a family of primitive models of water: three-, four and five-site models

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MOLECULAR PHYSICS, 1997, VOL. 90, NO. 3, 353± 372

A family of primitive models of water:three-, four and ® ve-site models

By IVO NEZBEDAE. HaÂ la Laboratory of Thermodynamics, Institute of Chemical Process

Fundamentals, Acad. Sci., 165 02 Prague 6± Suchdol, and Department of TheoreticalPhysics, Charles University, 180 00 Prague 8, Czech Republic

and JAN SLOVAÂ KDepartment of Theoretical Physics, Charles University, 1800 00 Prague 8,

Czech Republic

(Received 13 May 1996; revised version accepted 31 July 1996)

A family of primitive models of water, which di� er from one another in the number andlocation of interaction sites, is introduced and their properties examined by Monte Carlosimulations. In addition to the existing symmetric 5-site model, which has its origin in theST2 potential, asymmetric 3- and 4-site descendants of TIPS potentials are introduced alongwith an extended 5-site model which incorporates a short-range repulsion between the likesites. The structure of the ¯ uids de® ned by the primitive models has been investigated in detailby computing site± site correlation functions, both at high and low densities, and the angulardistribution of particles engaged in hydrogen bonding. For completeness, the thermodynamicproperties have also been computed. It transpires that the extended 5-site model, due to itsenlarged range of the hydrogen bond interaction, clearly is much better than all the othermodels. It is able to reproduce even semi-quantitatively the structure of real water, and thusseems well suited to all potential applications involving water, including perturbation theoriesusing the extended model as a reference ¯ uid.

1. Introduction

Very interesting and simultaneously very di� cult tointerpret are the so-called associating ¯ uids, i.e., those¯ uids whose molecules form hydrogen bonds (water,alcohols, etc.); of these, undoubtedly, the most in-triguing is water, exhibiting a number of anomalies [1].Because of its general importance, water has attractedthe attention of researchers for centuries, and the ® rstserious attempts to explain its properties at the molec-ular level date as far back as the end of the last century,when RoÈ ntgen [2]proposed a mixture model. Yet a ® rst-principles molecular theory of water is not yet available.

Hydrogen bonding (H bonding) results from a verycomplex interaction between molecules, and may becharacterized generally as a strong, short-range, andstrongly orientationally dependent attractive inter-action. It was well established long ago that preciselythis part of the total intermolecular interaction makesassociating ¯ uids di� erent from non-associating ¯ uids.A good description of H bonding is thus the key to ourunderstanding of the thermodynamic and structuralproperties of water and aqueous solutions. This maybe the reason why molecular models which have been

constructed to describe associating ¯ uids, have generallybeen focused to provide detailed descriptions of inter-molecular interactions and, as a consequence, have beenrather complex (for reviews see [3± 6]).

By the early 1980s, it was becoming evident that thelack of a suitable simple model which for associating¯ uids might play the same role as that of hard bodiesfor normal (i.e., non-polar) ¯ uids was a main obstaclehindering the development of a molecular theory ofassociating ¯ uids. A real breakthrough took place atthis time, when several simple models mimicking directlythe H-bond interaction and neglecting completely all thelong-range electrostatic forces were proposed: twospeci® cally for water [7, 8]and one for general moleculeswith two asymmetric bonding sites [9]. Simultaneously,Dahl and Andersen [10] and Wertheim [11, 12] devel-oped accurate theories speci® cally tailored to thesemodels. The viability of these models was later sup-ported by computer simulations [13], which showedthat the molecular structure of the ¯ uid with a cut-o�short-range potential is essentially identical to that ofthe ¯ uid with the full potential.

The original model of Bol [8] for water was

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reformulated by Kolafa and Nezbeda [14], and subse-quently extended to methanol and to ammonia [15].This type of model, called a primitive model (PM) ina similar spirit to the use of this term in the case ofprimitive models of electrolytes, has gradually becomea generally accepted simple model for associating ¯ uidsand has been used in a number of studies of the thermo-dynamic properties of associating ¯ uids and aqueoussolutions ([16± 20] and references therein).

A PM results from a realistic `parent’ model bysimplifying its essential features and neglecting theremaining ones, i.e., it pictures an associating ¯ uid asbeing composed of hard bodies that copy the essentialshape of the molecules and which contain embeddedinteraction sites of two kinds (corresponding to the loca-tion of hydrogen atoms and of localized’ electroncharges) which are engaged in the H bonding; the unlikesites on two di� erent molecules interact via a sphericallysymmetric, short-range attractive (usually square-well)potential, whereas the (repulsive) interaction betweenthe like sites is neglected. Henceforth we will con® neour considerations only to water.

A large number of intermolecular potential modelsfor water have been proposed. The PM of water usedso far [14] and denoted as PM5 (5-site model) hence-forth, has its origin in the Bjerrum concept [4] and itsparent model is the well known ST2 potential [21]. Anequation of state for this model has been derived in aclosed analytical form [22] using the extended thermo-dynamic perturbation theory of Wertheim [23]. Thisequation exhibits the same anomaly in the temperaturedependence of the isothermal compressibility as doesreal water. Also, the low value of the critical compres-sibility factor predicted by this equation is in close agree-ment with the experimental value for real water. These® ndings are undoubtedly a key to the success of themodel when applied, either directly or in an extendedform (by incorporating the van der Waals dispersionterm [18, 20]) to aqueous mixtures. However, themodel also has drawbacks, both theoretical and prac-tical. Four tetrahedrally arranged o� -centre sites forcethe molecules to adopt unique positions and orienta-tions when forming an H-bond network, which even-tually leads to an overpronounced tetrahedral spatialarrangement of the molecules. We remark in passingthat the same comment applies to the parent ST2model, whose importance seems to have diminished(mainly for this reason); more ¯ exible 3- and 4-sitemodels, like TIPS3 and TIPS4 [24], are predominantlyused currently. Furthermore, PM5 also poses severeproblems in applications. To satisfy the necessary con-dition of saturation of the H-bond interaction, the rangeof this interaction in the model must be very small,making the model quite singular’ . Consequently, com-

puter simulations have been con® ned to high tempera-tures, only marginally lower than the Boyle temperature,and the true liquid phase has not been simulated at all[14]. This also explains why the application of the modelhas been limited thus far to only qualitative estimates ofthe thermodynamic behaviour of aqueous solutions, andthe model has not been used in any other simulationstudies, e.g., water in con® ned geometries, etc. In addi-tion to these drawbacks, the main shortcoming of themodel is that it is not able to reproduce, even approxi-mately, the structure of real water in the same way as thehard sphere ¯ uid reproduces the structure of simple¯ uids. Consequently, its use as a reference term in aperturbation-type equation of state of pure water hasmet with only limited success [25].

