On independence of the solvation of interaction sites of a water molecule

JOURNAL OF CHEMICAL PHYSICS VOLUME 118, NUMBER 14 8 APRIL 2003

On independence of the solvation of interaction sites of a water moleculeMilan Predota and Arieh Ben-Naima)

E. Hala Laboratory of Thermodynamics, Institute of Chemical Process Fundamentals,Acad. Sci., 165 02 Prague 6—Suchdol, Czech Republic

Ivo Nezbedab)

E. Hala Laboratory of Thermodynamics, Institute of Chemical Process Fundamentals,Acad. Sci., 165 02 Prague 6—Suchdol, Czech Republic and Department of Physics,J. E. Purkyneˇ University, 400 96 U´ stı n. Lab., Czech Republic

~Received 11 November 2002; accepted 17 January 2003!

To support simplifying assumptions used in analytic theories of aqueous systems, we have usedcomputer simulations to examine correlations in the bonding of the individual sites of a watermolecule using two qualitatively different extended primitive models, EPM4 and EPM5. We havestudied these correlations not only for the fully interacting water molecule~considered as a solute!but also for a series of other solutes made from the water molecule by turning off some of itsinteraction sites. We have found that for the EPM5 solvent the local density of water moleculesbound to a specific site is independent of the state of the other sites being turned on or off; for theEPM4 solvent such an independence does not hold exactly, but the correlations have been found tobe very small. These facts fully justify previously used speculative approximations for thecalculation of the solvation Helmholtz free energy of a water molecule, and also lend support to thefirst-order thermodynamic perturbation theory of Wertheim. ©2003 American Institute of Physics.@DOI: 10.1063/1.1559687#

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I. INTRODUCTION

One of the main difficulties in pursuing a molecultheory of liquid water is the fact that many-body effects mplay a significant role in determining its properties and tthe water–water pair potential is strongly dependent onrelative orientations of the two molecules. Most progressthe theoretical treatments of liquids in general, and waterparticular, has been due to the assumption of pairwise ativity of the total potential energy,U, which is a good ap-proximation for simple spherical atoms~such as argon oeven methane molecules!, but it is known to be a poor approximation for liquid water.1 Nevertheless, this assumptiohas been applied almost always in theoretical as welsimulation calculations with the hope of understanding liqwater and aqueous solutions by constructing an effectivepotential. As a matter of fact, using effective pair potentiamany properties of liquid water have already been repduced. This may indicate which of the features of a ppotential may be responsible for the unique propertieswater and aqueous solutions.

An integral part of various efforts to construct an anlytical theory of aqueous systems are further various simfying approximations. A typical example is the assumptionthe independence of the interaction sites in commonly ufirst-order thermodynamic perturbation theory of Wertheim2

A similar assumption, independence of the interaction sof the water molecule engaged in hydrogen bonding~HB!,

a!On leave of absence from the Department of Physical Chemistry, HebUniversity, Jerusalem, Israel.

b!Author for correspondence; Electronic mail: [email protected]

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was used in calculations of the solvation free energy ofwater molecule in liquid water.3,4

In view of the importance of the above assumptionpursuing any analytical theory of liquid water, we hatested ‘‘experimentally’’ this assumption using computsimulations. We have used two qualitatively different primtive models, EPM4 and EPM5 models5 descending from theTIP4P6 and ST27 potentials, respectively. The idea underling this experiment is to consider a hypothetical soluwhich is an ordinary water molecule with some of its inteaction sites switched off, and to calculate then how the dsity of water molecules in the region where they can formHB around this solute changes when one successively toff the interaction sites. Physical quantities evaluated frsimulations to obtain the required information are averaenergy of the solute, probability distributions of bonds, aspatial distributions of water molecules around the solute

II. THEORETICAL BACKGROUND

To be consistent with the usual way in which contemprary both effective realistic pair potentials and simple mopotentials are written,5,8 it is assumed that the water molecuconsists of a core~oxygen site! and k off-center interactionsites of two kinds. A general form of such a potential is

u~X1 ,X2!5uOO~R12!1 (i , j 51

k

uss~r i~1! ,r j

~2!!, ~1!

whereR125uR22R1u, Rl is the position vector of the oxygen ~O! site of moleculel, andr i

( l ) is the position vector ofembedded sitei on moleculel. Further,uOO is a neutral~typi-

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6447J. Chem. Phys., Vol. 118, No. 14, 8 April 2003 Solvation of interaction sites of a water molecule

cally Lennard-Jones! interaction between the O sites, anduss

is a site–site interaction~usually Lennard-Jones plus Coulombic interaction!.

