Molecular multipole moments of water molecules in ice IhEnrique R. Batista, Sotiris S. Xantheas, and Hannes Jónsson Citation: The Journal of Chemical Physics 109, 4546 (1998); doi: 10.1063/1.477058 View online: http://dx.doi.org/10.1063/1.477058 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/109/11?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Multipole moments and polarizabilities of nonpolar diatomic molecules through quantum solvation in solidparahydrogen: The quadrupole moment of N 2 J. Chem. Phys. 125, 124507 (2006); 10.1063/1.2357596 The total molecular dipole moment for liquid water J. Chem. Phys. 117, 5290 (2002); 10.1063/1.1501122 A Generalization of the EffectiveCharge Concept: Dynamical Multipoles in Molecular Solids and Liquids AIP Conf. Proc. 626, 198 (2002); 10.1063/1.1499568 Multipole moments of water molecules in clusters and ice Ih from first principles calculations J. Chem. Phys. 111, 6011 (1999); 10.1063/1.479897 A systematic study of basis set, electron correlation, and geometry effects on the electric multipole moments,polarizability, and hyperpolarizability of HCl J. Chem. Phys. 108, 5432 (1998); 10.1063/1.475932

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Molecular multipole moments of water molecules in ice IhEnrique R. BatistaDepartment of Physics, Box 351560, University of Washington, Seattle, Washington 98195-1560and Department of Chemistry, Box 351700, University of Washington, Seattle, Washington 98195-1700

Sotiris S. Xantheasa)

Environmental Molecular Sciences Laboratory, Pacific Northwest National Laboratory,906 Battelle Boulevard, PO Box 999, MS K1-96, Richland, Washington 99352

Hannes JonssonDepartment of Chemistry, Box 351700, University of Washington, Seattle, Washington 98195-1700

~Received 9 April 1998; accepted 15 June 1998!

We have used an induction model including dipole, dipole–quadrupole, quadrupole–quadrupolepolarizability and first hyperpolarizability as well as fixed octopole and hexadecapole moments tostudy the electric field in ice. The self-consistent induction calculations gave an average total dipolemoment of 3.09 D, a 67% increase over the dipole moment of an isolated water molecule. Aprevious, more approximate induction model study by Coulson and Eisenberg@Proc. R. Soc. Lond.A 291, 445 ~1966!# suggested a significantly smaller average value of 2.6 D. This value has beenused extensively in recent years as a reference point in the development of various polarizableinteraction potentials for water as well as for assessment of the convergence of water clusterproperties to those of bulk. The reason for this difference is not due to approximations made in thecomputational scheme of Coulson and Eisenberg but rather due to the use of less accurate values forthe molecular multipoles in these earlier calculations. ©1998 American Institute of Physics.@S0021-9606~98!52335-5#

I. INTRODUCTION

An understanding of the properties of water and ice re-quires a quantitative description of the electric field createdcollectively by the water molecules and the influence thefield can have on the atomic scale structure and dynamics. Itis, for example, expected that the dipole moment of a watermolecule in liquid water is significantly larger than the di-pole moment of an isolated water molecule, which has beenexperimentally measured to be 1.855 D.1 The dipole momentof a water molecule in a condensed phase environment can-not be measured directly, but estimates based on the mea-sured dielectric constant of water and assuming the mol-ecules can be represented by dipole moments only indicatean increase of the molecular dipole moment by 0.6–1.2 D.2,3

The change in molecular electric properties due to the localenvironment is, in particular, important for the developmentof accurate intermolecular potential functions describingwater–water and water–solute interactions. While most ofthe attention has been focused on the dipole moment, it isclear that the quadrupole moment is also important for accu-rately describing the molecular interactions4 and the macro-scopic dielectric constant.5

Ice is a natural starting point for systematically studyingthese issues since, apart from the proton disorder in iceI h ,the molecular structure is known. Liquid water has theadditional complexity of irregular and poorly known order-ing of the molecules. Over thirty years ago, Coulson and

Eisenberg6 carried out an ingenious calculation of the dipolemoment in iceI h using an induction model. Their calculationgave a net molecular dipole moment of 2.6 D. Unfortunately,their numerical result has frequently been misrepresented.Several authors have incorrectly quoted the paper by Coul-son and Eisenberg as giving the molecular dipole moment inliquid water or, as giving the results of a measurement of thedipole moment in ice, or even as giving an experimentalmeasurement of the molecular dipole moment in liquid wa-ter. These misquotations seem to perpetuate in the watermodeling literature of recent years.

