The radial distribution functions of water and ice from 220 to 673 K ...

Chemical Physics 258 (2000) 121±137www.elsevier.nl/locate/chemphysThe radial distribution functions of water and ice from 220 to673 K and at pressures up to 400 MPaA.K. Soper *ISIS Department, Rutherford Appleton Laboratory, Chilton, Didcot, Oxon OX11 0QX, UKReceived 13 June 2000AbstractNeutron di€raction data for water and ice in the form of OO, OH and HH partial structure factors now exist over atemperature range 220±673 K, and at pressures up to 400 MPa. In order for these data to be useful for comparingwith di€erent computer simulations and theories of water, it is ®rst necessary to Fourier transform them to the correspondingsite±site radial distribution functions. The process of doing this is not straightforward because of the inherentsystematic uncertainties in the data, which arise primarily in the case of neutron scattering, from the inelasticityor recoil e€ects that can distort the experimental data. In this paper, it is shown that the empirical potential structurere®nement procedure, which attempts to ®t a three-dimensional ensemble of water molecules to all three partialstructure factors simultaneously, leads to improved reliability in the extracted radial distribution functions. There arestill some uncertainties, primarily associated with the hardness of the repulsive core of the intermolecular potential,which current data are not precise enough to resolve. The derived empirical potentials show some variability associatedwith particular experiments. General trends can be discerned however which indicate polarisation e€ects may be signi®cantwhen e€ective intermolecular potentials are used over a wide temperature and density range. Ó 2000 ElsevierScience B.V. All rights reserved.1. IntroductionIt is generally recognised that obtaining a reliableset of radial distribution functions for waterhas to be an essential prerequisite in any attemptto understand the properties of water and aqueoussystems at the atomic level. The reasons for thisare several. Firstly, reliable radial distributionfunctions are one of the crucial yardsticks againstwhich any computer simulation or theory of watermust be tested. Very few published computersimulations of water exist which do not at some* Tel.: +44-1235-445543; fax: +44-1235-445642.E-mail address: a.k.soper@rl.ac.uk (A.K. Soper).point make a comparison with some form of diffractionexperiment. Secondly, when attempting tounderstand the e€ect of dissolved substances onwater it is essential to be able to compare thestructure of the modi®ed solvent in the presence ofthe solute with that of the pure solvent. Thiscomparison is an important indication of how thesolute±solute interaction may be modi®ed whendissolved in water. Thirdly, the thermodynamicstate dependence of the structure of water is, atleast in principle, an indispensable guide to thenon-pairwise-additive terms in the interatomicpotential. Finally, the pronounced hydrogenbonding between water molecules, combined withthe particular molecular geometry of the watermolecule, gives rise to quite distinct forms for the0301-0104/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved.PII: S0301-0104(00)00179-8

122 A.K. Soper / Chemical Physics 258 (2000) 121±137OO, OH and HH radial distribution functions [1],in a way which is not reproduced by other molecularsystems. See for example, equivalent diffractionwork on the hydrogen halides [2±4],hydrogen sulphide [5], and ammonia [6]. Thestructure of water is therefore fundamentally importantfor characterising the nature of the hydrogenbond at the atomic level.Since the early 1980s, it has been shown that itis possible to extract the partial structure factorsfor water using the technique of hydrogen/deuteriumsubstitution in combination with neutrondi€raction [7,8]. This is because the hydrogen atomhas a marked di€erence in neutron scatteringlength compared to deuterium, and because theneutron is scattered directly from the nucleus.Thus, by making di€raction measurements onheavy water, light water and at least one mixtureof these two liquids, the corresponding nucleus±nucleus partial structure factors can be extractedby inverting the scattering matrix corresponding toeach di€raction dataset [1]. The fundamental assumptionbehind the neutron experiment is thatthe structures of heavy water and light water areisotope independent, whereas earlier quantumcomputer simulations [9] and now X-ray data [10]indicate that the two liquids may well be slightlydi€erent in structure. If this is the case, then itshould be noted that this does not automaticallyrule out the possibility of doing isotope substitutionon water. There is so far no indication that thetwo liquids are di€erent in structure by more thana few percent in the radial distribution functions,and it was shown previously that an isotope substitutionexperiment in that case would measurethe average structure of the two liquids [1]. Inpractice, the uncertainties in the neutron experimentare typically likely to be signi®cantly greaterthan the di€erence in structure due to quantume€ects, so the di€erences between the two liquidsare usually ignored. However, the possibility ofthese di€erences being present, and in particularbecoming more apparent at low temperature mustalways be borne in mind when considering neutrondi€raction experiments on water.The neutron data can in principle be supplementedby X-ray di€raction data, which, particularlywith modern synchrotron sources, can beextracted with signi®cantly better statistical precisionthan the neutron data. Analysis of the X-raydata is also non-trivial, however, since it requiresknowledge of the electron form factor of themolecule, which due to the highly polarised natureof the water molecule in the liquid, is not known toany accuracy. In addition, the X-ray experiment isprimarily sensitive to the O±O structure factor andso can give little direct information on the hydrogenbonding in water. This information can beobtained only by inference in the X-ray experiment,by analysing the behaviour of the secondpeak in the OO radial distribution function withchanges in pressure and temperature [11].By far, the greatest controversy in experimentaldeterminations of water structure arises from theprocess of converting the measured di€ractiondata to site±site radial distribution functions. Theconventional practice of performing a simpleFourier transform on the data is prone to introducingserious truncation and systematic errorsinto the resulting radial distribution functions, andthis has led to statements about water structureabove the critical point [12] being made whichwere subsequently shown to be controversial[13,14]. Pusztai has recently attempted to extractthe radial distribution functions from di€ractiondata by ®tting the total scattering patterns insteadof the partial structure factors [15] using a reverseMonte Carlo (RMC) procedure. His results so fardiverge signi®cantly from other attempts usingFourier transform methods on both the extractedpartial structure factors [1,7], and on the totalscattering patterns [16±18]. They also disagree withother recent attempts using Monte Carlo simulationprocedures [19,20].There are two primary sources of error in thedi€raction data on water. Firstly, especially withneutrons, and also with X-rays if energy analysis isused to remove the Compton scattering, the datacan only be obtained with ®nite statistical uncertainty.In general, the statistical uncertainty isworse at large values of the wave vector transfer,Q, which has the e€ect of making it dicult todetermine at precisely which value of Q thestructural oscillations disappear. In addition forthe OH and HH structure factors, the di€ractionpattern has oscillations due to the intramolecular

