J. Phys. Chem. 1995, 99, 3798-3805
3798
Hydroxide Ion in Liquid Water: Structure, Energetics, and Proton Transfer Using a Mixed Discrete-Continuum ab Initio Model Iiiaki Tuii6nJ Daniel Rinaldi, Manuel F. Ruiz-L6pez, and Jean Louis Rivail* Luboratoire de Chimie thiorique-UA CNRS 510,t Universiti Henri Poincari, Nancy I, B. P. 239, 54506 Vandoeuvre-les-Nancy Cedex, France Received: October 14, 1994@
The hydroxide solvation process is studied by means of discrete-continuum and discrete models. The structure and energetics are analyzed. Calculations have been carried out at the HF/6-31+G* and MP2/6-31+G*// HF/6-31+G* levels. In agreement with experimental studies, the first solvation shell is shown to be composed of three water molecules in a branched structure. Both discrete and discrete-continuum models predict similar geometric changes in the first shell upon solvation. Discrepancies in these changes between both models are related to the noninclusion of charge transfer beyond the cavity. In spite of some deficiencies, the discrete-continuum model is shown to be good enough to predict solvation free energies of charged species in agreement with experimental values. Finally, a possible mechanism for hydroxide migration in liquid water is discussed. It implies proton transfer from a water molecule of the first solvation shell to the hydroxide anion. Second solvation shell water molecules play an important role in this process.
1. Introduction Water dissociation is a fundamental process in chemistry. Some theoretical efforts has been made to study the ionic equilibrium 2(H20) H3O+ OH- in liquid water' and the structure and energetics of OH-(HZO),~~~ and H30+(H20),4-1 clusters. These studies are important to understand the properties of solvated ions and their role in acid- or base-catalyzed reactions in water. l2 Moreover, accurate computation of the solvation energy is a challenge for theoretical chemistry since, owing to the slow decrease of the electrostatic potential, longrange interactions have to be taken into account. In principle, several approaches can be envisaged. Statistical simulations of classically described solute and solvent molecules with refined intermolecular potentials can be p e r f ~ r m e d . ~Nevertheless, ~~'~ this approach is not usually well adapted to evaluate nonadditive contributions to the interaction energy that are extremely important in the ion-water system and have to be described quantum mechanically. Some authors15 proposed a quantum treatment of the ion while keeping a classical description of the solvent molecules. Still, in this case, cooperative phenomena cannot be accounted for and at least some solvent molecules should be included in the quantum subsystem, leading to very expensive computations, unless semiempirical methods are employed.16 If one is interested in static properties of particular configurations of the solute, Le., if the detailed dynamics of the whole system is not required, discrete-continuum models are very helpful. In this case, the ion and a limited number of solvent molecules are described quantum chemically and are assumed to be surrounded by an infinite polarizable dielectric medium. Short-range interactions are described at the quantum level, while long-range interactions are introduced through an electrostatic model. Cooperative phenomena due to interactions in the quantum part are treated rigorously and those due to interactions with the bulk solvent are correctly approximated.l7 The main limitation of this model is the neglect of charge
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+
t On leave from the Departamento de Quimica Fisica Universidad de Valencia, Valencia, Spain. Part of the Institut Nanceien de Chimie MolBculaire. @Abstractpublished in Advance ACS Abstracts, February 1, 1995.
*
0022-365419512099-3798$09.00/0
transfer between quantum molecules and the dielectric and the more or less arbitrary definition of an unphysical cavity surface. Both require a careful analysis of the errors they introduce. Its main advantage is that geometry optimizations can be easily carried out for the quantum subsystem, since the computation of the reaction field (i.e., the potential created by the polarized dielectric in the solute's domain) is straightforward for a given configuration. Accordingly, this approach allows a relatively easy estimation of the coordination number and solvation energy.18 This has been successfully used to study the hydronium ion solvation5 and here we apply it with the aim of investigating the structure of the hydroxide ion in water. Quantum molecular dynamics will probably be feasible in the near future and the results of simpler models, such as the discrete-continuum model used in this work, will be of considerable help in defining computational strategies. Little work has been devoted to OH-(HzO), clusters compared to H30+(H20), ones. In particular, the most stable structures of OH-(HzO), are not well established. Therefore, we shall fiist of all consider this point. Afterward, the coordination shell structure in liquid will be investigated. Both the energetics of cluster formation and solvation in water are compared to available experimental data. lg4 Finally, the proton transfer from a water molecule to hydroxide ion is also studied in the gas phase and in solution. This process is an alternative mechanism for the hydroxide transport in liquid water.
