A general methodology for quantum modeling of free-energy profile ofreactions in solution: An application to the Menshutkin NH3+CH3Clreaction in waterThanh N. Truong, Thanh-Thai T. Truong, and Eugene V. Stefanovich Citation: J. Chem. Phys. 107, 1881 (1997); doi: 10.1063/1.474538 View online: http://dx.doi.org/10.1063/1.474538 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v107/i6 Published by the American Institute of Physics. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

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A general methodology for quantum modeling of free-energy profileof reactions in solution: An application to the Menshutkin NH 31CH3Clreaction in water

Thanh N. Truong,a) Thanh-Thai T. Truong, and Eugene V. StefanovichDepartment of Chemistry and Henry Eyring Center for Theoretical Chemistry, University of Utah, SaltLake City, Utah 84112

~Received 4 March 1997; accepted 6 May 1997!

We present a general methodology for calculating free-energy profile of reaction in solution usingquantum mechanical methods coupled with the dielectric continuum solvation approach.Particularly, the generalized conductorlike screening model~GCOSMO! was employed in thisstudy, though any continuum model with existing free-energy derivatives could also be used.Free-energy profile is defined as the steepest descent path from the transition state to the reactantand product channels on the liquid-phase free-energy surface. Application of this methodology tocalculate the free-energy profile of the Menshutkin NH31CH3Cl reaction in water is discussed. Theefficiency of the GCOSMO method allows characterization of stationary points and determinationof reaction paths to be carried out at less than 20% additional computational cost compared togas-phase calculations. Excellent agreement between the present results and previous Monte Carlosimulations using a combined quantum mechanical/molecular mechanics~QM/MM ! potentialconfirms the accuracy and usefulness of the GCOSMO model. ©1997 American Institute ofPhysics.@S0021-9606~97!52130-1#

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I. INTRODUCTION

For the past half century, quantum chemistry has msignificant progress in prediction properties of gas-phasecesses. Theoretical efforts recently have been turning towsolution chemistry. Despite some progress has been madeveloping predictive models for equilibrium and spectscopic properties of molecules in solution, quantum moding of free-energy profile in solution still remainschallenge.1–3 Free-energy profiles are important for elucidaing mechanisms of reactions in solution and addressing tdynamics. The challenge arises from the requirementsadequate theoretical models must have accurate descriof both the bond-forming and -breaking processes andsolvent–solute interactions.

First, proper description of the bond-forming an-breaking processes requires an accurate correlated levab initio quantum mechanical theory. At the present timeis computationally unfeasible to perform fullab initio quan-tum simulations for reactions in solution due to the larnumber of solvent molecules required. Although rapprogress is being made for quantum~electronic degrees ofreedom only! simulations of equilibrium properties ocondensed-phase systems with the use of Car–Parrinapproach,4–6 in most theoretical studies so far, approximtions were made to reduce the computational demandcalculating the solvent–solute interactions. These stutreat the solvent either explicitly or implicitly.

A. Explicit treatments of solvent–solute interaction

Within this approach, there are two general methodogies. The supermolecule model7,8 treats a small number o

a!Author to whom correspondence should be addressed.

J. Chem. Phys. 107 (6), 8 August 1997 0021-9606/97/107(6)/18

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first-solvation-shell solvent molecules explicitly at the samquantum mechanical level as the solute. This model offestraightforward and accurate way to account for the localiportion of the solvent–solute interaction. However, it canninclude the long-range electrostatic interactions with the bsolvent. Another common methodology is to constructapproximate solvent–solute interaction potential either usclassical molecular mechanics force fields,9 empirical va-lence bond method,10 combined quantum mechanicamolecular mechanics~QM/MM ! method,11–14 or recent ef-fective fragment method.15 Then, for calculating reactionprofiles in solution, one performs classical free-energy simlations to evaluate the potential of mean force along alected reaction coordinate.