A.K. Soper / Chemical Physics 258 (2000) 121±137 123interferences which proceed to large Q. Measuringthese oscillations precisely is an essential prerequisiteto determine the intramolecular structure.Thus, ®nite counting statistics give rise to an effectivetruncation of the di€raction data, resultingin a loss of resolution in real space.The second main source of error comes fromthe need to estimate the single atom scattering forneutrons or the single electron (Compton) scatteringfor X-rays. This scattering does not containuseful information on the relative arrangement ofatoms or electrons, but it does constitute a significantpart of the di€raction pattern. It is the singleatom or single electron scattering about which theuseful interference signal oscillates. Therefore, anyerror in estimating this single particle scatteringcan give rise to potentially signi®cant errors inestimating the interference signal. In both X-raysand neutron scattering, the single atom or electronscattering is subject to substantial recoil distortionby the scattering radiation and attempts to estimatethis e€ect invariably prove to be approximateat best. The main features of the recoil or Comptondistortion however is that it is additive to thetotal scattering pattern, and that it is unlikely tohave the kind of oscillatory structure seen in theinterference signal. Therefore, it has to be regardedas a form of (generally unknown) background ontop of which the true oscillatory structure is added.A third aspect, relevant particularly to theneutron experiment, is that the nuclear recoil distortionis neutron energy dependent, rather than Qdependent. If the neutron measurements are madeusing an energy dispersive technique as at a pulsedneutron source such as ISIS, the same Q values arescanned over a range of di€erent neutron incidentenergies using neutron detectors at di€erent scatteringangles. This means that the recoil distortions,which are substantial for neutrons scatteredby light hydrogen, will to some extent be blurredwhen the di€erent detector arrays are combined ata given Q value, whereas the genuine interferencesignal will be ampli®ed in the combination.For all these reasons therefore, there is considerablescope to develop a practical process forextracting the radial distribution functions fromexperimental di€raction data on water. It will beessential that the nature and source of the errors beunderstood and taken into account. A premise ofthis paper and much of the work that has precededit is that there are errors in the data. If however weare careful enough in the way we set up ourstructure re®nement process, we should be able toprevent much of that error being carried throughto the calculated radial distribution functions.The rest of this paper therefore is devoted to theapplication of a particular computer simulationtechnique, called empirical potential structure re-®nement (EPSR), to the process of estimating theradial distribution functions of water. The methodhas been applied to all the available neutron dataon water and ice, from 220 to 673 K, some ofwhich have not hitherto been published, and theresults of this analysis are given here. The di€ractiondata used here have been derived entirelyfrom neutron di€raction experiments, and unlessotherwise noted, were obtained on the small angleneutron di€ractometer for amorphous and liquidsamples (SANDALS) at ISIS.2. Empirical potential structure re®nementThe object of this procedure is to set up a modelensemble of water molecules whose partial structurefactors agree as close as possible with thoseobtained from di€raction data [20]. This process isclosely analogous to what has been done withcrystallographic data for many years. Since thesystem is disordered, the only practical way tomodel the structure is via a computer simulation.In the case of water, the primary constraints on thedistribution of water molecules is that they musthave the correct density, the correct intramoleculargeometry, and of course the modelled distributionmust reproduce the di€raction measurements asclose as practical. EPSR is a technique forachieving that goal.2.1. Setting up the interaction potentialThe potential in the EPSR method [20] consistsof three principal terms. An intramolecularharmonic potential is used to de®ne and maintainthe molecular geometry. A reference intermolecularpotential provides the starting point for the