Let us consider now the process in which a water mecule~labeled as 0! is brought from infinity and inserted intobulk water at locationX0 . The Helmholtz free energy associated with this process~Helmholtz free energy of solvation!,DAW* , is given by

exp@2b DAW* #5êxp@2bBW#&W , ~2!

where b is the inverse temperature,b51/kBT, BW is thetotal binding energy of the water molecule atX0 with allotherN molecules of the system,

BW5(i 51

N

u~X0 ,Xi !, ~3!

and the symbol¯&W stands for an average over all watmolecules but that located atX0 . To determineDAW* it isuseful to decompose the above process into two steps~i!first insert a particle interacting with the water moleculesthe system only through the potentialuOO; and~ii ! only thenits sites are turned on to make the particle equivalent to owater molecules. This decomposition makes it possiblewrite DAW* as follows:

exp@2b DAW* #

5* exp@2b~BOO1Bss!#exp@2bUW#dVN

* exp@2bUW#dVN

5* exp@2bBOO#exp@2bUW#dVN

* exp@2bUW#dVN

3* exp@2b~BOO1Bss!#exp@2bUW#dVN

* exp@2bBOO#exp@2bUW#dVN

5êxp@2bBOO#&Wêxp@2bBss#&W1OO. ~4!

In these equations,BOO is the energy of the particle atX0 ,stripped off of all interaction sites, with all other molecule

BOO5(i 51

N

uOO~R0i !, ~5!

Bss is the interaction of the sites of the particle atX0 with thesites of all other particles,

Bss5(j 51

k

(l 51

N

(i 51

k

uss~r j~0! ,r i

~ l !!5(j 51

k

Bss~ j ! , ~6!

V denotes the volume of the configurational space ofparticle, and the symbol &W1OO stands for aconditionalaverage over all water molecules but that atX0 , providedthat there is a particle atX0 that interacts with the watemolecules through the potentialuOO. Thus, after making useof ~6!, we can finally rewrite the solvation free energy as

exp@2b DAW* #5exp@2b DAOO* #

3K )j 51

k

exp@2bBss~ j !#L

W1OO

. ~7!


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In two previous calculations of such average quantitiewas assumed that the water molecule possessed four inttion sites and that these sites actedindependently.4,9 Thismeans that the distribution of water molecules aroundsite, say sitea, is independent of the distribution of watemolecules around other sites. In means that the averagthe rhs of~7! could be factored into a product of averagquantities, i.e.,

K )j 51

k

exp@2bBss~ j !#L

W1OO

5)j 51

k

êxp@2bBss~ j !#&W1OO. ~8!

Thus, if one further assumes that the sites are equivalentsolvation free energy of a water molecule may be expresin a very simple form,

DAW* 5DAOO* 1k DAa* , ~9!

whereDAa* 52 logêxp@2bBss( j )#&W1OO is the solvation free

energy of an interaction sitea. The determination of the solvation energy thus reduces to the calculation of the averenergy of the individual interaction sites.

III. MODELS AND COMPUTATIONAL DETAILS

In order to adhere to the general interaction model givin ~1! and yet to operate on a simple level, we use in tstudy two qualitatively different extended primitive modeof water: the five-site model EPM5,10,11 which descendsfrom the ST2 potential,7 and the four-site model EPM412 thatdescends from the TIP4P potential.6 These models representwo basic geometries of water models~both complex andsimple! commonly used, i.e., the tetrahedral arrangemenfour off-center sites~EPM5! and a planar molecule with twotetrahedrally arranged hydrogens~positively charged sites!and one negatively charged site located on the bisector otetrahedral angle~EPM4!. The extended primitive modeladopt the geometry of their more complex ‘‘parent’’ modebut the site–site interactions are significantly simplified.

