we have estimated bH and bO in thegas phase using several values of P (Fig. 6). Even with anumber of beads as large as P = 256, a full convergenceof the b factor is not obtained. A similar trend is expectedfor b-factors calculated in the liquid phase of water wherewe have seen a difference of 50& at 300 K for the D/H

Fig. 6. Difference in & of the b factor for D/H (top) and 18O/16O(bottom) in the vapour phase of water. The curves correspond tothe difference of b estimated as jb256

beads � bXbeadsj where X corresponds

to 32 beads (black dots), 64 beads (blue triangles) and 128 beads(red squares) respectively. The inset shows the b factor calculated ineach case: 32 beads (blue cross), 64 beads (green diamonds), 128beads (red squares), 256 beads (black dots). (For interpretation ofthe references to color in this figure legend, the reader is referred tothe web version of this article.)

Fig. 8. Calculated D/H equilibrium fractionation factor in & forthe liquid/vapour system as a function of the size of the simulationcell. The black solid line corresponds to calculations carried outusing 256 water molecules. Red dashed line represents the valuesfound using a 32 water molecules cell. The inset shows the H kineticenergy at T = 300 K as a function of the number of watermolecules contained in the simulation cell. (For interpretation ofthe references to colour in this figure legend, the reader is referredto the web version of this article.)

210 C. Pinilla et al. / Geochimica et Cosmochimica Acta 135 (2014) 203–216

b-factor estimated using 32 and 64 beads in simulationscontaining 256 water molecules. The study of the variationof the b-factors as a function of number of beads providesan estimation of the precision of calculated b-factors. Asimilar procedure can be carried out for the quantity ofinterest i.e. the equilibrium fractionation factor a. The D/H fractionation factors between liquid and gas water calcu-lated using 32 and 64 beads on a model containing 256water molecules system differ by less than �5&, which issignificantly smaller than the range of variation of the D/H isotope fractionation factor (Fig. 7). Overall, our calcula-tions seems to show b-factors to differ more with the num-ber of beads than the a-factors however given that theformer are larger in magnitude than the later ones we findthe relative error in b and a factors to be similar. This sug-gests that 32 beads calculations are enough to provideacceptable results for H/D a-factors. These estimates arein line with the results of Ceriotti and Markland (2013).They have suggested the number of beads P needed forthe convergence of a PIMD calculation of water to be pro-portional to hmmax/kBT where mmax is the magnitude of themaximum vibrational frequency of the studied system (ifmmax � 3700 cm�1 the numbers of beads to use range from30 at 190 K to 12 at 600 K). Another important questionis related to the effect of the system size, i.e. the numberof molecules in the unit-cell imposed by periodic boundaryconditions, on the computed fractionation factors. Accord-ingly, we have performed PIMD calculations on modelscontaining 32, 64, 256 and 1000 water molecules per unit-cell. The calculated liquid–gas D/H fractionation factorobtained for the smallest system (32 water molecules) differsby less than 2& from that of 256 water molecules (Fig. 8).Consistently, the kinetic energy values obtained for varioussystem sizes at T = 300 K differ by less than 7 � 10�3 kJ/mol (Fig. 8 inset). Within the accuracy of our calculations,the kinetic energy and therefore the fractionation factor areweakly dependent on the system size. Even calculationsperformed using systems as small as the one containing32 water molecules give a good estimation of thefractionation factor. This suggests that isotopic fraction-ation in water is dominated by the local environment of

Fig. 7. Calculated D/H fractionation factor in & for the water–vapour system using 32 (squares) and 64 (circles) beads.

the water molecule depending mainly on the nearby bondsand hydrogen network. In addition, the little dependence ofthe fractionation on the system size opens the possibility ofcombining path integral methods with a description ofinter-atomic interactions at the density functional level.Such type of combination has been already carried outfor other properties of water (Benoit et al., 1998;Morrone and Car, 2008; Pamuk et al., 2012) and shouldlead to more accurate predictions of the isotopic fraction-ation factors.

