J. Phys. Chem. 1993,97, 11087-1 1091
11087
Proton Transfer between Water Molecules. A Theoretical Study of Solvent Effects Using the Continuum and the Discrete-Continuum Models Francisco R. Tortonda,' Juan-Luis Pascual-Ahuir, Estanislao Silla,' and Iiiaki Tuii6n Departamento de Quimica Fisica, Universidad de Valencia. 46100 Burjasot, Valencia, Spain Received: May 26, 1993'
Proton transfer between two water molecules both in the gas phase and in solution has been studied at the HF/6-311G** level. For the gas phase we have characterized the reduced potential energy surface. Proton transfer in solution has been modeled with a pure continuum model and with a mixed discretecontinuum model where the two water molecules between which the proton is transferred are surrounded with four more water molecules and a continuum. Both in the gas phase and in solution the proton transfer between the two water molecules presents a double-well potential with a barrier of 0.30 kcal/mol in the gas phase and, in solution, 1.54 kcal/mol in the continuum model and 0.99 kcal/mol in the mixed discretecontinuum model. The reaction path, within the reaction-solvent equilibrium hypothesis, is nearly coincident for both phases, and the transition structure hardly changes. Differences between the pure continuum model and the discrete+ontinuum model are rationalized by means of a double effect played by the discrete solvent molecules: an electrostatic effect equivalent to the continuum model and a charge delocalization effect which increases the proton acceptor character of the water molecules. By using the discrete model, it is also shown that solvent parameters can play an important role in the description of the proton transfer.
Introduction The proton-transfer process has been widely studied theoretically' because of its fundamental nature in many biological and chemical reactions.2 For instance, proton-transfer processes are involved in catalysis of bond cleavage,3 synthesis of ATP: ionic conduction mechanism of ice,5 storage of biochemical energy,6 and all those processes controlled by an acid-base mechanism. Among all the proton-transferreactions, one of the most studied theoretically7-*6as well as e~perimentally*~-l9is that taking place between water molecules, in particular the H5O2+ system. This proton-transfer process is perhaps the most important because of the widespread presence of water as solvent in biological and industrial processes. For the H502+ system, it seems wellestablished by quantum calculationsthat the proton transfer takes place along a double-well path with a considerable barrier for long 0-0distances, but this barrier is lower for shorter distances, and it remains an open question whether the absolute minimum of the potential energy surface is the proton-centered structure or whether this structure is a true transition state.I2 At the semiempiricallevel' as well as the HartreeFock level with small basis sets," the proton-centeredstructure appears as the absolute minimum in the potential energy surface. The influence of the calculation level used, Le., basis sets and level of correlation, has been thoroughly studied by Del Bene et al.,12-15who have determined that the proton-centeredstructure is a true transition state at the HartreeFocklevelof theory with large basis sets and the noncentered structures are the minima. However, when correlation energy is included, the proton-centeredstructure has an energy lower than that of the noncentered structures. The flatness of the potential energy surface in the zone of the reaction path seems to be the reason of these highly theory-dependent results. Because of this flatness, solvent effects on proton transfer could be expected to be relevant. In fact, an activation free energy of 2.4 kcal/mol has been experimentally obtained for the proton transfer between a hydronium ion and a water molecule in aqueous s0lution.1~Solvent effects have been incorporated into the study
* To whom correspondence should be addressed.
Permanent address: I. B. Monastil, Elda, Alicante, Spain. Abstract published in Advance ACS Abstmrs, October 1, 1993.
0022-3654/93/2097-11087304.00/0
of the HsO+ system in aqueous solution by Monte Carlo,20-21 molecular dynamics,22 and more recently by some of us with a quantum-continuum model.23 By means of a mixed discrete continuum model at the HF/6-31G* level, we obtained almost 95% of the total solvation free energy of the proton. In spite of the interest about hydronium solvation, only a few works have dealt with solvent effects on proton transfer between water molecules. At the HF/4-31G level, Scheiner found that the addition of more water molecules to the H502+ system had little effect on proton transfer24and that the proton-centeredstructure was the absolute minimum just as in the gas phase. Schuster et aLz5and Bertrln et a1.,26 by using a semiempiricalmethod, found that solvation with discrete water molecules stabilizes the noncentered structures. Karlstrom" has dealt with solvent effects on the H5O2+ system by using a continuum model. He used the image charge approximation28 to simulate the dielectric and a spherical cavity. In a limited search of the free energy surface he found a double-well potential with a barrier of 0.96 kcal/mol. However, both the cavity shape and the computational approach were not completely satisfactory. In this paper we make an analysis of the reduced potential energy surfacefor the proton transfer between two water molecules in the gas phase and of the reduced free energy surface in solution. This is a free energy surface in the sense that it incorporates the free energy of solvation but not the entropy due to the internal and translational degrees of freedom. The reduced potential energy surface is calculatedat the ab initiolevel without geometry constraints and without any a priori symmetry assumption. All the geometrical parameters are optimized for each set of values of the independent variables. We have used the HF/6-311G** level of calculation. Following this, the points on the reduced surface have been solvated without reoptimization by means of a quantum-continuum model, and in this way we have obtained the reduced free energy surface in solution. The main features of this free energy surface are the double-well potential and that the proton-transfer path coincides very largely with that in the gas phase. The main changes introduced by the solvent is the considerably higher barrier, and the positions of the minima are moved backward along the proton-transferpath. The conclusions obtained with the continuum model are compared with that obtained with a mixed discretecontinuum model. In this case 0 1993 American Chemical Society
Tortonda et al.