The EPM5 model pictures the water molecule as a hsphere of diametersW ~representing the oxygen O site! withfour tetrahedrally arranged embedded interaction sites ofkinds on its surface~see Fig. 1!: two H sites and two M siteswhich represent the positive and negative charges, restively. A short-range attractive interaction between the unlsites mimics H bonding and a repulsive interaction betwethe like sites mimics the interaction between the like charat close separations~for details see Ref. 5!. The intermolecu-lar potential of the EPM5 model thus assumes the [email protected]. ~1!#

u~X1 ,X2!5uHS~ROO;sW!1 (i , j 51

4

uss~r i~1! ,r j

~2!!, ~10!

where the first term stands for the hard sphere repulsiontween the oxygen sites, and the other term for the interacbetween the off-center sites,


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6448 J. Chem. Phys., Vol. 118, No. 14, 8 April 2003 Predota, Ben-Naim, and Nezbeda

(i , j 51

4

uss~r i~1! ,r j

~2!!5(i , j

uHS~r i~s1! ,r j

~2! ;sR!

1(i , j

uHB~r i~1! ,r j

~2! ;l!. ~11!

Here the summations in the first term on the rhs runs ovepairs of like sites, in the other term over the pairs of unlisites, subscript HS denotes the hard sphere interaction,

uHS~r 12;s!51`, for r 12[ur12r2u,s,

50, for r 12.s, ~12!

subscript HB stands for H bonding given by a square-winteraction,

uHB~r 12;l!52eHB, for r 12[ur12r2u,lsW ,

50, for r 12.lsW , ~13!

and the meaning of the remaining symbols is the same a~1!. This model withl50.4sW andsR50.8sW was shownto reproduce the structural properties of real water surpingly well.11 This choice of parametersl and sR preventsalso double bonding between two water molecules, i.e., thcannot be a pair of water molecules with two hydrogbonds between their two pairs of sites and, moreover, nocan create more than one bond~a condition of stericincompatibilities!.

FIG. 1. Schematic representation of the extended primitive models of wEPM5 ~top! and EPM4~bottom!, and interactions between the two kindstheir sites—repulsion between like sites and attraction between unlike s


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Similar to the EPM5 model, the EPM4 model models twater molecule as a hard sphere of diametersW ~represent-ing the O site!. It has also two tetrahedrally arranged H siton its surface but only a single M site located on the bisecH–O–H with uOM u50.15sW ~see Fig. 1!. The interactionsare modeled in the same way as in the EPM5 model wl50.7sW for the range of H bonding andsR50.8sW for thehydrogen–hydrogen hard sphere repulsion. There is nopulsion between the M sites of the EPM4 model becausehard core and the hydrogen–hydrogen repulsion preventM sites from getting close to one to another. The tetrahedstructure of the EPM4 liquid results from the mutual effecof the H–H repulsion and M–H attraction forcing the watmolecules bonded to the M site to be preferentially in twtetrahedral directions complementary to the two H sitTherefore, the M site forms typically two bonds and, whmaking a comparison with the M sites of the EPM5 modelshould be considered as a~degenerated! double M site. Thevalues of parametersl andsR prevent again double bondinbetween two water molecules as in the EPM5 model; hoever, there is a nonzero probability for the H site to form twbonds, and the M site can be engaged even in three boAs a result, the maximum number of bonds that the EPwater molecule can establish is not restricted by the numof sites, i.e., 4 as for the EPM5 model~note that we count theM site of the EPM4 molecule twice! but it can be also 5 or,in an extremely unlikely event, even 6.

Throughout the paper we will use henceforth units suthatsW5e/kB51, wherekB is the Boltzmann constant. It isalso worth remarking here that unlike complex point-chamodels, for which the definition of H bonding is rather arbtrary, the definition of a H bond is unique for the considerEPM4 and EPM5 models.