Since minority isotopes usually occur at very low con-centrations, it seems desirable to perform the computer sim-ulations with the most dilute configuration, in which onlythe mass of one atom is modified whilst the other atomshave the mass of the most common isotope. This impliesthat the kinetic energy of Eq. (7) has to be computed onlyfor this atom and represents a problem for the gathering ofstatistics and length of the simulations. In the case of waterit would be more efficient to consider several isotope substi-tutions in such a way that statistics can be collected fasterwith shorter simulations. One could then introduce a sys-tem for water where for each molecule one of the twohydrogen atoms has been substituted by the less commonisotope. Based on similar work by Ceriotti and Markland(2013), we show in Fig. 9 the kinetic energy of a D atom cal-culated using a system containing 32 water molecules whereNHDO represents the number of molecules where one Hatom has been substituted by a D. The kinetic energy isweakly dependent on this multiple isotope substitution.Similarly, the D/H fractionation factors between liquidand gas at 300 K calculated for a system of 32 water mole-cules containing only one HDO or 32 HDO molecules are68 ± 8& or 65 ± 3&, respectively. These results togetherwith the little dependence of the a-factor on the system size

Fig. 9. Average kinetic energy of a D atom as a function of thenumber of water molecules where one H has been substituted by aD atom (HDO) in a simulation box containing 32 water molecules.

Fig. 10. Equilibrium fractionation factor in & of D/H betweenliquid (ice) and vapour phases of water as a function of temper-ature. Top: The equilibrium fractionation factor calculated fromPIMD using the q-SPCFw+anh force field (solid line), DFT (semi-filled dots) as well as experimental results (plus) by Horita andWesolowski (1994), Merlivat and Nief (1967), Ellehoj et al. (2013).Bottom: comparison between the various calculation methods.

C. Pinilla et al. / Geochimica et Cosmochimica Acta 135 (2014) 203–216 211

corroborate the fact that the momentum-dependent observ-ables tend to be very local. In this work, we have used thecomplete substitution to evaluate isotopes effects usingPIMD.

3.4. Equilibrium fractionation factor of H and O isotopes in

water

We have calculated the D/H and 18O/16O equilibriumfractionation factors between the liquid (ice) and gas phaseof water (Figs. 10 and 11). In both cases, the PIMD modelis in good agreement with experiments. For D/H in waterthe PIMD correctly shows inverse fractionation above550 K where D more favorably occurs in the gas phase thanin the liquid one. Markland and Berne (2012) decomposedthe D/H fractionation factor in water, into contributionsfrom parallel and perpendicular atomic displacements withrespect to the molecular plane. They showed that this inver-sion is related to a cancellation of these contributions dueto the different rates at which these components reach theclassical limit when the temperature increases. Further-more, in their work these authors highlighted the impor-tance of incorporating anharmonic terms in the OH bondpotential to obtain an accurate estimate of the H/D frac-tionation factor in water. We have estimated the a factorusing the original q-SPCFw force field by Paesani et al.(2006) where the OH bond is described through a harmonicpotential and obtained a D/H fractionation factor rangingfrom 246& at a temperature of 300 K to nearly 48& at600 K. These values are much higher than experimentalresults and also higher than all the present results obtainedusing an anharmonic potential for the OH bond.

The PIMD method represents an exact approach wherequantum and anharmonic effects are correctly included.Both effects are important to obtain values of the a-factorcomparable with experimental results. In addition, thePIMD method has the advantage over methods based onclusters of atoms to allow for calculations using periodic sys-tems. A more realistic representation of extended condensedphases is thus obtained as a function of temperature andpressure. In Figs. 10 and 11 we have reported estimates of

the isotopic fractionation of H and O using the otherapproaches already mentioned (MD and RELAX). TheMD method displays the right asymptotic behavior at hightemperature, as compared to the reference PIMD result,and hints on the importance of including multipleconfigurations in the calculations. In the case of D/H, wherethe fractionation is large, this method overestimates thefractionation values. For heavier isotopes such as those ofoxygen, the method performs better. The RELAX methodprovides a description of the fractionation factor of H inthe liquid phase with the wrong asymptotic behaviour. Athigh temperatures this method seems unable to recover thesharp change in density that occurs at the coexistence ofliquid and vapour water and does not lead to any inversionof the D/H fractionation. The fact that the atomic positionsare optimized to minimize the atomic forces of the systemleads to a much narrower distributions of OH distancesand molecular angles that somewhat erases some of the dis-order present in the liquid phase. This effect is less evident inthe case of 18O/16O fractionation where the RELAX perfor-mances are comparable to those of the MD method in the

Fig. 11. Equilibrium fractionation factor in & of 18O/16O betweenliquid (ice) and vapour phases of water as a function of temper-ature. Top: The equilibrium fractionation factor calculated fromPIMD (solid line), DFT (semi-filled dots) as well as experimentalresults (plus) by Horita and Wesolowski (1994), Majoube (1970),Ellehoj et al. (2013). Bottom: comparison between the variouscalculation methods.