11088 The Journal of Physical Chemistry, Vol. 97, No. 42, 1993
Results and Discussion We have obtained the reduced potential energy surface (PES) for the proton-transfer process in the gas phase studying the dependence of the energy on the 01-H1 and 01-02 distances. Geometricparameters are shown in Figure 1,and PES is presented in Figure 2. To obtain this surface, more than 110 ab initio points were calculated. Minima and the transition structure were directly located and characterized on the PES with the optimization methods described before and are marked on the surface. The two minima, which are symmetrical, appear for a 01-02 distance of 2.406 A and a 01-H1 (or 02-H1) distance of 1.064 A, while the proton-centered structure is a true transition state with 01-02 = 2.359 A and 01-H1 = 1.180 A. The energy of the minima with respect to the H30+ H20 system is -33.38 kcal/mol. It can be seen how the proton-transfer path takes place in a very flat zone of the PES. The barrier height for the proton transfer is 0.30 kcal/mol. These features of the PES topology explain the difficulties of establishing whether the absolute minimum is the proton-centered structure or not and its dependence on the level of theory used. It must be pointed out that when no symmetry or geometry constraints are used, the proton transfer does not take place exactly on the line connecting the oxygen centers. However, deviation of the transition structure of the exactly symmetrical proton-centered structure is small. In this way the PES is not completely symmetrical, and there is another equivalent path for the proton transfer from the second water molecule to the first. In Figure 3 we present the reduced free energy surface (FES) obtained from the points calculated in the gas phase solvated with the continuum model described in the previous section. The stationary points together with that obtained in the gas phase are marked on the surface. The minima appear at a 01-02 distance of 2.450 A and a 01-H1 (or 02-H1) distance of 1.021 A, while the transition structure is nearly coincident with that in the gas phase, 01-02 = 2.370 A and 01-H1 = 1.185 A. It can be seen that, although very similar, the FES in solution is not as flat as the PES in the gas phase. In particular, the barrier height for the proton transfer is now 1.54 kcal/mol, Le., 5 times greater than in vacuo. For a better comparison, we show in the corner of the figure the curves obtained for a 01-02 distance of 2.40 8, both in the gas phase and in solution. It can be seen how the barrier is considerably higher in solution and how the minima appear more separated than in the gas phase. It is also worth noting how the proton-transfer reaction path in solution overlaps with the path in the gas phase. It can be seen on the FES that in solution the path is nearly the same but longer than in the gas phase. This fact can be explained because a dielectric reduces the ion-dipoleinteraction by a factor of E , and then, the equilibrium distance between the hydronium ion and the water molecule becomes greater. Thus, the energy of the minimum with respect to the H30+ H2O system is now only -12.06 kcal/mol. It can be concluded that the solvent, at least as dielectric, makes the proton transfer more difficult because of the greater stabilization of the isolated H30+ structure in comparison with the protoncentered structure. Table I resumes the main characteristics of the stationary structures found in the gas phase and in solution with the two models used: the pure continuum model and the discrete model. In this table the geometric parameters, the Mulliken charge on the transferred proton, the activation energy, and the electrostatic solvation free energy are given. The geometric parameters refer to Figure 1. As in the case of the continuum model, we also found in the discrete model a larger 01-02 distance for the minima than in the gas phase while the transition structure changes only slightly. The energetic barrier is more than 3 times that of the gas phase and twice if we remove the continuum. It is interesting to note how the distances to the first solvation shell change from the minimum to the transition state. In the minimum
+
Figure 1. Systems for which the proton transfer has been studied.
each of the two water molecules is solvated with two explicit water molecules. The minimum and transition states have been located and afterward solvated with a continuum. The barrier obtained in this way is close to that of the pure continuum model.