To obtain the properties of interest, we carried outseries of standard Metropolis Monte Carlo simulations inNVT ensemble,13 each withNW5215 water molecules andone solute molecule. The preferential sampling with tprobability distribution function,

f ~r SW!5~11D !/~r SW2 1D !, ~14!

was used to sample efficiently the vicinity of the solute. Her SW is the distance between the solute and a water molecandD was chosen so thatf (L/2)50.1, whereL is the lengthof the simulation box. Simulations were organized in cycleach cycle consisting ofN trial attempts. The number oequilibration cycles was 53105, followed by 183106 pro-ductive cycles. Errors were estimated from the standardviation of the block averages using ten blocks.

IV. RESULTS AND DISCUSSION

The primary motivation for the reported investigatiohas been to verify validity of Eq.~9! which means to verifyvalidity of the assumption on the independence of the sotion of the individual interaction sites. For this purpose wmay consider an arbitrarily chosen water molecule as aute, change the number of its interaction sites~by settingtheir interaction to zero!, and calculate its average energin dependence on the number of active interaction sicf. Eq. ~7!.

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TABLE I. Probability distributions of different solutes to formn bonds with the solvent molecules, and average energies for the EPM5 solvent.Pth(n) is thebinomial distribution withp50.9225, andPsim(n) is the simulation result;ES is the average energy of the solute,ES /Nsites is the average energy per site othe solute, andEW /NW is the average energy of solvent per water molecule. The solutes are labeled according to their activated sites.

n H HH HM HHM HHMM

0 0.08 0.006 0.006 531024 431025

1 0.92 0.14 0.14 0.02 0.002Pth(n) 2 ¯ 0.85 0.85 0.20 0.03

3 ¯ ¯ ¯ 0.79 0.244 ¯ ¯ ¯ ¯ 0.72

0 0.08 0.008 0.004 0.001 331025

1 0.92 0.15 0.17 0.02 0.003Psim(n) 2 ¯ 0.84 0.83 0.22 0.03

3 ¯ ¯ ¯ 0.76 0.234 ¯ ¯ ¯ ¯ 0.74

( nPsim(n)522ES 0.9260.01 1.8360.02 1.8360.02 2.7660.02 3.7060.0224ES /Nsites 1.8360.02 1.8360.02 1.8360.02 1.8460.02 1.8560.012EW /NW 1.84260.002 1.84360.002 1.84560.002 1.84360.002 1.84560.002

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The total internal energyE is given by the water–wateinteraction,EWW , and by the solute–water interaction,EWS.For the model used these energies are given directly bynumber of corresponding hydrogen bonds, i.e.,

E5EWW1EWS52NWW2NWS. ~15!

When splitting the total energyE into the energy of watermolecules,EW , and the energy of the solute,ES , we have totake into account that the interaction energy between wand solute has to be assigned equally both toEW and ES ,i.e.,

E5EW1ES , ~16!

where

EW5EWW1 12 EWS; ES5 1

2 EWS. ~17!

Another source of information on the independencebonding of the sites comes from the distribution of the nuber of bonds of the solute, i.e., the relative percentage ofoccurrence of a solute to create 0, 1,..., bonds. If the sitesindependently, then the probabilityPth(n) of a solute withNsites to have n bonds must be given by the binomidistribution,

Pth~n!5S Nsites

n D pn~12p!Nsites2n, ~18!

wherep is the probability of a single site to create a H bond.In the EPM5 model, the role of the H sites and M sites

fully symmetric and altogether six different molecules~sol-utes! can be obtained from the EPM5 water moleculeremoving some of its interaction sites, including the limitincases of the hard sphere and the EPM5 molecule itself.different infinitely diluted solutions were therefore consiered with the solutes labeled according to their turnedsites; e.g., solute ‘‘HM’’ means that one H site and onesite have been activated on the solute molecule; see TabThe solutes ‘‘M,’’ ‘‘MM,’’ and ‘‘HMM’’ are identical to theirsymmetric counterparts with all H and M sites interchangWhen constructing the solute by removing some of the s


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from the EPM5 molecule, both the attractive and repulsinteractions of the sites are turned off, i.e., as if the sitesnot exist at all.