2. Methodology Calculations have been carried out by using GAUSSIAN9OZ2 and GAUSSIAN9223 packages of programs. The basis set employed in all the calculations was 6-31+G*,N325 which includes polarization and diffuse functions on oxygen atoms. The 6-31G* basis set gave good results in the previous study of hydronium solvation? Diffuse functions on oxygen atoms are important because we are dealing with negatively charged complexes. Of course it might be desirable to include also polarization functions on hydrogen atoms, but because of the size of some of the systems studied here, this would increase considerably the computational effort. Basis set superposition 0 1995 American Chemical Society
Hydroxide Ion in Liquid Water errors (BSSE) have been estimated in several cases by computation of counterpoise corrections,26but because of their modest magnitude (with a typical value of around 2-3 kcaUmo1 for the OH-(H20)3 clusters) and the tendency of counterpoise corrections to overestimate the BSSE,27,28we finally decided not to include them. Geometry optimizationshave been carried out by means of the Berny's at the Hartree-Fock level. Analytical frequency calculations have been carried out to determine the nature of the stationary points found and to obtain contributions to the free energy by using standard procedures. In some cases we have recalculated the energy at the second order of Moller-Plesset perturbation theory (MP2/ /HF/6-31+G*). Solvation free energies have been obtained as the sum of electrostatic plus polarization and cavitation contributions:
A full description of the solvation process should also include the dispersion-repulsion or Lennard-Jones term, but this has not be considered in this work. Its possible influence on our results will be discussed. Electrostatic plus polarization contributions have been obtained by means of the cavity model. The liquid is assimilated to a macroscopic continuum characterized by the dielectric constant (78.4 in the case of water). The quantum system is then placed in a cavity surrounded by this continuum. The electrostatic free energy is30-32
where f l is a component of the multipole moment of order 1 and R;" is the corresponding component of the reaction field. In practice good convergence is obtained expanding the expression up to 1 = 6. This electrostatic interaction term can be included into a SCF method by introducing the corresponding operator. The analytical energy derivatives of the electrostatic leading to an efficient geometry term have been optimization procedure.34 We have used ellipsoidal cavities whose axes are related to the axes of inertia of a solid of uniform density limited by a corrected van der Waals surface.33 The volume of the initial cavity is usually constrained to be equal to the average molecular volume in the liquid. This molecular volume is obtained by means of an empirical equation relating the van der Waals and molecular volumes.34 However, this approach is no longer valid when one deals with large clusters as s o l ~ t e s . ~InJthis ~ ~case, ~~ the elliposidal cavity that we have used is the minimal ellipsoid that ensures that all the van der Waals surface remains inside. In practice this is made by looking for the proportionality factors of the axes of inertia of the solid of uniform density that fulfill this condition in the initial cavity. These definitions of the cavity allow the use of deformable cavities during the geometry optimization procedure and it has been previously tested that both definitions lead to nearly the same results for standard molecules such as water.5 Solvent effect calculationshave been carried out with an extra link36 added to the GAUSSIAN package. Cavitation energies have been obtained with the formula proposed in ref 37 that makes use of the surface tension of the solvent and of the solute's surface.38 All solvent effect calculations were carried out at 298 K. 3. Results and Discussion
3.1. Nuclear and Electronic Structure. First Solvation Shell. Experimental enthalpy sequenceslg for OH-(H20),