The advantage of approaches with explicit solvent is tthey provide detailed solvation structure enabling one to ecidate specific roles of solvent in reaction mechanisms. Tmain difficulty of these models is the definition of the reation coordinate. Due to the large number of solvent degrof freedom there is no unique way to define a reaction codinate. As a result, this approach relies on the gas-phasesimple user-defined reaction coordinate. As a result, parpation of solvent motions in the reaction coordinate isnored. Such practice works reasonably well for many simreactions. For complex reactions where competing reacpaths exist and solvent effects change the topology offree-energy surface, one can in principle, examine all psible free-energy paths but this would be too costly. Thare numerous examples of such complex cases. For instasolvent effects change the prefered conformation of the trsition state and the diastereofacial selectivity of the DieAlder reaction between cyclopentadiene and~-!-menthyl-acrylate.16–18 Solvent can also change the reaction mec

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1882 Truong, Truong, and Stefanovich: Free energy profile of reactions in solution

nism as seen in the ketene-imine cycloaddition in whichtwo-step mechanism is prefered in solution while the ccerted mechanism was predicted for the gas phase.19 Anotherexample is a significant shift of the transition state towathe reactant channel in the Menshutkin NH31CH3Cl reactionin aqueous solution.20

In general, the combined QM/MM approach in conjuntion with molecular dynamics or Monte Carlo free-enersimulations would be the best way to provide accuratedetailed picture of reactions in solution. However, determnation of the potential of mean force requires a large numof free-energy simulations, typically around 30–60 ruThis is quite time consuming and computationally expensSuch an approach is useful to gain detailed fundameknowledge for selected well-known reactions. It is, howevnot practical for gaining first insight into the roles of solvefor assisting synthetic design of new materials. Here oneeds acost effectivetheoretical tool to provide reasonabaccurate predictions to guide experiments.

B. Implicit treatments of solvent–solute interactions

Among several methodologies for implicit treatmentthe solvent, the dielectric continuum approach21–23offers thesimplest and yet reasonably accurate methodology. Indielectric continuum approach, the solvated system is meled as the solute~if necessary, with inclusion of severafirst-solvation-shell solvent molecules! inside a cavity sur-rounded by a dielectric continuum medium representedthe dielectric constante. Such model has several weaknessIn particular, it does not provide any information on the svent structure. In addition, the size and shape of the cahave no rigorous definitions. However, there are also sevimportant advantages. First, one can select a desired levquantum mechanical theory from a wide range ofab initiomolecular orbital~MO! and density functional theory~DFT!levels that is sufficiently accurate for modeling bonbreaking and forming processes. Second, the reaction cdinate is uniquely defined because solvent effects fromcontinuum medium are effectively included in the soluHamiltonian and do not increase the dimensionality ofsystem. Third, availability of analytical free-energy derivtives greatly enhances the ability to explore the free-enesurface and to characterize stationary points. With approate choice of van der Waals radii for cavity construction, ocan obtain reasonably accurate free energy of solvation fwide range of solutes. Overall, for modeling reactions inlution the dielectric continuum approach provides the mcost effective methodology.

Most of previous dielectric continuum calculationssolvent effects on reaction mechanisms were based on si~spherical or ellipsoidal! cavity models.21,23 Although theseself-consistent reaction field studies provide useful insigtheir accuracy is often questionable due to the uncertaintthe cavity size and shape for variable geometry of the reing system. For more realistic molecular-shape cavities, thmodels have shown some promise due to the availabilityanalytical free-energy derivatives. One is known as the

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larizable continuum model~PCM!.23–27 Second is the reaction field factors formalism which is based on the Kirkwoomultipole expansion model.28,29Third is the generalized conductorlike screening model~GCOSMO!30–35 developed inour group that is based on an approximate scheme for scascreening conductor surface charges originally proposedKlamt and Schu¨urmann.36 Implementation of the originaCOSMO approach to the density functional theory was adone by Andzelmet al.37 However, to date a general methodology for following reaction paths on the solution-phafree-energy surface has not been developed.