There is no symmetry of the H sites and M sites in tEPM4 model, which leads to a different set of solutes otained from the EPM4 molecule by removing some ofsites; see Table II. The recognition of the collaborative effof the H–H repulsions and H–M attractions needed forproper~i.e., with an approximate tetrahedral symmetry! cre-ation of H bonds by the M site requires a different processthe removal of interaction sites: the repulsive part of thesites must be maintained because without the H–H repulthe bonds established by the M site of the solute would lits directionality needed for a proper H bond. This is quevident if we consider the solute ‘‘M.’’ In this case the M siwould attract H sites of surrounding water molecules indirections, most easily in those closest to the M site, iclose to the position of the turned-off hydrogens where nobonds with the M site would have been created if all sihad been turned on. Therefore, in solutes with the M sturned on we always keep therepulsiveinteraction of the Hsites of the solute in effect, even when one or both H sitesturned off, i.e., we switch off only the attractive part of thinteraction of the turned-off H sites. In the solute ‘‘H,’’ onlthe repulsive and attractive interaction of the single siteturned on; in the solute ‘‘HH’’ all repulsive and attractivinteractions of the H site are, of course, turned on. Accouing for the different geometry of the EPM4 and EPM5 moels we find the above process of removing the individsites of the EPM4 molecule in direct analogy to that of tEPM5 model.

The density of primitive water is conveniently defineby the packing fractionh5(p/6)(N/V)sW

3 . We applied thisdefinition also to the system containingNW water moleculesand one solute molecule,h5(p/6)(N/V)sW

3 , where N5NW11, and used the valueshEPM550.3, bEPM5

51/TEPM555, and hEPM450.35, bEPM456, found in theprevious studies to give the best agreement betweenstructure of primitive water and that of real water.11,12There-



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TABLE II. The same as Table I for the EPM4 model. The probabilities of the creation ofn bonds fromsimulation are given separately for the individual sites,PH

sim(n) andPMsim(n), and for the entire solute,Psim(n).

The theoretical predictionPth(n) is given by the binomial distribution withp50.775.

n H M HH HM HHM

0 0.25 ¯ 0.24 0.22 0.22PH

sim(n) 1 0.75 ¯ 0.76 0.78 0.772 0.01 ¯ 0.01 0.01 0.01

Average 0.76 ¯ 0.76 0.79 0.79

0 ¯ 0.06 ¯ 0.05 0.041 ¯ 0.38 ¯ 0.36 0.34

PMsim(n) 2 ¯ 0.55 ¯ 0.57 0.60

3 ¯ 0.02 ¯ 0.02 0.01Average ¯ 1.53 ¯ 1.56 1.59

0 0.23 0.05 0.05 0.01 0.0031 0.78 0.35 0.35 0.12 0.04

Pth(n) 2 ¯ 0.60 0.60 0.41 0.183 ¯ ¯ ¯ 0.47 0.424 ¯ ¯ ¯ ¯ 0.36

0 0.25 0.06 0.05 0.01 0.0031 0.75 0.38 0.36 0.12 0.042 0.01 0.55 0.57 0.40 0.17

Psim(n) 3 ¯ 0.02 0.01 0.45 0.404 ¯ ¯ 0 0.02 0.375 ¯ ¯ ¯ 0 0.02

( nPsim(n)522ES 0.7660.02 1.5360.03 1.5460.04 2.3460.03 3.1560.0424ES /Nsites 1.5260.03 1.5360.03 1.5460.04 1.5660.02 1.5760.022EW /NW 1.5460.01 1.5460.01 1.5560.01 1.5660.01 1.5560.01

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fore, the volume of the periodic simulation box may be tsame for all solutes. Moreover, in our studies14,15 of hydro-phobic hydration of a hard sphere of variable size in primtive water, we showed that replacing a single water molecby a hard sphere with the identical coresW had a negligibleeffect on the total pressure of the system. For solutes winonzero number of turned-on interaction sites the differemust be even smaller because such a solute is, qualitaticloser to the molecule of water. This means that the preswas, within simulation uncertainties, the same for all sotions considered.