124 A.K. Soper / Chemical Physics 258 (2000) 121±137simulation to get the molecules into a realistic regionof con®gurational space. A perturbation,called the empirical potential, is then added to thereference potential to drive the simulation as closeas possible to the experimental data.A large number of computer simulations ofwater are of course available in the literature andwill not be referenced here in any detail. The majorityof water simulations are performed withwater molecules in which the atoms are preciselylocalised with respect to the molecular axes, buteven when the molecules are not rigid, the simulationsalmost invariably treat the intramolecularmotion using classical mechanics. This immediatelyruns into a problem when attempting tocompare with di€raction data, since the intramolecularstructure of molecules in the real liquid isdominated by quantum mechanical zero-pointdisorder which has no classical analogue, andcertainly must be included in any simulation whichis to be compared directly with the di€ractiondata.The starting point of the EPSR simulationtherefore is to build an ensemble of moleculeswhose internal structure reproduces that whichcan be obtained from the di€raction experiment.Essentially, each intramolecular distance is characterisedby an average distance, d ab , and a width,w ab , and the intramolecular structure is establishedby assuming the atoms in each molecule interactvia a harmonic potential. The total (dimensionless)intramolecular energy of the system is representedbyU intra ˆ C X iX …r aibi d ab † 2; …1†ab6ˆaw 2 abwhere r aibi is the actual separation of the atoms a, bin molecule i, w 2 ab ˆ 1= pM a M b , M a is the mass ofatom a in atomic mass units, and C is a constantdetermined from the di€raction data. All the simulationsreported here used C ˆ 40 A 2 , determinedby comparing the simulated structurefactors with the measurements at large Q. Thesame value of C was used for all temperatures,since the zero-point disorder was not anticipatedto be strongly temperature dependent, and therewas little evidence for a temperature dependence inthe data. The intramolecular distances used wered OH ˆ 0:976 A and d HH ˆ 1:550 A, which valueswere determined to give the best ®t to the data.Again no signi®cant temperature dependence onthese intramolecular distances was found. It mustbe emphasised that the inclusion of non-localisedatoms within the molecule does not constitute adegree of polarisability into the model: as describedbelow intramolecular atom moves aremade without reference to the intermolecular potentialenergy function. The intramolecular potentialis used here only as a device to mimic themeasured average molecular geometry. It does notcontribute to the overall system Hamiltonian.When sampled with an exponential function (seeSection 2.2) it reproduces the correct temperatureindependentGaussian disorder expected of zeropointmotion.The intermolecular reference potential is typicallybased on an intermolecular potential forwater drawn from the literature. In principle, thispotential can be as complicated as desired, butmost of the simulations reported here used thesame Lennard-Jones and charge parameters as thesimple point charge extended (SPC/E) potential[21]. A few of the present simulations have used amodi®ed version of this, as described below. Thispotential was derived for rigid molecules and sowill not necessarily be appropriate to the kind ofdisordered molecules used in EPSR simulations.Nonetheless it gave a reasonable description of thelocal order in water (Figs. 1±3) and so was a usefulpoint from which to start the structure re®nement.The intermolecular reference potential energy isgiven by an expression of the form8U inter ˆ 1 XX

A.K. Soper / Chemical Physics 258 (2000) 121±137 125Fig. 1. Computer simulation of the OO, OH and HH partial structure factors of water at 298 K and 1 bar, using the SPC/E (a) andSPC10 (b) parameterisation of the reference potential as described in the text. The lines show the simulated structure factors, while thecircles show the residual di€erence between neutron di€raction data and the simulated structure factors. Signi®cant oscillations can beseen in these residuals, although the SPC10 simulation appears to give smaller residual oscillations than the SPC/E parameterisation.Note that all the structure factors in this paper are shown as per atom. Thus, they are three times greater in amplitude than if they wereshown per molecule.q O ˆ0:8476e, and q H ˆ‡0:4238e. For a few ofthe simulations (labelled SPC10), the parametersof the reference potential were set to n ˆ 10,r OO ˆ 3:42 A, q O ˆ1:0e, and q H ˆ‡0:5e.The perturbation to the reference potential,U epi , is determined for each partial structure factorby taking the di€erence between experimentalstructure factor and simulated structure factor.The di€erence is Fourier transformed to r-space,using the minimum noise method [22] to ensure aminimum of truncation e€ects are transferred tothe perturbation in r-space. Provided the perturbationat each iteration is not too large, this differencein r-space is equivalent to the di€erencebetween the potential of mean force for the experimentand that for the simulation. Having ®rstequilibrated the simulation with the reference potentialalone, the simulation is then run again usingthe reference potential plus perturbationpotential until a new equilibrium is reached. Theprocess is repeated iteratively, accumulating theperturbations to form the empirical potential, untilsuch time as no further improvement in the ®t tothe data can be achieved. Because the re®nement issimultaneously ®tting all the available data at thesame time, it is less likely to introduce e€ects in theradial distribution functions related to truncation,counting statistics and systematic errors in thedata.All intermolecular potentials, both referenceand empirical, were truncated at half the box dimensionwith a function of the form T …r† ˆ1= ‰ 1 ‡ expf20… r=r c 1†gŠ,withr c ˆ 0:35D, whereD is the box dimension and r is the interatomicseparation. This truncation function ensures thatthe intermolecular potential is essentially zero