We describe here more accurate and detailed calcula-tions that lead to a significantly larger molecular dipole mo-ment in ice than the one obtained by Coulson and Eisenberg.Our calculations include several higher order terms in themultipolar expansion, and do not make some of the approxi-mations invoked in the induction calculations of Coulson andEisenberg. The self-consistent multipole iterative schemethat we employed as well as the input parameters for themolecular multipoles are described in Sec. II. Our results arepresented in Sec. III. We address the differences between ourcalculations and those of Coulson and Eisenberg in Sec. IV.In view of the widespread confusion in the literature abouttheir work, we describe their calculations in considerable de-tail. Our conclusions are presented in Sec. V.

II. MULTIPOLE ITERATIVE METHOD

The starting point of our calculations is the constructionof a hexagonal lattice of water molecules representing iceI h .For the oxygen–oxygen separation we used 2.76 Å, a value

a!Author to whom correspondence should be addressed. Electronic mail:sotiris.xantheas@pnl.gov

JOURNAL OF CHEMICAL PHYSICS VOLUME 109, NUMBER 11 15 SEPTEMBER 1998

45460021-9606/98/109(11)/4546/6/$15.00 © 1998 American Institute of Physics

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consistent with the measured density at 273 K.7 It is wellestablished that while the oxygen atoms are ordered on alattice, the protons are to a certain extent disordered, but stillfollow the ‘‘ice rules,’’8,9 namely that~1! there is one andonly one hydrogen atom between each pair of adjacent oxy-gen atoms, and~2! each oxygen atom has two hydrogens at aclose distance~;1.0 Å, covalently bonded! and two hydro-gens at a larger distance~;1.75 Å, hydrogen bonded!.

The calculations were carried out as follows. A configu-ration of 2,592 water molecules was generated following theice rules, filling a 40.6346.9344.2 Å3 simulation box. Pe-riodic boundary conditions were imposed in all three direc-tions to simulate a bulk ice environment. We generated anice lattice with 96 molecule unit cell with proton disorder,zero total dipole moment, and periodic boundary conditionssatisfied in all three directions. We followed the same proce-dure as Kroes.10 This cell was periodically repeated threetimes in each direction to generate the simulation box. Thedegree of randomness of the ice lattices generated by thismethod and by other methods was studied by Hayward andReimers.11 In the notation of Hayward and Reimers we gen-erated a ‘‘33232’’ lattice with constraints of the ‘‘C2’’type. The randomness is characterized by the dipole–dipoleautocorrelation coefficientsfn for thenth coordination shellthat has radiusr n , defined as

fn5( i 51

N ( j 5 ii

P ~ i !•P ~ j !d~Ri j 2r n!

( i 51N ( j 5 i

i d~Ri j 2r n!, ~1!

whereRi j is the distance between dipoles andP is the di-pole moment vector. In our simulation cellf150.227,f2

520.03,f3520.179, andf4520.015.The electric field at a molecule is the sum of the fields

produced by its neighbors. In order to systematically studythe effect of distant neighbors, only those closer than a givencutoff distancer cut were included in the calculation of theelectric potential. The full calculation was then repeated withdifferent values ofr cut, up to 20 Å. It turns out~see Sec. III!that it is sufficient to user cut57 Å.

When calculating the electric field, each molecule wasrepresented with a point dipole, quadrupole, octopole, and ahexadecapole moment tensor at the center of mass. The elec-tric field at a molecule due to its neighbors then induces botha dipole moment and a quadrupole moment in the molecule.The i th component of the induced dipole moment is givenby12

DP i5a i j Ej11

3Ai , jk

]Ej

]r k1

1

2b i jkEjEk , ~2!

whereE is the total electric field~cf. Appendix!, a i j is themolecular dipole polarizability,Ai , jk the dipole–quadrupolepolarizability, andb i jk the first hyperpolarizability. The re-peated indices are to be summed over.

The induced quadrupole moment of the molecule is12

DQ i j 5Ak,i j Ek1Ci j ,kl

]Ek

]r l, ~3!

whereCi j ,kl is the quadrupole–quadrupole polarizability.

Equations~2! and ~3! are implicit equations ofP i andQ i j . A given molecule polarizes its neighbors and theseneighbors in turn induce extra dipole and quadrupole. Sincethe effect is nonlinear, an iterative procedure is used to solvethe equations.