In this study, we present a general methodology for cculating free-energy profiles in solution by following thminimum free energy path~MFEP! on the free-energy surface. Using the GCOSMO solvation model, we have carrout a systematicab initio study of the MenshutkinNH31CH3Cl↔NH3CH3

11Cl2 reaction in aqueous solutionThis is a type IISN2 reaction which is different from thetype I SN2 charge transfer reactions, suchCl21CH3Cl→ClCH31Cl2, in several aspects. First, for thCl21CH3Cl→ClCH31Cl2 reaction solvent effects significantly decrease the reaction rate due to charge delocalizaand decrease of the free energy of solvation at the transstate. In contrast, the rate of the Menshutkin reactionbeen found to increase dramatically with increasing solvpolarity which favors charge separation and thus stabilithe transition state. Second, the Cl21CH3Cl→ClCH31Cl2

reaction is symmetric, thus one expects very little solveffect on the transition state structure. In contrast, the Mshutkin reaction is an unsymmetric charge separation prowith endothermicity of 110 kcal/mol in the gas phase but itexothermic by about234610 kcal/mol in aqueous solutiondue to the stabilization of the H3CNH3

1 and Cl2 ions by thesolvent.20 According to the Hammond postulate,38 solventeffects would shift the transition state toward the reactchannel, thus solvent has a direct participation in the reaccoordinate. Such role of solvent challenges the validity ofstatic equilibrium solvation treatment which assumes solvto be in equilibrium with the chemical system at each poalong the gas-phase reaction coordinate. For this reasonMenshutkin reaction provides a challenging test to any svation theory attempting to model solvent effects on trantion state structure and reaction profile.

Gao and Xia20 recognized that for this reaction it is morappropriate to determine the MFEP in solution rather thanfollow the gas-phase reaction coordinate. To do this, Gand Xia have performed elaborated QM/MM simulationsmap out a 2D free-energy surface. Transition state strucand the MFEP were then estimated from this 2D surfaAlthough these authors used rather approximate AHamiltonian for the solute, their QM/MM description osolvent–solute interactions was quite accurate and the Abarrier height happened to agree well with the free energactivation for the gas-phase reaction. These results willused below for assessing the accuracy of the GCOSMOvation model. Another Menshutkin reaction, NH31CH3Br,was studied by Solaet al.39 using both the supermoleculand PCM approaches. The PCM results for the free energ

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1883Truong, Truong, and Stefanovich: Free energy profile of reactions in solution

activation appeared to be too low. This disagreement ledand Xia20 to question the accuracy and applicability of tdielectric continuum approach for studying reactions of ttype. Both PCM and QM/MM studies confirmed qualittively the large solvent effect on the transition state. Hoever, the transition state in solution has not been fully omized and the full dimensional MFEP has not becalculated. In the present study, using the GCOSMO dietric continuum methodology, we fully optimized the transtion state of the Menshutkin NH31CH3Cl↔NH3CH3

11Cl2

reaction in aqueous solution and determined the minimfree-energy path in all degrees of freedom of the solute. Psible sources of error in previous PCM calculations39 areanalyzed, and applicability of the dielectric continuum aproach for modeling reaction free-energy profiles is estlished.

II. METHODOLOGY

Let us consider anA→B reaction. The standard freenergies of reaction in the gas phase and in solutiondenoted asDGg

0 andDGs0, respectively. Associating with th

reactantA and productB are free energies of solvation denoted asDGsolv

0 . From the thermodynamic cycle given belo

the standard free energy of the reaction in solution canwritten as

DGs05DGg

01@DGsolv0 ~B!2DGsolv

0 ~A!#, ~1!

where the gas-phase free energy is given by

DGg05DE2RT lnS QB

QAD . ~2!

HereDE is the reaction energy;R is the Boltzmann constantT is the temperature;QA andQB are the total partition func-tions evaluated with the zero of energy set at the bottomeach respective potential well.

We can generalize this expression for the free-eneprofile of reaction in solution. In particular, the standard frenergy at a pointR(s) along the reaction coordinates rela-tive to that of the reactantA is expressed as

DGs0~s!5DGs

0@A→R~s!#

5VMEP~s!2RT lnH Q@R~s!#

QA J1DGsolv

0 @R~s!#2DGsolv0 ~A!, ~3!

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whereVMEP(s) is the potential energy along the gas-phaminimum energy path with the zero of energy set atreactant. Equation~3! indicates that in order to obtain accurate free-energy profile for reaction in solution, one requito have not only accurate free energy of solvation but aaccurate gas-phase free-energy profile. This point has obeen overlooked in many previous studies.

