Ab initio parameterization of an all-atom polarizable and dissociable forcefield for waterCarlos Pinilla, Amir H. Irani, Nicola Seriani, and Sandro Scandolo Citation: J. Chem. Phys. 136, 114511 (2012); doi: 10.1063/1.3691603 View online: http://dx.doi.org/10.1063/1.3691603 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v136/i11 Published by the American Institute of Physics. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

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THE JOURNAL OF CHEMICAL PHYSICS 136, 114511 (2012)

Ab initio parameterization of an all-atom polarizable and dissociableforce field for water

Carlos Pinilla,1,a) Amir H. Irani,1 Nicola Seriani,1 and Sandro Scandolo1,2

1The Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, I-34151 Trieste, Italy2INFM/Democritos National Simulation Center, Via Beirut 2-4, I-34014 Trieste, Italy

(Received 9 December 2011; accepted 14 February 2012; published online 20 March 2012)

A novel all-atom, dissociative, and polarizable force field for water is presented. The force field isparameterized based on forces, stresses, and energies obtained form ab initio calculations of liquidwater at ambient conditions. The accuracy of the force field is tested by calculating structural anddynamical properties of liquid water and the energetics of small water clusters. The transferabilityof the force field to dissociated states is studied by considering the solvation of a proton and theionization of water at extreme conditions of pressure and temperature. In the case of the solvatedproton, the force field properly describes the presence of both Eigen and Zundel configurations.In the case of the pressure-induced ice VIII/ice X transition and the temperature-induced transi-tion to a superionic phase, the force field is found to describe accurately the proton symmetriza-tion and the melting of the proton sublattice, respectively. © 2012 American Institute of Physics.[http://dx.doi.org/10.1063/1.3691603]

I. INTRODUCTION

Water is ubiquitous in our environment, and plays a fun-damental role in a variety of chemical, physical, biologi-cal, and geological processes. Understanding the microscopicmechanisms of these processes has proven difficult also forthe complexity of the inter-molecular interactions in the liq-uid phase. In this context, classical molecular dynamics (MD)simulations are useful for their ability to reveal atomistic pro-cesses in detail given the large simulation time and systemsize that can be used. However, it relies heavily on the ac-curacy of the force field (FF) used for the description of thesimulated system.

The development of water force fields has been contin-uously pursued for several decades and still goes on.1 Thesedevelopments have spanned from rigid water models2, 3 up tocomplex force fields based on the accurate description of thephysics of water molecule or water clusters,4, 5 also includingdispersion corrections.6

Efforts in the development of water force fields have pri-marily focused on non-dissociative models aimed at describ-ing the behavior of water at conditions where dissociation isirrelevant. However, there is growing interest in the micro-scopic study of phenomena where water dissociation is cru-cial, including proton hydration and diffusion,7 water-inducedcorrosion,8, 9 water splitting at oxide surfaces,10 and waterbehavior under geological and planetary conditions.11 Forcefields developed so far to model proton hydration treat the ex-cess proton as a distinct particle with respect to the protons ofthe water molecules, and thus do not allow the study of pro-ton exchange and diffusion.12 Ab initio simulations have shedlight on the phase diagram at extreme conditions of pressureand temperature, where water is found to dissociate, however,

a)Electronic mail: cpinilla@ictp.it.

a careful study of the phase boundaries, including melting andionization, would benefit enormously from the availability ofan accurate force field to complement ab initio simulations inthe determination of free energies and phase boundaries.13 Inaddition, the availability of a dissociative force field can be ofgreat advantage in hybrid quantum-classical simulations,14–16

where water molecules leaving the quantum region mighthave arbitrary configurations, including dissociated states.

