LETTERS

Multistate Empirical Valence Bond Model for Proton Transport in Water

Udo W. Schmitt and Gregory A. Voth*Department of Chemistry and Henry Eyring Center for Theoretical Chemistry,UniVersity of Utah, Salt Lake City, Utah 84112

ReceiVed: April 8, 1998; In Final Form: May 22, 1998

A multistate empirical valence bond (MS-EVB) model for describing proton transport in aqueous systems ispresented. In this approach the electrostatic interaction of the solvent water molecules with an exchangecharge distribution is explicitly included in the off-diagonal elements of the EVB Hamiltonian. The MS-EVB model is parametrized to reproduce geometrical and energetic quantities of selected H3O+‚(H2O)N clusters.The obtained geometries, formation energies, and energy barriers are in excellent agreement with resultsfrom high-level ab initio calculations. It is furthermore applied in a classical molecular dynamics simulationof condensed-phase water with an excess proton in order to estimate the proton-transfer rate.

1. Introduction

Proton-transfer (PT) reactions represent a unique class ofprocesses of paramount importance in many different fields inphysics and chemistry and for that reason have achievedextensive experimental and theoretical attention over severaldecades.1-5

From the theoretical point of view, the accurate modeling ofPT remains a challenge for mainly two reasons. First, owingto the small mass of the transferring particle, quantum effectssuch as tunneling or zero-point energy can have a majorinfluence on the microscopic mechanism and calculated ob-servables. The second challenge in the modeling of PT is tofind a reliable and accurate description of the potential energysurface (PES) describing the ongoing formation and breakingof the participating hydrogen bonds. Since in PT reactions thedifferences in energy, e.g., reaction energy or barrier heights,are only about a few kcal/mol and the rate constants are stronglysensitive to it, the underlying PES needs to be represented withsufficient precision in order to calculate various quantitiesreliably.

One possible route to model the reactive PES is given by theempirical valence bond (EVB) model,6 which has been suc-cessfully used in numerical studies of various charge-transfer

reactions in solution or enzymes,6 proton transfer in gas-7 andcondensed-phase systems,8 and to construct the PES in reactivescattering calculations.9 Lobaugh and Voth8 first introduced atwo-state EVB model in order to describe the ground-state PESfor the migration of a proton between two water molecules inthe liquid state. While this approach was able to model thedynamical delocalization of the proton between two watermolecules and, e.g., reproduced the IR absorption spectrumcharacteristics for strong acids in aqueous solutions, it does notallow for explicit proton hopping between several watermolecules. There has recently been an effort10 to improve uponthe Lobaugh-Voth two-state EVB model through a higher levelof ab initio calculations as the basis of the empirical modeling.Since these calculations do not include the effects of solutepolarization, it is unclear whether they improve upon the originalmodel.

Very recently, Vuilleumier and Borgis extended the Lo-baugh-Voth approach by introducing a nonadditive valence-bond force field11,12to model the migration of an excess protonin water clusters and bulk water, which goes beyond a two-state EVB description. The model has been used to calculatevibrational spectroscopic properties of small protonated waterclusters and the transfer of an excess proton in bulk-phase water,

© Copyright 1998 by the American Chemical Society VOLUME 102, NUMBER 29, JULY 16, 1998

S1089-5647(98)01813-6 CCC: $15.00 © 1998 American Chemical SocietyPublished on Web 06/29/1998

but it differs in an important way from the model describedherein as will be described below.

In this Letter we present a methodology to describe the PESfor PT reactions based on a multistate empirical valence bond(MS-EVB) picture, which allows one to model the migrationof a single proton in aqueous environments, e.g., protonatedwater clusters, proton wires, and aqueous bulk-phase solutions.This modeling is accomplished in a numerically efficient butaccurate fashion, and so it enables the dynamical treatment oflarge aqueous systems and provides a basis for includingquantum effects stemming from the nuclei. This approach canbe regarded as a multistate extension of the two-state Lobaughand Voth8 EVB model in order to allow for unrestricted PTalong three-dimensional hydrogen-bonded networks present inaqueous systems.