In principle, a primitive model can be constructedfrom any reasonable realistic site± site parent model,and it is the primary goal of this paper to explore thispossibility further. We consider a family of primitivemodels which are descendants of the TIP models, andwhich di� er one from another in the number andlocation of interaction sites. Furthermore, primitivemodels can also be modi® ed by incorporating a repul-sive interaction between the like sites, which brings themeven closer to reality.

We have performed extensive Monte Carlo simula-tions on a number of models to obtain data on theequation of state, on the structure, and on the occur-rence of n-mers of di� erent sizes. The latter results havemade it possible to discriminate between di� erent mod-els, and to assess their potential for possible futureapplications. It turns out that the extended 5-sitemodel, i.e., the original symmetric model with an addi-tional hard-sphere repulsion between the like sites(denoted as EPM5) not only has several advantagesover the remaining models but also may even performqualitatively di� erently.

The presence of the repulsive interaction between thelike sites makes it possible to lift the restriction imposedon the range of the H-bond interaction, which makes theEPM5 model quite ¯ exible while retaining all its speci® cwater-like properties. As a result, this model is able toreproduce even semi-quantitatively the structure of realwater. The simplicity of the model suggests further thatalso it may be treated theoretically using the thermo-dynamic perturbation theory of Wertheim [23] toexpress a number of its properties analytically and,ultimately, to form a reference system in a rigorous per-turbation theory of water. These results are reportedseparately.

2. Primitive models

Modern realistic intermolecular water± water poten-tials typically involved a rigid water monomer which is

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represented by interaction sites of three kinds: sites withpositive and negative charges, and a site generating theshort-range repulsive and the weak long-range disper-sion forces (typically the Lennard-Jones interaction)[24, 27]. The H-bond interaction, i.e., the strongly orien-tation-dependent short range attractive interaction, thenresults as a net e� ect of the Coulombic interactions. At asimpler level, the H-bond interaction may be explainedas the attraction between the positively charged `bare’nucleus of hydrogen on one molecule and a localized’electron charge on the other molecule shielded by thepresence of other nuclei and electrons. It is easy to seethat, qualitatively, the same attractive e� ect can beobtained if we place interaction sites of two kinds (letus called them + and - ) within the body of the particleand let the unlike sites on two molecules interact via astrong and short-range attraction. Thus, the H-bondinteraction between molecules 1 and 2 may be modelledby

uHB(1,2) =i,j

[uattr(r(1)i+ , r(2)

j- ) + uattr(r(1)j- , r(2)

i+ )], (1)

where r(k)i+ is the position vector of site i+ on molecule k.

For the ® nal e� ect, the actual choice of the attractivepotential uattr is quite immaterial; it is usually taken as asquare-well interaction,

uattr(r1, r2) º uSW(|r1 - r2|) = - e

f or r12 º |r1 - r2| < s W,= 0

for r12 > 0, (2)

where e measures the strength of the H-bond interactionand ¸ and s W de® nes its range. However, it is importantto bear in mind the fact that the latter parameters can-not be speci® ed arbitrarily. The H-bond interaction isknown to exhibit so-called saturation. Only one bond

can be established between a pair of molecules, and agiven interaction site can be engaged in establishing onlyone bond. ¸ and s W must therefore be speci® ed so as tomaintain this condition, which is achieved readily bysimple geometric considerations.

The primitive model of water and of associating ¯ uidsin general results from adding the above HB term to ahard core repulsion,

uPM(1,2) = uHC(1,2) + uHB(1,2), (3)

where uHC stands for the hard-core (hard-body) interac-tion which, in the case of water, may be considered asthe hard-sphere interaction:

uHC(R1,R2) º uHS(R12) = + ¥f or R12 º |R1 - R2| < s W,

= 0

for R12 > s W, (4)

where Ri stands for the radius vector of the centre, C, ofmolecule i and which may coincide with the - ’ site. Forsimplicity we will henceforth use units such thats W = e /kB = 1, where kB is the Boltzmann constant.

The above idea, to model the H-bond interaction bymeans of a simple, short-range attraction with simul-aneous neglection of the repulsion between the likesites, forms the basis of all simple models of associating¯ uids used since the early 1980s. As regards water, theonly primitive model used so far has been that which isan o� spring of the well known ST2 potential [21], thePM5 model [14] depicted in ® gure 1. However, ST2 isnot the only realistic simple potential, and in the lastdecade potentials with three (TIPS3 [24], SPC [28])and four (TIPS2 [29], BF [30]) interaction sites havebeen shown to estimate the actual water± water inter-actions better, and hence also the structural andbulk properties of water, and have therefore become

n-Site primitive models of H2O 355

Figure 1. Family of n-site primitive models of water: (a) PM3; (b) PM4-l and (c) PM5-l.

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preferred over the ST2 potential. These models employa di� erent arrangement of interaction sites, and eachmodel evidently can give rise to a corresponding primi-tive model. The associated primitive models, denoted asPMi, where i stands for the number of all interactionsites, are also depicted in ® gure 1; the parameters de® n-ing the models are summarized in table 1. Concerningthe numerical values of the parameters, it is seen fromthe table that we do not strictly retain the geometrygiven by the original TIP models. Speci® cally, for thesake of simplicity, we use in all the models a tetrahedralarrangement of sites (i.e., x + C+ = arccos (- 1 /3) =109.47ë , instead of 104.52ë for PM3 and PM4).

Second, we recall that in the parent ST2 model the - ’sites are submerged towards the centre by several percent, to obtain a better agreement with experimentaldata. In the original PM5 model [14] we retained thisarrangement but found that its only e� ect was to intro-duce greater complexity in the computations. We there-fore use an equivalent placement of the (+ ) and (- ) sitesin the PM5 models considered in this paper. Once thegeometry of the model is speci® ed, the largest possiblerange of the H-bond interaction, ¸max, maintaining thecondition of saturation can be determined. These are

¸max = 12[5 - 2(3)1 /2]1 /2 = 0.6197 for PM3, (5)

¸max = 12[5 - 2(3)1 /2]1 /2 - l- f or PM4, (6)

¸max = 12[5 - 2(3)1 /2]1 /2 - 1

2 f or PM5, (7)

We set the ¸’s to slightly smaller rounded values, seetable 1.