The central point needs to be discussed here is thetermination of the reaction coordinates. The simplest ap-proach, adopted in most studies to date, is to defines as thedistance along the minimum energy path on the gas-phpotential-energy surface. Then, using the static equilibrisolvation approach the free-energy profile in solution canestimated by adding free energy of solvation to the gas-phfree-energy profile along this reaction coordinate. As mtioned earlier, this approach can be erroneous if solvasignificantly changes the topology of the free-energy surfrelative to the gas-phase potential-energy surface. Incase, one should include solvent effects in the determinaof the reaction coordinate.

Following the reaction path on the solution-phase freenergy surface, as defined in Eq.~3!, is a difficult task. Themajor difficulty arises from the necessity to perform normmode analysis at every point on the gas-phase potentialface in order to calculate vibrational partition functions. Tcircumvent this problem, we first assume that the gas-phBorn–Oppenheimer potential-energy surfaceE(R) has simi-lar topology to the gas-phase free-energy surface alongreaction coordinate. In this case, we then define a pseufree energy surfaceG* (R) in the solute nuclear coordinateR that is related toE(R) by the following expression:

G* ~R!5E~R!1DGsolv~R!. ~4!

The pseudo-free energy surface defined above allows onutilize advanced computational methods that have beendeveloped for following reaction paths in the gas phaAnalogous to the gas phase, the reaction coordinates in so-lution is defined as the distance along the minimum frenergy path~MFEP! which is the steepest descent path frothe transition state toward both the reactant~s! and product~s!on the pseudo-free energy surfaceG* (R). To obtainfree energy profile of reaction in solution we correct tpseudo-free energy profileDG* (s) for the gas-phase2RT ln(Q(s)/QA) term along the solution-phase reaction cordinate. The central approximation here is that in the demination of the solution-phase reaction coordinate2RT ln(Q(s)/QA) term does not have significant contribution thus can be ignored.

Within the dielectric continuum solvation methodologwe can write the free energy of solvation as

DGsolv~R!5DGel~R!1DGdis~R!1DGrep~R!

1DGcav~R!, ~5!

where DGel constitutes solvation terms of electrostatic nture; DGdis andDGrep are the solvent–solute dispersion arepulsion interactions, respectively;DGcav is the work re-quired to create the cavity. Note that contributions from t

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1884 Truong, Truong, and Stefanovich: Free energy profile of reactions in solution

solute internal motions, such as shifts in the solute vibtional frequencies, are effectively included by fitting the caity size to experimental free energies of hydration.

For polar solvents such as water, the electrostatic tmakes the largest contribution to the solvent effects. Forterm, we used the GCOSMO model30–35which was general-ized from the COSMO approach which was originally impmented at the semiempirical MO level36 for a more accurateand general description for the solute charge density withusing the multipole expansion and for inclusion of noneltrostatic contributions. The essence of the GCOSMO mois to determine first the surface chargess~r ! on the surface(S) of the cavity in a screening conductor~the dielectricconstante5`! from a boundary condition that the electrstatic potential on the surfaceS is zero

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ur2r 8ud2r 850,

~6!

wherer is on S; r is the solute electron density; andzi andRi are the nuclear charge and position vector of atomi . Fora dielectric medium specified by the dielectric constante, thesurface charges are then determined approximately by sing the screening conductor surface charges by a factor off (e)5(e21)/e to satisfy Gauss’ theorem. This is a gooapproximation to the exact boundary condition used inPCM model. Average unsigned differences in hydration fenergies between two models are less than 0.2 kcal/moneutral solutes and 0.9 kcal/mol for ions.31

Within the boundary element approach, the cavboundary is defined byM surface elements with area$Su%. The surface charge density at each surface elemeapproximated as a point charge,$qu%, located at the center othat element,$tu%. From the above boundary condition, thtotal electrostatic free energy of the whole syste(solute1surface charges! is then given by

Gel5(mn

Pmn~Hmn1 12Gmn!2 1

2f ~e!z1B1A21Bz1Enn ,

~7!

whereEnn is the solute nuclear repulsion andz is the vectorof N nuclear charges. The solvent contributions to the oand two electron terms of the Fock matrix~Hmn and Gmn ,respectively! are expressed as

Hmns 52 f ~e!z†B†A21Lmn , ~8!