Methodology Calculations have been carried out at the Hartree-Fock level with the MONSTERGAUSS program29 by using the 6-31 1G** basis set.30 This basis set is flexible enough to give interaction energies for the H5O2+ system in good accordance with larger basis sets.15 Geometries have been fully optimized with the gradient technique of Davidon” up to the total gradient was less than 5 X 10-4 mdyn A-l. Appropriate curvature for minima and the transition structure were checked with the VA05 Solvent calculationshave been carried out with the PCM model of Tomasi’s group33 by using a dielectric constant of 78.4, corresponding to that of water at 298 K. The cavity chosen for the continuum calculations was the solvent-excluding surface as defined by Richards34 calculated by means of the GEPOL p~ogram.35J6As usual, the radii of the initial spheres (Ro = 1.68 A and RH = 1.44 A) were 20% larger than the van der Waals ones. Calculations were also carried out with a modified MONSTERGAUSS program at the HF/6-3 1 1G** level. Nonelectrostatic contributions to the solvationfree energy (cavitation and dispersion-repulsion) have not been considered. As we have recently shown,37these contributions are surface area dependent, and as long as the maximum difference in the areas of the minimum and the transition state is less than 1.6%, their differentialcontribution to the free energy surface should be small. Minima and the transition structure in the continuum model were located varying the oxygen-xygen and hydrogen-oxygen distances by 0.002 A. In the discretecontinuum model two equivalent water molecules were added, bonded by hydrogen bonds, to each of the two water molecules involved in the proton transfer, as can be seen in Figure 1. An extra water molecule solvating the lone pair of the oxygen atom of the hydronium ion has been shown not to be strongly bonded in solution.23 The internal geometry of these new four water molecules has been kept fixed to their optimized values ( r = 0.9410 A, 0 = 105.46O). The rest of the variables of the system have been optimized to locate the minima. The transition structure has been characterized using the VA05 subroutine. Without further reoptimization,these structures have been solvated with the continuum model described above. For these calculations we have also used the HF/6-311G** level of theory and the PCM model.
+
Proton Transfer between Water Molecules 2.70
The Journal of Physical Chemistry, Vol. 97, No. 42, 1993 11089
0.95 1.00 1.05 1.10 1.15 1.20 1.25
1.30 1.35 1.40
1.45 1.50 1.55 1.M)
1.65
1.70 1.75 1.80 I
2.60
2.50
2.40
2.30
2.20
0.95
1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75 1.80
d(O1-El)
(A)
Figure 2. Potential energy surface for the proton transfer in the gas phase. Minima (A) and transition structure (X) are marked on the surface. Isoenergetic lines are in kcal/mol. 2.70
0.95
1.00 1.05 1.10 1.15
095
100
1.35
1.44 1.45 1.50
1.20
1.25
1.30
120
125
1 3 0 135 1.44
1.55
1.60
1.65
1.70
1.75
1.80
155
1.60
1.65
170
175
180
2.60
2.50
2.40
2.33
2.20
105
1 1 0 115
145
150
d(Ol-H1)
(A)
Figure 3. Free energy surface for the proton transfer in solution (e = 78.4). Minima (A),transition structure (X), and stationary points of the gas phase (0)are marked on the surface. Isoenergetic lines are in kcal/mol. In one corner the curves in the gas phase and in solution for an oxygen4xygen
distance of 2.40 A are shown.
the solvation shell is closer to the hydronium ion than to the water molecule because of the charge distribution, while in the transition structure the first solvation shell is at an intermediate distance. This large variation of the solvation parameters along the reaction suggests that they play an important role in the description of the reaction coordinate. In Table I1 the most significant components of the transition vector are given. The largest components are associated with the movement of the transferred proton from one water molecule to the another. However, the solvent parameters
are also very important components of the transition vector. So, the proton-transfer reaction could also be regarded as promoted by fluctuations of the first solvation shell. If the solvation shell of the water molecule becomes closer to the system than the solvation shell of the hydronium ion, then the proton can be transferred. In fact, if we solvate only the proton acceptor water molecule of the HsOz+system with two solvent molecules and we leave the system to be relaxed, the proton is spontaneously transferred to the solvated water molecule. The role of solvent
Tortonda et al.