A first order correction to the dipole moment of eachmolecule is induced by the total electric field~see Appendix!of the neighboring unpolarized molecules

DP i~1!5a i j Ej

~0!11

3Ai , jk

]Ej~0!

]r k1

1

2b i jkEj

~0!Ek~0! , ~4!

and also a first-order correction to the quadrupole moment isinduced

DQ i j~1!5Ak,i j Ek

~0!1Ci j ,kl

]Ek~0!

]r l. ~5!

The first-order induced dipole moments create an addi-tional electric fieldEd

(1) and the induced quadrupole mo-ments generate,Eq

(1) . The total first order correction field,E(1)5Ed

(1)1Eq(1) induces a second-order correction to the di-

pole moments of

DP i~2!5a i j ~E! j

~1!11

3Ai , jk

]Ej~1!

]r k1

1

2b i jkEj

~1!Ek~1! , ~6!

and a second-order correction to the quadrupole moments of

DQ i j~2!5Ak,i j Ek

~1!1Ci j ,kl

]Ek~1!

]r l. ~7!

The new induced multipoles in turn, create an additionalelectric field E(2). This procedure was continued until themagnitude of thenth induced dipole

DP i~n!5a i j Ej

~n21!11

3Ai , jk

]Ej~n21!

]r k

11

2b i jkEj

~n21!Ek~n21! , ~8!

and quadrupole moments

DQ i j~n!5Ak,i j Ek

~n21!1Ci j ,kl

]Ek~n21!

]r l, ~9!

became smaller than a given tolerance, 1026 D and1026 D Å, respectively. Thus, self consistency was obtainedup to this level of accuracy. An ensemble average was ob-tained by averaging the induced dipole on each of the 2,592water molecules in the simulation cell. Due to the protondisorder, both the magnitude and the direction of the localelectric field at a molecule varies, as discussed below.

Ab initio calculated multipole moments for a water mol-ecule up to hexadecapole as well as experimental values forthe dipole1 and quadrupole moments13 are given in Table I.The second column corresponds to second-order Moller–Plesset~MP2! level calculations with the aug-cc-pVQZ basisset14 at the minimum energy configuration~O–H distance0.9590 Å andH–O–H angle 104.28 degrees!.

The column labeled GC in Table I correspond to thevalues of Glaeser and Coulson15 which were used in theinduction model calculation of Coulson and Eisenberg.6 The

4547J. Chem. Phys., Vol. 109, No. 11, 15 September 1998 Batista, Xantheas, and Jonsson

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GC multipoles were originally reported with respect to theoxygen atom, but in order to make the comparison with theother multipole values easier, we have translated them to acoordinate system with origin at the center of mass~see theAppendix!. The GC multipole moments were taken from acalculation by McWeeny and Ohno16 using a very limitedbasis set, consisting of just seven atomic orbitals: An oxygencore orbital, two lone pair orbitals, two hydrogen orbitals,and two bond orbitals. At the time Coulson and Eisenbergdid their calculations, experimentally measured values werenot available.

It is evident from Table I that the results of these earlyab initio calculations are in poor agreement with the experi-mentally measured values especially for the quadrupole mo-ment. The numerical values of the multipoles used as inputin the induction model calculations are, of course, very im-portant in determining the resulting induced dipole moments,as discussed in Sec. IV.

Our induction model calculations made use of the ex-perimentally measured dipole and quadrupole moments. Forthe octopole and hexadecapole moments we used the valuescomputed at the MP2/aug-cc-pVQZ level of theory, listed inTable I. The agreement of the dipole and quadrupole mo-ments at the MP2/aug-cc-pVQZ level of theory with the ex-perimental ones justifies the use of the higher moments~oc-topole and hexadecapole! computed at this level of theory. Inaddition to these, the induction calculations made use of theexperimentally measured molecular dipole polarizability,a i j .17 Results of previousab initio calculations were usedfor the values of the dipole-quadrupole polarizability,Ai , jk,18

the quadrupole–quadrupole polarizability,Ci j ,kl,18 and the

first hyperpolarizability,b i jk .19 The values that we used forthe polarizabilities are given in Table II.