A. The EPM5 solvent

Since this primitive model has been constructed so asatisfy the conditions of steric incompatibilities, the numbof bonds of any molecule cannot exceed the number ointeraction sites. The average energy of a water molecEW /NW , the average energy of the solute,ES , and the av-erage energy per site of the solute,ES /Nsites, are given inTable I; the ratioES /Nsites has been multiplied by 4 in thetable to get the quantity directly comparable to the averenergy of the water molecule. It is seen that the averenergy of the water molecule remains constant for all tyof solutes, confirming that the presence of the solute hasmarginal effect on the properties of water. The energy ofsolute increases, in the absolute value, with increasing nber of sites turned on; however, the energy per site ofsolute remains constant, which lends support to the origintuitive assumption. For the solute that is identical to twater molecule, the energyES must be equal toEW , which

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is satisfied within simulation uncertainties. In this table walso give the probability distributions of bonds. The proability for a site to create a H bond is22ES /Nsites and canbe most accurately determined from the average energymolecule in pure water, i.e.,p50.9225. As it is seen, theobserved probabilitiesPsim(n) of the solute to haven bondsand the predicted probabilityPth(n) according to~18! are invery good agreement.

To further support our findings, we have to examinestructural property that would give more detailed information the bonding of the individual sites in different soluteLater, in Fig. 3 we plot therefore the angular distributionwater molecules around the solute that are bonded tosolute. Anglea ~for a definition see Fig. 2! is measured inthe plane defined by the center of the solute~O site!, one Hsite, and another off-center site that is either an empty site

FIG. 2. The definition of anglea, used for the calculation of the anguladistribution of water molecules around the solute, as the angle betweeOHW vector of the solute molecule and the projection of the solute–solvOOW vector onto the reference plane of the solute.


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solute ‘‘H,’’ or M site for solute ‘‘HM,’’ or another H site inother cases. The distancer OO of the bonded water moleculefrom the solute can take on any value between 1~the mini-mum separation due to the hard core repulsion! and 1.4~themaximum distance for which a bond can be established wl50.4!. In the upper part of Fig. 3 we plot the angular dtribution regardless of the height of the water molecabove or below the solute’s reference plane; in the lowerwe consider only those water molecules lying in the slabthickness 0.25 around the reference plane to distinguishmolecules bonded mostly to in-plane versus off-plane siThe width of the slab defining the in-plane molecul~60.25! has been chosen as a compromise to include anificant portion of the water molecules bonded to theplane sites needed for a good statistics and, at the sameto exclude most of molecules bonded to the off-plane siThe peak ata50° corresponds to molecules bonded to thesite; the peak at the tetrahedral anglea5109.5° correspondsto another site in the reference plane; the peak ata5234.7°is formed by the off-plane sites located above and belowreference plane.

Figure 3 confirms that there is almost no correlationtween the density of the water molecules bonded to thedividual interaction sites. The first peak ata50° is the samefor all solutes, the second peak is also very similar forsolutes when this site is turned on. The third peaka5234.7° is different in shape because the correspondsites are off plane, but its area is the same as that of theor second peaks for the solute with one off-plane site,solute ‘‘HHM,’’ and twice as big for the water molecule because of two off-plane sites. The peaks are well localizedwas already shown in Fig. 5 of Ref. 12, for the case of pwater. The minimum ata555° for the ‘‘HM’’ solute isdeeper than that for the ‘‘HH’’ solute, because the M sitethe water molecule is not repelled at all by the solute wtwo H sites, while there is a partial repulsion when the t

FIG. 3. The angular distribution of bonded molecules around various sofor the EPM5 model. The definition of the in-plane molecules is given intext.


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solute sites are of a different type. The independence of sis proved both for the in-plane water molecules and all wamolecules.