11090 The Journal of Physical Chemistry, Vol. 97, No. 42, 1993
TABLE I: Distances (A), Charges on the H1 au), Energies, and Electrostatic Solvation Free Energies (kcal mol) for the
I
StatiOMW ShUChlWS StUdied
gas phase min TS d(Ol-02) 2.406 2.358 d(O1-H1) 1.064 1.182 d ( 0 1-03) d(02-05) 0.490 0.495 Q(H 1)
AE or AG 0.0
GI 0
0.297
continuum min TS 2.450 1.021 0.474 (0.470)O 0.0 -70.02
2.370 1.185
discretbcontinuum min TS 2.430 1.034 2.655 2.796 0.455 (0.453) 0.0
0.507 (0.499) 1.54 (0.07) -50.63 -68.55
2.353 1.174 2.723 2.726 0.478 (0.469) 0.99 (0.67) -50.3 1
Values in parentheses are obtained removing the continuum.
TABLE Ik Main Components of the Transition Vector for the Discrete Model d ( 0 1-H 1) d(02-H 1) d(01-03)
0.65 1 -0.651 0.144
d ( 0 1-04) d(02-05) d(02-06)
0.144 -0.144 -0.144
parameters is so important that the same result is obtained by adding only one solvent molecule to the proton acceptor water molecule. The solvation parameters can also be used as independent variables. If the values of the solvent parameters of the proton donor water molecule and of the proton acceptor water molecule are interchanged and fixed and the rest of the system is reoptimized, the proton is transferred without any potential barrier. Thus, the overall process can be considered as if the proton adjusts its position according to the fluctuations in the distribution of the solventmolecules. This solvent-inducedproton transfer has been interpreted as a protonic polarizati~n,~~ and it has recently been studied by means of molecular dynamics separating the protonic motion from the rest of the motions of the system.39 The influence of solvent fluctuations on chemical reactions has been also studied in the framework of the continuum mode1.40~41 Returning to Table I, it is interesting to compare the predictions of the continuum model and of the discrete-ntinuum model. Regarding the geometry of the minimum, it can be seen that the change is intense when going from the gas phase to solution, the predictions of both the pure continuum model and the discrete model being very close. The oxygen-oxygen distance lengthens nearly 0.03 A in the discrete model and nearly 0.05 A in the continuum model while the 01-H1 distance shortens more than 0.03 and 0.04 A, respectively, while the position of the minima moves backward in the proton-transfer reaction path. This fact is a consequence of the reduction of the ion-dipole interaction caused by the solvation of both the ion and the dipole. In a pure electrostatic continuum model of the ion-dipole interaction, this is reduced by a factor o f t . Then the Hammond postulate4zis fulfilled because it is accompanied by an increase in the barrier height. A deeper insight into the different effects involved in the solvation with a pure continuum model and a discrete model can be obtained comparing the lengthening observed for the 0 1 - 0 2 distance in the minimum, when we add four discrete water molecules, with the trends obtained in our previous study of hydronium solvation.23 In that case the addition of a second solvation shell produced a shortening of the distances between the oxygen of the hydronium ion and the oxygens of the first solvation shell waters. That was explained because the second solvation shell water molecule increased the proton acceptor character of the first shell water molecules by means of a charge delocalization mechanism. In our present system when we add two water molecules to the hydronium ion we are stabilizing its excess charge and then diminishing the interaction with the water molecule. This is an electrostatic effect that is also included in the continuum model. However, when we add two water molecules
to the original water molecule of the HsO2+ system, we are increasing its proton acceptor character, and this would try to reduce the 01-02 distance. As said before, this comes from a charge delocalization mechanismz3and it is not included in the continuum model where the electron cloud is constrained to stay inside the cavity. The first of these two effects dominates in the discrete calculations, and we obtain an overall lengthening of the 01-02 distance of 0.024 A. 111fact, if we only add two water molecules to the hydronium ion and none to the proton acceptor water molecule (i.e., we study the HaO+(H20)3 system), we get an 0 1 - 0 2 distance of 2.589 A, about 0.16 A longer than in our present case. On the contrary, as said before, if we add one or two water molecules only to the proton acceptor water molecule of the H$Oz+system and not to the hydronium ion, the proton is spontaneously transferred as a consequence of the increase in the acceptor capability of the water molecule. These two effects of the discrete solvent molecules explain the differences between the pure continuum model and the discretecontinuum model. Since the continuum model is only able to give the charge (and dipole) stabilizing or electrostatic effect and not the increase of the proton acceptor character of the water molecule, the 01-02 distance obtained with this model is 0.02 A longer than that obtained with the discretecontinuum model. The transition structures in solution and in the gas phase are nearly the same with both models. Since the charge is now more delocalized over the entire system, the electrostatic effect is somewhat less intense than for the minima. Thus, the continuum model gives only a slight lengthening of the 01-02 distance. In the discrete model, the increasingof the proton acceptor character is now the dominant effect, and so a slight shortening is obtained. In fact, it can be seen from the Mulliken charge of the transferred proton that, while the pure continuum model favors a slightly more ionic mechanism,just the opposite happens in the discretecontinuum model. Small changes of the transition structure by solvation have also been found for the equilibrium solvation of other symmetrical reaction~.4.4~.