126 A.K. Soper / Chemical Physics 258 (2000) 121±137Fig. 2. Site±site radial distribution functions for the simulationsshown in Fig. 1. Note that the second peak in the OO g(r)is not well reproduced by the SPC/E simulation.beyond half the box dimension. No correction wasmade for longer range forces in order that thesimulation would proceed as quickly as possible,and because there was little information in thedi€raction data on interactions greater than thisdistance. It is also generally believed [23] that longrange corrections have little e€ect on the radialdistribution functions of water.2.2. Running the simulationAll the simulations were performed with 550water molecules in a cubic box. The densities,temperatures and box sizes used in the simulationsare given in Table 1. The standard minimum imageconvention was applied to calculate energies andradial distribution functions. Atomic and molecularmoves are accepted or rejected on the basis ofthe conventional Metropolis Monte Carlo method[24].There are three di€erent kinds of moves withinthe EPSR simulation, namely individual atommoves where atoms are moved relative to oneanother within the molecule, molecular rotationsand molecular translations, the latter two kinds ofmove being whole molecule moves. For morecomplicated molecules other moves can be envisaged,such as rotations of particular headgroups,or moves which keep the intramolecular distancesthe same but which change the symmetry of themolecule (e.g. left-handed to right-handed, etc.),but these are not relevant to the water simulationsdescribed here.For the intramolecular moves, only the changein the intramolecular potential energy, DU intra ,isused to accept or reject the move. Thus, theprobability of acceptance of an intramolecularmove within the Monte Carlo scheme is based onthe value of exp‰ DU intra Š, i.e. there is no thermalfactor in this sampling, in order to simulate zeropointdisorder which is temperature independentto ®rst approximation. If the data warranted atemperature dependence to this function (for example,in the event that intramolecular vibrationalfrequencies are altered radically by the changes intemperature and pressure) then this could be includedin the value of C in Eq. (1). In fact, nosigni®cant temperature dependence has been observedin the present di€raction data.For whole molecule moves, the usual Boltzmannthermal factor is used outside the intermolecularpotential energies. In principle, wholemolecule moves should not involve the relativemovement of atoms within the molecule. However,computing round-o€ errors can accumulate over anumber of such moves, so the intramolecular potentialis included when calculating the total energychange. Thus, the probability of acceptanceof awhole molecule move is based on the value ofexp DU intra ‡ 1=k B T DU ref ‡ DU emp , wherefor whole molecule moves DU intra is normally expectedto make only a small contribution to thetotal energy di€erence before and after the move.Since the intramolecular moves involve disorderingindividual molecules without reference to thesurrounding molecules there are typically 50 wholemolecule moves for each intramolecular move. Inthis way, the zero-point disorder of the moleculesis simulated, while maintaining a realistic localmolecular order.

A.K. Soper / Chemical Physics 258 (2000) 121±137 127Fig. 3. EPSR of the OO, OH and HH partial structure factors for water at 298 K, 1 bar, using the SPC/E reference potential. Thegraph on the left corresponds to pulsed neutron time-of-¯ight di€raction data (a) [18], whereas that on the right corresponds the earlierreactor neutron di€raction data (b) [1].One feature of EPSR simulation which is differentfrom conventional simulation procedures isthat the con®gurational energy and pressure of thesimulation can be arbitrary. This is because theintermolecular potential energy function is beingmodi®ed in a piecewise fashion without referenceto the expected energy and pressure. With perfectdata extending over a wide Q range, one mighthope that the EPSR procedure would arrive atenergies and pressures which were close to theirmeasured values. With the imperfect data that areactually available this is never the case, althoughthe intermolecular con®gurational energy valuesobtained in the present instance, as reported inTable 1, are at least of the correct sign and of thecorrect order of magnitude. Equally, the pressuresobtained in the EPSR simulations (not shownhere) were typically within 200 MPa of the expectedpressure.In fact, a lower limit of ±100 kJ mol 1 wasplaced on the con®gurational energy to prevent theempirical potential becoming excessively large ornegative. In practice, this limit was only breachedin the simulation of the ice data. In the event thatthe limit of ±100 kJ mol 1 was exceeded the empiricalpotential was reduced in amplitude, withoutchanging its shape, by a proportionate factor, inorder to raise the energy of the simulation abovethis threshold.The partial structure factors and site±site radialdistribution functions derived from running thesimulation with the intramolecular and referenceintermolecular potentials alone (i.e. no empiricalpotential re®nement) at T ˆ 298 K are shown inFigs. 1±3, comparing them with the correspondingdatasets. It can be seen that they give a sensiblerepresentation of the short range structure thoughsome details in the partial structure factors areclearly incorrect. Given that the data may contain,as discussed above, some form of slowly varying,additive, systematic error, it is important to inspectthe residual, that is the di€erence between data and