J. Phys. Chem., Vol. 99, No. 11, 1995 3799 clusters show a break after n = 3, indicating that this is probably the coordination number in the first solvation shell of the hydroxide ion. For this reason, we have mainly explored this possibility, although we also considered a f i s t solvation shell with a fourth water molecule. In a previous work Newton et aL2 analyzed several structures for a first solvation shell with three water molecules in a limited ab initio geometry optimization. We have considered the most stables of these structures and some related geometries, which are shown in Figure 1. The first three structures are variants of a branched structure with the water hydrogens trans relative to the hydroxide hydrogen (structure I), in cis conformation (structure II), and in gauche conformation obtained by relaxing the C3" symmetry to C3 (structure 111). In the later, water hydrogen atoms are slightly oriented toward the oxygen atom of the neighboring water molecule, but no bond is observed between them when looking at Mayer bond indices.39 Structure IV is cyclic and structure V is a planar chain. We have also investigated the possibility of coordination with a fourth water molecule approaching along the C3 axis of the branched OH-(H2O)3 ion (structure VI). Several configurations of the OH-(H20)3 ion have been tested for approaching the fourth water molecule, but no minimum has been found for any of them. Structure I was found as the most stable in the work of Newton et al.? but a frequency analysis shows that this structure is a third-order saddle point, like structures I1 and V. Only structures I11 and IV are true minima on the potential energy hypersurface, structure I11 being the absolute minimum (2.52 kcal/mol more stable than structure IV). When considering free energies the difference is somewhat larger (5.25 kcaUmo1). In the studied structures, the distances between the hydroxide oxygen atom and hydrogens of water molecules range between 1.57 and 1.77 A. These bond lengths are shorter than typical hydrogen bonds between water molecules34but are similar to those found in hydronium ~olvation.~ Thus, it seems that hydrogen bonds with both negatively and positively charged ions are stronger than with neutral molecules and similar at least in their geometrical description. Due to the formation of hydrogen bonds with the hydroxide ion, the oxygen-hydrogen (02H2) distance of the water molecules is lengthened in all the cases. Since the hydrogen bond formed by a water molecule with the hydroxide anion is stronger than that formed with another water molecule, the lengthening is considerably larger than in the water dimer.34 It is also interesting to analyze the dipole moments (computed with respect to the center of mass) of the various structures. We can see in Figure 1 that structure I1 presents the largest dipole (5.72 D) and structure III, Le., the absolute minimum in gas phase, the smallest one (1.10 D).As will be shown, this is of great importance when considering the structure of the OH-(H2O)3 ion in solution. Although a full multipole moment expansion is necessary to compute accurate electrostatic solvation free energies, the large value of the dipole moment for structure 11 makes it the most stable structure in solution. Second Solvation Shell. Because of the size of the OH-(H2O)3 ion, the number of water molecules surrounding this structure in solution is expected to be quite large. In order to have a second solvation shell model accessible to ab initio calculations, selection of those second solvation shell water molecules playing a more active chemical role is essential. From the results of the previous section, water molecules approaching the OH-(HZO)~ion along the C3 axis are not expected to be strongly bonded in solution. Thus, it seems reasonable to select for our model those water molecules which are hydrogen bonded to the first solvation shell water molecules, compensating the
3800 J. Phys. Chem.,VoI. 99, No. 11, 1995
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0
1.946
I E=2.57 (3) y=3.46
I1 E= 5.76 (3) p=5.72
IV E=2.52 ( 0 ) p1.91
V E= 6.91 (3) p=2.65
VI No minimum found
Figure 1. Structures studied for the hydroxide ion in the gas phase with three and four water molecules. Distances are given in angsmms and angles in degrees. Relative energies are in kcaUmol and dipole moments (with respect to the center of mass) in debyes. The number in parentheses indicates the number of imaginary frequencies.