~ However, the nature of the transition state can be changed considerably by solvation when there is an important difference in the polarity of reactives and products43or by nonequilibrium solvation effects.@~4*~* Both solvation models predict a higher barrier in solution than that obtained in the gas phase. The barrier with the discrete continuum model is 0.99 kcal/mol and in the pure continuum model is 1.54kcal/mol. The higher barrier obtained in solution comes from the different solvation energies of the minimum and the transition structure with the solvent. In the pure continuum model the minimum has a solvation energy of about 1.47 kcal/ mol larger than that of the transition structure, and this difference can be attributed to the larger charge delocalization found for the transition structure. Thus, more than 95%of the free energy bamer height in the pure continuum model (1.54kcal/mol) comes from the interaction of the HsOZ+system with solvent (1.47 kcal/ mol), and the rest comes from the gas-phase energies of the stationary structures obtained with the continuum model (0.07 kcal/mol). In the discrete+ontinuum model, the influence of the continuum is not so high because the cavity is now considerably larger. Thus, the interaction energy of the minimum with the continuum is now only 0.32 kcal/mol larger than that of the transition state. This differential interaction energy of the stationary points with the continuum (0.32 kcal/mol) contributes about 33% to the total barrier height (0.99 kcal/mol).
conclusions We havestudied the proton-transfer process between two water molecules at the HF/6-311G+* level both in the gas phase and in solution. The solvent effects have been simulated with two models: a purecontinuum model and a mixed discretecontinuum model. The PES for the proton transfer in the gas phase has been studied, and the critical points have been identified. By using the
Proton Transfer between Water Molecules continuum model of Tomasi’s we have solvated all the points of the PES obtaining the FES for the proton transfer in solution. Finally, we havealsoinvestigated theeffects of a discrete solvation shell adding four water molecules hydrogen bonded to the HsOz+ system, and the resulting structures were solvated with the continuum model. It has been found that while the existence of a double-well potential for optimized oxygen-oxygen distances is not completely established in the gas phase,12 it seems that equilibrium solvent effects clearly favor this. With the two models employed to simulate solvent effects, a higher potential barrier than in the gas phase has been found. The transition structures and also the reaction paths have been found to be very similar in the gas phase and in solution within the equilibrium solvation hypothesis because of the nearly symmetrical natureof the reaction. On thecontrary, the positions of the minima on the reaction path are changed considerably, lengthening the 01-02 and 01-H1distances. This fact can be explained by the stabilizing effect of the ion and the dipole by the solvent and is more pronounced in the pure continuum model than in the discrete model because in the latter there is also a charge delocalization effect due to the new water molecules that increases the proton acceptor character of the original water molecule.23 So, the differences between the pure continuum and the discrete-continuum models can be rationalized by means of the double effect played by the discrete solvent molecules: an electrostatic effect included also into the continuum model and a charge delocalization effect. It has also been shown that the solvation parameters play an important role in the correct description of the reaction coordinate. Thus, fluctuations of the solvation shells of the hydronium ion and of the water molecule of the H~02+ system can provoke the proton transfer. In fact, if the solvation shell of the hydronium ion expands and that of the water molecule contracts, the proton is transferred spontaneously without any potential barrier.
Acknowledgment. Calculations were made on a VAX4000 and on a RISC-6000/550of the Departamentode Quimica Hsica de la Universitat de Valencia (SEUI Project OP90-0042 and with a grant of the Conselleria de Educaci6 i Ciencia de la Generalitat Valenciana). I.T. thanks the Ministeriode Educaci6n y Ciencia (Spain) for a doctoral fellowship. This work was supported in part by DGICYT (Project PS90-0264). References and Notes (1) Scheiner, S.; Redfern, P.; Hillebrand, E.A. In?. J. Quantum Chem. 1986, 29, 817. (2) Jeffrey, G. A.; Saenger, W. In Hydrogen Bonding in Biological Structures; Springer-Verlag: Berlin, 199 1. (3) Polgar, L.; Halasz, P. Biochem. J. 1982, 207, 1. (4) Negrin, R. S.;Foster, D. L.; Fillingname, R. H. J.Biol. Chem. 1980, 55, 5643.
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