Whereas non-dissociative force fields describethe properties of liquid water at standard conditionssatisfactorily,1, 2, 17–20 the success of dissociative force fieldshas been much more limited, and often involves the useof several parameters to model temperature and pressuredependence explicitly.21–23

The potential by Mahadevan and Garofalini22 describesbulk water and water at surfaces rather accurately. However,charges are rigid and the parameters have an explicit depen-dence on pressure and temperature conditions. The ReaxFFapproach allows charges to fluctuate in order to equilibratetheir chemical potential but the complexity of the model leadsto a heavy computational cost.24 Dissociable and polarizablemodels in principle overcome some of the above problems.25

In a seminal paper by Halley et al.,21 a polarizable anddissociable model for water was constructed by fitting itsparameters to experimental data. Major advances have beenmade in the last decade in the development of all-atom po-larizable force fields for bulk oxides, with parameters fittedto density functional theory (DFT) calculations.25, 26 The em-ployment of DFT calculations makes it possible to use vari-ables such as forces, polarization, and stresses as quantities ofthe fitting and to include configurations generated at differentpressures and temperatures and so enhancing the final forcefield towards transferability. The functional form of the forcefields developed for SiO2, MgSiO3, Al2O3, and other bulk ox-ides, contain short-range terms and long-range electrostatics,

0021-9606/2012/136(11)/114511/9/$30.00 © 2012 American Institute of Physics136, 114511-1

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114511-2 Pinilla et al. J. Chem. Phys. 136, 114511 (2012)

including induced dipoles. Extension of these models to wateris not straightforward, due to complexity of the energy land-scape arising from the molecular nature of water.27

In this work, we present a polarizable and dissociableforce field for water obtained from force matching to ab initiocalculations on bulk liquid water and check its performanceto reproduce some of the known properties of the system bothin the undissociated and dissociated state. More precisely, weinvestigate its ability to describe the free molecule, liquid wa-ter at standard conditions, the hydrated proton in solution, andsome aspect of the dissociated state at extreme conditions. Fi-nally, we give an overview and summary of the present work.

II. COMPUTATIONAL METHODS

The form of the force field that describes the interac-tion between atoms is based on the form originally proposedby Tangney and Scandolo25 for the description of amorphousSiO2. It can be divided into short-range and electrostatic con-tributions. In order to be able to describe the intra- and theinter-molecular length scales with comparable accuracy, wemodified the short-range interaction term by adding a secondMorse-stretch-like term,

UMS(rij ) = D1[eγ1[1−(rij /r01 )] − 2eγ1/2[1−(rij /r0

1 )]]

+D2[eγ2[1−(rij /r02 )] − 2eγ2/2[1−(rij /r0

2 )]], (1)

where rij = |ri − rj | is the distance between atoms i and j andD1, D2, γ 1, γ 2, r0

1 , and r02 are fitted parameters characteristic

of the interaction between atomic species.The electrostatic part contains charge-charge, charge-

dipole, and dipole-dipole interactions,

Ues = 1

2

∑i

∑j �=i

[qiqj

rij

− 2qj

rij · pi

r3ij

+ pi · Tij · pj

](2)

+∑

i

p2i

2α,

where qi and αi are the charge and polarization of atom i,respectively. Tij is the dipole propagator, a second rank tensor

Tij =[

I − 3rij rT

ij

r2ij

]1

r3ij

. (3)

Finally, the effect of neighboring atoms on the dipole of a par-ticular atom is included by using the short-range correctionsas described by Wilson et al.,28, 29

pshorti = αi

∑i �=j

qiri

r3ij

fij (rij ), (4)

where

fij = cij e−bij rij

4∑k=0

(bij rij )k

k!(5)

and bij and cij are fitting parameters. The total dipole is calcu-lated by minimizing Ues self-consistently at every step.25

The force-matching process was carried out by minimiz-ing with respect to a set of parameters {η} a function � mea-suring the distance between force field and ab initio forces,

stresses, and energies, as described in Ref. 25. A straightfor-ward use of the force matching procedure reported in Ref. 25did not lead to satisfactory results as far as the intermolecularpart of the interaction is concerned. The force acting on eachindividual atom of a molecule is in fact dominated by the in-tramolecular component, and we found that a generalized fit-ting procedure based on the fit of individual forces leads toaccurate results for the intramolecular component but unac-ceptable results for the intermolecular component. We there-fore modified the original expression for the function � by in-troducing different terms for the intra- and the inter-molecularcomponents of the force