proton domor or proton acceptor character of the later. Because of the geometry of structure Ill this can be already reached with only three additional water molecules. We have not made a systematic search of the potential energy surface because of the large number of degrees of freedom of the whole system but several configurations have been explored. With this purpose we have canied out geometry optimizations where the secondshell water molecules were placed in such a way that they can be hydrogen bonded either to the outer hydrogens or to the oxygens of the fmt-shell water molecules. In addition, several orientations of the outer hydrogens of the cennal OH-(HzO), ion were tested. Only one stationary point was found in these calculations. The resulting structure VII is shown in Figure 2. The second-shell water molecules form proton-donor bridges among the oxygen atoms of the first solvation shell. The distances of the second-shell water hydrogens to the first-shell
oxygens are typical of hydrogen bonds between water molecule~.'~These distances are closely similar to those found in structure IV which presents a similar arrangement of water molecules and can be seen as a simplified model of the complete discrete model with two solvation shells. Note that the central OH-(H?O), ion has the same configuration as structure II with the water hydrogens in cis respect to the central hydrogen atom. However, frequency analysis of this two-shell structure shows that it is a minimum on the potential energy hypersurface. It is interesting to examine in some detail how the geometry of the central OH-(H20), ion changes after solvation with a second shell. By comparing structures ll and VI1 we can see that both O l H l and 01H2 distances are slightly shortened, while 02H2 and 02H3 distances are slightly lengthened. The pyramidalization angle increases substantially (from 109.4' to 116.9O). These results are very similar to those obtained in the
Hydroxide Ion in Liquid Water
W
VI1 Figure 2. Structure of the hydroxide ion with two explicit solvation shells. Distances are given in angstroms and angles in degrees.
VI11 Figure 3. Structure of the hydroxide ion with three water molecules in a continuum ( E = 78.4). Distances are given in angstroms and angles in degrees
study of the hydronium solvations and we will discuss them below in comparison with the results obtained with the discretecontinuum model. Discrete-Continuum Model. Structure III,i.e., the absolute minimum in gas phase for the hydroxide ion solvated with three water molecules, bas been placed in an ellipsoidal cavity surrounded by a continuum of dielectric constant 78.4 as explained in the Metholodology section. Geometry optimization has been carried out at the HF/6-31+G* level. The resulting structure VI11 is shown in Figure 3. The configuration is very similar to that of structure Il in the gas phase although it exhibits slightly different geometric parameters. The largest solvation energy of this configuration, in which the dipole moments of the hydroxide and water molecules are nearly parallel, is the reason for this change of contiguration when going from gas
J. Phys. Chem., Vol. 99, No. 11. 1995 3801 phase to solution. In fact, the difference in the electrostatic solvation free energy at fixed geometry between structures II and I11 (5.2 kcallmol) is mainly due to the difference in the dipolar term (5.0 kcal/mol). This example shows the importance of considering geometry reoptimization in solution. Moreover, this configuration is in agreement with that found for the central OH-(HzO), ion in the two-shell discrete model, showing the ability of continuum models to correctly predict the main geometrical changes induced by the solvent on the solute even when specific interactions are important. A deeper analysis of bond angles and distances shows some interesting features. Both discrete and discrete-continuum models predict a slight shortening of the O l H l distance compared to structure II in the gas phase, going from 0.949 to 0.947 and 0.948 A, respectively. They also agree to predict a lengthening of the OH bonds of the first solvation shell water molecules (02H2 and 02H3). The lengthening of the 02H3 bond is somewhat larger in the discrete-continuum model than in the discrete model because in the later the hydrogen H3 is not explicitly solvated. These geometrical changes illustrate the importance of the cooperative effects between water molecules, where the formation of hydrogen bonds between the first and second solvation shells reinforces those established between the first shell and the hydroxide anion. Our results confirm the ability of the continuum model to correctly predict the tendencies of these pben0mena.3~We will further analyze this below. The main difference between the two models of solvation considered here appears in the description of the distance between the hydroxide oxygen and water hydrogens (01H2). The distance in