A.K. Soper / Chemical Physics 258 (2000) 121±137 129methods. In addition, the systematic errors in areactor experiment are quite di€erent to those in atime-of-¯ight experiment [7] and so this comparisonwill help to establish the likely accuracy of theresults.Two of the datasets, the one at 298 K and 210MPa, and the data on ice at 220 K, have not beenpublished before, though the former were referredto in a previous publication [25]. The samplecontainer used for pressurised water is the same asthat described previously for superheated water[27].Ice presents a signi®cant challenge for theanalysis in the manner prescribed here, since it isimportant to have di€raction data on an absolutescale of di€erential scattering cross-section. Thetraditional method of using a powdered crystal isnot appropriate for this since it makes determinationof the relative scattering cross-section forthe di€erent isotopes of water unreliable. On theother hand, simply freezing liquid water in a containeris also not suitable on its own since thatmethod invariably leads to a high degree of preferredorientation in the sample, making a validpowder average impossible. The method adoptedhere to circumvent these problems was to freezethe water to 220 K, make the di€raction measurements,then thaw the sample. This cycle wasrepeated several times. After combining the datafrom all elements of the large detector array onSANDALS and from all freeze-thaw cycles, theresult was e€ectively a powder average. The containerin this case was a 1 mm thick slit in a 3 mmthick slab of zirconium±titanium alloy.In all cases, the data analysis proceeded usingthe ATLAS data analysis suite [28], and the subsequentanalysis of di€erential cross-section toHH, OH and OO partial structure factors used thepolynomial method for subtracting the single atomscattering [29]. These data were then ®t directlywith a series of EPSR simulations under the conditionsgiven in Table 1. Figs. 3±5 show ®ts tosome of the di€raction data at 298, 573 and 673 K,whereas Figs. 6±9 show the derived OO, OH andHH radial distribution functions for all the statesstudied. Figs. 10±12 show the OO, OH and HHempirical potentials for the same states. Fig. 13shows the ®ts to the ice data, whereas Fig. 14shows the corresponding radial distribution functions.For this ice simulation, the starting pointwas an ensemble of molecules in the liquid stateand there was only slight evidence that this simulationhad started to form a crystalline structure,even after a lengthy simulation run, as discussedbelow.Numerical tabulations of all these estimatedradial distribution functions can be found at thewebsite http://www.isis.rl.ac.uk/disordered.4. Results4.1. The reference potentials aloneFigs. 1 and 2 demonstrate that either of thechosen reference potentials produced a qualitative®t to the data, but disagreed in the detail. TheSPC10 potential was used in an attempt to softenthe repulsive core of the SPC/E potential. There issome evidence from the residual in the OO partialstructure factor that the core is too hard in theLennard-Jones term of this potential, making the®rst peak in the OO radial distribution undulyhigh and sharp. Changing the hardness of the core,meant that both the charges and the core diameterneeded to be re-estimated to give the closest representationof the data. In the event howeverchanging the hardness of the core potential from a12th order power to 10th does not seem to a€ectthis peak height unduly.The con®gurational energy of the SPC10 potentialwas lower than that for the SPC/E parameters,but since con®gurational energies are notaccurate in any of the present simulations this wasnot felt to be a serious de®ciency. In general, the®ts with SPC10 on its own are equivalent to thosewith the SPC/E parameters, although the SPC10potential appears to capture the second peak in theOO radial distribution better than for SPC/E whenthe molecules are disordered. This is no doubt aconsequence of the fact that the SPC/E potentialwas originally parameterised with localised atoms,and with slightly longer OH bonds than have beenused here. It seems that once the atoms are delocalised,larger point charges are needed to compensatefor the loss of intramolecular order.

130 A.K. Soper / Chemical Physics 258 (2000) 121±137Fig. 4. EPSR ®ts to two datasets for water at 573 K, both at the same atomic number density of 0.072 atoms per A 3 and pressure of100 bar, which were measured at di€erent times. These are given in Table 1 as 573a (a), and 573b (b). A considerable degree ofsimilarity exists between the two datasets, albeit with some small di€erences.Clearly, therefore, there is still scope for improvementsto e€ective potential models such asthe SPC/E model. Inclusion of a representation ofintramolecular zero-point disorder will certainlybe needed if a better ®t to di€raction data is to beachieved.4.2. Cold and ambient waterThe radial distribution functions for cold waterunder pressure can be compared with those ofambient water. As has been discussed elsewhere[11,26], the primary e€ect of increasing waterpressure is to modify the second water shell arounda central water molecule, pulling it inwards, whilechanges to the ®rst shell are relatively small. Whenthis shell is pulled in it appears that it involvesbreaking the hydrogen-bonded network, allowingthe H bonds that remain to become more linear.This is seen from the fact that the OO distanceactually appears to increase slightly on increasingthe pressure, whereas the OH intermolecular distancebecomes smaller and better de®ned. If thehydrogen bonds were simply being bent by theapplication of pressure then the opposite to thesetendencies would be expected.Also of interest in the ambient water data is thenew analysis of the old reactor neutron datacompared to the same analysis of the more recentpulsed neutron data. The pulsed data were measuredto much larger Q values than the reactordata. The residuals for the two datasets are different,in fact, the residuals for the reactor data aregenerally larger than for the time-of-¯ight data.Nonetheless, the resulting radial distributionfunctions from the two datasets are surprisinglyclose. This helps to establish the extent of systematice€ects in these experiments. They are obviouslynon-trivial, particularly for the reactorexperiment, but the computer simulation approach

A.K. Soper / Chemical Physics 258 (2000) 121±137 131Fig. 5. EPSR ®ts to two datasets for water at 673 K, at atomic number densities of 0.066 atoms per A 3 (673b (a)) and 0.073 atomsper A 3 (673c (b)), which were measured at di€erent times.to estimate the radial distribution functions servesto prevent the worst artifacts in the data beingtransferred to the calculated functions.4.3. Superheated and supercritical waterAs the temperature is raised above the normalboiling point, the water network becomes progressivelymore disordered, so that by 573 K it haslargely disappeared, and the hydrogen bond itselfis considerably weakened. This can be seen fromthe disappearance of the 4.5 A peak in the OOfunction, and the weakening of the ®rst intermolecularpeaks in the HH and OH functions. By 673K the OH peak has weakened to the point of becominga shoulder rather than a distinct peak, andin an earlier publication, we claimed this implied acomplete breakdown of the hydrogen bond underthese conditions [12]. Although the revised analysisof the supercritical data, both here and previously[18], gives more intensity to this peak than wasoriginally indicated [12], it still shows up as ashoulder rather than the distinct peak that othersimulations have indicated [30]. In fact, an angularde®nition for the hydrogen bond indicates that thelevel of hydrogen bonding between molecules hasdropped to about 20% of its level in ambient water[31]. Other di€raction estimates indicate that adistinct OH peak may still be present [32] in supercriticalwater, but given the nature of possibletruncation and systematic e€ects a ®nal resolutionof this matter remains elusive at present. Certainly,it is fair to claim that once the temperature is takenabove the critical point, the hydrogen bonding inwater becomes considerably depleted and this depletionis not fully reproduced by existing e€ectivewater potentials. Simulations using the Car±Parinelloapproach apparently give better agreementwith the experiment than those with e€ective intermolecularpotentials [33].Two of the runs at 573 K, namely 573a and573b, were measured at the same thermodynamic

