J. Phys. Chem. 1996, 100, 16495-16501
16495
Kinetics of the Proton Transfer in X‚‚‚(H2O)4 Clusters (X ) H2O, NH3, H2S, and HCl): Evidence of a Concerted Mechanism Marc Planas,† Chengteh Lee,‡ and Juan J. Novoa*,† Departament de Quı´mica Fı´sica, Facultat de Quı´mica, UniVersitat de Barcelona, AV. Diagonal 647, 08028-Barcelona, Spain, and Premis Corporation, 15301 Highway 55 West, Plymouth, Minnesota 55445 ReceiVed: March 14, 1996; In Final Form: June 27, 1996X
The kinetics of proton transfer of acids, bases, and water in aqueous solutions has been studied at the ab initio level, using a nonlocal exchange and correlation density functional and an extended basis set. As a first step toward a full comprehension of the process, we have used as model system the X‚‚‚(H2O)4 clusters (X ) NH3, H2O, H2S, and HCl). These are the smallest water clusters for which the neutral form, B‚‚‚(H2O)3‚‚‚AH, and the double ionic form resulting from the proton transfer, BH+‚‚‚(H2O)3‚‚‚A-, are minimum-energy structures in the potential energy surface of the X‚‚‚(H2O)4 cluster. Our results show that for these clusters the transfer takes place following a concerted mechanism. A transition state is found for all clusters except for the H2O‚‚‚(H2O)3‚‚‚HCl one, which shows a strong tendency to dissociate into the more stable double ionic form without barrier. For the other clusters, there is a small transition state displaced toward the double ionic products, due to the higher stability of the neutral cluster. Therefore, these clusters present a weak acid or base behavior, except the H2O‚‚‚(H2O)3‚‚‚HCl cluster which follows one of a strong acid.
Introduction Proton transfer reactions are one of the most important chemical and biological processes.1,2 These reactions involve the transfer of one proton from a proton donor substance AH (also called the acid) to a proton acceptor substance B (also called the base), according to the following scheme:
AH + B h A- + BH+
(1)
Previous theoretical studies on various systems3 have indicated that such reactions do not take place in the gas phase, even for strong acids4 such as HCl, and only take place in aqueous media.5,6 The lack of reaction in the gas phase can be understood by looking at the energetics of reaction (1) in isolated 1:1 complexes between AH and B: the A- + BH+ ions resulting from reaction 1 lie much higher in energy than the neutral reactants. At the same time, it is known that some solid monoand dihydrates1,7 do not show the presence of the ionic A- + BH+ products. Apparently, one needs a large number of water molecules to allow the formation of stable hydrated ions. The importance of proton transfer reactions makes appealing the study of the proton transfer mechanism at the molecular level using ab initio methods. However, it is necessary to have large enough water clusters so that the hydrated ions are minimum-energy structures in the potential energy surface of the whole cluster. In the absence of such model systems, there have been two different approaches in the literature. The first one6 was based on the use of a continuum model of the solvent and allowed to show the presence of a minimum for the double ionic products of HCl when the dielectric permittivity of the medium was that of water. The other approach,5 also carried out on the HCl, was to perform Monte Carlo simulations in a cube containing 250 water molecules to locate the configurations in which the solvated reactants and solvated products have the †
Universitat de Barcelona. Premis Corporation. X Abstract published in AdVance ACS Abstracts, September 15, 1996. ‡
S0022-3654(96)00789-7 CCC: $12.00
same energy. At these geometries, in the fixed field of the solvent molecules, the proton transfer is studied using ab initio methods. An alternative for the study of the proton transfer mechanism is to use small water clusters for which the neutral and double ionic products of reaction 1 are minimum-energy structures. If one can find reasonably small water clusters of this type, one can study the process in these clusters and look at the changes caused by the inclusion of more water molecules in the mechanism. We can use initially the B‚‚‚(H2O)3‚‚‚AH clusters, which present minimum-energy structures for the neutral and double ionic forms in the potential energy surface of the AHB-(H2O)3 system.8-10 These clusters are not large enough as to allow a full description of the reorganization generated by the proton transfer process in the structure of the solvent. In water solutions these clusters would be surrounded, at least, by a first and second solvation shell whose structure will change during the proton transfer process. However, the AH-B(H2O)3 clusters can provide useful information to understand the nature of the process in the absence of these solvation shells and to understand the importance of the solvent reorganization by comparing the results obtained here with those for larger solvated clusters. Stable double ionic BH+‚‚‚(H2O)3‚‚‚A- clusters have been found up to now for three families of compounds:8-10 The first is that formed by AH ) FH, HCl, and H2S and B ) H2O, which can serve as prototypes for strong and weak acid ionization. The second is for the A ) B ) H2O case, a good starting model for the study of the autoionization of water. The third is for A ) NH3 and B ) H2O, the prototype of a weak basic ionization. In all these cases, the geometrical structure of the BH+‚‚‚(H2O)3‚‚‚A- clusters is similar, that is, one in which three water molecules are located between the positive and negative contraions in the equator of a trigonal bipyramid. In this study we will focus our attention on the proton transfer kinetics in the following clusters: NH3‚‚‚(H2O)3‚‚‚H2O, H2O‚‚‚(H2O)3‚‚‚H2O, H2O‚‚‚(H2O)3‚‚‚H2S, and H2O‚‚‚(H2O)3‚‚‚HCl. All of these clusters have an overall formula X‚‚‚(H2O)4 (X ) NH3, H2O, © 1996 American Chemical Society
16496 J. Phys. Chem., Vol. 100, No. 41, 1996
Figure 1. Geometrical conformation of the double ionic BH+‚‚‚ (H2O)3‚‚‚A- clusters: (a) NH4+‚‚‚(H2O)3‚‚‚OH-, (b) H3O+‚‚‚ (H2O)3‚‚‚OH-, (c) H3O+‚‚‚(H2O)3‚‚‚SH-, and (d) H3O+‚‚‚(H2O)3‚‚‚Cl-.