structure I1 (1.767 A) is shortened when a second solvation shell is included through a discrete model (1.675 A) but lengthened when the discrete-continuum model is employed (1.830 A). An opposite trend is also found between the two models for the pyramidalization angle (H101H2), which increases in the discrete model hut decreases in the discrete-continuum model. Nearly the same similarities and discrepancies between the two models were found in the study of hydronium solvation.5 In order to understand the differences between the two models, it is interesting to compare the electronic structure of Structures VI1 and VIII with that of structure II. For this purpose, we have calculated the potential derived charges for the three structures discussed. The CHELP derived chargesM for these structures appear in Table 1. The net charges of both hydroxide atoms decrease from II to either Vll or VIII, the effect being more pronounced in the discrete model. A quite important polarization of the first-shell water molecules is obtained with the discrete-continuum model but not with the discrete model. This difference can be rationalized because of the lack of complete solvation of these water molecules in the discrete model and, as we shall see, because of the lack of specific interactions in the discrete-continuum model. The analysis of the total charges on water molecules evidences a key difference between these models. In both cases, negative charge is transferred from the hydroxide anion to the solvent molecules, so that the total electron density on the central ion decreases. However, in the discrete-continuum model, where the charge transferred to the solvent cannot be delocalized beyond the first solvation shell, the charge transfer (W1 = -0.324) is less than the charge transfer obtained in the discrete model (WI Wz = -0.408). Moreover, in this last model, most of the charge transferred to the solvent rests on the second solvation shell and the fist-shell charge decreases when compared with structure III. One then expects nonnegligible differences in the interaction between hydroxide anion and its fist solvation shell
+
3802 J. Phys. Chem., Vol. 99, No. 11, 1995
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TABLE 1: CHELP Derived Charges (in au) at the HF/6-31+G* Level on Atoms and Sum on Hydroxide Anion (OH)and All the Water Molecules of a Solvation Shell (Wi, i = Solvation Shell) structure
01
11 VII
-1.040 -0.729 -0.919
m
02 -0.959 -0.698 -1.078
03 -0.948
H1 0.258 0.137 0.244
TABLE 2: Mayer Bond Indices at the HF/6-31+G* Level structure OlHl 01H2 02H2 02H3 II 0.756 0.065 0.469 0.75 1 Vll 0.741 0.090 0.401 0.700 m 0.737 0.075 0.470 0.681 ~
SCHEME 1: Thermodynamic Cycle Considered for Calculating Solvation Free Energies of the Hydroxide Ion
in Water AGZ OH (8) + "HZOI,,
in both the discrete-continuum and discrete models, as indeed happens. Due to this charge delocalization, the presence of a second shell of water molecules allows the first-shell water molecules to get closer to the central ion. As this effect is not present in the discrete-continuum model, the trend of the dielectric is to separate the ion and the dipoles in order to solvate them better. This is a general feature of solvent effects which tend to counterbalance direct coulomb interactions by separating the charges and thus increasing the electrostatic solvation free energy. Cooperative effects can be better discussed in terms of bond strength. Mayer bond indices39of the three structures, 11, VII, and VIII are presented in Table 2. It can be seen that in both solvation models, in spite of their differences in bond lengths, the hydrogen bonds between the hydroxide and the fast-shell water molecules (01H2) are strengthened by the presence of the solvent, although this reinforcement is somewhat larger in the discrete model. In general, the discrete and the discretecontinuum model exhibit the same trends for all the bonds in comparison with the isolated structure 11, except for the 02H2 bond, which is unaltered in the discrete-continuum model (Vm) but weakened in the discrete model (VII). This difference can be due to the explicit formation of new hydrogen bonds of the 0 2 oxygen with the second-shell water molecules. To complete this analysis its is worth noticing that one of the main differences between the two models of solvation is that in the discrete model the second shell forms hydrogen bonds with the oxygen atoms of the fast shell, while in the discretecontinuum models, the dielectric is closer to the hydrogen atoms and then one is simulating virtual hydrogen bonds with these atoms but not with the oxygens. In this way, both solvation models have a complementary nature, letting us explore different aspects of the solvation process. 3.2. Solvation Free Energies. Solvation free energies have been obtained from the thermodynamic cycle presented in Scheme 1. Thus solvation energies are calculated as AGs = AG2
+ AG3 + nAG,