132 A.K. Soper / Chemical Physics 258 (2000) 121±137Fig. 6. Site±site radial distribution functions for water at 298K, 1 bar, as derived from the two simulations used in Fig. 3.The simulations of the reactor data (Fig. 3(b)) are shown asdashed lines. It can be seen that although the systematic andtruncation errors are di€erent for the two datasets, the resultingradial distribution functions are closely similar.state point but separated in time by more than oneyear. Small di€erences between the re®nements ofthese datasets are visible (Fig. 5), and thesetranslate to di€erences in the estimated radialdistribution functions (Figs. 7±9). These di€erencesare an indication of the likely uncertaintiesin the extracted radial distribution functions. Forsuperheated water systematic di€erences from onerun to the next appear to have an e€ect at the 10%level in the estimated radial distribution functionsat short distances, and these di€erences almostcertainly account for the reported discrepanciesbetween di€erent datasets. It will be seen that thediscrepancies occur not only on peak amplitudes,but also on peak positions. The problem oftransferred systematic error is particularly acutefor superheated water, because it is possible tohave only a relatively small amount of sample inFig. 7. Oxygen±oxygen, site±site radial distribution functionsfor water for the 17 state runs given in Table 1. Each curve isshifted by unity above its predecessor, except for run 573bwhich is shown as a dashed line on top of run 573a, becausethese two datasets were taken at di€erent times but at the samethermodynamic state point. The comparison helps to establishthe overall reliability of the reconstructions.the beam (about 20% of that for the ambient data),while at the same time the inelasticity distortionsare enhanced by the increased temperature.In spite of these reservations, some generaltrends are seen in the radial distribution functionsof Figs. 7±9. In particular, a general movement ofthe OO peak to larger distances is seen as thetemperature is raised and the density lowered. Thesame peak also becomes broader and less wellde®ned. At the same time, the ®rst OH intermolecularpeak broadens out and also moves to largerdistances on average, as does to a lesser extent the®rst intermolecular HH peak. However, for thesecond peak in the OH and HH functions thereappears to be little movement with thermodynamicstate point.

A.K. Soper / Chemical Physics 258 (2000) 121±137 133Fig. 8. Same as in Fig. 7, but for the OH radial distributionfunctions.4.4. Empirical potentials, 268±673 KFigs. 10±12 show the empirical potentials obtainedfrom the structure re®nements of water.These are the overall perturbations that have to beadded to the SPC/E parameterisation of the referencepotential to reproduce the radial distributionfunctions shown in Figs. 7±9. Considerablevariation between the di€erent potentials is apparent,particularly in the OO and OH potentials.In particular, it may be discerned that the potentialsderived from data measured in the same batchof experiments tend to have similar features whichare often signi®cantly di€erent from those derivedfrom another batch of experiments. The runs thatcan be grouped into batches of data taken within afew days of each other are [268a,268b,268c];[423a,423b,573f]; [573b,573c,573d,573e]; [573a,673b]; and [673a,673c,673d]. By comparison, theseparation between the batches of runs is oftenseveral months or even a few years in some cases.Fig. 9. Same as in Fig. 7, but for the HH radial distributionfunctions.It is clear therefore that considerable uncertaintyremains about the true nature of the empiricalpotential for water. The OO empirical potential isperhaps the most uncertain of the three. This nodoubt re¯ects the general diculty in obtaining areliable OO partial structure factor for water,particularly under conditions of high pressure andtemperature. It should be borne in mind howeverthat the amplitudes of these empirical potentialsare typically an order of magnitude or moresmaller than the corresponding Coulomb potentials,so they do represent relatively small perturbationsto the overall intermolecular potential.Perhaps, it is not overly surprising therefore thatthey have a high degree of uncertainty.It is possible however to make a few commentsabout some general trends that can be discerned inthese ®gures. At temperatures up to 423 K, the OHempirical potential is generally attractive (positivegradient) at the normal hydrogen bond distancearound 1.9 A. At 673 K, it tends to be repulsive

134 A.K. Soper / Chemical Physics 258 (2000) 121±137Fig. 10. Oxygen±oxygen empirical potentials derived from theEPSR simulation of the 17 datasets of Table 1 for water, usingthe SPC/E parameterisation of the reference potential. Thepotentials are shown in ®ve bands corresponding to the 5temperature bands of Table 1, namely 268 K (three pressures),298 K (two pressures), 423 K (two pressures), 573 K (®vepressures, six experiments), and 673 K (four pressures). Forclarity, each band is shifted upwards by 20 kJ mol 1 from thepreceding band. Within each band, each pressure is separatedfrom its neighbour by 10 kJ mol 1 with the highest pressure atthe bottom of each band. The curve labelling corresponds tothat used in the table. As with the radial distribution functions,the potential for run 573b is shown as a dashed line overlappingthat for run 573a because these two runs corresponded to thesame thermodynamic state point.Fig. 11. Same as in Fig. 10, but for the OH empirical potentials.(negative gradient) at this distance range, while at573 K itÕs slope is not well de®ned. Although thereis greater variability, the OO empirical potentialshows a similar trend at the near neighbour distance,2.8 A. Meanwhile the HH potential showsno obvious variation, being generally slightly repulsiveat the near-neighbour intermolecular distance.There is therefore some evidence here thatpolarisation e€ects are important in the waterpotential. At lower temperatures and higher densities,the OH attraction tends to be stronger thanthat given by the charge model, resulting in aslightly attractive empirical potential. At highertemperatures and lower densities, where the degreeof polarisation is reduced, the OH potential becomesmildly repulsive compared to the ®xedcharge model. It will be interesting to see whetherthese general trends can help to establish a usefulmodel for the e€ects of polarisation in the intermolecularwater potential.In addition, the fact that the OO potential tendsbe attractive at short distances and at lower temperaturesand densities suggests that the assumedrepulsive core potential is too hard under theseconditions, and a softer core is required. Alternatively,this could equally be regarded as anotherindication of polarisation e€ects: a larger chargeon the hydrogen atoms will lead to a strongerhydrogen-bond than for the reference potential onits own, pulling the molecules slightly closer together.