212 C. Pinilla et al. / Geochimica et Cosmochimica Acta 135 (2014) 203–216

liquid/gas and ice/gas cases. We have also included data forthe equilibrium fractionation factor of H and O isotopesobtained from DFT-optimized configurations. This DFTapproach is comparable to the RELAX method and DFTresults are indeed consistent with those of the RELAXmethod using the classical force field. However, given thelack of description of nuclear quantum effects andanharmonic effects, the good agreement between DFT andexperiments in the case of H/D ice-vapour fractionation

Table 2Equilibrium fractionation factor aliquid–ice in & at T = 273 K as estimate

103ln(a) PIMD MD RELAX

D/H 15.34 0.60 14.7318O/16O 1.20 2.51 1.30

a Experimental data correspond to values from Arnason (1969).b Experimental data correspond to values from O’Neil (1968).c Experimental data correspond to values from Merlivat and Nief (196d Experimental data correspond to values from Ellehoj et al. (2013).

could be fortuitous. We believe this agreement with experi-ments to be due to a cancellation of errors stemming fromthe accuracy of the model used for the description of thewater system and from errors inherent to the methodologyused for the estimation of the fractionation factor.Similar fortuitous cancelation between anharmonicity andthe use of the generalized gradient approximation to theexchange–correlation functional has been previouslyobserved on the stretching frequency of OH groups in min-erals (Balan et al., 2007). For a given method (RELAX),the difference between RELAX and DFT makes possibleto assess the effect of using different potentials to describethe same water system. Finally, the BEAD method leads toa factor that are largely overestimated in all cases.

It is important to note that at very low temperaturesatoms perform small displacements around the equilibriumposition. In particular, the MD method for ice Ih shouldconverge towards the RELAX values (i.e. |aMD(T)� aRELAX(T)|! 0, when T! 0 K). In the case of D/H(18O/16O) fractionation between ice and vapor, we havefound this difference to increase when temperaturedecreases, with a maximum value at 100 K. The differencedecreases down to zero below this temperature. This behav-ior is related to the contribution of low frequency modes tothe isotopic fractionation that becomes important at lowtemperatures. The analysis of the vibrational density ofstates shows that below 100 K, no imaginary frequenciesappear in the ice snapshots and the MD fractionationfactors converge to the RELAX ones. Above 100 K, thepresence of a small set of imaginary frequencies due to asmall structural frustration leads to a divergence of MDand RELAX results. This effect becomes less importantwhen temperature increases, where the contribution fromhigh frequency modes becomes dominant. It almost van-ishes for the ice system at 265 K. This effect is irrelevantat the temperatures of the liquid phase existence, at whichthe main contributions to the fractionation factor stemsfrom the high frequency modes.

Finally, the values from Figs. 10 and 11 allow us to esti-mate the liquid/ice equilibrium fractionation factor at thecoexistence temperature of 273 K (Table 2). In general,the values are in fair agreement with the exact PIMDresults. Besides the MD method all the other methods per-form relatively well for H and O ice–liquid isotope fraction-ation. Overall, the performance of a method againstanother depends on its ability to describe equally well iso-tope effects in the solid and liquid phases as well as onthe accuracy obtained within reasonable computationaltime limits.

d from the different computational approaches.

BEAD DFT EXP

22.27 10.46 20.58a, 19.31b, 22.8c, 17.2d

1.81 0.60 3.0b, 3.4c, 2.7d

7).