The primitive models di� er from one another not onlyin the placement of sites but also in certain topologicalaspects, which have important consequences. The PM5model has all its interaction sites engaged in H bondingplaced symmetrically and these are thus topologicallyequivalent. Since the centre of the molecule is notinvolved in the establishment of an H bond, the rota-tions of bonded molecules are considerably restricted(which helps to satisfy the conditions of saturation).On the other hand, the PM3 and PM4 models are asym-

metric with respect to the placement of their interactionsites. When the bond between a pair of molecules isestablished, the molecule which donates its (- ) site tothe bond can still completely freely rotate around itscentre (as is the case of PM3 where the (- ) site andthe centre of the molecule coincide) or its rotation isonly partially hindered (PM4 model), which meansthat its (+ ) site can establish a second bond with the(- ) site of the other molecule: the pair then becomesdouble bonded, see ® gure 2. We see that this phenom-enon is unrelated to the range of the H bond, and toprevent its occurrence one must introduce a repulsiveinteraction between the like sites in these models. Wemay thus claim that the PM3 and PM4 models are, toa certain extent, more realistic than PM5 because sucha repulsion is always present in realistic interactionmodels, and its absence in PM5 is only an artefact ofthe symmetry of the model. To maintain a level of sim-plicity, we will model the repulsive interaction betweenthe (+ ) sites (the like sites in general) by a hard sphereinteraction, and the complete pair potential of the pri-mitive models thus assumes the general form

uPM(1,2) = uHC(R12) + uHB(1,2)

+i,j

u+HS(|r(1)

i+ - r(2)j+ |)

+i,j

u-HS(|r(1)

i- - r(2)j- |), (8)

where u+HS and u-

HS are the potentials of hard spheresof diameters s + and s - , respectively, which must bespeci® ed so as to prevent double bonding. For thegiven ¸, the smallest values of s + satisfying this con-dition are given by

s + = (2¸2 - 1

2)1/2 for PM3, (9)

s + = (2(¸ + l- )2 - 12)

1 /2 for PM4. (10)

Although the introduction of the additional inter-action between the like sites is necessary for the PM3

356 I. Nezbeda and J. SlovaÂ k

Table 1. Geometric parameters de® ning the primitivemodels of water considered in this paper.

Model L + l- ¸ s + s -

PM3 0.5 0 0.615 0.5064 0PM4-4 0.5 0.04 0.575 0.5064 0PM4-8 0.5 0.08 0.535 0.5064 0PM5 0.5 0.5 0.15 0 0EPM5-2 0.5 0.5 0.2 0.4 0.4EPM5-3 0.5 0.5 0.3 0.6 0.6EPM5-4 0.5 0.5 0.4 0.8 0.8 Figure 2. Example of double bonding in the PM3 model

when no repulsion between the + sites exists. Dashedlines denote the bonds.

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and PM4 models, for the PM5 model it is optionaland thus o� ers a possibility for removing (or at leastweakening) its singularity’ : adding a hard-sphererepulsion between like sites makes it possible to enlargeconsiderably the range of the H-bond interaction; thecondition of saturation will then be governed by thediameters s 6 . It is easy to show that for the given ¸

the smallest s 6 maintaining the steric incompatibilitiesare given by

s + = s - = 2¸. (11)We will call the PM5 model modi® ed in this way theextended PM, and denote it as EPM5. The largestpossible value of ¸ of this model, ¸max = 0.422, ® xesthe maximum centre-to-centre separation at which theH bond can be established at RCC = 1.42, comparablewith the experimental value, Rexp < 1.4 [31]. The PM3and PM4 models give RCC = 1.12 at best, making theminferior on these grounds.

3. Basic relationships and computation details

Low density properties are given by the properties ofa pair of isolated molecules, which can be calculateddirectly from the interaction potential model. For site±site intermolecular potential models the most importantlow density properties are the average Boltzmannfactors (the (000)-spherical harmonic expansion co-e� cients), in the respective site± site frames,

k eij l º eij000(rij) =

rij= constexp[- u(1,2) /kBT ]d(1) d(2),

(12)where T is the temperature, and rij is the separationbetween the interaction sites i and j chosen as the refer-ence points [32]. The function k eij l provides directly theexact low density limit for the i- j site± site pair correla-tion function gij , de® ned by

gij(rij) =rij = const

g(1,2) d(1) d(2), (13)

where g(1,2) is the full pair correlation function. Ifeij(rij) is integrated over rij , the second virial coe� cientis obtained:

B(T ) = - 2p [k eij(rij) l - 1]r2ij drij , (14)

regardless of the choice of the pair of reference sites.When performing the integration in equation (12), it

is important to realize that the integrand takes on onlythree distinct values in its dependence on the separationof the reference sites and the orientation of the mole-cules: 0 if the particles overlap (including the overlapassociated with the additional repulsion between the

like sites), 1 when there is no overlap and no H bond,and exp (1 /T ) when the bond is established. Thisproperty makes it possible to factor our the tem-perature-dependent term and express the averageBoltzmann factor by means of two terms only: One,denoted as S, which is proportional to the fraction ofcon® guration space with an H bond, and the other,denoted as a, which gives the fraction of con® gurationspace in which neither overlap nor H bonding takesplace. Thus,

e000(r) = a(r) + [e1 /T - 1]S(r), (15)

where a(r) provides the high temperature limit to e000

determined solely by the hard core repulsion. For thepurposes of this paper we determined the functions aand S using the Conroy integration [33] method with25000 sampling points. The correctness and accuracyof the integration were checked by integration using adi� erent number of sampling points (up to 200 000 forone selected model) for each i- j pair considered.

For any site± site interaction model, the equation ofstate can be expressed in the form [34]:

P = q kBT - 2p q2

3 k,l

dukl

drQkl(r)r2 dr, (16)

where P is the pressure, q is the number density, ukl

denotes the site± site interaction, and

Qkl(rkl ) = rijrkl

rklgkl(rkl) º k rij m kl l gkl(rkl). (17)

Here, i and j denote reference points chosen in therespective particles, k and l denote interaction sites,and k l denotes an ensemble angle average over all con-® gurations with the separation rkl kept constant.

Due to the step-wise character of the interactions, theintegral in equation (16) reduces to the sum over thevalues of the involved functions at their points of dis-continuity. Placing the reference points at the inter-action sites, the compressibility factor for the PM3model assumes the form

Pq kBT

= 1 + 4h {gCC(1+ ) + 4k R m +- l r= ¸¸2

´ [g+- (¸+ ) - g+- (¸- )]+ 1 + 4k R m + + l r= s +

s 2+ g+ + ( s + + )}

= 1 + 4h {gCC(1+ ) + 4k R m +- l r= ¸¸2g+- (¸- )

´ [exp (- b ) - 1]+ 4k R m + + l r= s +

s 2+ g+ + ( s + + )} (18)

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and for the EPM5-¸ model, recalling that s + º s - , weget a similar expression as

Pq kBT

= 1 + 4h {gCC(1+ ) + 8k R m +- l r= ¸¸2g+ - (¸- )

´ [exp (- b ) - 1]+ 8k R m + + l r= s +

s 2+ g+ + ( s + + )}, (19)

where R stands out for the centre-to-centre separation,and h is the packing fraction, h = ( p /6) q s 3. Evaluationof the compressibility factor thus requires an extrapola-tion to contact of the values of g and Q. Since thesefunctions vary rapidly in this region, the extrapolationmust be made with care. For reasons discussed in detailin the following section, no explicit expression for thecompressibility factor for the PM4 model is given here.