III. RESULTS

The average molecular dipole moment obtained in theself-consistent induction calculation is shown in Fig. 1 as a

function of the cutoff radius,r cut. Clearly, the calculationhas converged atr cut57 Å. The converged value of the di-pole moment is 3.09 D. The error bars shown in Fig. 1 cor-respond to the variation among the molecules in the sample.Due to the proton disorder, no two molecules are in exactlythe same environment. For a 20 Å cutoff, the standard de-viation in the dipole moment is 0.03 D.

Figure 2 shows the magnitude and direction of the vari-ous components of the electric field at a typical molecule.The length of an arrow represents the magnitude of the elec-

FIG. 1. Convergence of the calculated molecular dipole moment as a func-tion of the cutoff distance used in summing up the electric potential due tothe neighbors. The error bars correspond to the fluctuations due to the dif-ferent environments seen by the various molecules in the proton disorderedice I h . The results show that it is sufficient to include only neighbors thatare closer than 7 Å when evaluating the electric field at a given molecule.

TABLE I. Multipole moments of a water molecule used in the self-consistent induction calculations and comparison with values used in previ-ous calculations. The moments are computed using the definitions given inthe Appendix. The origin of the coordinate system is located at the center ofmass of the molecule. The experimental values for the quadrupole momentare from Ref. 13. GC denote the multipole moments of Glaeser and Coulson~Ref. 15! ~which were used by Coulson and Eisenberg in their calculations!.

Exp MP2 GC

Dipole P 3 21.855 21.86 21.76 310218 e.s.u. cmQuadrupole Q33 20.13 20.1328 0.142 310226 e.s.u. cm2

Q11 2.63 2.6135 0.961Q22 22.50 22.4807 21.103

Octopole O 333 1.3565 0.470 310234 e.s.u. cm3

O 113 22.3288 20.851O 223 0.9723 0.381

hexadecapole H3333 21.3637 310242 e.s.u. cm4

H1133 1.6324H2233 20.2687H1111 20.3575H1122 21.2749H2222 1.5436

TABLE II. Values of the polarizabilities used in the self-consistent induc-tion calculations. All quantities are in atomic units (@a i j #5a0

3, @b i jk #5a0

6/e a0 and@Ai , jk#5a04 and@Ci j ,kl#5a0

5). The coordinate reference frameof the molecule was defined as:e1 along the bisector of the molecule,axis e2 perpendicular toe1 and on the plane of the molecule, and axise35e13e2 .

a11 10.31160.088a22 9.54960.088a33 9.90760.02

a isotropic 9.922

b111 5.4715b122 0.5445b133 10.029

A1,11 21.694A1,22 5.943A1,33 24.249A2,12 210.323A3,13 3.096

C11,11 11.930C11,22 26.829C11,33 25.101C12,12 7.862C13,13 13.142C22,22 11.685C22,33 24.856C23,23 7.045C33,33 9.957

4548 J. Chem. Phys., Vol. 109, No. 11, 15 September 1998 Batista, Xantheas, and Jonsson

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tric field component averaged over the molecules in thesample: The induced quadrupole moment produces 20% ofthe quadrupole field while the induced dipole moment ac-counts for 40% of the total field generated by the dipoles.The effect of the self-consistency in the calculation is signifi-cant: The dipole after the first iteration is only 2.70 D and itincreases by 14% in subsequent iterations. The dipolar andquadrupolar fields increase by 15% and 6%, respectively, inthe second and subsequent iterations.

A large value for the induced quadrupole moment wasobtained. The principal axes of the total quadrupole momentare almost the same as those of the free molecule. The anglebetween the original axes and the new principal axes is al-ways less than 0.5°, with an average value of 0.15°. Theaverage eigenvalues of the quadrupole moment are(Q11,Q22,Q33)5(3.29,23.14,20.15)310226 e.s.u. cm2, anincrease of 25%, 26%, and 14% over the value for an iso-lated molecule.

In order to quantify the effect of the dipole-quadrupolepolarizability Ai , jk , the quadrupole–quadrupole polarizabil-

ity Ci j ,kl, the first hyperpolarizabilityb i jk , and the hexade-capole field, we calculated the induced dipole moment ignor-ing these contributions one at a time. We find that the dipole-quadrupole and the quadrupole–quadrupole polarizabilitieshave a significant and opposite effect. IfAi , jk andCi j ,kl arenot taken into account, i.e., when there are no induced quad-rupole moments, the total dipole moment is predicted to be3.05 D, which is 0.04 D smaller than the result of the fullcalculation. After includingAi , jk the total dipole is 2.99 Dand after including the quadrupole–quadrupole polarizabil-ity, Ci j ,kl, the result is 3.09 D. Ignoring the nonlinear effectsintroduced byb i jk we obtain a dipole moment of 3.10 D,showing that the effect of this correction is very small. Sup-pressing the hexadecapole moments had a small but signifi-cant effect. The total dipole moment predicted with a multi-pole expansion up to octopole is 3.04 D.