�({η}) = wintra�intra + winter�inter + wS�S + wE�E,

(6)where �intra, �inter, are the differences between force fieldand ab initio forces within (�intra) and between (�inter) thewater molecules, and are given by

�intra =√∑Nc

k=1

∑NH

i=1

∑α

∣∣�Fff

iα − �Faiiα

∣∣2√∑Nc

k=1

∑NH

i=1

∑α(�Fai

iα )2, (7)

�inter =√∑Nc

k=1

∑Nj=1

∑α

∣∣Fff

CM,jα − FaiCM,jα

∣∣2

√∑Nc

k=1

∑Nj=1

∑α(Fai

CM,jα)2, (8)

where �Fiα is the Cartesian component α of the differencebetween the force acting on proton i and the force acting onthe center of mass of the molecule, and FCM,jα is the forceacting on the center of mass of molecule j. NH is the numberof protons in the cell, N is the number of water molecules,and Nc is the number of configurations used in the fit. �S and�E are the difference on total cell stress and configurationalenergies, respectively, as defined in Ref. 25. The weight pa-rameters were fixed to wintra = 0.7, winter = 0.3, wS = 0.1, andwE = 0.01.

Ab initio simulations were performed within densityfunctional theory and a 32 water molecule supercell usingthe BLYP functional30, 31 as implemented in the QuantumESPRESSO code.32 We used norm-conserving pseudopoten-tials for H and O.33 The electronic wave functions have beendescribed using a plane-wave basis set with an energy cutoffof 150 Ry. Such high cutoff guarantees convergence on thestress tensor. The integration over the Brillouin zone was car-ried out with the gamma point for the Car-Parrinello molecu-lar dynamics, and with a 2 × 2 × 2 grid otherwise.

III. RESULTS AND DISCUSSION

We started the force field generation by selecting tenatomic configurations from a 20-ps long Car-Parrinellomolecular dynamics simulation at 300 K. Electronic densityminimizations were then performed on these configurations toextract converged values for the forces, energies, and stresses.These quantities were used to generate a preliminary forcefield based on the fitting procedure described above. The re-sulting force field was used to perform a 50-ps long MD sim-ulation from which ten more configurations were extracted

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114511-3 Polarizable interatomic force field for water J. Chem. Phys. 136, 114511 (2012)

TABLE I. Optimized parameters for a classical force field for water (a.u.).

qO qH αO

− 1.19950 0.59979 4.08675bOO bOH

2.24850 3.92473cOO cOH

6.86766 −2.04454DOO DOH DHH

−1.26367×10−4 3.77059×10−5 8.71664×10−1

EOO EOH EHH

3.08029×10−4 8.67779×10−6 1.93115×10−7

γ 1OO γ 1

OH γ 1HH

13.09317 15.20544 13.13611γ 2

OO γ 2OH γ 2

HH

13.96521 12.38136 16.13997

r0,1OO r0,1

OH r0,1HH

7.473237 3.08451 0.38486r0,2OO r0,2

OH r0,2HH

7.03818 5.63316 8.12431

and new DFT calculations were carried out to obtain forces,stresses, and energies. These new sets of data were added tothe set of configurations used for the fit. This process was it-erated five times, i.e., until the function � did not show anysignificant reduction. Overall, the force field generation wascarried out by including 50 configurations. The final valuesof the error functions are �intra = 0.23, �inter = 0.24, �E

= 0.24, and �S = 0.07. The final set of parameters is givenin Table I. Although the purpose of our work is to gener-ate a force field for water at conditions where dissociationis important, the parameters of the force field were generatedbased on configurations obtained at ambient conditions, i.e.,at conditions where water is undissociated. We will focus inSecs. III C and III D on the performance of our approach inthe description of dissociated water. In Sec. III A, we beginwith an analysis of the performance of the force field at stan-dard conditions. The analysis is deliberately restricted to thecomparison of our force field with DFT results and not withexisting force fields for water, since the purpose is to validatethe fitting procedure rather than to prove that our force fieldis competitive with other force fields specifically designed toreproduce the properties of water at ambient conditions.