Gmns 52 f ~e!c†A21Lmn . ~9!

whereA, B, andc are M3M , M3N, andM31 matrices,respectively, with matrix elements defined by

Auv51

utu2tvufor uÞv, and Auu51.07A4p

Su,

~10!

Bui51

utu2Ri u, ~11!

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andPmn is the density matrix element.For the dispersion and short-range repulsion contri

tions, we adopted Floriset al. method.40 For the cavity for-mation term, we employed the scaled particle fluid theoryPierotti,41 which was transformed by Huron and Claverie42

into an atom–molecule-type formalism. These methods hbeen found to be sufficiently accurate from several previstudies30,40,43–48including our own.

The solvent excluding surface49 was used for definingthe cavity boundary due to its smooth nature. It was cstructed using the GEPOL algorithm.50 The atomic radii~N:1.74; C: 2.10; H: 1.17; Cl: 1.75 Å! used in this study havebeen optimized at the HF/6-31G(d) level to reproduce freeenergies of hydration for a representative set of small mecules and ions.31 By optimizing the atomic radii for determining the cavity size we effectively account for other cotributions that were not explicitly considered in the modOne of such contributions is the outlying charge effect. Teffect arises from the fact that implementation of any clascal continuum solvation model within a quantum mechanilevel will lead to a small portion of the electronic densidistribution being outside of the cavity regardless of the city size. Recent studies51,52 have proposed several post-SCcorrection schemes accounting for this effect. In our viesuch schemes add an order of complication when onequires to include the outlying charge effect in free enerderivatives for consistency as in the determination of retion path, yet the atomic radii are still needed to be optimizto account for other effects. As pointed in previostudies,51,52 the outlying charge effect is large for anionsuch as Cl2 in this study and shows a noticeable basisdependence. Such effect is an interesting subject for invegation on its own. Since our focus here is on free-eneprofile of reaction in solution, we will discuss the outlyincharge effect in a forthcoming paper. For this study, hoever, as seen in Table I, using the 6-31G(d,p) basis set ouratomic radii yield free energies of hydration of ions in lethan 6% difference with the experimental data. This is withthe experimental uncertainty for solvation free energiesions. Since the outlying charge effect is expected to be sowhat smaller at points along the reaction path than at theproducts, it would not be totally canceled in calculationsfree-energy profile. One would expect an error in the freenergy profile of less than 6% in this case.

III. COMPUTATIONAL APPROACH

Calculations of a reaction profile first required optimiztions of the reactants, products, and transition state in stion. These were done using analytical first derivatives ofelectrostatic free energy@Eq. ~7!#.33 For application given

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olution.

1885Truong, Truong, and Stefanovich: Free energy profile of reactions in solution

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TABLE I. Geometries~distances in Å, angles in degrees!, dipole momentsm ~Debye!, and hydration freeenergies~kcal/mol! of the reactants, products, and transition state of the NH31CH3Cl↔H3NCH3

11Cl2 reactionin the gas phase and in aqueous solution

Gas phase Aqueous solution

BH&HLYP MP2 BH&HLYP MP2

NH3

NH 1.004 1.013 1.010 1.015/HNH 108.7 106.1 105.4 104.7m 1.696 1.924 2.237 2.337DGhyd 26.5(24.3)a 26.4

CH3ClCCl 1.785 1.776 1.800 1.788CH 1.081 1.084 1.080 1.082/HCCl 108.5 110.0 107.8 108.3m 2.129 2.212 2.757 2.865DGhyd 22.2(20.6)a 21.7

H3NCH31

CN 1.502 1.507 1.484 1.490NH 1.017 1.023 1.019 1.025CH 1.081 1.084 1.082 1.084/HNC 111.5 111.5 111.5 111.5/HCN 108.3 108.1 108.4 108.3DGhyd 269.8(268)a 269.9

Cl2

DGhyd 279.3(275)a 279.0@H3N•••CH3•••Cl#‡ TS

CN 1.798@1.66#b 1.793 2.215@1.96#b 2.156CCl 2.484@2.09#b 2.443 2.183@1.94#b 2.170CH 1.070 1.073 1.069 1.073NH 1.010 1.017 1.009 1.015/HNC 110.5 110.8 111.5 111.7/HCN 99.4 98.8 85.6 85.9

aValues in parentheses are experimental hydration energies taken from Ref. 29.bValues in square brackets are for AM1 transition state geometry in the gas phase and in aqueous sTaken from Ref. 20.