Gas-phase free energies have been computed at the MP2//
HF level. Solvation energies include electrostatic plus polariza-
H2 0.509
0.344 0.483
H3 0.377 0.314 0.487
H4 0.426
OH -0.782 -0.592 -0.676
Wl
wz
-0.218 -0.120 -0.324
-0.288
tion and cavitation. In addition, because the gas free energies refer to the 1 atm process but solvation energies refer to the 1M 1M process, an expansion term of the form PAV has been added. The electrostatic plus polarization free energies have been also calculated at the MP2//HF level. The results obtained for three and six water molecules are presented in Table 3. Gas-phase free energies are divided into enthalpic and entropic contributions for better comparison with experimental values. For structure III (the gas-phase absolute minimum with three water molecules) the enthalpy and the entropy of formation are slightly overestimated in absolute value. However, the errors nearly cancel in the free energy, which is in almost perfect agreement with the experimental value. The solvation energy AG3 corresponds to the energy difference between the optimized structures in solution and in the gas phase (structures VIII and 111, respectively). For the cluster with six water molecules the trends are the same although the errors are somewhat larger. As was said before, we have not exhaustively explored the potential energy surface in this case and so we are not completely sure that the experimental measurements correspond to the structure that we propose. The free energy of solvation of the cluster with six water molecules has been obtained by single-point calculations; Le., no geometry reoptimization has been carried out with the continuum around it. The estimated free energy of solvation of the hydroxide ion in the case of n = 3 compares very well with the experimental value with an error of about 7%, very similar to that found in other cases using a similar meth~dology.~J*The lack of dispersion energy in our calculations could be responsible for part of the deviation with respect to the experimental value. By means of the use of MP2 calculations, the dispersion contribution from the first solvation shell is included in our results. The dispersion energy coming from the interaction with the second and other solvation shells has not been considered. Because of the R-6 decay of this term, this contribution is expected to be small but its negative sign would reduce substantially the discrepancy with the experimental value. For the case in which six water molecules are considered, the error is considerably larger. This fact can be due to several reasons. Since the gasphase free energy is underestimated only by 4.2 kcallmol, the main source of error must be in the solvation free energy calculation. The main reasons for this error could be the fact that no geometry optimization has been carried out and the neglect of charge transfer beyond the cavity. However, analyzing the results obtained for the OH-(H20)3 ion, charge transfer does not seem to play an essential role in the energy calculation and, in addition, this effect should be smaller for a two-shell model. Therefore, the absence of geometry optimization could be responsible for this error. Nevertheless, one should keep in mind that these results refer to a static treatment of the system. If one considers the free energy of cluster formation in the gas phase, which amounts to ca. 3 kcal/mol for each water molecule added in the second solvation shell, these clusters are likely to be very labile. Therefore, temperature effects for this second shell could be expected to be important in solution, favoring the exchange of water molecules with the solution bulk and reducing the loss of entropy calculated from the 0 K structure. Statistical treatment of the second solvation shell would be
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J. Phys. Chem., Vol. 99, No. 11, 1995 3803
Hydroxide Ion in Liquid Water
TABLE 3: Solvation Free Energies (in k d m o l ) and Intermediate Steps As Shown in Scheme 1 for the Two Models of Solvation Here Studied" AG2 AG, AGs' -TA% AG2 -94.44 (-102.7)" -41.47 (-41.0)' -57.14 3 -64.52 (-62.7)' 23.04 (21.7)d -81.26 6 -103.39 (-96.7)' -50.03 (-54.2)d -39.87f 54.36 (42.V I am. e Values calculated at the MWiJ3F/6-31+G* level. Experimental values in parenthesis. This includes the expansion work from 1 M 'Final solvation energies refer to the 1 M 1 M process. dReference 19. e Reference 21. e Single-point calculation. n8Glb 4.32 8.64
n
AH2
-
-
TABLE 4 Imaginary Frequencies (in em-') and Energy Barriers (in keSVmol) for the proton Transfer hetween Water and Hydroxide at the HF/6-31+G* and MP2//HF/ 6-31+G* Levels of Theorv
e=l.o
IX
.95 I
~=78.4
X
: 1.775
1.8511
HF
MP2m
structure v A@ @ A@ AEd @ A@ 1304.31 2.78 -0.37 0.37 0.37 -2.78 -2.04 M X" 1707.61 6.17 2.65 3.92 3.15 -0.37 0.90 0.36 0.42 XI 1101.01 4.67 1.47 1.53 3.55 a Energies include electrostatic plus polarization and cavitation contributions.