The internal energy is another macroscopic measur-able property. The energy per particle, which is equiv-alent to the average number of OÐ H bonds permolecule, equals 2 when the H-bond network is fullydeveloped and is less than 2 otherwise. The energy isalso a useful quantity to monitor during simulationruns, to ensure the development of the system is undercontrol [35]. Another such quantity is the autodi� usioncoe� cient Di :

Di = limk ® ¥

d i(k)6k

, (20)

where d i(k) = k [ri( s j+ k) - ri( s j)]2 l 1/2 for the ith mole-cule. In equation (20), ri( s j) denotes the position vectorof particle i in the jth Monte Carlo (MC) step. In MCexperiments, in addition to the ensemble average,averaging over all molecules is performed. If the abovelimit converges rapidly to a non-zero value, the system isin a ¯ uid phase. The mobility of molecules in the glassystate is much lower and in such a case equation (20)converges slowly or not at all, while it ¯ uctuates arounda constant value if the molecules are trapped (vibrating)in cages.

An important structural property providing indirectevidence about the packing of molecules around achosen particle is the running coordination numberN(r) which is de® ned with respect to the centre of themolecule (oxygen atom) by

N(R) = 4p q

R

0gCC(r)r2 dr. (21)

If for R in equation (21) the location of the ® rst mini-mum on gCC is taken, R = Rm, then there is a sharpdi� erence in N(Rm) and its temperature dependencebetween associating and non-associating ¯ uids. The run-ning coordination number for associating ¯ uids is quitelow (typically between 4 and 6) and increases with

increasing temperature. For normal ¯ uids under normalconditions, N < 12 and decreases with increasing tem-perature.

Details of the orientational correlations may beobtained, in general, from the spherical harmonicexpansion coe� cients [37]. However, for the modelsconsidered in this paper they do not seem to be veryuseful, because a large number of them would have tobe calculated to obtain a reasonable representation ofthe orientational correlations. Furthermore, the mainquantity of interest as regards the spatial arrangementof particles is in this case the arrangement of bondedparticles around an arbitrary chosen central partical.We have therefore measured directly the number ofbonded molecules in all directions around a centralmolecule, using the bisector +C+ as a reference axisand body angles µ and u , cos µ Î (- 1,1), u Î (0,2p ),where u is measured from the +C+ plane.

We performed standard Metropolis Monte Carlosimulations in an NV T ensemble using 512 particles.Starting from the regular cubic lattice with randomlychosen orientations we generated, depending on thethermodynamic conditions, between 0.5 ´ 106 and1.0 ´ 107 one-step con® gurations to reach the equi-librium liquid state. We then performed 104 measure-ments of quantities of interest, with a span of 104

con® gurations between consecutive measurements. Inaddition to the standard quantities mentioned above,we also stored after every 50 subsequent measurementsinformation on the network structure, i.e., a list ofbonded molecules. Analysis of this informationenables us to obtain better knowledge of the structure,i.e., the numbers of hydrogen-bonded clusters ofmolecules.

To estimate the error in the energy of the system, weused a general method taking into account correlationsbetween measured values. We computed the autocorre-lation coe® cients q k of the measured energy seriesde® ned as q k = ck /c0, where

ck = cov (Ei,Ei+ k) = k (Ei - k E l )(Ei+ k - k E l ) l , (22)

and k E l stands for the expectation value of E. Becauseof the stationarity (we assume a stationary correlatedtime series Ei) cov (Ei,Ei+ k) does not depend on i. The® nal result for the variance of E is

var (E) =1n

var (Ei) +2n2

n- 1

k= 1

n- k

i= 1cov (Ei,Ei+ k)

=c0

n+

2n2

n- 1

k= 1

(n - k)ck < c0

n(1 + 2¿), (23)

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where E = (1 /n) ni= 1 Ei , and ¿ = n- 1

k= 1(1 - kn) q k. For

more details on this procedure see [38].

4. Results and discussion

4.1. L ow density propertiesA knowledge of low density properties provides a ® rst

estimate of the possible behaviour of the system underconsideration, and may also aid in understanding highdensity properties. Moreover, they are also very oftenrequired as necessary input for various theoretical treat-ments, e.g., in perturbation theories.

The average Boltzmann factors (BFs) in the mostimportant site± site frames for the four models, PM3,PM4± 8, EPM5± 3, and EPM5± 4 are plotted in ® gures3± 6 for temperatures T = 1 and 1/3, and also for thehigh temperature limit, T = + ¥ , which makes it pos-sible to estimate roughly the e� ect of association on thestructure. The results for the remaining two modelsconsidered, PM4± 4 and EPM5± 2, are not knownbecause di� erences would scarcely be detectable withinthe scale of the graphs.

As one could expect, the basic shape of the averageBFs is determined by the properties of the underlyinghard bodies on which the H-bond interaction is super-imposed. The main feature of the hard-body contri-bution a is the existence of cusps, i.e., slopediscontinuities, which are of purely geometrical originand are associated with changes in the excluded volume.At high temperatures the average BFs copy these cusps,whereas at low temperatures, though they remainpresent, they need not be easily detectable. Similarly,the average H bond contribution S , exhibits slope dis-

continuities at separations where the bond between acertain pair of sites can no longer be established (orcome into existence. We remark in passing that theunderlying hard body of the PMs forms the so calledpseudomolecular ¯ uid model discussed at length in [36],and we refer the reader to this paper for details on thecusps and other properties.

The simplest behaviour of all k e000 l functions is foundin the centre-to-centre frame. Due to symmetry, a is freeof cusps and there exists only one range of particleseparations where the bond can be established. Thelargest probability for establishing the bond at low tem-peratures for all the models but EPM5-4 is at contact,and this probability decreases rapidly with increasingseparation. k eCC l of the EPM5-4 model exhibits asmooth peak outside the contact separation, whichclearly di� erentiates it qualitatively from the remainingones. The range rmax of S , beyond which it is identicallyzero, is given directly by the appropriate value of¸ : rmax = 1 /2 + ¸ for PM3, rmax = 1 /2 + ¸ + l- forPM4-l, and rmax = 1 + ¸ for EPM5-¸.

In the remaining frames the behaviour of k e000 l maybe rather complex. The main feature of the average BFsin the non-central frames is the existence of at least tworanges with H bonds of qualitatively di� erent origin;moreover, these ranges may even be separated by arange with no H bond at all.