H2S, and HCl), and when necessary, we will identify them as the NH3, H2O, H2S, and HCl clusters, respectively. They can serve as models to study proton transfer for a weak base, pure water, a weak acid, and a strong acid, respectively. Figure 1 shows the geometry of these clusters for the double ionic conformations. Why three water molecules and not a smaller number in the equator of these clusters? The question is important methodologically because there have been in the literature proposal of models which use just one water molecule as a bridge between the two contraions. The answer is that three is the minimum number of waters we have to use in the equator to stabilize the double ionic form of the cluster. We have explicitly shown this10 for the NH4+‚‚‚(H2O)3‚‚‚OH- cluster, but the reasons are also valid for the other clusters studied here. In effect, full geometry optimization of the geometry of the NH4+‚‚‚ (H2O)2‚‚‚OH- and NH4+‚‚‚(H2O)1‚‚‚OH- clusters, starting form the optimum geometry of the NH4+‚‚‚(H2O)3‚‚‚OH- cluster by erasing one and two waters, gives rise to the neutral structures after a proton is transferred from the NH4+ into the OH- group. Such a fact was rationalized by comparing the molecular electrostatic potential (MEP) map11 of the double ionic mono-, di-, and trihydrated clusters. The MEP maps showed10 that for the three hydrated cluster the most attractive region is located on the water oxygens and the OH- potential is totally shielded by the waters, while for the other two clusters the OH- looses part of the shielding, the OH- region becomes the most stable region for a positive charge, and the potential around the water oxygens is significantly less stable. As a consequence, the proton present in the BH+ contraion is transferred back to the OH- fragment. Given the similar geometries of all the BH+‚‚‚(H2O)3‚‚‚A- clusters and the nearly identical features of their MEP maps in the region of the equatorial waters, one can expect that the same explanation will hold for all the clusters studied here. The aim of this paper is to study the kinetics of the proton transfer process which takes place between the neutral and doubly ionic forms of the BH+‚‚‚(H2O)3‚‚‚A- clusters as a first step toward a full understanding of the proton transfer in a water solution. As mentioned above, we are using a small cluster model which cannot describe all the solvent recombination features probably present in the bulk. Consequently, our model can help to understand the nonrecombination factors. The
Planas et al.
Figure 2. Geometrical conformation of the neutral and ionic forms of the AH-B-(H2O)3 clusters for AH ) SH2 and B ) H2O: (a) B‚‚‚(H2O)3‚‚‚AH, (b) B‚‚‚(H3O)+(H2O)2‚‚‚A-, (c) BH+‚‚‚(H2O)2 (OH-)‚‚‚AH, and (d) BH+‚‚‚(H2O)3‚‚‚A-.