The above results are summarized in Table III for thevarious levels of molecular multipolar expansions~P , Q , O ,H! and increasing levels of response to multipolar fields.

The anisotropy of the dipole polarizability was tested byusing only the isotropic component of the polarizability. Av-eraging the induced dipole moment over all possible direc-tions of the electric field gives the average of the three ei-genvalues of the anisotropic tensor. The anisotropy of thepolarizability is quite important. The dipole moment inducedwith the isotropic part is 3.16 D, larger by 0.07 D than thedipole induced with the full anisotropic polarizability.

In the calculations of Coulson and Eisenberg, discussedbelow, the simplifying approximation is made that the dipo-lar component of the total electric field at each moleculepoints in the direction of the bisector of the molecule. Wetested this assumption in our calculations and the results areshown in Fig. 2. For each molecule in the sample, we definedthe polar axis as the bisector of the molecule and evaluatedthe polar angle of the various electric field vectors. The av-erage values of the polar angles are indicated by the arrowsin Fig. 2. The electric field due to the dipoles alone is asmuch as 16° off the bisector axis~with an average polarangle of 7°!. However, the total electric field at a moleculelies very nearly along the direction of the bisector, with amaximum polar angle of 1.7° and a mean value of 0.7°. Eventhough the direction of the individual components deviatesfrom the direction of the bisector of the molecule, the totalfield, and the induced dipole moment are to a good approxi-mation pointing along the bisector.

IV. THE METHOD OF COULSON AND EISENBERG

We now review the induction calculation carried out byCoulson and Eisenberg. As noted earlier, the results of theircalculation are still very widely referred to but are frequentlymisrepresented.

Coulson and Eisenberg included a molecular dipole mo-ment, quadrupole moment, and octopole moment as well asan isotropic dipole polarizability in their model. The totalelectric field at a molecule was divided into two parts: Thefield produced by the dipoles,Ed , and the field produced bythe higher order multipoles~quadrupoles and octopoles!, Eh .The total field at a given molecule due to neighbors within a9.6 Å radius induced a dipole moment given by

FIG. 2. Various components of the local electric field at a typical moleculeshowing the magnitude and direction of the field with respect to the bisectorof the molecule. The length of each arrow represents the magnitude of theelectric field components averaged over all molecules in the ice sample. Thedipole field vector is labeledEd , the quadrupole fieldEq , and the field dueto higher multipolesEh ~octopolar plus hexadecapolar field!. The anglebetween the arrows and the bisector of the molecule is the average of thepolar angle of the electric field component when choosing the bisector aspolar axis. The smaller superimposed arrowsEd

(ind) andEq(ind) are the electric

fields due to the induced parts of the dipoles and quadrupoles. Even thoughthe direction of the individual electric field components deviates somewhatfrom the direction of the bisector, the total field~and, therefore, the induceddipole moment! are to a good approximation pointing along the bisector.

TABLE III. Calculated values for the molecular dipole moment of water inice I h at the various levels of the multipolar expansion and increasing levelsof response to multipolar fields within the self-consistent induction scheme.All the dipole moments are expressed in Debye.

Expansion a a, A a, A, C a, A, C, b

P , Q 2.94 2.89 2.98 2.97P , Q , O 3.00 2.94 3.04 3.04P , Q, O , H 3.05 2.99 3.10 3.09

4549J. Chem. Phys., Vol. 109, No. 11, 15 September 1998 Batista, Xantheas, and Jonsson

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DP ~1!5aE5a~Ed1Eh!, ~10!

wherea is the isotropic molecular polarizability. A value ofa51.59 Å3 was used in their calculations~induced quadru-poles or hyperpolarizabilities were not taken into account!.An interesting simplifying assumption was made by Coulsonand Eisenberg that enabled them to analytically sum up theiterative induction calculation to infinite order. The total di-pole fieldEd at a molecule was assumed to be parallel to theunpolarized dipole moment (P (0)) of the molecule

Ed5cP ~0!, ~11!

c being a constant of proportionality. The first-order correc-tion to the dipole then becomes

DP ~1!5a~cP ~0!1Eh!5acS P ~0!1Eh

c D . ~12!