A.K. Soper / Chemical Physics 258 (2000) 121±137 135Fig. 12. Same as in Fig. 10, but for the HH empirical potentials.Fig. 13. EPSR ®ts to the OO, OH and HH partial structurefactors of ice Ih at 220 K, 1 bar. The lines show the measureddata, while the circles are the EPSR ®ts.4.5. Fits to the ice dataThe ®ts to the ice data are shown in Fig. 13,with the corresponding radial distribution functionsin Fig. 14. Obviously, the crystalline natureof ice means that a complete representation of thedi€raction data cannot be achieved with the EPSRmethod in its present form. There is some evidencethat the simulated ensemble of water molecules hasstarted to crystallise, as seen by the sharpness ofthe peaks in Q space. But clearly the true longrange order of ice is not correctly represented here.This is a subject for a more detailed discussionelsewhere, but it is important to emphasise that infact the short range structure out to about r ˆ 5 Ais accurately represented by the EPSR radial distributionfunctions. Fourier transform of the residualsshown in Fig. 14 reveals little missingstructure in any of the three site±site radial distributionfunctions out to this distance range. Thefact that the method does not reproduce the sharpBragg peaks is a consequence of the smallness ofthe box used. A box of dimension at least ®vetimes bigger than the present one would be neededif a more accurate representation of the Braggpeaks were to be achieved. At this point, such alarge simulation has not been attempted. A keyquestion of course would be to see whether itwould crystallise into the correct structure for iceIh.What is clear from this simulation is that signi®cantshort range disorder is prevalent in ice at220 K: this can be seen from the signi®cant widthof the ®rst intermolecular peaks in both the HH(2.2 A) and OH (1.7 A) peaks. Also, after the ®rstpeak in the OO distribution there is non-zero intensitynear 3.4 A indicating some residual interstitialatoms in the lattice. (The density in thisregion would go to zero if the lattice were perfect.)The use of partial structure factors in this casedemonstrates that direct crystallographic methodsmay miss some of the local detail in the lattice.

136 A.K. Soper / Chemical Physics 258 (2000) 121±137Fig. 14. Site±site radial distribution functions (±±) from theEPSR simulations of Fig. 13. The dashed lines show the Fouriertransform of the di€erence between di€raction data and EPSR®t. They indicate that most of the intermolecular structure inthe region 2±5 A has been captured by the EPSR simulation,although the longer range structural oscillations are too weakcompared to what would be required to reproduce the observedBragg peaks.5. DiscussionAccording to the results presented here, ourearlier calculation of the OH and OO radial distributionfunctions in ambient water [1] appearedto overestimate the sharpness and height of the®rst intermolecular peak in each of these functions.This is because a Gaussian function was used torepresent each of these peaks, the height and widthof which was adjusted to give the best ®t to thedata, which extended to a maximum Q of about 10A 1 . The more recent time-of-¯ight di€ractiondata extend to 40 A 1 and were ®t to 20 A 1 in thepresent simulations. The smallness of the Q rangeof the reactor data appears to introduce a signi®cantlimitation on the accuracy to which peakheights can be estimated. However, the discrepanciesbetween the present estimates and the earlierpublished results does give an indication of theuncertainty in the height and width of these peaks.It emphasises again the diculty of comparing acomputer simulation from a ®nite box of moleculesin real space with experimental di€ractiondata in reciprocal space. The two activities domeasure formally distinct quantities, and anystatements to the e€ect that one simulation ®ts thedata better or worse than another simulation canbe really only be sustained if the comparison isdone in experimental reciprocal space.The present simulations demonstrate that it ispossible to make this comparison for liquid waterprovided the simulation box is large enough. Themethod fails in the event the correlations becominglonger ranged than the size of the box used in thesimulation. This was seen here in the case of ice,and would probably occur for amorphous solidwater if partial structure factor data were available.However, even when long range order exists,the short range order in the material can still apparentlybe represented in these cases.One remaining question to be discussed concernsthe likelihood of improving the quality of thedata to resolve the remaining discrepancies. Thiswill require di€raction data with a statistical precisionat least 3±6 times better than is currentlyavailable at high Q values, requiring a 10- to 30-fold increase in source intensity. In principle, X-rays should be able to obtain better statistics thanneutrons. However, the realities of needing to estimateor measure the Compton scattering, and toknow the electron form factors to very good accuracy,has so far precluded any serious attempt toimprove the quality of the data at large Q. In anycase, X-rays will measure the OO structure factorto good accuracy, but yield little information onthe OH and HH functions. A new neutron sourceand associated instrumentation with the necessary¯ux advantage will not be available for 5±10 years.In this situation, the present estimates, using thecomputer simulation approach to reduce the contributionof artifacts in the data, are probably thebest estimates of the radial distribution functionsof water available at the present time.The EPSR method also appears to be able discernsome general trends in the intermolecular