2. Multistate EVB Approach

The EVB methodology, pioneered by Mulliken,13 allows ingeneral the construction of potential energy surfaces (PES) forspecies undergoing chemical reactions in an accurate andnumerically efficient fashion. It is based on the assumptionthat the ground-state PES of a reactive chemical system can beobtained from the ground-state energy of an effective EVBHamiltonian

where the matrix elementshij(Q) are a function of the set ofnuclear degrees of freedomQ of the molecular system underconsideration only. The diagonal elementshii(Q) are mostlyassumed to have a simple functional dependence on the nucleardegrees of freedom and so can be represented for example byestablished molecular mechanics force fields.14,15 The functionaldependence of the off-diagonal elements ofQ has to be chosento reproduce the PES in the reaction region where the ap-proximations employed in the standard force fields break down.The PES is obtained by solving the eigenvalue problem

and by calculating the expectation value ofH(Q)

with respect to its ground-state vectorc0. Owing to itsunderlying simplicity and resulting numerical efficiency, theEVB methodology has been used to model various chemicalreactions in the gas and condensed phase, and it seems to be avaluable tool for describing reactive PES in condensed systems,where the large number of atoms involved in the chemicallyreactive event prohibit a high-level electronic structure calcula-tion.

Lobaugh and Voth8 employed the EVB method to model theground-state PES for the migration of a proton between twowater molecules by using a linear combination of two EVBstates, where the excess proton is bound to one distinct watermolecule in each state, respectively. The diagonal elementswere expressed as the sum of intramolecular contributions ofthe distinct molecular species (H3O+ and H2O) plus anintermolecular nonbonded interaction term. The off-diagonalelements were taken to be a function of a specified subset ofthe nuclear coordinates, i.e., the distance between the twooxygens in the H5O2

+ dimer.

To properly treat proton migration along a three-dimensionalhydrogen-bond network in aqueous solutions, it is necessary toextend the aforementioned two-level EVB approach to ageneralized MS-EVB model. This results from the equivalenceof all three hydrogens atoms bonded to the hydronium oxygen,which should be allowed to migrate to the water moleculesconstituting the first solvation shell. For example, the statesnecessary to correctly treat the proton migration in the Eigencation H9O4

+ are shown in Figure 1. While the centralhydronium ion (state|3⟩) can act as a proton donor to threeH2O molecules, there are in total four states to be included. Ingeneral, a multistate EVB description results in anN × N EVBHamiltonian (eq 2), where again the expectation value ofHwith respect to the eigenvector corresponding with the lowesteigenvalue ofH gives the electronic ground-state PES.

While the MS-EVB approach presented in this Letter alsoextends the Lobaugh-Voth two-state EVB description througha MS-EVB model, the implementation and parametrization ofthe off-diagonal matrix elementshij(Q) entering into the MS-EVB Hamiltonian differs considerably and importantly fromthe approach presented in ref 11 in that they depend explicitlyon the solvent configurations. To our best knowledge, theapplications of the EVB approach so far have neglected thisdependence, which is a natural consequence of the quantum-mechanical nature of the VB states (see below).

As pointed out by Kim and Hynes,16 a molecular systemdescribed by a VB electronic Hamiltonian that is immersed ina polarizable solvent interacts with the solvent molecules in twoways. The first is due to the charge distributionFii(QS)corresponding to the diagonal VB states|i⟩ of the solute, i.e.

whereQS is the set of nuclear degrees of freedom of the solutecomplex only and

is the coordinate representation of the VB state|i⟩. The secondinteraction comes from an exchange (or transition) chargedistributionFij(QS), i.e.,

H(Q) ) ∑i,j

|i⟩hij(Q)⟨j| (1)

H(Q)c ) E(Q)c (2)

E(Q) ) ∑∑i,j

ci0cj

0hij(Q) (3)

Figure 1. Schematic picture of the four valence-bond states necessaryto describe the H9O4

+ complex in an MS-EVB approach.

Fii(QS) ) -eΨi*(QS)‚Ψi(QS) (4)

Ψi(QS) ) ⟨QS|i⟩ (5)

5548 J. Phys. Chem. B, Vol. 102, No. 29, 1998 Letters

resulting from the quantum-mechanical wave character of theelectronic degrees of freedom of the solute. When one treatsthe electronic degrees of freedom as being described by a purelyclassical charge distribution, the transition charge distributionFij and the corresponding exchange electric field

are not present,16 and this is not consistent with the quantummechanics of the system.