Assuming that all the molecules in the bulk are equivalent,the same dipole moment is induced in all the molecules. Theinduced dipole moment in all the surrounding molecules thengenerates an extra dipole fieldEd

(1) proportional to the first-order correction of the dipole moment

Ed~1!5cDP ~1!, ~13!

that induces a second-order correction to the dipole moment

DP ~2!5aEd~1!5~ac!2S P ~0!1

Eh

c D . ~14!

The nth order correction to the dipole moment is

DP ~n!5aEd~n21!5acDP ~n21!5~ac!nS P ~0!1

Eh

c D , ~15!

and the total induced dipole moment is

DP 5 (n51

`

DP ~n!5S P ~0!1Eh

c D (n51

`

~ac!n, ~16!

DP 5ac

12ac S P ~0!1Eh

c D5a

12ac~Ed1Eh!, ~17!

DP 5uP ~0!uaE

uP ~0!u2auEdu. ~18!

Therefore, by making a simplifying assumption about thedirection of the induced dipole, Coulson and Eisenberg wereable to calculate analytically the total induced dipole mo-ment in a bulk molecule in terms of the multipoles of theisolated molecule and the dipole polarizability.

After averaging over the possible orientations of eachmolecule in a configuration of molecules satisfying the icerules, Coulson and Eisenberg predicted a total dipole mo-ment of 2.6 D for the water molecule in iceI h . This numberrepresents an average over different arrangements of protons,with some orientations resulting in molecular dipole mo-ments as low as 1.9 D and others as high as 3.1 D. This resultis significantly smaller than the value 2.96 D that we obtainusing a more elaborate scheme and the question naturallyarises what the important difference is.

We repeated our numerical induction calculation usingthe same input parameters as Coulson and Eisenberg~multi-

poles and molecular polarizability! and neglecting dipole-quadrupole polarizability, hyperpolarizability, and the hexa-decapole moment of the molecule. Using a system sizesimilar to theirs that includes 121 molecules and averagingover 50 configurations, we obtained a net dipole moment of2.6560.08 D in good agreement with the result of Coulsonand Eisenberg. This shows that the simplifying assumptionmade in their calculation, namely that the total dipole field isparallel to the permanent dipole moment of the molecule, Eq.~11!, is in fact good enough. This is also evident from theanalysis of our results, presented in Sec. III, which show thatalthough the field does not lie exactly along the bisector ofthe molecule, the deviations are effectively small.

We then repeated the calculation, now including the ex-perimentally determined dipole and quadrupole momentsand dipole polarizability, and the MP2 calculated octopoleand hexadecapole moments as given in Tables I and II. Theresulting net dipole moment was 3.0460.04 D. This showsthat the reason for the low value obtained by Coulson andEisenberg is a result of the numerical values of the multipolemoments they used as input in their model.

V. CONCLUSIONS

We have used a self-consistent induction model to studythe electric field in ice and found that the best estimate forthe dipole moment of a water molecule in iceI h is 3.09 D.This value represents a 67% increase over the dipole momentof an isolated water molecule. Our estimate is significantlyhigher than the one reported earlier by Coulson and Eisen-berg who also used a similar but more approximate inductionmodel. The main reason for the difference between our re-sults and the earlier estimates lies in the numerical values forthe quadrupole moment of the isolated molecule. Coulsonand Eisenberg relied on the best estimates of multipole mo-ments from first principle calculations at the time which in-cluded a very small basis set. We have used the experimentalvalues for the dipole and quadrupole moments~the latterobtained after Coulson and Eisenberg carried out their calcu-lation! and the results of accurateab initio calculations forthe higher octopole and hexadecapole moments. When thesame set of multipoles are used as input, the method of Coul-son and Eisenberg gave very similar results as our more de-tailed model, indicating that the approximations used byCoulson and Eisenberg to simplify the induction calculationsare quite valid.