A. Liquid water at ambient conditions

We performed a 300 ps molecular dynamics run with 64molecules and the force field at Tav = 298 K and constant vol-ume. The density has been fixed to the experimental densityin order to facilitate the comparison with previous ab initioresults. This corresponds to a volume of 29.98 Å3/molecule.In order to be consistent with ab initio calculations we calcu-lated statistical averages over 20 ps segments as well as overlarger periods and did not found significant differences.

Considering that the functional form of the force fielddid not contain any explicit information about the molecu-lar state of water—with the exception of the two length scalesintroduced in the pair term—the finding that it spontaneouslyyields the molecule as a stable building block of liquid water

TABLE II. Thermodynamic properties of liquid water at ambient conditionscalculated with the new force field, with BLYP and from experiments. dOH isthe distance between the O and the H atom, θHOH is the average molecularangle, μ is the molecular dipole (in Debye), D is the diffusion coefficient andε the dielectric constant.

FF BLYP Expt.dOH (Å) 0.98 0.99 (Ref. 34) 0.97 (Ref. 35)

θHOH (deg) 105.5 105.5 (Ref. 34) 106 (Ref. 35)μ (D) 3.4 3.1 (Ref. 34) 2.8-3.2 (Ref. 36)

D (cm2/ps) 1.80 1.10 (Ref. 37) 2.40 (Ref. 38)ε 70.0 67 (Ref. 39), 86 (Ref. 34) 78.4 (Ref. 40)

is a first remarkable achievement. The structure of the watermolecule is reproduced fairly well (Table II), including the av-erage value of the HOH angle, whose value is the result of anequilibrium between proton-proton repulsion and induced po-larization on the oxygen ion. The average value of the dipolemoment is also in good agreement with the DFT calculationsand with experimental data.

A comparison between the radial distribution functionsobtained with the force field and those from DFT and ex-periment (Figs. 1 and 2) shows that the force field re-produces fairly well the radial distribution functions fromBLYP at ambient conditions. The main difference is in the

FIG. 1. O–O radial distributions functions for water at Tav = 298 K (top)and 400 K (bottom) obtained with the new force field (black line); BLYPcalculations (Refs. 31 and 42) (red dashed) and experiments (Ref. 43) (blue-dotted line).

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114511-4 Pinilla et al. J. Chem. Phys. 136, 114511 (2012)

FIG. 2. H–O (top) and H–H (bottom) radial distribution function of water atTav = 298 K obtained with the new force field (black line); BLYP calcula-tions (Refs. 31 and 44) (red dashed) and experiments (Ref. 43) (blue-dottedline).

intensity of the first peak of the O–O radial function, a quan-tity that is notoriously sensitive to differences in the inter-molecular potential, including between DFT calculations withdifferent functionals.41 The force field gives a less structuredwater than BLYP. Although the reduction of the intensity ofthe O–O peak brings the force field in better agreement withexperiments, it must be remarked that this can only be the re-sult of a cancellation of errors (including neglecting quantumeffects on the nuclei), as the force field was parameterized onthe BLYP calculations. Similar considerations apply to the O–H and H–H radial distribution functions. Figure 1 also showsthe O–O radial distribution function at 400 K. Similar to theresults at ambient conditions, there is a loss of structure char-acterized by a lowering of the first and second peaks with aslight increase of the second maximum location, as reportedin other simulations.22, 45 Again, our force field gives a radialdistribution function closer to the experiments than to BLYPcalculations.42