The above results prove the independence of the dibution of bondedwater molecules to the interaction sitesthe solute, which means that the density of bonded molecis an additive function of the number of sites turned on. Trequirement imposed on the primitive models of watnamely that no double bonding between two molecules mtake place, leads to localized regions of the bonded mecules to each site. The localization of these regions isnecessary to prevent double bonding, but guarantees it; hever, the localization of the bonding regions does not guantee the independence of the interaction of the individsites of different solute molecules. It can be onlya prioriassumed that the number of bonded water molecules tosolute with a single site would be the same as the numbemolecules bonded to any of the four sites of the water mecule because of the different density of water molecuaround the solute and possible changes in the structurwater molecules around different solutes. That this indehappens is demonstrated in Fig. 4, where the angular dibution of neighbor molecules around different solutes is pted. Anglea is defined as in the previous case of bondmolecules, but nowall molecules~i.e., both bonded and nonbonded! within the distance of 1.4 from the solute are cosidered. The distribution of neighbor water molecules aroudifferent solutes is not the same and it is not, of course,additive function of the number of sites. The peak aroua50 is similar for all solutes because about 70% of neighmolecules in this region is bonded, as it is clear from tcomparison of the curves in Figs. 3 and 4. However,density around the turned-off sites results from the combition of the waterlike and hard-sphere-like nature of the sutes and it is rather a complex function.

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FIG. 4. The angular distribution of neighbor molecules around the solufor the EPM5 model. All solvent molecules within ther OO distance 1.4 fromthe solute are included.


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Gradual changes that take place in the structure arothe solute as it undergoes the transition from the neat hsphere to the full water molecule are also demonstrateFig. 5, where we show the solute–solvent pair correlatfunction for all six types of solutes considered. The shapethe correlation function remains hard-sphere-like when oone or two identical sites are turned on, but for two and msites of a different kind, the contact value decreases sigcantly and the correlation function becomes waterlike.

B. The EPM4 solvent

In comparison with EPM5 water, the behavior of thEPM4 model is more complex because the H sites and thsite are not equivalent by symmetry as in the EPM5 moand both types of sites can create multiple bonds. Noneless, the double bonding between two water molecules isallowed. The average energy of a water molecule,EW /NW ,the average energy of the solute,ES , and the average energper site of the solute,ES /Nsites, are given in Table II; incalculations of the average energy of the solute per site,M site was treated as a double site~see Sec. III!: Nsites

HHM54,Nsites

HM 53, NsitesM 5Nsites

HH 52, NsitesH 51. The distribution of the

number of bonds of the individual H and M sites,PHsim(n)

andPMsim(n), as well as of the entire solute,Psim(n), are also

given in Table II and shown in Fig. 6. The first thing w

FIG. 5. The solute—solvent pair correlation function for all solutesscending from the EPM5 water molecule.

FIG. 6. The probabilities of a creation ofn bonds for the H site~left! and Msite ~right! in different solutes descending from the EPM4 water molecu


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point out is that for the solute identical to the EPM4 wamolecule, i.e. the ‘‘HHM’’ solute, the average numberbonds of the H site is one-half of the average numberbonds of the M site. If the averages are taken over all mecules~pure solvent!, this statement must be valid exactbecause, by definition, there is one H site and one Mparticipating in any H bond. However, when considering sutes different from the water molecule, e.g., the ‘‘HM’’ soute, there is no such relation between the average numbbonds of the H site and M site, because the solute is diffefrom the solvent and the solvent compensates for the discancy between the number of H bonds originating in thesite or the M site of the solute. On the other hand, ifsolute and solvent molecules were, e.g., ‘‘HM,’’ then the aerage number of bonds of the H site and the M site had toequal. This demonstrates that the average numbers of bfor all solutes are, except for EPM4 water itself, charactetics of the solute at an infinite dilution in EPM4 water, whithe numbers given for the EPM4 solute are subject toabove formulated restriction and are characteristics ofpure solvent. This feature takes place also in the EPmodel, but the symmetry of the H sites and M sites mathe probability of bonding of any site of the solute constato a high degree of accuracy in the EPM5 model.