A.K. Soper / Chemical Physics 258 (2000) 121±137 137water potential with temperature and density. Onthe basis of the evidence in Figs. 10±12, it appearsthat polarisation e€ects on the intermolecularpotential may be signi®cant for determining thelocal structure at short distances. Simple pointcharge models of water are obviously good over arange of state conditions, but it is likely that anaccurate potential will have to include a manybodydependence, such as a polarisability term.6. ConclusionThe method of EPSR has been used here toestimate the site±site radial distribution functionsfor water for a series of thermodynamic statepoints ranging from ice at 220 K to supercriticalwater at 673 K, and including pressures up to 400MPa. The inputs to this structure re®nement arethe measured HH, OH and OO partial structurefactors as obtained in neutron di€raction experimentsusing hydrogen/deuterium isotope substitution.Comparison of structure re®nements ondi€erent datasets measured at di€erent times butunder the same thermodynamic states gives anindication of the likely uncertainties in the radialdistribution functions obtained from di€ractionexperiments. Using this method of analysis, which®ts all three partial structure factors simultaneously,the estimated functions appear more robustthan earlier estimates. Likely improvementsin the data can be considered, but qualitativechanges to the intensity of current neutron and X-ray sources will be needed before the remainingcontroversies can be resolved.References[1] A.K. Soper, M.G. Phillips, Chem. Phys. 107 (1986) 47.[2] C. Andreani, M. Nardone, F.P. Ricci, A.K. Soper, Phys.Rev. A 46 (1992) 4709.[3] C. Andreani, F. Menzinger, M.A. Ricci, A.K. Soper, J.Dreyer, Phys. Rev. B 49 (1994) 3811.[4] C. Andreani, M.A. Ricci, A.K. Soper, J. Chem. Phys. 107(1997) 214.[5] G. Santoli, F. Bruni, F.P. Ricci, M.A. Ricci, A.K. Soper,Mol. Phys. 97 (1999) 777.[6] M.A. Ricci, M. Nardone, F.P. Ricci, C. Andreani, A.K.Soper, J. Chem. Phys. 102 (1995) 7650.[7] A.K. Soper, R.N. Silver, Phys. Rev. Lett. 49 (1982) 471.[8] A.K. Soper, Physica 136B (1986) 322.[9] R.A. Kuharski, P.J. Rossky, J. Chem. Phys. 82 (1985)5164.[10] J.H. Root, P.A. Egelsta€, A. Hime, Chem. Phys. 109(1986) 437.[11] A.V. Okhulkov, Yu.N. Damianets, Yu.E. Gorbaty, J.Chem. Phys. 100 (1994) 1578.[12] P. Postorino, R.H. Tromp, M.A. Ricci, A.K. Soper, G.W.Neilson, Nature 366 (1993) 668.[13] A. Chialvo, P.T. Cummings, J. Chem. Phys. 101 (1994)4466.[14] G. Lo‚er, H. Schreiber, O. Steinhauser, Ber. Bunsen-Ges.Phys. Chem. 98 (1994) 1575.[15] L. Pusztai, Phys. Rev. B 60 (1999) 11851.[16] M.-C. Bellisent-Funel, Hydrogen Bonded Liquids, in: J.C.Dore, J. Teixeira (Eds.), NATO ASI Ser C 329 (1991)117.[17] M.-C. Bellisent-Funel, R. Sridi-Dorrez, L. Bosio, J. Chem.Phys. 104 (1996) 1.[18] A.K. Soper, F. Bruni, M.A. Ricci, J. Chem. Phys. 106(1997) 247.[19] P. Jedlovsky, I. Bako, G. Palinkas, T. Radnai, A.K. Soper,J. Chem. Phys. 105 (1996) 245.[20] A.K. Soper, Chem. Phys. 202 (1996) 295.[21] H.J.C. Berendsen, J.R. Grigera, T.P. Straatsma, J. Phys.Chem. 91 (1987) 6269.[22] A.K. Soper, C. Andreani, M. Nardone, Phys. Rev. E 47(1993) 2598.[23] I. Nezbeda, J. Kolafa, Mol. Phys. 97 (1999) 1105.[24] M.P. Allen, D.J. Tildesley, Computer Simulation ofLiquids, Oxford University Press, Oxford, 1989.[25] R. Leberman, A.K. Soper, Nature 378 (1995) 364.[26] A.K. Soper, M.A. Ricci, Phys. Rev. Lett. 84 (2000)2881.[27] P. Postorino, M.A. Ricci, A.K. Soper, J. Chem. Phys. 101(1994) 4123.[28] ATLAS-Analysis of Time-of-Flight Di€raction Data fromLiquid and Amorphous Samples, in: A.K. Soper, W.S.Howells, A.C. Hannon (Eds.), Rutherford Appleton LaboratoryReport RAL-89-046, 1989.[29] A.K. Soper, A. Luzar, J. Chem. Phys. 97 (1992) 1320.[30] A.G. Kalinichev, J.D. Bass, J. Phys. Chem. 101 (1997)9720 and references therein.[31] A.K. Soper, Farad. Disc. 103 (1996) 41.[32] T. Tassaing, M.-C. Bellisent-Funel, B. Guillot, Y. Guissani,Europhys. Lett. 42 (1998) 265.[33] E.S. Fois, M. Sprik, M. Parinello, Chem. Phys. Lett. 223(1994) 411.