The equilibrium density found from an MD run in theNPT ensemble from our force field at ambient conditions (T= 298 K and P = 1 atm) is 0.91 g/cm3, which is 9% lower thanthe experimental value. However, this value is in line with

those calculated at the DFT level,37, 46 where the water densityhas been found to be lower than experiments by around 19%(with BLYP). Interestingly, the inclusion of van der Waals in-teractions into DFT functionals has been proven to increaseconsiderably the density37 and suggests that the use of theforce-matching method on ab initio forces containing disper-sion corrections could produce an improved force field.47

The infrared vibrational spectrum is obtained as a Fouriertransform of the time correlation function of the total dipolemoment in the cell M(t). In the case of polarizable forcefields, the total dipole is composed by permanent and inducedcontributions. Furthermore, the absorption coefficient in thepresence of these moments can be calculated from the imag-inary part of the total dielectric function (I(ω) ≈ ωIm(�(ω)))(Refs. 29 and 48) as

�(ω) − 1 = 4π

3V kBT[〈P (0)2〉 + iω〈P · P〉ω

+ 2〈P · J〉ω + i

ω〈J · J〉ω], (9)

where 〈A · A〉ω =∫ ∞0 dteiωt 〈A(t) · A(0)〉, J(t) = ∑N

i=1 qivi(t)

is the charge current, and P(t) = ∑Ni=1 pi is the total dipole

composed by permanent, induced and short-range contribu-tions. In Eq. (9), ω is the frequency, V the volume, kB Boltz-mann’s constant, and T the temperature.

The vibrational spectra of liquid water and deuteratedwater are compared in Fig. 3 with BLYP calculations andexperimental data. The IR spectrum calculated with theforce field is found to agree reasonably well with that cal-culated with the BLYP functional30, 34, 42 as well as withexperiments.49 However, there are evident differences in theposition of some peaks as well as in their broadening. In par-ticular, the peak corresponding to the bending mode of the wa-ter molecule (at about 1600 cm−1 in H2O) is considerably red-shifted for both H2O and D2O. A comparison with the BLYPresults on D2O shows that the redshift is of about 20%.

FIG. 3. Infrared spectra for H2O (top) and D2O (bottom) at Tav = 298 Kand zero pressure, calculated with the new force field. Dashed line shows thespectra from experiments (Ref. 49) (top) and BLYP calculations (Ref. 50)(bottom).

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114511-5 Polarizable interatomic force field for water J. Chem. Phys. 136, 114511 (2012)

FIG. 4. Dielectric constant of water as a function of temperature as calcu-lated from the force field (dots) and from experiment (triangles) (Ref. 52).Error bars are small (≈0.5) and so are not shown in the plot.

To further validate our model we also calculated the di-electric constant ε◦. The static dielectric constant in the caseof a polarizable polar fluid is related to the total dipole mo-ment fluctuations of the system as48, 51

ε◦ − ε∞ = 4π

3V kBT(〈P 2〉 − 〈P 〉2), (10)

where P is the total dipole moment of the simulation cell, ε∞is the high-frequency limit of the permittivity, T is the tem-perature, kB is the Boltzmann constant, and V is the volume.We obtain a value of ε − ε∞ = 64.6. Since the most accu-rate value of ε∞ lies between 4.9 and 5.0,51 our calculationyields a value of ε ≈ 70.0. This value is slightly lower thanthe experimental and BLYP values but nevertheless in goodagreement. The temperature dependence of the dielectric con-stant is also well reproduced, as shown in Figure 4. The dif-fusion coefficient D for water was determined from the meansquare displacement of the water molecules using the Einsteinrelation.53 The mean square displacement was obtained froma 200 ps NVT simulation. The value found at T = 298 K isof 1.8 × 10−5 cm2/s, which is in fair agreement with the ex-perimental one of 2.3 × 10−5 cm2/s and higher than the 1.1× 10−5 cm2/s found from BLYP calculations under similarconditions.42, 44 The results are consistent with the observationthat the force field yields a less structured water than BLYP,and a more structured water than experiments.