The probabilities of the H site to create 0, 1, or 2 bonare dictated by the geometry of the model. On the othand, the M site is treated as two individual bonding sicollapsed into a single geometric point. If each of these sacted independently, or, in other words, if there were no crelations between the two regions in which a bond canestablished with the M site, the distribution of bonds of tM site would be binomial with two sites and the bondinprobability would be equal to one-half the average numbebonds of the M site, as given in Table II. The predictprobabilitiesPM

th(0), PMth(1), andPM

th(2) of the M site wouldbe 0.06, 0.36, 0.59, respectively, for the solute ‘‘M,’’ 0.00.34, 0.61 for the solute ‘‘HM,’’ and 0.04, 0.33, 0.63 for thsolute ‘‘HHM,’’ i.e., in fair agreement with the simulationdata. This finding further supports the idea of treating thesite as degenerated double site. The probability distribuof the number of bonds of the entire solute, given in Tablecan be theoretically predicted with the simplest predictionthe binomial distribution assuming independence of all fosites and an equal probabilityp of binding of the H sites andM sites ~which is, however, considered as two degenerasites!. The probabilityp50.775 was taken as one-half of thenergy of EPM4 water molecule.

The average number of bonds of the M site for the ‘‘Mand ‘‘HM’’ solutes is, respectively, by 4% and 2% lower thain pure water. The average number of bonds of the H sitesolute ‘‘HM’’ is close to that in pure water, for the ‘‘H’’ and‘‘HH’’ solutes it is lower by 4%. When comparing the aveage energies per the total number of sites,ES /Nsites, thisratio is by less than 3% lower in the solutes with one or msites turned off than in pure water, i.e., constant within simlation uncertainties.

For the EPM4 solute, the independence of the interacsites does not hold exactly, as is clear also from Fig. 7, whthe angular distribution of bonded water molecules is plot

-

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for all solutes. The stronger bonding of water molecule retive to other solutes is evident. Note also that the areawhich water molecules can establish bonds with the Mof the solute is much broader than for the H sites or thesite in the EPM5 model, as it can be inferred from the cufor the ‘‘M’’ solute. The definition of the anglea and thedistinction of the molecules as ‘‘in-plane’’ molecules is thsame as in the case of the EPM5 model. As it can be eadetermined from the geometry of the EPM4 molecule,maximum O–O separation,r OO, of two bonded moleculescan be 1.35.

The solute–solvent pair correlation function for all typof solutes descending from the EPM4 water moleculeshown in Fig. 8. The contact value increases significan

FIG. 7. The angular distribution of bonded molecules around various sofor the EPM4 model.

FIG. 8. The solute–solvent pair correlation function for all solutes desceing from the EPM4 water molecule.


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with an increasing number of the turned-on sites. This isopposite trend to that observed for the EPM5 molecule anclosely related to the position of the M site which requirmore bonded molecules to be close to the minimseparation.

V. CONCLUSIONS

We have used computer simulations and two extenprimitive models of water to investigate the probabilitythe formation of bonds between a hypothetical solute~a wa-ter molecule with a variable number of interaction sites! andwater molecules in dependence on the number of activasolute’s interaction sites. The two models of water usEPM5 and EPM4, represent two qualitatively different typof models because of their different geometry~tetrahedralversus planar!, the number of interaction sites~central1foursites versus central1three sites!, and bonding~sites of theEPM5 model cannot form multiple bonds, while thoseEPM4 can!. By studying these two models we thus hacovered rather a broad range of representations of waterecules on a simple level. For the EPM5 solvent we hashown that the probability of creating an H bond withavailable interaction site on the solute does not dependwhether other sites of this~solute! molecule have alreadybeen engaged in H bonding or not. For the EPM4 modsuch an independence does not hold exactly, but the cortions are very small, which justifies the assumption of indpendence for practical applications.

The findings of this paper fully confirm the plausibilitof the assumption of independence of the interaction sand have thus important implications regarding the develment of an analytical theory of water because they may csiderably simplify various considerations. For instance, thmay reduce the calculation of average quantities over upquadruplet distribution function to calculations of averagover pair distributions only. This is clearly a drastic simpfication which we hope will render the development ofanalytical theory of water~and aqueous systems in gener!feasible.

ACKNOWLEDGMENTS

This research was supported by the Grant Agency ofAcademy of Sciences of the Czech Republic~Grant No.A4072908! and by the Grant Agency of the Czech Repub~Grant No. 203/01/0464!. One of the authors~A.B.-N.!would like to thank the Academy of Sciences of the CzeRepublic for its hospitality and support.

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es

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~2002!.


On independence of the solvation of interaction sites of a water molecule

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