The k e+- l functions exhibit a sharp peak at smallseparations, which corresponds to direct bondingbetween the reference sites. For separations larger than¸, this bond can no longer exist, but other sites may now


Figure 3. Average Boltzmann factors of the PM3 model in di� erent site± site frames as a function of temperature: T = ¥( Ð Ð ); T = 1/3 (Ð Ð Ð ); and T = 1 ( ± ± ± ). Long ticks on the r axis denote the location of cusps.

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establish a bond, either with the reference site orbetween themselves.

On k e+- l º k eC+ l for the PM3 model two cusps areobserved, separated by a cusp which is not present onthe underlying a. The ® rst occurs at r+ - = 1 /2 + s + . Atthis distance the possibility for an overlap between the(+ ) reference site and any (+ ) site of the other moleculeceases to exist. The second cusp at R+ - = sin( x /2) + ¸

belongs to S, and corresponds to the distance where thenon-reference (+ ) site of the molecule whose other (+ )site is the reference site can no longer form a bond withthe reference (- ) site (i.e., the centre) of the other mole-cule. The third cusp occurs at r+- = 1.5, and ® nally thecontribution to S vanishes at rmax = 1 + ¸.

The k e+- l function for the EPM5-¸ models issimilar to that of the PM3 model, with the exceptionthat at intermediate separations there is a range,r+- Î [ ,1 - cos ( x - a )], where cos a = 0.7, withinwhich no bond can be established. After it reaches alocal maximum, the function S exhibits a cusp atr+- = sin ( x /2) + ¸, whose origin is the same as thatof the second cusp of PM3. The next cusp belongs toa, and is located at r+- = sin ( x /2) + s - . The next cuspon a, undetectable within the scale of the graph, occursat r+- = 2, and the maximum separation beyond whichS vanishes is rmax = 2sin ( x /2) + ¸.

As regards the cross-average BF for the PM4-lmodels, one can consider both k eC+ l and k e+- l ; the for-mer BF is shown in ® gure 4. This BF is characterized bya very small contribution of the H bonding. This resultre¯ ects the fact that the (- ) site lies on the bisector of theangle formed by the centre and two (+) sites; althoughthe range of the H-bond interaction associated with the

(- ) site goes well outside the oxygen sphere around thebisector, its e� ective range is strongly diminished by therepulsion between the like sites; on the other hand, therange of this intersection over the reverse side of thecentral sphere is only very narrow.

As a consequence of di� erent geometries of themodels, k e+ + l for PM3, PM4-8, and EPM5-4 has asigni® cant S contribution at contact, whereas this func-tion increases from a zero value at contact for EPM5-3.k e+ + l for PM3 and EPM5-¸ also exhibits a localminimum at larger separations. The origin of thisminimum, as well as of the cusps indicated on the graphsand of other features, is easy to explain using the samearguments as above.

From the above analysis of the average BFs, it ispossible to conclude that the EPM5-4 model di� ersqualitatively not only from the remaining qualitativelydi� erent models but also from the models within its ownEPM5-¸ family; to a certain extent, it combines theproperties of its family with those of the PM3 model.

From the known average Boltzmann factors thesecond virial coe� cient is obtained via equation (14).The results are:

B(T ) /V = 4.56624 - 0.10788 e1/T PM3, (24)B(T ) /V = 4.47708 - 0.02844 e1/T PM4-4, (25)B(T ) /V = 4.45464 - 0.00612 e1/T PM4-8, (26)B(T ) /V = 4.00216 - 0.00216 e1/T PM5, (27)B(T ) /V = 4.36092 - 0.02016 e1/T EPM5-2, (28)


Figure 4. As for ® gure 3 but for the PM4-8 model.

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B(T ) /V = 5.87280 - 0.10560 e1/T EPM5-3, (29)B(T ) /V = 9.36348 - 0.23100 e1/T EPM5-4, (30)

from which the following Boyle temperatures, TB,de® ned as B(TB) = 0, are obtained:

TB = 0.2665( b B = 3.7523) for PM3, (31)TB = 0.1977( b B = 5.0589) for PM4-4, (32)TB = 0.1517( b B = 6.5901) for PM4-8, (33)

TB = 0.1329( b B = 7.5230) for PM5, (34)TB = 0.1860( b B = 5.3767) for EPM5-2, (35)TB = 0.2488( b B = 4.0184) for EPM5-3, (36)TB = 0.2701( b B = 3.7022) for EPM5-4, (37)

where b = 1 /T and V = p s 3W /6 is the volume of the

molecule. The Boyle temperature is the only relevantreference temperature when the second virial coe� cientsof di� erent models are compared. In ® gure 7 we there-fore compare B(T ) of the above primitive models as a


Figure 5. As for ® gure 3 but for the EPM5-3 model.

Figure 6. As for ® gure 3 but for the EPM5-4 model.

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function of T /TB. In addition to the primitive modelresults, we also show in ® gure 7 the second virial coe� -cient of the spherical square-well ¯ uids with ¸ = 1.15,which is the approximate range of the H-bond interac-tion in the primitive models, and of the ¯ uid of dipolarhard spheres with the experimental value of the dipolemoment of water, ¹ = 1.8 D (D = debye < 3.335 64 ´10- 30 C m). The di� erence between the models withand without association is remarkable and re¯ ects thedi� erences in the strength of the attractive forces. Theplotted second virial coe� cients also clearly di� erentiatebetween di� erent primitive models. The steepest tem-perature dependence is found for the original 5-sitemodel, which corresponds to its earlier mentionedsingularity’ . The remaining models seem lesssingular’ , at least as far as one can judge from thecourse of B(T ). Examination of ® gure 7 reveals twotrends of temperature dependence with respect to thegeometry of the primitive models: (i) within the 5-sitemodels, the increase of the range of the H bond makesB(T ) less steep (i.e., less singular), and (ii) displacementof the (- ) site o� the centre (cf. the models PM3± PM4-4± PM4-8) makes B(T ) steeper.

4.2. Structural and thermodynamic propertiesFollowing the details given in the preceding section,

we performed extensive MC simulations on the PM3,EPM5-2, EPM5-3 and EPM5-4 models, and severalsimulations on the PM4-8 model.