In the MS-EVB model presented in this Letter, the interactionof the transition charge distributionFij with the solvent isexplicitly taken into account in describing the off-diagonalelements in the EVB Hamiltonian (eq 2). This adds additionaldegrees of freedom to the parameter set used in modeling thefunctional form of the individual matrix elementshij in eq 2.The additional terms in the Coulomb interaction between asolute moleculek and a solvent moleculel

can perhaps be seen more explicitly from the expectation valueof the charge distribution operator with respect to the MS-EVBground state:

The second term on the right-hand side of eq 9, whichexclusively results from the exchange charge distributions, canbe modeled within reasonable physical limits in order to allowfor a more accurate description of the exact PES within an EVBdescription. For example, in bulk-phase PT studies a properlychosen exchange charge distribution serves to compensate forthe strong localization tendency of a distinct EVB state owingto the solvent orientational relaxation, the latter normallyresulting in a decrease of the PT rate compared to the gasphase.11,17

For a specified adiabatic or diabatic electronic state,Fij canin general be calculated using established electronic structuremethodologies to obtain partial charges fixed on atomiccenters.16,18 However, no attempt was made to do so in thepresent study owing to the implicit treatment of the electronicdegrees of freedom invoked in the EVB framework. Thetransition charge distribution, approximated as a sum of pointcharges localized at the solute’s atomic centers (interaction sitemodel), was modeled as

and then fitted self-consistently with all other parametersentering the MS-EVB Hamiltonian in order to reproduce theab initio gas-phase PES of the H5O2

+ dimer and the geometriesof the global minima for selected H3O+‚(H2O)N clusters (N )1, ..., 3).

While a detailed description of the functional forms employedand parameters used in our parametrization will be given inforthcoming publications,17 the potential energy expressionsemployed for the matrix elementshij in eq 2 are briefly

summarized in the following and listed in Table 1. It shouldbe noted that this is a preliminary study and the parameter setwill evolve as the model is refined and new effects (e.g., solventpolarization) are included.

As any MS-EVB state can be regarded as being composedof one hydronium plusNH2O water molecules (Figure 2), thediagonal element

can be modeled as a sum of intramolecular contributionsstemming from the individual species H3O+ and H2O, respec-tively, and two intermolecular terms describing the nonbondedinteraction between all molecular species as approximated bystandard electrostatic and Lennard-Jones type expressions. Theterm VH3O+

intra is expressed as

Fij(QS) ) -eΨi*(QS)‚Ψj(QS) (6)

E(Q)S ) ∇∫dQ′SFij(Q′S)

|Q′S - QS|(7)

VCkl ) ∫∫dQkdQl

Fk(Qk)Fl(Ql)

|Qk - Ql|(8)

Fk(Qk) ) ⟨Fk(Qk)⟩ ) ∑∑i

ci2 Fii

k(Qk) + 2 ∑∑i<j

cicj Fijk(Qk)

(9)

Fij(QS) ) ∑k

Nc

qijk δ(QS - QS

k) (10)

TABLE 1: Parameters Used in the MS-EVB Model

DOH 266.3 kcal mol-1 rOO0 1.92 Å

aOH 1.285 Å-1 â 4.50 Å-1

ROH0 0.98 Å ROO

0 3.14 ÅkR 73.27 kcal mol-1 rad-2 DOO 2.875 ÅR0 116.0 deg P 0.27εO′ 0.155 kcal mol-1 k 11.5 Å-2

σO′ 3.164 Å γ 1.85 Å-2

qO′ -0.5e ú 1.5 rad-2

qH′ +0.5e ω0 174.3°B 2.295 kcal mol-1 Vconst

ij -32.925 kcal mol-1

b 3.50 Å-1 qOexch -0.1334e

dOO0 2.50 Å qH

exch 0.05336eR 15.0 Å

Figure 2. Illustration of the H5O2+, H7O3

+, and H9O+4 clusters in their

global PES minima. The numbers given are the O-O distance andO-H distances of the central oxygen atom, respectively, obtained withinthe MS-EVB approach. The bond lengths in parentheses are distancesobtained by high-level ab initio (MP2/6-311G*) geometry optimization.The formation energies in parentheses are taken from refs 19 and 20,respectively.