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below, the electrostatic term is dominant, thus neglectderivative contributions from the nonelectrostatic termsexpected to yield negligible errors. As discussed below,present results confirm this expectation.

The minimum energy path~MEP! on the gas-phasepotential-energy surface and the minimum free-energy p~MFEP! on the liquid-phase pseudo-free energy surface wdetermined by using the second-order reaction path folling method developed by Schlegel and co-workers.53 Step-size of 0.1 amu1/2 bohr was used. Reaction path calculatiorequire the normal mode analysis at the transition state.the reaction in solution, such analysis has been done bymerical differentiation of the free-energy gradient. Impmentation of the analytical second free-energy derivatifor the equilibrium GCOSMO solvation approximation isprogress. To calculate the temperature depend2RT ln(Q(s)/QA) term we performed gas-phase Hessian cculations for a small number of points along the solutiophase reaction coordinate.

The choice of the basis set and calculation methoddiscussed in detail in the next section. All calculations wdone using our locally modifiedGAUSSIAN92/DFT program.54

It is important to note that GCOSMO calculations in soluti

J. Chem. Phys., Vol. 107

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are only 10%–20% more time consuming than gas-phmolecular calculations.

IV. RESULTS AND DISCUSSION

A. Accuracy comparison between BH&HLYP and MP2methods

In our previous studies30,31,34for a large set of moleculesand ions, we found that free energies of solvation obtainfrom HF, MP2, and DFT methods are very similar. Therfore, the choice of the adequate method should be basetheir ability to accurately represent the gas-phase potenenergy surface. Nonlocal DFT methods are particularlytractive since they are computational less demanding teven MP2, especially for large systems.

Investigations on the accuracy of DFT methods habeen an active area of research. The B3LYP methodbeen the most popular choice for many applications dueits good performance for equilibrium properties.55 Reactionrate constants, however, are rather sensitive to the transstate region and the shape of the potential-energy suralong the MEP. Therefore, inspection of these propertiealso important for the choice of appropriate method. Fr

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1886 Truong, Truong, and Stefanovich: Free energy profile of reactions in solution

Downloaded 02 May 2013 t

TABLE II. Reaction energies and barrier heights~kcal/mol! of the Menshutkin NH31CH3Cl↔H3NCH3

11Cl2 reaction in the gas phase and in aqueous solution.

MP2 MP4~SDTQ! BH&HLYP Expt.

Gas phaseDE 121.0 114.0 116.5DG 128.3 121.3 124.0 110a

DV‡ 38.4 32.1 32.6DG‡ 51.4 45.1 45.7

Aqueous solutionDG 219.8 216.5 234610b

DG‡ 31.4 24.8 23.5c

aTaken from Ref. 20.bEstimated from experimental free energies of hydration and standard free energy of formation~see also Ref.20!.

cExperimental data for the NH31CH3I↔H3CNH311I2 reaction in water~Ref. 65!.

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several our studies on the accuracy of DFT methods foraction profiles~includingSN2Cl21CH3Cl reaction32! and re-action rates,56–59 we found that among existing DFT methods, the combination of the hybrid Becke’s half-anhalf for exchange and Lee–Yang–Parr for correlat~BH&HLYP !60,61 gives the best overall performance.