The multipolar expansion for the electric field in the icecrystal was carried out to what seems to be reasonably goodconvergence. As an indicator of the effect of various terms inthe expansion, we have focused on the net dipole and quad-rupole moment per molecule obtained in the self-consistentcalculation. The inclusion of dipole–quadrupole andquadrupole–quadrupole polarizability has small but signifi-cant effect on the dipoles; it increases the dipole moment ofthe water molecules by 0.04 D. The polarizability has a moredramatic effect on the quadrupoles increasing the magnitudeby 14% to 25% depending on the component. It is importantto go beyond the quadrupole and include also the~fixed!octopole moment of the molecules. This increases the net

4550 J. Chem. Phys., Vol. 109, No. 11, 15 September 1998 Batista, Xantheas, and Jonsson

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dipole moment by 0.07 D. The hexadecapole moment, how-ever, has a small effect on the dipole moment~increasing itby 0.05 D!.

A remarkable feature in the self-consistent calculationsis how little effect distant neighbors have on the field at agiven molecule. It is sufficient to include the electric fieldfrom neighbors within a 7 Å radius in order to get the con-verged, large system value of the induced dipole moment to99%~see Fig. 1! as was already made by Coulson and Eisen-berg. This is, of course, not true when the molecules aremodeled by point charges, as is most frequently done insimulation studies of water and water solutions.

ACKNOWLEDGMENTS

We wish to thank Geert-Jan Kroes for the computer codeto generate configurations of iceI h with proton disorder, andProfessor A. J. Stone for helpful suggestions. Part of thiswork was performed under the auspices of the Division ofChemical Sciences, Office of Basic Energy Sciences, U.S.Department of Energy under Contract DE-AC06-76RLO1830 with Battelle Memorial Institute, which operates thePacific Northwest National Laboratory. Computer resourceswere provided by the Division of Chemical Sciences and bythe Scientific Computing Staff, Office of Energy Research, atthe National Energy Research Supercomputer Center~Berke-ley, CA!.

APPENDIX: DEFINITIONS OF MULTIPOLE MOMENTS

Several different definitions of the multipole moments ofa charge density are in use in the literature. The differenceamong them are signs, constants of proportionality and somebeing linear combinations of the others. In this appendix wegive the definitions of the multipoles in Cartesian coordi-nates used in Table I.

The total charge density of the moleculer~r ! is

r~r !5re~r !1(i

qid~r2r ~ i !!, ~A1!

where re is the electronic charge density,qi is the ioniccharge of thei th ion ~one proton charge for each of thehydrogen atoms and eight for the oxygen!, r ( i ) is the positionof the i th nucleus,d~r ! is the Dirac delta function, and thesum is over all the nuclear charges. The molecular multipolemoments are obtained by integrating over the charge densityof the molecule. The electric dipole moment is defined as

P i5E d3rr~r !r i . ~A2!

The quadrupole moment is defined as

Q i j 51

2E d3rr~r !~3r i r j2r 2d i j !, ~A3!

whered i j is the Kroenecker delta. We chose to measurerfrom the center of mass of the molecule. The octopole mo-ments are defined as

O i jk51

3!E d3rr~r !@15r i r j r k23r 2~r id jk1r jdki1r kd i j !#,

~A4!

and the components of the hexadecapole as

H i jkl 51

4! E d3rr~r !@105r i r j r kr l215r 2~r i r jdkl

1r i r kd j l 1r i r ld jk1r j r kd i l 1r j r ld ik1r kr ld i j !

13r 4~d i j dkl1d ikd j l 1d i l d jk!#. ~A5!

The electric field can be obtained from the gradient of Eq.~24!.

U5P i r i

r 3 1Q i j r i r j

r 5 1O i jk r i r j r k

r 7 1H i jkl r i r j r kr l

r 9 . ~A6!

With these definitions, the electrostatic potential gener-ated by a molecule is

E~r !5Ed~r !1Eq~r !1E0~r !1Eh~r !, ~A7!

where the dipole field is

Ed~r !n53P i r i

r 5 r n2P n

r 3 , ~A8!

the quadrupole field is

Eq~r !n55Q i j r i r j

r 7 r n22Q inr i

r 5 , ~A9!

the octopole field is

E0~r !n57O i jk r i r j r k

r 9 r n23O ni j r i r j

r 7 , ~A10!

and the hexadecapole field is

Eh~r !n59H i jkl r i r j r kr l

r 11 r n24Hni jkr i r j r k

r 9 . ~A11!

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4551J. Chem. Phys., Vol. 109, No. 11, 15 September 1998 Batista, Xantheas, and Jonsson

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