Discrete
XI Figure 4. Transition structures for the proton transfer from a water molecule to a hydroxide ion in the gas phase, in a continuum (6 = 78.4) and with two explicit water molecules. Distances are given in angstroms and angles in degrees. necessary for an accurate discription of the hydroxide ion in liquid water. 33. Hydroxide Migration in Water. We also analyzed the possibility of hydroxide migration in water by means of the proton transfer from a water molecule of the first shell to the hydroxide anion. For this purpose we have analyzed the proton transfer from a water molecule to the hydroxide anion in three different surroundings: the process in the gas phase, in a continuum, and on addition of two explicit water molecules. The transition smctures obtained for these three models (structures IX,X, and XI, respectively) are given in Figure 4. Both in gas phase and in a continuum (structures M and X) the hydrogen is centered between the two oxygen atoms in the transition structures. The main geometrical change between
both structures is the conformation of the outer hydrogens that are gauche in the gas phase and trans in solution. Although the dipole moment of the resulting structure is zero, the quadrupole moment increases significantly. The rest of the geometrical parameters remain essentially unchanged, as found also for the proton transfer between hydronium ion and water molecnle.6 In the discrete model (structure XI), because of the total negative charge of the system, the two additional water molecules are solvating the oxygen atoms. The 02H1 and OlHI distances are now very different, although the mean value (1.204 A) is similar to that of the transition stmcture in a continuum. The geometrical parameters of the two solvating water molecules appreciably participate in the transition vector. In fact, this transition structure leads to two symmetrical minima whose smcture corresponds to that of structure N where the hydroxide are the 0 1 and H2 atoms or 0 2 and H3 atoms and one of the two additional water molecules is placed in the second solvation shell. Thus, in this process, second-shell water molecules play an important role because they belong to the first solvation shell of the new hydroxide ion once the proton has been transferred. As was said before, structure IV can be seen as a minimal model of the whole system with two solvation shells. Comparing the structure of the minimum and of the transition state, it can be noticed that the two water molecules which do not participate directly in the reaction suffer a considerable rearrangement during the process. The hydrogen bond with the proton acceptor oxygen is lengthened from 1.634 to 1.775 A, while the hydrogen bond with the proton donor oxygen is shortened from 2.055 to 1.851 A. The hydrogen bond between the two water molecules is also increased from 2.055 to 2.301 A. Imaginaty frequencies, energy barriers, enthalpies and free energies for the three models are given in Table 4. The reaction barrier in the gas phase is very small and it disappears when considering zero-point energies. The loss of entropy in the normal model responsible of the reaction makes the free energy barrier slightly positive in the gas phase at the HF level, but it remains negative when correlation is included (MPUlHFI calculations). Similar results for the energy barriers are found for the proton transfer between hydronium ion and water in the gas phase? Solvation, both with a continuum and with two discrete solvent molecules, increases the energy barriers, because of the trend of the solvent to separate charges. Energy barriers obtained with the continuum and the discrete models are similar, mainly at the MPUIHF level. The energy barriers obtained with
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TABLE 5: Mayer Bond Indices of Transition Structures at the HF/6-31+G* Level structure OlHl 02H1
Ix X XI
0.267 0.256 0.269
0.267 0.256 0.241
these two solvation models compare well with experimental estimations that range from 2.1 kcal/mo14' to 4.8 kcaYmo1.42 Activation free energies in these two models are also very small, above all when electron correlation is included (MW/HF results). Thus, it seems that hydroxide migration can take place easily by means of a proton transfer from a water molecule of its first solvation shell. Note that the entropic contribution is nearly zero in the discrete model. This results from the compensation of the loss of entropy in the reaction normal mode and the gain of entropy in the two water molecules that are not so strongly bonded in the transition structure as in the minima. It should be pointed out that the increase of the barrier height by solvation is accompanied by the increase of the distance between the minima along the proton-transfer reaction path. The 0 1 0 2 distance for the minima in the gas phase, in the discrete model, and in the continuum is 2.614, 2.737, and 2.803 A, respectively. Thus, the perturbation introduced by the solvent in both models verifies the Hammond postulate.43 The Mayer bond indices for the broken and formed OH bonds of the three structures are given in Table 5 . In spite of the slight shortening of the O l H l and 02H1 bonds, these are weakened when going from gas phase to the continuum. This is due to the polarization of the electronic cloud toward the cavity surface. In the discrete model (structure XI) these bonds are no longer equal because of the asymmetry of the surrounding water molecules. However, the mean value of the bond indices (0.255) is very close to the value found in the continuum model (structure X). Therefore, it seems that our continuum model for the proton transfer reproduces the main trends of the discrete model. Obviously, the continuum model keeps the symmetry of the solvated species, so that the asymmetry observed in the discrete model cannot be reproduced.