analysis of the possible mechanisms for the proton transfer indicates that such transfer can take place though a concerted and a step-wise pathway. Therefore, a proper answer to the question of what mechanism is followed is only possible if we know the potential energy surface along the two pathways. Therefore, we have structured the study in the following steps: We will start by systematically computing the region of the potential energy surface of the cluster which contains the pathways of interest. Then, we will analyze the shape of these surfaces using diabatics to define the allowed pathways, therefore extending and systematizing a previous study on the H3O+‚‚‚(H2O)3‚‚‚OH- cluster,12 in which a concerted transition state connecting the neutral and ionic forms was found by testing a small part of the potential energy surface along the reaction coordinates. Then, we will fully optimize the geometry of the reactants neutral clusters and the products double ionic forms, as well as the transition states present between them. Finally, on the fully optimized geometries we will perform a vibrational analysis to properly characterize all the stationary points. At this point we will compare the mechanisms for all the proton transfers studied here. All the computations will be carried out at the ab initio level, within the density functional methodology, using the nonlocal Becke exchange13 and Lee-Yang-Parr correlation14 functionals. Computational Details To define the area of the potential energy surface which has to be mapped out, we have to fully understand the main characteristics of the concerted and stepwise pathways of the potential energy surface of the cluster. These two pathways are defined hereafter for the type of clusters employed in this study. The stepwise pathway is always a two-step process which involves two consecutive proton transfers and requires the existence of a stable intermediate. Depending on the order in which these two steps take place, we have different intermediates. If the proton is transferred first from the AH fragment of the neutral B‚‚‚(H2O)3‚‚‚AH cluster (Figure 2a) to the equatorial water which is hydrogen-bonded, the B‚‚‚(H2O)2 (H3O+)‚‚‚Aintermediate (Figure 2b) is obtained. This intermediate has to be stable in order to allow for the presence of a second step
Proton Transfer in X‚‚‚(H2O)4 Clusters (which competes with the reverse of the first step) in which one proton from the H3O+ fragment of the B‚‚‚(H2O)2 (H3O+)‚‚‚A- intermediate is transferred to the B fragment, to form the double ionic BH+‚‚‚(H2O)3‚‚‚A- cluster (see Figure 2d). Alternatively, if the first proton transferred is from the central water to the attached B fragment, one obtains the BH+‚‚‚(H2O)2 (OH-)‚‚‚AH intermediate (Figure 2c). In this case, the final double ionic cluster is obtained after transferring a proton from the AH fragment of the BH+‚‚‚(H2O)2 (OH-)‚‚‚AH intermediate into the OH- ion. The existence of a stepwise pathway requires that at least one of the intermediates mentioned above is stable. The presence of an stable intermediate implies the existence of two transition states, located in the pathways which connect the intermediate with reactants and products. The concerted or synchronous pathway is a one-step process in which the two proton transfers take place simultaneously, and no stable intermediate is involved. Now, while the AH fragment transfers the proton to one of the equatorial waters attached to it, simultaneously another proton is transferred from the same water to the B fragment. This process takes place through a transition state which is located around the point in which the two protons are in the middle of their transfer process. However, as will be discussed below, the exact position of this transition state and the barrier height strongly depends on the energy difference between the neutral and double ionic forms of the cluster. In some cases, when the energy difference is very large, the barrier can even disappear, and only one minimum is found in the potential energy surface of the cluster. We can cover in a systematic way all the possible pathways between the initial (neutral) and final (double ionic) forms of the clusters by mapping the potential energy surface as a function of the degree of transfer of the two protons involved in the process, optimizing at the same time all other geometrical parameters of the cluster. We can measure the degree of transfer of the two protons by looking at R1 and R2, the O-H distance in the H2O‚‚‚B contact and the A-H distance in the A-H‚‚‚OH2 contact, respectively (see Figures 1 and 2). In the E(R1,R2) energy map, the starting neutral cluster is located in the lower left corner, the final double ionic cluster is on the upper right corner, the B‚‚‚(H2O)2 (H3O+)‚‚‚A- intermediate is on the upper left corner, and the BH+‚‚‚(H2O)2 (OH-)‚‚‚AH intermediate is on the lower right corner. In the E(R1,R2) potential energy maps, the stepwise pathways connect the lower left and upper right corners