C. Pinilla et al. / Geochimica et Cosmochimica Acta 135 (2014) 203–216 213

4. CONCLUSIONS

We have estimated the equilibrium fractionation factorof D/H and 18O/16O in the liquid–gas and ice Ih-vapourphases of water using PIMD approach. Fractionation fac-tors are in good agreement with experiments. We havefound the fractionation factors to be little dependent onthe simulation cell size or multiple isotope substitution.This implies that isotope fractionation is a local effect thatdepends only on the local environment surrounding thefractionated isotope. Also, the weak dependence on systemsize carries the possibility to combine PIMD calculationswith a more accurate description of inter-atomic interac-tions, for example using dispersion-corrected density func-tional theory.

In addition, we have compared the performance of thePIMD results with a set of approximate approaches basedon the determination of the harmonic vibrational proper-ties of the system. Namely, the MD, RELAX and BEADapproaches. In the case of liquids the inclusion of config-urational disorder, as present in the MD method, is a keyingredient to obtain the right dependence of the a-factoras a function of temperature. In the case of D/H fraction-ation, the MD method over-estimates the a-factor becauseanharmonic and quantum effects are not treated. How-ever, for heavier isotopes such as 18O/16O, the MD andRELAX methods provide an acceptable description ofthe fractionation as function of temperature. This suggeststhat the MD and RELAX methods are good approxima-tions when anharmonic and quantum effects such as zeropoint energy are expected to be small, for example in thecase of heavy ions. The BEAD method has been foundnot to be a good approach for the estimation of isotopefractionation. Overall, PIMD represents an advance inthe study of fractionation properties of species in liquidsystems and opens the door for its application in the studyof more complex and inhomogeneous geochemicalsystems.

ACKNOWLEDGEMENTS

This work was performed using HPC resources from GENCI-IDRIS (Grant 2013-i2013041519). This work has been supportedby the French National Research Agency (ANR, project 11-JS56-001 “CrIMin”).

APPENDIX A. THERMODYNAMIC INTEGRATION

AND THE GENERAL VIRIAL ESTIMATOR

The isotope fractionation can be seen from the point ofview of the kinetic isotope effects, where isotopes Y and Y*

differ by their masses. In addition, a look at Eq. (2) showsthat we can use thermodynamic integration (Frenkel andSmit, 2002) to evaluate the relationship:

ðQðAY n�1Y �i ÞÞaðQðAY n�1Y �i ÞÞb

¼ exp �hZ 1

0

dkdF ðkÞ

dk

� �ðA:1Þ

where F ðkÞ ¼ � 1h lnðQðkÞÞ is the free energy and

h = (kBT)�1 (notice that we have chosen h instead of thecommon letter b to avoid any confusion with the fraction-ation b-factor described throughout the paper). k is aparameter, which provides a smooth transition between Y

and Y*. One can consider a continuous isotope changeapplied to k in such a way that the transitional mass canbe written according to the equation (Zimmermann andVanicek, 2009):

miðkÞ ¼ ð1� kÞmi þ km�i ; ðA:2Þ

where mi and m�i are the masses of isotopes Y and Y* and kvaries from 0 to 1. Eqs. (A.2)–(A.8) provide then, a way totransform continuously from a system containing a Y spe-cies in a lattice site to a similar one with a Y* on the same

lattice site and therefore estimate the ratioðQðAY n�1Y �i ÞÞaðQðAY n�1Y �i ÞÞb

.

Since

dF ðkÞdk

¼ � 1

hd lnðQðkÞÞ

dk¼ � 1

hdQðkÞ=dk

dQðkÞ ðA:3Þ

Eq. (A.1) represents a thermodynamic average that canbe computed using either Monte Carlo or MolecularDynamics simulations. Classically, the ratio of Eq. (A.1)can be easily evaluated using several computational meth-ods. However, when quantum effects are important thissimplification cannot be used. To describe quantum ther-modynamic effects one can use the path integral formula-tion (Feynman and Hibbs, 1965). In the path integralformalism the quantum partition function is calculatedthrough a discretization of the integral representing thedensity matrix. The implementation relies on the isomor-phism between the quantum particle and a classical onerepresented by a ring polymer of P classical particles con-nected by harmonic springs (Feynman and Hibbs, 1965;Ceperley, 1995; Tuckerman, 2010). This method has beenextensively used in the past for the study of quantum prop-erties of water and ice (Benoit et al., 1998; Morrone andCar, 2008), calculation of heat capacities (Ramirez et al.,2012), rate constants (Yamamoto and Miller, 2005) and iso-topic effects (Vanicek and Miller, 2007; Suzuki et al., 2008;Chialvo and Horita, 2009; Herrero and Ramirez, 2011;Markland and Berne, 2012; Zeidler et al., 2012). Here weaim to give a brief introduction to the relevant expressionsfrom which this work is based but a more detail descriptionon PIMD methods can be found elsewhere (Feynman andHibbs, 1965; Ceperley, 1995; Tuckerman, 2010).