hii ) VH3O+intra + ∑

k

NH2O

VH2Ointra,k + ∑

k

NH2O

VH3O+,H2Ointer,k + ∑

k<k′

NH2O

VH2Ointer,kk′ (11)

Letters J. Phys. Chem. B, Vol. 102, No. 29, 19985549

which is a sum of O-H stretch terms modeled by Morsefunctions and harmonic H-O-H angle contributions, while forVH2O

intra and VH2Ointer,kk′ a standard, flexible TIP3P14 force field is

assumed. The intermolecular interactionVinter of the H3O+ andH2O species with each other are described by a standard,pairwise 12-6-1 methodology plus an additional repulsive termVrep(ROO) between the hydronium and water oxygens

The coupling elements between the diabatic VB states areassumed to be of the form

with Vconstij being a constant coupling term. Here,Vex

ij repre-sents the electrostatic interaction (like the second term in eq13) of the exchange charge distributionFij (eq 10) of a distinctH5O2

+ dimer spanned by VB states|i⟩ and |j⟩ with theremainingNH2O -1 water molecules (so being a function ofallcoordinates in the system). The exchange charges used in thisinteraction are given by the last entries in Table 1. Thefunctional form ofA(ROO,q, ω) is taken as a product of functionssuch that

are chosen to damp out the coupling in case of unfavorablegeometries. HereROO represents the O-O distance,q theasymmetric stretch coordinate andω the bend angle forO-H-O in the distinct H5O2

+ dimer spanned by states|i⟩ and|j⟩, respectively.

3. Numerical Results

Figure 2 parts a-c depict the stable structures correspondingto the global minimum on the PES and the formation energyfor the H5O2

+, H7O3+, and H9O4

+ clusters, respectively. Theenergy values in parentheses following the given MS-EVBformation energy are ab initio results performed at an SCF/4-31G19 and MP2/pTVZ+20 level, respectively. The geometrieshave been obtained by simulated annealing MD runs of 5 ps.

As can be seen in Figure 2a, the O-O (2.4 Å) and O-H(1.2 Å) bond lengths of H5O2

+ are in quantitative agreementwith the high-level ab initio results at a post Hartree-Focklevel.20 The formation energy of-33.7 kcal/mol is in excellentagreement with the reported ab initio results.20 Although the

ab initio calculation reveals a geometry with aC2 symmetryand the MS-EVB model predictsCSsymmetry, it has to be notedthat the energy difference∆E ) ECS

min - EC2

min is found to be onthe order of 0.1 kcal/mol, so internal conversion between thistwo structures is expected to occur even at very low tempera-tures.

On solvating the H5O2+ dimer with an additional H2O

molecule, two important effects can be observed in the stablestructure (Figure 2b) of the resulting H7O3

+ cation. In the MS-EVB minimization the O-O distance from the central oxygenand the two hydrogen-bonded water molecules is elongated to2.52 Å, as the O-H distance is shortened to 1.07 Å comparedto the H5O2

+ cluster. An MP2/6-311G* calculation performedto obtain the stable structure reveals 2.51 and 1.02 Å for theO-O and O-H bond lengths, respectively, again in almostperfect agreement of the MS-EVB results. The MS-EVBformation energy for the H7O3

+ cluster is given by-57.0 kcal/mol, and there is no reliable ab initio result with which tocompare. Furthermore, the structure of the H7O3

+ speciesexhibits the correctC2V symmetry.

After two water molecules are added to the H5O2+ dimer, the

computed global minimum of the underlying PES reveals thegeometry given in Figure 2c. The supermolecule has aC3

symmetry, with an oxygen-oxygen bond length of 2.57 Å anda oxygen-hydrogen bond length of 1.04 Å. The ab initiocalculation of ref 20 revealed the O-O distance to be 2.57 Åand a O-H bond length of 1.01 Å. Again, the O-O distancefrom the central oxygen is reduced on micro-solvation with oneadditional water molecule, and all three strongly hydrogen-bonded water molecules turn out to be described equivalentlywithin the MS-EVB approach. The MS-EVB formation energyis given by-79.5 kcal/mol, being close to the reported high-level ab initio results20 (-77.4 kcal/mol).