B. Water clusters

The transferability of the force field to conditions dif-ferent with respect to those at which the force field hasbeen parameterized, has been tested first in the case of smallwater clusters. Small clusters are an important test for theforce field, as energetics and dynamics are different from thebulk case.54, 55 Atomic configurations for clusters with up to21 molecules have been taken from the Cambridge clusterdatabase56 and fully relaxed. The cohesive energy are com-pared in Fig. 5 to the results of BLYP calculations carried outby Silvestrelli.57 The cohesive energies calculated with ourforce field differ by less than 0.1 meV/molecule from those

FIG. 5. Cohesive energies for water clusters. N is the number of moleculesin the cluster, the energy is expressed in eV/molecule. Black solid line: newforce field; blue-dashed line are values from BLYP (Ref. 57).

calculated with BLYP and in general show the same trend asa function of the cluster size. It is important to mention thatthe structure of the clusters is retained upon relaxation.

C. The solvated proton

One important advantage of a fully dissociable force fieldis its ability to describe charged species such as the hydro-nium complex as well as the classical dynamics of the proton.Attempts to model the hydronium complex with force fieldshave so far considered the excess proton as a distinct parti-cle interacting with molecular water. An obvious drawback ofsuch models is the inability of the excess proton to exchangewith protons belonging to water molecules. Proton exchangemechanisms are at the basis of charge diffusion in liquid wa-ter. In this section, we model the solvation of an excess protonby assuming that the proton is the result of a dissociation eventwhere the OH− moiety is removed from the supercell. Thesupercell therefore contains N oxygens and 2Nmol + 1 hydro-gens described by the interaction parameters listed in Table I.The number of molecules was fixed to N = 64. We performed80 ps of molecular dynamics within the NVT ensemble at Tav

= 298 K and volume corresponding to water at ambient con-ditions. We found several exchanges of the excess proton withthe surrounding molecules, with a mean residence time of≈10 fs. Figure 6 shows the interaction distance between theexchanging proton and its two closest O atoms. We see thatthe distance between the acceptor and the proton decreasesand at the same time that the distance from the donor watermolecule to the proton increases. The exchange occurs in lessthan 1 fs. With our model most of the H3O+ complexes dif-fuse through proton exchange.

The structure of solvation of the hydronium complex hasbeen determined by calculating the radial distribution func-tion concentrating only on two complexes seen in our simula-tions. Namely, the Eigen complex ((H9O4)+) where a H3O+

complex is bonded to three neighboring H2O molecules andthese forming hydrogen bonds with neighboring moleculesof the bond network as seen in Fig. 7 and a second Zundel

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114511-6 Pinilla et al. J. Chem. Phys. 136, 114511 (2012)

FIG. 6. Variation of H–O distance for the reaction of an adjacent H+ ionexchange with a nearby water molecule. Black line shows the initial H3O+configuration where H+ is bonded to O1. Red line corresponds to the adjacentO2 where the H+ is exchanged several times in 6 ps. The O1–O2 distances(blue line) is also shown for comparison.

complex ((H5O2)+) where two H2O molecules bind the ex-cess proton between them with each proton forming a hydro-gen bond with other water molecules embedded in the net-work (see bottom of Figure 7). These structures have beenextensively described in the literature on proton solvation.7, 58

Figure 8 shows the radial distribution function of thecomplexes (H9O4)+ and (H5O2)+ in the liquid environmentcalculated by taking the central hydronium oxygen O* asreference.

In the case of (H5O2)+, there are two of such kind of oxy-gens. We have followed Tuckerman et al.58 and defined O*as the oxygen bonded to three neighboring H atoms within acutoff of 1.3 Å. For comparison we have included the radialdistribution functions for pure water as presented earlier inFig. 1. From these description we obtain a gO∗O and a gO∗H

where the structure of the H3O+ complex can be readily iden-tified, with the rO∗O ≈ 2.5 Å distance much shorter than therOO = 2.8 Å found in pure water. A look at the gO∗H showsa broader first peak than the one calculated for pure waterand thus suggests of a longer and less localized O*–H bond(≈ 0.15 Å longer) in agreement with previous BLYP works inbulk water and protonated clusters.58, 59 As an important char-acteristic of cation solvation58–60 we have the absence of the1.8 Å peak in the gO∗H which indicates that protons of the wa-ter molecules forming the first solvation shell point outwards.