In this study, we compare the accuracy of the MP262 andBH&HLYP methods using the 6-31G(d,p) basis set. First,we optimized geometries of the reactants, products and tsition state inC3v symmetry in the gas phase and in aqueosolution using both methods. Note that on the gas-phpotential-energy surface, there exists a stable compH3NCH3•••Cl in the product channel. We found that at thBH&HLYP/6-31G(d,p) level, the linearlike complex~C3vsymmetry! has negligible stabilization energy relative to ttransition state. Gordon and Webb63 pointed out that due to azwitterionic character, the bendlike complex~the angleNCCl is less than 180°! is much more stable. In water, thzwitterion, if exists, would prefer to have a linearlike struture due to larger solvation free energy.31 However, wefound that no such stable complex exists in aqueous soluand thus no further attention is given for determination ofmost stable complex on the gas-phase potential surface

Geometrical parameters, dipole moments, and hydrafree energies of the reactants, products, and transitionare listed in Table I. As expected, hydration free energcalculated from both methods are very similar with the laest difference of only 0.5 kcal/mol. To investigate the acracy of these methods for the gas-phase potential surfaceperformed benchmark single point MP4~SDTQ! calculationsat the MP2 optimized geometries using a largaug-cc-pVDZ64 basis set. This is the most accurate calcution of this system to date. Reaction and activation freeergies computed by all three methods are comparedavailable experimental data in Table II. As compared to Mcalculations, MP2 overestimates the classical reaction ba(DV‡) by 6.3 kcal/mol while BH&HLYP agrees to within0.5 kcal/mol. Although all three methods yield too largeaction free energiesDG ~the same order of magnitude fothis error was also observed in Gao and Xia’s Mcalculations20 using a smaller basis set!, the BH&HLYP re-sult is closer to both MP4 and experiment than the M

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result. In addition, an excellent agreement was foundtween MP4 and BH&HLYP calculated free energies of acvation DG‡.

In agreement with our previous finding, we concluthat the BH&HLYP method is computationally cheaper ayields more accurate results than MP2. Consequently,the method of choice for further calculations and the remaing discussion will be based on the BH&HLYP results.

B. Solvent effects on geometry and electronicstructure

Aqueous solvent has noticeable effects on the equirium geometries of the polar reactants molecules and proions as shown in Table I. For NH3, solvent water moleculescompress the HNH angle from 108.7° to 105.4° and elongthe NH bond from 1.004 to 1.010 Å. Similar effects are sefor CH3Cl, where the CCl bond is elongated by 0.015 Å athe HCCl angle is compressed by less than 0.7°. Howeveis different for H3CNH3

1 ion, where the CN bond is shortened by 0.018 Å upon solvation. These results indicatein addition to the solvent polarization, the solute polariztion, i.e., the feedback of the solute to the solvent polarition field, is also important.

In Fig. 1 we plotted the CN and CCl bond distances athe HCN angle along both the gas-phase MEP and solutphase MFEP. These internal coordinates show the larchanges as functions of the reaction coordinate. For reacin water, they are shifted in the product direction. Thmeans solvent effects advance the reaction coordinateparticular, at the transition state solvent effect elongatesCN bond by 0.42 Å, shortens the CCl bond by 0.30 Å areduces the HCN angle by 14°. These trends are consiswith previous results from PCM model39 and semiempiricalAM1/MM simulations.20 This can be understood from thfact that the Menshutkin reaction is a charge separationcess. Aqueous solvent facilitates charge transfer andcreases the dipole moment by gaining favorable free eneof solvation. This is also seen from Figs. 2 and 3 whereplot the dipole moment of the solute and the Mulliken charof the Cl atom along the reaction coordinate.

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1887Truong, Truong, and Stefanovich: Free energy profile of reactions in solution

C. Solvent effects on reaction profile

Heats of reaction and activation free energies are lisin Table II. Free-energy profiles in both the gas phaseaqueous solution are also plotted in Fig. 4. As mentionearlier, our BH&HLYP calculations overestimate the gaphase free energy of reaction by 14 kcal/mol. As conquence, our calculated free energy of reaction in wa

FIG. 1. Bond distances NC and CCl and angle HCN as functions ofreaction coordinates for the Menshutkin NH31CH3Cl reaction in both thegas phase~g! and aqueous solution~aq!. Origin of the reaction coordinate isat the saddle point in each case.

FIG. 2. Plot of the dipole moment of the reacting system as function ofreaction coordinate.

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(216.5 kcal/mol) is in good agreement with the AM1/MMresult of 218 kcal/mol, but both are too high comparedthe experimental estimate of234610 kcal/mol.20 If experi-mental gas-phase free energy of reaction were used,GCOSMO and AM1/MM free energies of reaction in watwould fall within the experimental uncertainty.