The general tendency toward a stabilization of the H3O+

species observed on successive solvation of the H5O2+ dimer

with water molecules is reproduced within the MS-EVB model.This effect is related to the nonadditive character of the H3O+-H2O interaction potential, which stems from the strong mixingof the electronic degrees of freedom when both species approacheach other. Althoughno electronic wave function needs to becalculated in the MS-EVB model, the effect is quantitativelyreproduced within this methodology.

As stated previously, the strong dependence of the barrierheight for the proton exchange on the O-O distance is acharacteristic feature of hydrogen-bonded systems and shouldbe accurately reproduced by any simplified model used indescribing the PES. For the H5O2

+ dimer, there are a largenumber of ab initio calculations reported in the literaturefocusing on this issue (see, for example, ref 21 and referencestherein), which basically reveal that the barrier in H5O2

+ in thegas phase is very low to nonexistent for O-O separations lessthan 2.50 Å. This barrier rises exponentially for larger O-Odistances. The barrier heights are sensitively influenced by thebasis set size and by electronic correlation effects,22,23 and inorder to achieve accurate results large basis sets allowing forpronounced electronic structure reorganization and post-Har-tree-Fock methods such as second-order perturbation theory(MP2) or higher-order coupled cluster approaches (CCSD(T))must be employed.

In this study, the barriers computed within the MS-EVBmodel were compared with the results obtained by a coupled-cluster calculation including up to triple excitations using a large,correlation-consistent, valence triple-ú (cc-pVTZ) Gaussian basisset reported in Ref 22. The barrier heights were calculated at

VH3O+intra ) ∑

j

3

DOH[1 - e-aOH(ROHj -ROH

0 )]2 +1

2∑

i

3

kR(Ri - R0)2 (12)

VH3O+,H2Ointer,k ) 4ε[( σ

ROOk)12

- ( σROOk

)6] +

∑n,m

qmqn

Rmn+ Vrep(ROOk

) (13)

Vrep(ROOk) ) B(1 - tanh[b(ROOk

- dOO0 )]) (14)

hij ) (Vconstij + Vex

ij )‚A(ROO, q, ω) (15)

A(ROO, q, ω) ) f(ROO)‚g(q)‚h(ω) (16)

f(ROO) ) (1 + P exp(-k(ROO- DOO)2))‚

(12(1 - tanh[â(ROO - ROO0 )]) + exp[-R(ROO - rOO

0 )]) (17)

g(q) ) exp(-γq2) (18)

h(ω) ) exp(-ú(ω - ω0)2) (19)

5550 J. Phys. Chem. B, Vol. 102, No. 29, 1998 Letters

fixed O-O separation between 2.5 and 2.8 Å, where for eachfixed O-O distance the difference in energy was obtained fromthe geometry with all four hydrogens allowed to relax and theexchanging proton fixed to the geometric center of the O-H-Ocomplex. The MS-EVB approach reproduces the high-level abinitio results almost perfectly with deviations on a 0.1 kcal/mol scale.

To determine the applicability of the MS-EVB approach forthe condensed phase, it was employed in a preliminary classicalmolecular dynamics (MD) study to model the proton-transferdynamics in aqueous solution consisting of 255 water moleculesplus one excess proton with periodic boundary conditionsintroduced to mimic the infinite liquid water sample. Thesimulation was performed with an average kinetic temperatureof about 300 K in the microcanonical ensemble (NVE). TheEVB states included were chosen by a geometric criterion thatchecks for all chemically plausible (so energetically accessible)EVB states, and so the number of statesNEVB in the MS-EVBapproach was a dynamical quantity throughout the simulation,typically ranging between 6 and 10. The dominant state waschosen to be the one having the largest amplitudeci

2 in theground-state EVB eigenvector, and can be regarded as theoxygen atom showing the most hydronium-like character of allthe oxygens.