along the sides of the map, while the concerted pathway approximately goes along the diagonal. At each (R1,R2) point of the E(R1,R2) surface the geometry was optimized with respect to the other geometrical parameters using standard optimization techniques. The computations were done within the density functional theory framework. The Becke13 and Lee-Yang-Parr14 nonlocal exchange and correlation functionals15 (BLYP) were used as it has been shown that they are adequate for a proper description of hydrogen-bonded complexes16 and reproduce the structures and energetics obtained using the MP2 method for the neutral and double ionic forms of the (H2O)5 clusters.12 In all computations we used the recently developed TZ94+P Gaussian basis set, which includes diffuse and polarization functions.17 This basis set is a (5s,1p) set of primitives contracted to a [3s,1p] set for the H atoms, a (10,6p,1d) primitive set contracted to a [4s,3p,1d] set for the O and N atoms, and a (13s,9p,1d) primitive set contracted into a [5s,4p,1d] set for the Cl and S atoms. Once the important stationary points have been located on the potential energy surface, we have fully optimized the geometries of the clusters to define their structure, and
J. Phys. Chem., Vol. 100, No. 41, 1996 16497 vibrational analyses were carried out on the fully optimized points to confirm their minimum energy or transition state nature. All the computations were carried out using the appropriate options in the DGAUSS18 and GAUSSIAN-9419 codes. These two codes do not implement the solution of the Khom-Sham equations14 in the same form, and consequently, the total energy given for the same system by one is not always identical to that given by the other. However, both give very similar values for the interaction energies, optimum geometries, and vibrational frequencies. Consequently, we consistently computed the interaction energies on each potential energy surface, and no distinction will be made about the code employed. Results and Discussion The E(R1,R2) potential energy surfaces for the proton transfer computed using the BLYP method and the TZ94+P basis set are shown in Figure 3 for the NH3, H2O, H2S, and HCl clusters. Except for the H2O‚‚‚(H2O)3‚‚‚HCl cluster, the shape of the E(R1,R2) potentials is pretty much the same. Two minimumenergy structures, for the neutral and double ionic forms, are found in the NH3, H2O, and H2S clusters, and no stable intermediates are found along the stepwise pathways (i.e., along the sides). There is only one possible transition state in these surfaces, the one which connects the neutral and double ionic forms through the concerted pathway. For the H2O‚‚‚ (H2O)3‚‚‚H2O cluster this transition state is similar to the one already reported in the literature.12 Our potential energy surface shows that it is the only possible one for this cluster. We also found that the neutral form of these clusters is the most stable one, but the energy differences between the neutral and double ionic forms are not large, and given the small size of the barrier in these surfaces, one can expect to find some of the clusters to be in their double ionic form at room temperature. Therefore, these X‚‚‚(H2O)4 clusters behave as weak acids or bases, depending on the relative acidity of the X molecule with that for water. The potential energy surface for the H2O‚‚‚(H2O)3‚‚‚HCl cluster is different from all others, and there is no minimum for the neutral form, the double ionic form being the more stable one. According to this surface, the H2O‚‚‚(H2O)3‚‚‚HCl clusters should evolve to the dissociation without a barrier in all cases; that is, these clusters should be always doubly ionized. These results agree with the fact that the HCl molecules in water solution have the behavior of a strong acid. One can rationalize the shape of the E(R1,R2) potential energy surfaces in terms of diabatics and their crossing. The E(R1,R2) adiabatic surface can be divided into many regions, each one representing one of the possible proton transfer steps in our mechanisms: four associated with the two stepwise mechanism (the four sides of the surfaces) and one with the concerted one (the diagonal of the potential surface). Each of these steps is of the type
R-H + R′ h R + R′-H
(2)
and involves the rupture of a R-H bond and the creation of a R′-H bond, R and R′ being the proper combination of the adequate A-, OH-, H2O, and B fragments. The adiabatic of each proton transfer step is the result of the interaction of the R-H and R′-H diabatics. The transition state is located near the crossing point of these two diabatics.20 If the R-H and R′-H systems have the same relative stability, the transition state is located around the middle of the transfer process. If R-H is much more stable than R′-H, the transition state is shifted to R′-H, and its size is decreased. When the energy
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Figure 3. Potential energy surface as a function of R1 and R2 (see text for definition) for the various clusters studied here: (a) NH3‚‚‚(H2O)3‚‚‚H2O, (b) H2O‚‚‚(H2O)3‚‚‚H2O, (c) H2O‚‚‚(H2O)3‚‚‚H2S, and (d) H2O‚‚‚(H2O)3‚‚‚HCl.