Our calculated gas-phase free energy of activation atroom temperature~45.7 kcal/mol! is consistent with Gao and

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FIG. 3. Plot of the Mulliken charge on the Cl atom as function of treaction coordinate.

FIG. 4. Potential energy profile along the gas phase MEP and free-enprofile along the solution-phase MFEP. QM/MM free energy of activationtaken from Ref. 20.

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1888 Truong, Truong, and Stefanovich: Free energy profile of reactions in solution

Xia’s best estimate of 46.7 kcal/mol. This activation eneris only relevant to the formation of the gas-phaH3NCH3•••Cl complex from the reactants. For formationion products H3CNH3

1 and Cl2, the reaction is 110 kcal/moendothermic. Thus, the dynamical bottleneck is located fathe exit channel. Solvent effect changes the reaction eneics to noticeably exothermic thus shifts this bottleneck snificantly to the entrance channel and decreases the freeergy of activation to 24.8 kcal/mol. This free energyactivation agrees well with Gao and Xia’s AM1/MM simulations ~26.3 kcal/mol! and experimental value for a similaMenshutkin reaction NH31CH3I in water ~23.5 kcal/mol inRef. 65!. The agreement between GCOSMO results aAM1/MM simulations leads to an important conclusion thelectrostatic solvent–solute interaction not specific hydrobond interaction as infered by Gao and Xia makes the mcontribution to the transition state stabilization in the typeSN2 reactions. This conclusion is further supported by exaining individual contributions to the solvation free eneralong the reaction coordinate as shown in Fig. 5. Furthmore, the relatively small contributions of the dispersiorepulsion and cavitation to the free energy of hydration copared to the electrostatic term support the use of oelectrostatic free energy derivative in our structure deternations as mentioned earlier. In conclusion, simple~evenwithout explicit water molecules! dielectric continuum mod-els should be applicable for studying these reactions.

Previous dielectric continuum calculations done by Set al. for the NH31CH3Br reaction in water yielded the activation energy of 8.3 kcal/mol. This value appears to below. In our opinion, this large error does not necessareflect the inaccuracy of the dielectric continuum approanor indicate the importance of specific solvent–solute inactions but is rather due to several other facts. First, con

FIG. 5. Individual contributions, i.e., electrostatic, dispersion–repulsand cavitation, to the free energy of hydration along the reaction coordin

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butions from the solute motions@the2RT ln(Q‡/QA) term inEq. ~3!# was not included. From our present study for tNH31CH3Cl reaction, this term increases the free energyactivation by 13.1 kcal/mol. Second, the calculated gphase barrier~21.5 kcal/mol! is probably too low due to neglect of the electron correlation and use of the small 3-1G(d) basis set. Finally, these authors did not discuss thselection of atomic radii for cavity construction. Impropradii may significantly contribute to the total error.

V. CONCLUSION

We have presented a general methodology for calcuing free energy of activation and reaction profile in solutiusing the dielectric continuum solvation approach. The avability of efficient free energy derivatives of the GCOSMmodel makes it an attractive tool for studying solvent effeon transition state structures, modeling reaction mechaniin solution and addressing dynamical solvent effectsthe reaction coordinate. Using this model, we have carrout a detailed study of the Menshutkin NH31CH3Cl→H3CNH3

11Cl2 reaction in aqueous solution. An excelleagreement between our results and previous elaboQM/MM simulations confirms the accuracy of thGCOSMO model and its usefulness. In particular, for tMenshutkin reaction, we found that aqueous solvent stalizes charged products which are unstable in the gas phAs result, the reaction becomes exothermic in aqueous stion and has the free energy of activation at the room teperature of about 24.8 kcal/mol. This result is consistent wavailable experimental data on a similar NH31CH3I reactionin water. In addition, aqueous solvent shifts the transitstate significantly to the reactant channel and advancesreaction coordinate. Nonequilibrium dynamical solventfects are expected to have smaller contribution to the retion rate for this reaction. It is however a fundamentally important aspect of reaction in solution and will be addressea future study. Reaction path determination discussed inpaper is a first important step in this direction.

ACKNOWLEDGMENT

This work is supported in part by the National ScienFoundation via a Young Investigator Award to T.N.T.

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