A further look into gO∗O and gO∗H for the (H5O2)+ showsa splitted first peak, indicating that the distances of the threehydrogens surrounding O* and the O*O are not the same. Infact, the distance O*–O* is shorter than any O*–O distance by0.1 Å in agreement with previous ab initio calculations.59, 62

Finally, we have calculated the mean square displacementof the hydronium ion core (O*) as shown in Fig. 9. From theslope of the linear regime we predict a diffusion coefficientof 0.76 Å2/ps, which is in good agreement with the experi-mental value of 0.67 Å2/ps (Ref. 63) and the recent value of0.80 Å2/ps from first principles molecular dynamics usingBLYP and based on a discrete variable representation func-tions for the basis set.61

FIG. 7. Snapshots of the reaction of H3O+ with neighboring watermolecules. Black lines represent hydrogen bonds. Top, a reacting H ion splitbetween three water molecules (H9O+

4 complex). Bottom, reacting H splitbetween two oxygens hydrogen bonded to adjacent water molecules forminga H5O+

2 complex.

Overall, these results show the ability of the new forcefield to describe correctly the structural properties of theH3O+ complex in a water environment as well as its evolu-tion on time.

D. Water at extreme conditions

We further explore the performance of the force fieldin situations where the molecule dissociates by consideringthe behavior of water at extreme conditions of pressure andtemperatures, where the molecule undergoes a transition toionized phases. We focus in particular on the ice VIII – iceX transition under high pressure64–66 as well as on the su-perionic transition for water predicted to occur at planetaryconditions.11

The transition between ice VIII and ice X is characterizedby the centering of the proton position along the line con-necting two oxygens and is therefore an excellent test case to

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114511-7 Polarizable interatomic force field for water J. Chem. Phys. 136, 114511 (2012)

FIG. 8. O*–O (top) and O*–H (bottom) radial distribution function for the(H9O4)+ and (H5O2)+ structures of H+ in water. The distribution functionsfor the (H9O4)+ and (H5O2)+ complexes are represented by the black andred-dashed lines, respectively. The radial distribution function of pure wateris also given for comparison (blue-dotted line).

FIG. 9. Mean square displacement for the O* atom calculated with the forcefield (black solid line) and from CPMD-BLYP (Ref. 61)(blue dotted-dashedline).

FIG. 10. Evolution of the average RO-O and RH-O bond lengths for the iceVIII–ice X transition. Circles correspond to values found with the new forcefield and squares correspond to ab initio data with a classical treatment of thenuclei (Ref. 64). Inset panel shows oxygen distance as a function of pressurefor the force field model (circle) and ab initio with classical nuclei (squares).Dashed line located at the transition distance is used as reference.

study the transferability of our force field. Full structural re-laxation of the two structures at different pressures between 0and 400 GPa yields the O–O and H–O distances reported inFig. 10. The results are compared to BLYP calculations64 inwhich the proton has been treated classically. The force fieldreproduces well the critical value (∼2.3 Å) of the O–O dis-tance at which symmetrization takes place (this correspondsto the point where the H–O distance becomes exactly a half ofthe O–O distance). The inset of Fig. 10 shows that althoughthe force field does a good job in describing the transition, itfails to give the right value of pressure at which the transitiontakes place, and in general it reproduces the equation of statewith a considerable deviation with respect to BLYP results.We stress however that the transition takes place at a densitywhich is about twice that of ice at ambient conditions, i.e.,very far from the conditions where the parameters of the forcefield have been determined. We believe that the deviation onthe equation of state can be corrected by repeating the fittingprocedure based on configurations closer to the transition.