difference is very large, the shift is so large and the decrease in the barrier size so important that the barrier can disappear, and R′-H is no longer a stable minimum. On the other hand, if R′-H is much more stable than R-H, the transition state is shifted to reactants, lowering its barrier. If the energy difference becomes extreme, R-H can no longer be a minimum-energy structure in the adiabatic surface. Using the diabatics, we can rationalize the absence of a minimum-energy structure for the B‚‚‚(H2O)2 (H3O+)‚‚‚A- and BH+‚‚‚(H2O)2 (OH-)‚‚‚AH intermediates. These two intermediates present in their structure one positive and one negative contraion which are directly bonded. The electrostatic interaction energy between these two neighboring contraions is very large, and consequently, the formation of these two intermediates must be necessarily a highly endothermic process compared to the neutral fragments. Therefore, the neutral and intermediate diabatics cross near the minimum of the intermediate diabatic, the transition state disappears, and no minimum is found in the adiabatic for the intermediates. An analysis of the stability of the intermediates with respect to the double ionic forms allows
us to reach the same conclusion: in the double ionic cluster the electrostatic field created by one ion is strongly shielded by the equatorial water molecules while in the intermediates the two contraions are directly linked and have a much stronger interaction; the large repulsion gives rise to a crossing point for the two diabatics close to the minimum of the intermediate diabatic, and no minimum should be expected in this region of the surface. At this point we can focus our attention on understanding why the HCl cluster presents a different stability than the neutral B‚‚‚(H2O)3‚‚‚AH and double ionic BH+‚‚‚(H2O)3‚‚‚A- forms. As indicated above, the neutral form is the most stable one for the H2O, NH3, and H2S clusters by about 12, 5, and 4 kcal/ mol. For the H2O‚‚‚(H2O)3‚‚‚HCl cluster, no minimum is present in this region of the potential energy surface. However, one can estimate the energy difference between the double ionic minima and one representative point of the neutral region. This point was selected as the one with R1 ) 1 Å and R2 ) 1.3 Å, and the other geometrical parameters were optimized. The neutral-double ionic energy difference can then be estimated
Proton Transfer in X‚‚‚(H2O)4 Clusters to be -10 kcal/mol. To rationalize the presence of these two opposite behaviors, we carried out a decomposition of the total energy of the neutral and doubly ionic clusters as a function of the total energy of all its fragments (the intrafragment total energy, Eintra) plus the interaction energy between these fragments (Einter). According to this decomposition scheme, the total energy of any neutral B‚‚‚(H2O)3‚‚‚AH cluster can then be written as
E(neutral) ) E(AH) + E(B) + E(W3) + E(AH-B) + E(AH-W3) + E(B-W3) + Ep(neutral) ) E(neutral)intra + E(neutral)inter (3) Here, E(I) represents the total energy of each isolated I fragment in their cluster geometry, E(I-J) the interaction energy between the fragments I and J at their cluster geometry, and Ep(neutral) the rest of the energetic contributions (three-body interactions, etc.). The sum of the three E(I) components constitutes the E(neutral)intra term, while the sum of the rest of the terms constitutes the E(neutral)inter term. Each fragment is represented by its name and the three equatorial waters which are indicated by the symbol W3. The total energy of the double ionic BH+‚‚‚(H2O)3‚‚‚A- cluster can be written as
E(d-ion) ) E(A-) + E(BH+) + E(W3′) + E(A--BH+) + E(A--W3′) + E(BH+-W3′) + Ep(d-ion) ) E(d-ion)intra + E(d-ion)inter (4) Note that W3 and W3′ are different because the geometries of the three-water fragment is different in the neutral and double ionic clusters. The energy difference is then
∆E ) E(d-ion) - E(neutral) ) (E(A-) - E(AH)) + (E(BH+) - E(B)) + (E(W3′) - E(W3)) + (E(d-ion)inter E(neutral)inter) ) ∆Eintra + ∆Einter (5) The ∆Eintra term (the sum of the first three terms in parentheses) represents the changes in the energy of the isolated fragments, and the ∆Einter term (the last term in parentheses) represents the change in the interfragment interactions between the neutral and double ionic forms. Our computations show that the size of the (E(W3′) - E(W3)) term is similar for all the clusters studied here (about 8 kcal/mol). The values of ∆Einter for the HCl and H2S clusters are -192.6 and -196.2 kcal/mol, respectively, where the negative sign and the size reflect the much stronger interfragment interactions within the double ionic cluster. In the HCl and H2S clusters B ) H2O and, consequently, (E(BH+) - E(B)) is similar for the two clusters (164.2 and 164.5 kcal/mol, respectively). Therefore, the main difference is found in the values of the (E(A-) - E(AH)) terms: for the HCl cluster this term is equal to 337.4 kcal/mol, while for the H2S cluster it amounts to 357.0 kcal/mol. The roughly 20 kcal/mol decrease in energy outweighs the -4 kcal/mol contribution associated with the intramolecular terms and is the main reason for the higher stability of the double ionic form of the HCl cluster. Once the shape of the E(R1,R2) potential energy surfaces has been rationalized, we focused our attention on the stationary points found in these surfaces, to allow a precise computation of the energy difference between the minima and of the barriers. Therefore, we started by fully optimizing each stationary point. On the fully optimized geometries we carried out a vibrational analysis to test their stationary point nature, looking at the number of negative eigenvalues of the Hessian. Our fully
J. Phys. Chem., Vol. 100, No. 41, 1996 16499 optimized results confirmed the presence of the stationary points found by inspection of the surfaces: one neutral and double ionic minimum and a transition in the concerted pathway connecting these minima. The interatomic distances confirmed the nature of the fragments in each point and a Mulliken population analysis the net charges on each fragment:21 the neutral forms are composed of neutral fragments, while the double ionic forms and the transition state forms show the presence of A- and BH+ charged fragments. At the BLYP level, the net charge associated with the A- and BH+ fragments is -0.8 and +0.8, numbers which show the presence of a small charge transfer between these fragments. A MP2 computation using the BLYP geometry and basis set gave as a result -0.9 and -0.9, thus confirming the BLYP conclusions but with a smaller degree of charge transfer. The fully optimized results indicate the presence of a concerted mechanism of the proton transfer for the clusters studied here. This conclusion does not agree, for the HCl case, with the one reached before by Hynes et al. in their combined ab initio-Monte Carlo study using larger water clusters.5 These authors obtained a stepwise mechanism also with low activation barriers. Preliminary computations carried out by us indicate that the difference is probably due to the lack of the water molecules which form the first solvation shell of the cluster which allows for solvent reorganization. However, extensive computations are required to confirm this point. A detailed analysis of the properties of the stationary points and the proton transfer mechanism requires some numerical data. For such a purpose, we have included in Table 1 the most important properties of each stationary point for all the clusters in relation to the dynamics of the proton transfer: (1) R1 and R2, which define the geometry and the degree of advance of the proton transfer reaction on the E(R1,R2) surface, (2) the total energy (E), (3) the interaction energy (∆E) with respect to the dissociated neutral fragment, that is AH, B, and three water molecules, (3) the zero-point vibrational energy (ZPE), and (5) the zero-point corrected interaction energy (∆EZPE). These interaction energies have not been corrected by the possible basis set superposition error (BSSE). Given the similarities in the structure and bonds between the transition state and the double ionic products, we do not expect that the BSSE is going to affect the existence of a barrier connecting the neutral and double ionic forms. This is in good agreement with previous reported results for proton transfer barriers.22-24 In any case, we confirmed this fact computing the size of the BSSE on the transition state and double ionic points, and a value of 7.6 kcal/mol was found in the two cases at the BLYP/TZ94+P level. Lets start by looking at the energetics. First of all, all the stationary points are stable against dissociation into the corresponding neutral molecules, and consequently, the proton transfer does not imply a rupture of the cluster. The energy difference between the neutral and double ionic forms is 4.56, 11.83, and 4.41 kcal/mol for the NH3, H2O, and H2S clusters, respectively. The size of the barrier for formation of the double ionic form from the neutral one is not very large (5.91, 11.89, and 5.64 kcal/mol for the NH3, H2O, and H2S clusters). These energies are within the range of energies due to thermal activation at room temperature, and consequently, some clusters can transform into their double ionic form, although most clusters will remain in the more stable neutral one. Therefore, the NH3, H2O, and H2S clusters have a weak acid or base character. The HCl cluster, giving its strong tendency to dissociate, behaves as a strong acid. These conclusions, obtained on such small clusters, agree well with the experimental acid-base behavior of the NH3, H2O, H2S, and HCl molecules