Under the path integral formalism, the partition func-tion of a system containing N particles in a volume V andtemperature T takes the form (Tuckerman, 2010)

QðN ; V ; T Þ ¼YNi¼1

miP

2ph�h2

� �dP=2 Z YNi¼1

drð1Þi . . . dr

ðPÞi e�hUðfrðkÞgÞ:

ðA:4Þ

where each particle is characterised by a path in d dimen-

sions specified by points frð1Þi ; rð2Þi ; . . . ; r

ðP Þi g. Likewise,

rðkÞi ¼ fr

ðkÞ1 ; r

ðkÞ2 ; . . . ; r

ðkÞN g represent a set of 3N coordinates

defining a system like our quantum system of interest at

214 C. Pinilla et al. / Geochimica et Cosmochimica Acta 135 (2014) 203–216

the kth bead. P represents the total number of beads in thediscretized path integral (P = 1 gives classical mechanics,P!1 gives quantum mechanics). mi is the mass of theith particle and U({r(k)}) is the effective potential at kthbead:

UðfrðkÞgÞ ¼XP

k¼1

XN

i¼1

miP

2h2�h2ðrðkþ1Þ

i � rðkÞi Þ

2þ h

PUðrðkÞ1 ; . . . ; r

ðkÞN Þ

" #

ðA:5Þ

Here, each bead k interacts with two beads representing thesame nucleus in adjacent bead k � 1 and k + 1 through a

harmonic potential. UðrðkÞ1 ; . . . rðkÞN Þ represents the interac-

tion of a particle with other particles representing othernuclei within the same bead. From Eq. (A.3) it is easy tosay that a differentiation of Eq. (A.4) with respect to k will

produce the required quantity dF ðkÞdk to use in (A.1). This will

produce a function usually called the thermodynamic esti-mator given by

dF ðkÞdk

¼XN

i¼1

dmi

dkdP2mi� 2p2P

h2H

XP

k¼1

rðkÞi � r

ðkþ1Þi

� �2* +

; ðA:6Þ

The function in Eq. (A.6) often has a problem regardingthe growth of the statistical error with the number of timeslices, which is not very convenient for computer simula-tions (Herman et al., 1982).

In this work we have used the thermodynamic estimatorproposed by Vanicek and Miller (2007) and Herman et al.(1982) where the estimator is based only in terms of thepotential and its derivatives using the virial theorem(Herman et al., 1982; Zimmermann and Vanicek, 2009).This has the advantage of avoiding increasing statisticalerror when the number of time slices P is increased. In thiscase the thermodynamic estimator takes the form

dF ðkÞdk¼�1

h

XN

i¼1

dmi=dkmi

3

2þ h

2P

XP

k¼1

ðrðkÞi � rðCÞi Þ �

@UðrðkÞi Þ@rk

i

* +" #;

ðA:7Þ

where

rðCÞi ¼ 1

P

XP

k¼1

rðkÞi ðA:8Þ

is called the centroid coordinate and can be seen as an aver-

age position of the nucleus taken on all the beads.� @UðrðkÞi Þ@birðkÞi

represents the total force on the k-th bead as shown inFig. 1. Finally, if one takes into account only the transfor-mation of one atom from specie Y to Y* and the relation-ships from Eqs. A.1, A.7, 1, and 2 the b fractionationfactor can be expressed by Eq. (6) where the kinetic energyis sampled within the PIMD scheme using Eq. (7).

APPENDIX B. SUPPLEMENTARY DATA

Supplementary data associated with this article can befound, in the online version, at http://dx.doi.org/10.1016/j.gca.2014.03.027.

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Associate editor: Jean Francois Boily