Along the trajectory the proton is either localized on a distinctwater molecule or is oscillating between two water molecules(corresponding to successive switching of the oxygen label) forsome time interval. For a simulation time of 100 ps the totalnumber of hopping events turns out to be about 25, where theproton remains at the new oxygen for at least 0.5 ps, therebysuggesting a hopping rate of approximately 0.25 ps-1. Thisrate is about a factor of 3 too low compared to the experimentalresult of 0.7 ps-1.24 It should be noted that a hopping rate ofaround 0.1 ps-1 was obtained in the most recent work ofVuilleumier and Borgis12 in which the solvent dependence ofthe off-diagonal MS-EVB matrix elements was not treated asin the present work. The neglect of this dependence leads totoo much trapping of the hydronium MS-EVB statesan effectthat we verified by by setting the termVex

ij equal to zero in eq15. One must also bear in mind that our reported simulationswere performed within a classical framework, so quantizationof the nuclei will enhance the hopping rate through zero-pointmotion and tunneling. In fact, preliminary quantum mechanicalcentroid molecular dynamics (CMD)25 simulations using ourMS-EVB model have found a hopping rate of around 0.5 ps-1.More extensive CMD studies of the bulk phase and other protontransport systems are currently underway in our group.17

Note Added in Proof: Very recently, Vuilleumier and Borgis(J. Phys. Chem.1998, 102, 4261) have presented work in which

their model of refs 11 and 12 is reparameterized to fit a nuclearquantized version of the H5O2

+ dimer. Classical trajectoriesare then run with this reparameterized model, evidently to mimica CMD simulation (ref 25). The theoretical basis for such a“prequantization” procedure seems unclear, as do its underlyingerrors. Furthermore, this model still does not contain the solventdependence of the off-diagonal EVB matrix elements which isshown to be important in the present work.

Acknowledgment. This work was supported by the NationalInstitutes of Health (GM-53148) and the National ScienceFoundation (CHE-9712884). U.W.S. gratefully acknowledgessupport by the Deutsche Forschungsgemeinschaft. We thankDan Berard and Amir Karger for carefully reading the manu-script.

References and Notes

(1) Bell, P. The Proton in Chemistry; Chapman and Hall: London,1960.

(2) Schuster, P.; Zundel, G.; Sandorfy, C.The Hydrogen BondsRecentDeVelopments in Theory and Experiment; North-Holland Publishing Co.:Amsterdam, 1975.

(3) Jeffrey, G. A.; Saenger, W.Hydrogen Bonding in BiologicalStructures; Springer: Berlin, 1994.

(4) Bountis, T.Proton Transfer in Hydrogen-Bonded Systems; PlenumPress: New York, 1992.

(5) Muller, A.; Ratajczak, H.; Junge, W.; Diemann, E.Electron andProton Transfer in Chemistry and Biology, Elsevier: Amsterdam, 1992.

(6) Åqvist, J.; Warshel, A.Chem. ReV. 1993, 93, 2523.(7) Hinsen, K.; Roux, B.J. Chem. Phys.1997, 106, 3567.(8) Lobaugh, J.; Voth, G. A.J. Chem. Phys.1996, 104, 2056.(9) Chang, Y.-T.; Miller, W. H.J. Phys. Chem.1990, 94, 5884.

(10) Sagnella, D. E.; Tuckerman, M. E.J. Chem. Phys.1998, 108, 2073.(11) Vuilleumier, R.; Borgis, D.J. Mol. Struct.1998, 436, 555.(12) Vuilleumier, R.; Borgis, D.Chem. Phys Lett.1998, 284, 71.(13) Mulliken, R. S.J. Chem. Phys.1964, 61, 20.(14) Brooks, B. R.; Bruccoleri, R. E.; Olafson, B. D.; States, D. J.;

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Cheatham, T. E.; Ferguson, D. M.; Seibel, G. L.; Singh, U. C.; Weiner, P.K.; Kollman, P. A.AMBER 4.1; University of California: San Francisco,1995.

(16) Kim, H. J.; Hynes, J. T.J. Phys. Chem.1990, 94, 2736; (b)J.Chem. Phys.1990, 93, 5194; (c)1990, 93, 5211.

(17) Schmitt, U. W.; Voth, G. A., in preparation.(18) Ando, K.J. Chem. Phys.1997, 107, 4585.(19) Newton, M. D.J. Chem. Phys.1977, 67, 5535.(20) Ojamae, L.; Shavitt, I.; Singer, S. J.Int. J. Quantum Chem, Quantum

Chem. Symp.1995, 29, 657.(21) Xie, Y.; Remington, R. B.; Schaefer, H. F. III.J. Chem. Phys.1994,

101, 4878.(22) Pavese, M.; Chawla, S.; Lu, D.; Lobaugh, J.; Voth, G. A.J. Chem.

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Letters J. Phys. Chem. B, Vol. 102, No. 29, 19985551