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TABLE 1: Values of the R1 and R2 Distances (in Å), the Total Energy (E, in au), the Zero-Point Vibrational Energy (ZPE, in kcal/mol), the Interaction Energy (∆E, in kcal/mol) Relative to the Dissociation of the B‚‚‚(H2O)3‚‚‚AH Cluster into the Corresponding Neutral Molecules (AH, B, and Three Water Molecules), and the Zero-Point Corrected Interaction Energy (∆EZPE, in kcal/mol) for all the Stationary Points Found on the Potential Energy Surface of the B‚‚‚(H2O)3‚‚‚AH Clusters Studied Here stationary points cluster
parameter
neutral
transition state
double ionic
NH3‚‚‚(H2O)3‚‚‚H2O
R1 R2 E ZPE ∆E ∆EZPE R1 R2 E ZPE ∆E ∆EZPE R1 R2 E ZPE ∆E ∆EZPE R1 R2 E ZPE ∆E ∆EZPE
1.034 1.020 -362.370756 81.82 -34.61 -24.39 1.018 1.018 -382.258974 74.45 -36.05 -25.10 1.009 1.405 -705.175261 68.75 -26.04 -16.97
1.429 1.258 -362.361334 80.54 -28.70 -19.76 1.417 1.440 -382.240026 72.20 -24.16 -15.46 1.204 1.832 -705.166282 67.89 -20.40 -12.19
1.706 1.557 -362.363480 82.25 -30.05 -19.40 1.531 1.539 -382.240122 72.80 -24.22 -14.92 1.536 2.125 -705.168241 70.15 -21.63 -11.16 1.537 2.084 -766.619050 66.20 -42.03 -30.59
H2O‚‚‚(H2O)3‚‚‚H2O
H2O‚‚‚(H2O)3‚‚‚H2S
H2O‚‚‚(H2O)3‚‚‚HCl
in water solutions. On the other hand, the size of the barrier for the decomposition of the double ionic form to return to the starting neutral one (1.35, 0.06, and 1.23 kcal/mol for the NH3, H2O, and H2S clusters, respectively) is smaller than the barrier for the formation of the double ionic clusters (see above). When the zero-point corrections are included, the return barrier seems to disappear, and consequently, the double ionic forms would no longer be stable. However, one has to consider that the zeropoint corrections are obtained from the harmonic vibrational frequencies and that the intermolecular modes in our clusters involve intermolecular hydrogen bonds which are known to be strongly anharmonic.25 When this is the case, the anharmonic frequencies can be much smaller than the harmonic ones, and the zero-point corrections computed using anharmonic frequencies would be smaller than the harmonic zero-point correction.25 Therefore, one has to consider the zero-point corrections included in Table 1 as upper limit values of the real corrections. In any case, given the small size of the uncorrected barrier, one should expect short lifetimes for the double ionic forms. Now, we can shift our attention to the geometries. Lets begin by looking at the structural changes produced in these clusters during the proton transfer and analyzing the changes in the values of R1 and R2, the two geometrical parameters which present the main changes during the process. From our previous definition, R1 and R2 are the O-H distance in the H2O‚‚‚B contact and the A-H distance in the A-H‚‚‚OH2 contact, respectively. Their numerical values are different in each cluster, and consequently, one should analyze their variations during the proton transfer and understand the physical meaning of the variations. Small values of R1 are found in the neutral form of the cluster, and we talk of a O-H bond, while large R1 values are found in the double ionic forms and are an indication that we have an O‚‚‚H hydrogen bond. At the equilibrium geometry, a strong O-H or O‚‚‚H bond will have shorter values of R1. Therefore, the relative R1 values tabulated in Table 1 for the various forms of the clusters are an indication of the strength of the O-H or O‚‚‚H bonds in the H2O‚‚‚B contact. Similarly, the value of R2 is an indication of the strength of the
A-H (small R2 values) and A‚‚‚H (large R2 values) bonds found for the A-H‚‚‚OH2 contact in the neutral and double ionic forms of the clusters. Obviously, R2 is more cluster dependent. Thus, for the neutral forms, R2 is 1.405 Å for the H2S cluster and 1.018 Å for the H2O cluster and reflects the stronger nature of the O-H relative to the S-H bond. On the other hand, R1 is 1.018 Å in the H2O‚‚‚(H2O)3‚‚‚H2O cluster and 1.034 Å in the NH3‚‚‚(H2O)3‚‚‚H2O cluster. For the double ionic forms, R1 reflects the polarizing effect of the B group in the BH+‚‚‚O hydrogen bond and R2 the strength of the A-‚‚‚H hydrogen bond. Substituting the NH4+ group by the H3O+ group in a double ionic form, one induces a change of nearly 0.2 Å in the value of R1, while in the same type of cluster, R2 changes from 2.125 to 2.084 Å or to 1.557 Å when the A- group is changed from HS- to Cl- or to OH-. The transition state always has a geometry in between that of the neutral and double ionic forms. Concluding Remarks The overall picture that emerges from our study about the way in which the proton transfer takes place in the B‚‚‚ (H2O)3‚‚‚AH clusters is one of an equilibrium between the neutral and double ionic forms for all the clusters except the HCl one. Such equilibrium is strongly displaced to the neutral reactants but allows for the presence of a small amount of the double ionic forms, that is, the dissociated H3O+ and A-, when AH is a stronger acid than water, or of BH+ and OH-, when B is a stronger base than water. Therefore, the clusters present the same behavior as the equivalent weak acid or base in a water solution. In particular, H2S follows a weak acid behavior and NH3 a weak base behavior, in good agreement with the experimental data in bulk water solutions. Pure water clusters also present double ionic forms in a small amount, and the barrier for the formation of the double ionic forms is larger than that for H2S and NH3. Therefore, one has to expect a smaller amount of dissociated water than double ionized H2S and NH3, in good agreement with the experimental data on the ionization constants of these three systems. Finally, for HCl, our model