4. Conclusions The main features of hydroxide solvation in water, structure, energetics, and proton transfer from a water molecule, have been analyzed by means of discrete and discrete-continuum models. Regarding the structure, our calculations show that hydroxide anion has a fvst solvation shell composed of three water molecules, in agreement with experimental work^.'^,*^ In the gas phase, the minimum energy structure for the OH-(H20)3 ion (structure III) is a branched one, where the outer water hydrogens are slightly oriented toward the neighboring water oxygens. To our knowledge, this is the first time that this structure is considered. The MP2//HF formation free energy of this species agrees very well with experimental estimates.19 Distance between water hydrogens and hydroxide anion are similar to that found for the hydronium ion5 but shorter than hydrogen bonds between water molecules.3l This behavior of the intermolecular distances can highlight the problem of the definition of the cavity size to be used for anions in the continuum model, supporting the use of smaller cavities both for positively and negatively charged species, with respect to the neutral ones, in spite of the fact that the electronic cloud of the anions is expected to be rather diffuse.44 Solvation, both with discrete water molecules and with a continuum, modifies the geometry of the central OH-(H20)3 ion by favoring a branched structure with the outer hydrogens
trans with respect to the hydroxide hydrogen. This structure is stabilized by a larger solvation energy mainly due to the increase of the dipolar term. Both solvation models agree in the description of the variation of the geometrical parameters upon solvation, except for the distance between the hydroxide oxygen and water hydrogens: the gas phase distance is shortened in the discrete model but lengthened in the discrete-continuum model. The neglect of the charge transfer beyond the cavity seems to be responsible for this discrepancy. Other differences between both models can be rationalized by taking into account the fact that in the discrete model, second-shell water molecules mainly interact with the oxygen atoms of the fist-shell water molecules, while in the continuum model the dielectric interacts more efficiently with the outer atoms, here hydrogens. In spite of these differences, it has been shown that the continuum model is able to reproduce cooperative effects, such as the strengthening of hydrogen bonds by the formation of virtual hydrogen bonds with the continuum. The discrete-continuum model employed here is shown to be a good tool to estimate solvation free energies. In agreement with previous works5Js the error is about 7%. The lack of dispersion contribution in our computations could be the reason for the discrepancy. The use of a discrete model with six explicit water molecules plus a continuum does not lead to a better evaluation of the solvation energy. The fact that no geometry reoptimization has been carried out in this case and the lack of a statistical treatment of the second solvation shell could be the reasons for the discrepancies found. Finally, we have also investigated a possible mechanism for proton migration in water by means of proton transfer from a first-shell water molecule. Although solvation increases the activation energy for this process, consideration of activation enthalpies and free energies led us to conclude that this is a feasible alternative for the simple diffusion migration of the hydroxide ion in water. The use of the discrete model shows that in this proton transfer solvent coordinates strongly participate in the transition coordinate and cause an asymmetric transfer of the proton that cannot be found in the continuum model. Second-shell water molecules are necessary to correctly describe this process.
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