The quantities of main interest have been the site± sitecorrelation functions, the equation of state, the average

spatial arrangement of the bonded molecules around arandomly selected molecule and the occurrence of var-ious k-mers, and the dependence of these quantities onthe thermodynamic conditions. For the models con-sidered, the most important correlation function is thecentre-to-centre correlation function gCC (which for thePM3 model coincides with g- - ), from which the co-ordination number is derived. Other important site±site correlation functions are g+- and g+ + ; the latterfunction coincides with g- - for the EPM5 models. Themost important question which the simulation resultsshould answer is to what extent the considered primitivemodels are able to reproduce the main structural prop-erties exhibited by real water, manifested primarily by alow value of the coordination number and the locationof the second peak of the centre-to-centre correlationfunction. In the tables presenting the thermodynamicproperties, we have included also values of the com-pressibility factor obtained from the perturbed virialexpansion,

b P / q = ( b P/ q )HS + (B - BHS) h , (38)

using the ¯ uid of hard spheres (of diameter 1) as a refer-ence system. Since it has been well established that thisexpansion provides quite an accurate estimate of thegas-phase properties over the entire temperature range(and for supercritical temperatures up to intermediatedensities), comparison of the simulation and perturbedvirial expansion results provides additional informationon the thermodynamic state of the simulated system.

The simulations were performed under thermo-


Figure 7. Second virial coe� cient ofdi� erent primitive models of waterand their comparison with thesquare-well model (SW; ¸ = 1.15)and the dipolar hard-sphere model(DHS; ¹ = 1.8 D).

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dynamic conditions which may be relevant for realwater, and at which reasonable convergence of the simu-lations might be expected. The packing fraction ofspheres arranged in a hexagonal ice-like crystal is0.340 and the packing fraction was thus con® ned, ingeneral, to the range (0.25, 0.375) [14]. We recall,however, that due to the presence of the additionalhard sphere repulsion between the like sites the actualexcluded volume, and hence the packing fraction, arelarger than those de® ned solely by the s W sphere. Arough estimate of the relevant temperature range canbe obtained from the Boyle temperature. At tempera-tures below the Boyle temperature the convergence ofthe simulations for the primitive models deterioratesrapidly, re¯ ecting the formation of large polymericclusters. We recall that in the simulations on the originalPM5 model, we were able to perform the simulation onthe ¯ uid phase only for temperatures not much di� erentfrom the Boyle temperature, namely T > 0.934 TB( b < 0.071 b B). We therefore always started the simu-lation for a given model at a high temperature, and bygradually cooling the system tried to reach as low atemperature as possible under the constraint that thesystem remained in the homogeneous ¯ uid phase.

4.2.1. PM3 modelSimulations on this model were performed for inverse

temperatures b < 1.866 b B. Typical results for the corre-lation functions g+ + and g+- are shown in ® gure 8 andfor the centre-to-centre correlation function in ® gure 9.All the results are for systems at the same packing

fraction, h = 0.30, and at three representative tempera-tures spanning the temperature range considered.

As is seen from ® gures 8 and 9, despite the highdensity the correlation functions exhibit the same quali-tative shape as the corresponding average BFs, cf. ® gure3; the only di� erence is in their more pronounced localextrema. Qualitative di� erences between the results atdi� erent temperatures seem negligible; quantitatively,the extrema become more pronounced with decreasingtemperature, as expected.


Figure 8. Site± site correlation functions for the PM3 model at h = 0.3 and di� erent temperatures b = 3 ( ± ± ± ), b = 5 (Ð Ð ),and b = 7 (´´´´´ ).

Figure 9. As for ® gure 8 but for the centre-to-centre corre-lation functions.

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The centre-to-centre correlation functions are of thegreatest interest. For the relatively high temperature,T = 1 /3 < TB, the ® rst minimum is located at the dis-tance rmin < 1.7, and the function qualitatively followsthe corresponding average BF, despite its high density.Consequently, the coordination number is about 10. Asthe temperature is decreased, the location of the ® rst

minimum moves to rmin = 0.5 + ¸ = 1.115 and thecoordination number drops to about 4, which wouldbe in agreement with real liquid water. However, thesecond maximum occurs at rmax < 2, which is a strongindication of a rather hard-sphere-like arrangement ofthe molecules. Deeper insight into the spatial arrange-ment of molecules is gleaned from ® gure 10, where the


Figure 10. Average spatial arrange-ment of the bonded moleculesabout a central molecule for thePM3 model at b = 5 and h = 0.3.

Table 2. Simulation results for the compressibility factor, internal energy per particle, and occurrence of various k-mers asfunctions of the thermodynamic conditions for the PM3 model.

Mass concentration

b h b P / qa - U / e N 1-mer 2-mer 3-mer 4-mer Maxb

3.0 0.25 2.14 6 0.06(2.67)

0.669 6 0.005 0.204 0.112 0.080 0.057 141

0.30 2.64 6 0.05(3.49)

0.837 6 0.006 0.128 0.063 0.042 0.029 383

0.35 3.42 6 0.08(4.64)

1.013 6 0.006 0.070 0.030 0.016 0.011 466

5.0 0.25 0.48 6 0.05(- 0.41)

1.411 6 0.012 0.013 0.004 0.002 0.001 510

0.30 0.59 6 0.06 1.550 6 0.015 0.006 0.001 0.0 0.0 512(0.06)0.35 1.09 6 0.06 1.668 6 0.012 0.002 0.0 0.0 0.0 512(0.34)

7.0 0.25 - 0.55 1.834 6 0.022 0.0 0.0 0.0 0.0 512(0.91)0.30 - 0.47 1.886 6 0.014 0.0 0.0 0.0 0.0 512(0.94)0.35 - 0.08 1.935 6 0.006 0.0 0.0 0.0 0.0 512(0.96)

a Values in parentheses are the results of the perturbed virial expansion about the hard sphere reference.b Maximum k-mer detected during the simulation. Value in parentheses is mass concentration of the k-mer.

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average density of the bonded molecules about aselected molecule is shown. We see that distinct maximaoccur only in the directions of the (+ ) sites of the centralmolecule. This means that the excluded volume e� ectsare insu� cient to force the (+ ) site donating moleculesto form a bond with the central (- ) site in a preferreddirection. Consequently, the resulting structure exhibitsonly a partial orientational arrangement, and is notwater-like.

When the temperature is decreased further the be-haviour of the system changes gradually, as can bededuced from the behaviour of various quantitiesmonitored during the simulation. The mobility of mole-cules decreases, as witnesed by the behaviour of theautodi� usion coe� cient and the mean-square displace-ment. Also, the autocorrelation coe� cients increase con-siderably. Thus, whereas at the inverse temperatureb = 5 the system was still in the ¯ uid phase, equilibriumat lower temperatures was much more time-consuming,with a possibility of simulating a metastable (glassy)state rather than a stable ¯ uid one; this seems to bethe case at b = 7.

The thermodynamic properties of the model and theoccurrence of various k-mers are summarized in table 2.The perturbed virial expansion result for b = 5 indicatesthat the system is de® nitely in the liquid phase. The highvalues of the internal energy indicate that the H bondnetwork is well established, which also is in agreementwith the analysis of the k-mer occurrence.