We also investigated the performance of the force field athigh temperatures, and in particular, in the region where wa-ter has been predicted by ab initio molecular dynamics simu-lations based on the BLYP functional to undergo a transitionto a superionic phase characterized by the melting of the hy-drogen sublattice.11 We performed calculations starting fromice VIII at 100 GPa and ice X at 250 GPa and slowly increas-ing the temperature at constant pressure. The total duration ofthe simulation at a given pressure was 100 ps. The occurrenceof sublattice melting was monitored by calculating the diffu-sion coefficient of the two species at each temperature (seeFig. 11). At both pressure the diffusion coefficient of the hy-drogen species departs from the value expected for a solid atabout 1200 K while the oxygen sublattice remains crystallineup to the highest temperatures investigated in our study (3500K). We therefore conclude that our force field correctly repro-duces the superionic transition, although the value of the tran-sition temperature (1200 K) is underestimated with respect to

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114511-8 Pinilla et al. J. Chem. Phys. 136, 114511 (2012)

FIG. 11. Diffusion coefficient of oxygen (square) and hydrogen (circle) asa function of temperature for ice VIII at 100 GPa and ice X at 250 GPa.Diamond correspond to transition temperature found by Cavazzoni et al.(Ref. 11).

the one reported in the ab initio study (2000 K). It must berecalled however that the ab initio study was based on veryshort simulations (temperature ramps of 1000–2000 K/ps)and might therefore be affected by considerable hysteresis ef-fects. In this regard, we notice that, in spite of the higher tran-sition temperature, the value reported in the ab initio studyfor the diffusion coefficient of the protons at 2500 K is abouttwice the value found in this work at the same temperature.We stress once again that there is room for improvement in theparameterization, for example, by using configurations gener-ated in the superionic phase.

IV. SUMMARY

A novel atomistic, dissociative, and polarizable forcefield for water has been developed, fitted to ab initio (BLYP)forces, stresses, and energies of configurations of liquid wa-ter at ambient conditions. With the exception of the introduc-tion of two length scales in the two-body part of the forcefield, the functional form of the force field does not explic-itly consider the molecule as a fundamental structural unit.Information about the molecular state of water is entirely pro-vided through the parameterization. The force field is ableto reproduce the structure of the water molecule and the ra-dial distribution functions of liquid water, although the O–Oradial distribution functions are slightly less structured thanthose obtained from BLYP, and therefore closer than BLYPto the experimental results. Several properties of liquid wa-ter at room temperature have been calculated, such as averagedipole moment, dielectric constant, diffusion coefficient, andinfrared spectra and are all found to be in similar agreementwith experiment than BLYP results, with the exception of aredshift of the vibrational peak associated with the bendingmode of the water molecule. Furthermore, the force field re-produces the molecular structure and the cohesive energy ofneutral water clusters with up to 6 molecules with less than5% error with respect to BLYP calculations. In addition, theforce field is able to describe the structure of a solvated pro-

ton, including the local structure of the H9O+4 (Eigen) and

H5O+2 (Zundel) complexes and the dynamics of proton dif-

fusion. Finally, we have tested our force field to describe theice VIII/ice X and the superionic phase transitions of water atextreme conditions and found it to provide a reasonably gooddescription of these phenomena. Because the force field wasfitted to liquid water at ambient conditions, we consider thecapability of the force field to reproduce dissociated statesto be remarkable and believe that the transferability of theforce field could be substantially improved by including in thetraining set configurations containing dissociated water. Thedevelopment of improved all-atom force fields for water atextreme conditions and their combination with ab initio sim-ulations may pave the way to the accurate description of thethermodynamics and phase diagram of pure water and water-bearing substances which is presently hindered by the limitedtime scales and sizes of ab-initio methods.

ACKNOWLEDGMENTS

The authors acknowledge financial support by the Euro-pean Commission under the EU-FP7-NMP Grant No. 229205“ADGLASS”. The authors thank Y. Liang, P. Tangney, C. Mi-randa, F. Inam, S. Bhattacharya, and O. Akin-Ojo for fruitfuldiscussions.

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