4.2.2. EPM4-8 modelOne can expect intuitively that the properties of the

PM4-l models should not be very di� erent from those ofthe PM3 model. This is correct, but only for very smallvalues of l. Due to the location of the fourth interactionsite on the bisector +C+ , and due to the short range ofthe attractive interaction mimicking the H bonding, theprobability of establishing a bond on the inverse side ofthe bisector in the PM4 models rapidly decreases to zerowith increasing l. This can also be deduced from thebehaviour of k eC+ l shown in ® gure 4. Such a modelcan hardly be expected to reproduce correctly the struc-ture of dense liquid water, and this has also been con-® rmed by simulations.

The results for the centre-to-centre correlation func-tion at b = 7 and h = 0.30 is shown in ® gure 11, and theorientational arrangement of the bonded moleculesaround a central molecule for the same system isshown in ® gure 12. Regarding the gCC function, we seethat the result corresponds to a typical normal-¯ uid-likestructure. The orientational arrangement of the bondedparticles exhibits two additional peaks (in comparisonwith the PM3 model) but these peaks lie within the conede® ned by the CÐ (+ ) lines, which is an artefact of the

geometrical arrangement of the sites and of the short-range attractive potential between the unlike sites.Therefore, we have the same situation as for the PM3model, with no preferred direction for forming a bondwith the (- ) site on the reverse side of the oxygen sphere.Consequently, the PM4-l models do not seem to be areasonable alternative to the original PM5 model, andthis is also the reason why no formula for the EOSof this model has been given in section 3 and why nothermodynamic properties have been presented anddiscussed.

4.2.3. EPM5-¸ modelsSimulations for the EPM5-2 model were performed

for inverse temperatures b < 1.488 b B, for EPM5-3 forb < 1.742 b B, and for EPM5-4 for b < 1.351 b B.

As expected from the analysis of the average BFs, the® ndings for the EPM5-2 and EPM5-3 models are verysimilar, and therefore only the centre-to-centre corre-lation function of the former model is shown anddiscussed. As also is indicated by the behaviour of theBFs, the EPM5-4 model, due to its larger H-bondingrange, exhibits certain di� erences from the above twomodels.

The site± site correlation functions for EPM5-3 andEPM5-4 models are shown in ® gures 14, 17 and 18.As for the PM3 model, these functions qualitativelyfollow the shape of the average BFs. Their shapechanges only slightly with both temperature and density.The only qualitative change one can observe for the g+ -function of the EPM5-4 model is that for the lowest


Figure 11. Centre-to-centre correlation function for thePM4-8 model at h = 0.3 and b = 7.

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density the ® rst, cusp-like peak occurs at r = ¸; as thedensity increases this peak becomes smoother andmoves to shorter separations.

The centre-to-centre correlation functions are shownin ® gures 13, 15, 17 and 18, including that of theEPM5-2 model. In contrast to the above ® ndings, thebehaviour of the centre-to-centre correlation function

for EPM5-2 and EPM5-3 is a� ected by both b and h .At high temperatures (T > TB) the gas phase occurs,and the coordination number is therefore about 10. Adecrease in temperature causes the location of the ® rstminimum to move from r < 1.7 to about 1.2, and thistrend evidently is accompanied by a sharp decrease ofthe coordination number from r < 10 to about 4. When


Figure 12. As for ® gure 10 but for thePM4-8 model at b = 7 and h = 0.3.

Figure 13. Centre-to-centre correlation function for the EPM5-2 model at h = 0.25 ( ± ± ± ), h = 0.3 ( Ð Ð ), and h = 0.35(´´´´´ ), and di� erent temperatures (a) b = 4, (b) b = 6, and (c) b = 7.

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b is kept ® xed and h is increased we can see that gCC

becomes ¯ atter and the extrema less pronounced.Thesae ® ndings are similar to those observed for thePM3 model. However, unlike the PM3 model, thesecond maximum occurs at r < 1.7- 1.8 which corre-sponds to the OÐ O separation in a trimer with tetra-hedral symmetry, and this is in agreement with theoriginal PM5 model and, qualitatively, with realwater.

In ® gure 16 we show the average spatial arrangement

of the bonded molecules about a central molecule forone set of thermodynamic conditions for the EPM5-3model; similar results were obtained for all input para-meters used in the simulations. There is no doubt con-cerning the tetrahedral arrangement of these molecules.The four peaks correspond to the bonding directionsde® ned by the location of the interaction sites on themolecules. The peaks are relatively broad, whichdenotes a not too rigid structure.

Regarding the gCC function of the EPM5-4 model, the


Figure 14. Site± site correlation functions for the EPM5-3 model at h = 0.3 and di� erent temperatures: b = 6 (Ð Ð ); and b = 7(´´´´´ ).

Figure 15. Centre-to-centre correlation functions for the EPM5-3 model at h = 0.25 ( ± ± ± ), h = 0.3 ( Ð Ð ), h = 0.35 (´´´´´ ),and h = 0.375 (- ´ - ´ - ), at di� erent temperatures (a) b = 4, (b) b = 6, and (c) b = 7.

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results correspond to the di� erences already found forthe average BF. In contrast to the EPM5-2 and EPM5-3models, the ® rst maximum on the gCC does not lie atcontact but occurs at a slightly larger distance,r < 1.04- 1.10, depending on the thermodynamic condi-tions. In contrast, the second maximum is shifted to ashorter separation, about 1.5± 1.7, at least for higherdensities. Due to the location of the ® rst peak outsidecontact, the ¯ uid has in fact a higher e� ective packing

fraction than h , and we can consider h < 0.3 to be asu� cient value for the approximate upper densitylimit. Surprisingly, there is no great qualitative di� er-ence between gCC for the two temperatures, as onemight expect from the results for the EPM5 models.The main di� erence (rather quantitative) concerns thedepth of the ® rst minimum. For lower temperatures theminimum is much more pronounced and deeper. Thecoordination number is slightly larger than 4 for


Figure 16. As for ® gure 10 but for theEPM5-3 model at b = 6 and h = 0.3.

Figure 17. Site± site correlation functions (a) gCC, (b) g+- , and (c) g+ + , for the EPM5-4 model, at b = 3 and h = 0.2 (´´´´´´ ),h = 0.25 ( ± ± ± ), and h = 0.3 ( Ð Ð ).

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h = 0.3, and decreases with h . The spatial arrangementof the bonded particles for this model is not shown.Qualitatively, it is identical to that for the EPM5-3model shown in ® gure 16; the only di� erence is thatthe maxima are even broader, indicating increasing ¯ ex-ibility of the molecules at bonding.

The shape of the gCC correlation function resemblesthe same function obtained from di� raction experiments

on real water so strongly that it is tempting to comparethe two results directly, see ® gure 19. For real water weused the data by Soper and Phillips [39] because theolder experimental data [40± 42]show signi® cant scatter.Furthermore, it has not been the purpose of this paperto match the properties of real water, and no attempttherefore has been made to adjust the parameters of themodel to some speci® c values: we have used the data forthe lowest temperature and highest density considered,b = 5 and h = 0.3. The agreement between the EPM5-4model and the experimental results is striking.

The thermodynamic properties and the analysis of theH-bond network are given in tables 3± 5. The con-clusions which can be drawn are nearly the same asthose for the PM3 model, and the same applies also totechnical aspects of the simulations on the EPM5models.

5. Conclusions

We have introduced and examined a family of primi-tive models which are descendants of realistic (TIPS)water± water potentials: PM3, PM4-l, and EPM5-¸models. The models di� er from the PM5 model, theonly one considered previously, not only in the numberand location of the interaction sites, but also in the exist-ence of repulsive interaction (modelled by the hard-sphere potential) between like sites. This additionalinteraction itself, naturally exhibited in realistic modelsvia Coulombic interaction, forces the molecules to adopta certain water-like orientational arrangement at shortseparations and makes it possible to increase the rangeof the square-well interaction mimicking the H-bondinteraction while maintaining simultaneously the con-


Figure 18. As for ® gure 17 but for b = 5.

Figure 19. Comparison of the centre-to-centre correlationfunction of the EPM5-4 model at h = 0.3 and b = 5(± ± ± ) with that obtained experimentally [3] (Ð Ð ).

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dition of saturation of the hydrogen bonding; the un-realistically small values of this range in the originalPM5 model have been its main drawback, limiting con-siderably its potential applications.

Monte Carlo simulations were performed on themodels to obtain their structural and thermodynamicproperties at high densities; low density properties

were determined by computing the average Boltzmannfactors in di� erent site± site frames, and the secondvirial coe� cients. The simpli® ed interaction mimickinghydrogen bonding and the geometry of the interactionsites make the descendants of the TIPS3 and TIPS4models inferior to the EPM5-¸ models. While stillmaintaining the conditions of saturation of the


Table 3. Simulation results for the compressibility factor, internal energy per particle, and occurrence of various k-mers asfunctions of the thermodynamic conditions for the EPM5-2 model.

Mass concentration


4.0 0.30 3.30 6 0.04(3.75)

0.500 6 0.007 0.312 0.178 0.115 0.078 53

0.35 4.31 6 0.03(4.94)

0.613 6 0.008 0.233 0.134 0.092 0.068 172

6.0 0.25 1.28 6 0.06(1.13)

1.095 6 0.021 0.043 0.020 0.010 0.005 491

0.30 1.88 6 0.06(1.64)

1.221 6 0.015 0.026 0.009 0.004 0.002 502

0.35 2.84 6 0.06(2.48)

1.324 6 0.015 0.017 0.005 0.001 0.001 509

7.0 0.15 0.39 6 0.05(- 1.36)

1.111 6 0.023 0.042 0.020 0.010 0.007 486

0.25 0.74 6 0.06 1.402 6 0.002 0.009 0.002 0.0 0.0 512(0.02)0.30 1.32 6 0.05 1.533 6 0.021 0.004 0.001 0.0 0.0 512(0.07)0.35 2.39 6 0.06 1.572 6 0.012 0.004 0.001 0.0 0.0 512(0.14)0.375 3.05 6 0.06 1.577 6 0.015 0.004 0.0 0.0 0.0 512(0.11)

8.0 0.30 1.01 6 0.07 1.648 6 0.038 0.001 0.0 0.0 0.0 512(0.49)



Mass concentration


4.0 0.30 3.03 6 0.04(2.80)

1.226 6 0.010 0.023 0.007 0.004 0.002 505(0.01)

0.35 4.51 6 0.04(3.84)

1.359 6 0.010 0.012 0.003 0.001 0.000 509(0.03)

0.375 5.49 6 0.05(4.53)

1.411 6 0.009 0.008 0.002 0.001 0.000 512(0.01)

6.0 0.15 0.19 6 0.04(- 4.21)

1.344 6 0.017 0.013 0.004 0.002 0.001 508(0.01)

0.25 0.67 6 0.04 1.622 6 0.010 0.001 0.000 0.000 0.000 512(0.41)0.30 1.76 6 0.05 1.701 6 0.010 0.001 0.0 0.0 0.0 512(0.63)0.35 3.42 6 0.05 1.750 6 0.010 0.0 0.0 0.0 0.0 512(0.88)0.375 4.35 6 0.06 1.772 6 0.008 0.0 0.0 0.0 0.0 512(0.89)

7.0 0.25 0.23 6 0.05 1.747 6 0.016 0.0 0.0 0.0 0.0 512(0.91)0.30 1.63 6 0.06 1.810 6 0.010 0.0 0.0 0.0 0.0 512(0.94)0.35 2.99 6 0.05 1.844 6 0.014 0.0 0.0 0.0 0.0 512(1.00)


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H-bond interaction, the range of this interaction in thelatter models can be made comparable with experimen-tal values for real water. As a result, EPM5-¸ modelsenjoy a considerable ¯ exibility, and seem able to repro-duce not only qualitatively but even semi-quantitativelythe structure of real water in the same way as complexrealistic potentials. Because such agreement is a neces-sary condition imposed on any system which might beconsidered as a reference system in perturbation theory,EPM5-¸ models appear to be the ® rst simple systemswhich might play this role in the quest for developinga molecular theory of water.

This research was supported by the Grant Agency ofthe Academy of Sciences of the Czech Republic (GrantNo. A-4072607) and by the Grant Agency of the CzechRepublic (Grant No. 203/96/0585).

References[1] Stilling er, F. H., 1975, Adv. chem. Phys., 31, 1.[2] Roïntgen, W. C., 1892, Ann. Phys. , 45, 91.[3] Horne, R. A., 1972, W ater ad Aqueous Solutions (New

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Mass concentration


3.0 0.20 3.13 6 0.04(2.55)

1.070 6 0.008 0.045 0.023 0.012 0.008 483

0.25 4.89 6 0.04(3.26)

1.332 6 0.008 0.012 0.004 0.001 0.001 509

0.30 7.96 6 0.07(4.19)

1.565 6 0.008 0.002 0.0 0.0 0.0 512(0.37)

5.0 0.20 1.21 6 0.05(- 3.38)

1.576 6 0.009 0.001 0.0 0.0 0.0 512(0.43)

0.25 2.89 6 0.04 1.730 6 0.009 0.0 0.0 0.0 0.0 512(0.93)0.30 6.27 6 0.05 1.837 6 0.008 0.0 0.0 0.0 0.0 512(0.99)


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372 n-Site primitive models of H2OD

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a family of primitive models of water: three-, four and five-site models

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