Proton Transfer in X‚‚‚(H2O)4 Clusters cluster predicts that all the clusters are fully ionized, as the double ionic cluster is the only minimum in the potential energy surface. This is also in good agreement with the strong acid nature of HCl in water solutions. Therefore, our model clusters, although too small to describe the solvent reorganization during the proton transfer process, are adequate to describe the known qualitative properties in solution of the neutral and ionized forms of the molecules studied here. Acknowledgment. The authors thank CESCA and CEPBA for their generous allocation of computer time. This work was also supported by the DGICYT and CIRIT under Projects PB92-0655-C02-02 and GRQ94-1077, respectively. M. Planas thanks the Spanish Science and Education Department for his doctoral grant. References and Notes (1) For a general introduction see for instance: Bell, R. P. The Proton in Chemistry, 2nd ed.; Chapman and Hall: London, 1973. Proton Transfer in Hydrogen-Bonded Systems; Bountis, T., Ed.; Plenum: New York, 1993. (2) Eigen, D. Angew. Chem., Int. Ed. Engl. 1964, 3, 1. Albery, W. J. Prog. React. Kinet. 1967, 4, 353. Crooks, J. E. Compr. Chem. Kinet. 1977, 8, 197. (3) Somasundram, K.; Amos, R. D.; Handy, N. C. Theor. Chim. Acta 1986, 69, 491. Del Bene, J. E. J. Phys. Chem. 1988, 92, 2874. (4) Theoretical: Bacskay, G. B. Mol. Phys. 1992, 77, 61. Experimental: Legon, A. C.; Willoughby, L. G. Chem. Phys. Lett. 1983, 95, 449. (5) Ando, K.; Hynes, J. T. In Structure and ReactiVity in Aqueous Solutions; ACS Symposium Series Vol. 568; Cramer, C. J., Truhlar, D. G., Eds.; American Chemical Society: Washington, DC, 1994, and references therein. Ando, K.; Hynes, J. T. J. Mol. Liq. 1995, 64, 25. (6) Rivail, J. L.; Antonczak, S.; Chipot, C.; Ruiz-Lopez, M. F.; Gorb, L. G. In Structure and ReactiVity in Aqueous Solutions; ACS Symposium Series Vol. 568; Cramer, C. J., Truhlar, D. G., Eds.; American Chemical Society: Washington, DC, 1994. (7) Waldron, R. D.; Hornig, D. F. J. Am. Chem. Soc. 1953, 75, 6079. (8) Lee, C.; Sosa, C.; Novoa, J. J. J. Chem. Phys. 1995, 103, 4360. (9) Lee, C.; Sosa, C.; Planas, M.; Novoa, J. J. J. Chem. Phys. 1996, 104, 70. (10) Lee, C.; Fitzgerald, G.; Planas, M.; Novoa, J. J. J. Phys. Chem. 1996, 100, 7398.
J. Phys. Chem., Vol. 100, No. 41, 1996 16501 (11) Scrocco, E.; Tomasi, J. AdV. Quantum Chem. 1978, 11, 115. Politzer, P.; Murray, J. S. ReV. Comput. Chem. 1991, 2, 273. (12) Tozer, D. J.; Lee, C.; Fitzgerald, G. J. Chem. Phys. 1996, 104, 5555. (13) Becke, A. D. Phys. ReV. A 1988, 38, 3098. Becke, A. D. J. Chem. Phys. 1992, 96, 2155. (14) Lee, C.; Yang, W.; Parr, R. G. Phys. ReV. B 1988, 37, 785. (15) Parr, R. G.; Yang, W. Density-Fuctional Theory of Atoms; Oxford Science Publications: New York, 1989. (16) Kim, K.; Jordan, K. D. J. Phys. Chem. 1994, 98, 10089. Novoa, J. J.; Sosa, C. J. Phys. Chem. 1995, 99, 15837. (17) Lee, C.; Stahlberg, E.; Fitzgerald, G. J. Phys. Chem., submitted. (18) DGauss is a software product available from Cray Research Inc., a part of the UniChem software package. For details of the Dgauss methodology see: Andzelm, J.; Wimmer, E. J. Chem. Phys. 1992, 96, 1280. (19) Gaussian 94, Revision C.3: Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Gill, P. M. W.; Johnson, B. G.; Robb, M. A.; Cheeseman, J. R.; Keith, T.; Peterson, G. A.; Montgomery, J. A.; Raghavachari, K.; Al-Laham, M. A.; Zakrzewski, V. G.; Ortiz, J. V.; Foresman, J. B.; Ciolowski, J.; Stefanov, B. B.; Nanayakkara, A.; Challacombe, M.; Peng, C. Y.; Ayala, P. Y.; Chen, W.; Wong, M. W.; Andres, J. L.; Replogle, E. S.; Gomperts, R.; Martin, R. L.; Fox, D. J.; Binkley, J. S.; Defrees, D., J.; Baker, J.; Stewart, J. P.; Head-Gordon, M.; Gonzalez, C.; Pople, J. A., Gaussian, Inc., Pittsburgh, PA, 1995. (20) Bernardi, F.; Robb, M. A. AdV. Chem. Phys. 1987, 67, 155. (21) There is some ambiguity in defining the fragments at the transition state geometry, because the proton being transferred can be associated with any of the two fragments to which it is connected. To solve this ambiguity, we have assigned the proton to the fragment to which is closer. As in the clusters studied here, the transition states are all closer to the products with structures similar to that for the product, the positive charge is located on the proton being transferred, and the transition state is doubly charged. (22) Abkowicz, A. J.; Latajka, Z.; Scheiner, S.; Chalasinski, G. J. Mol. Struct.: THEOCHEM 1995, 342, 153. (23) The BSSE was also be computed for the neutral fragment and is 2.1 kcal/mol. However in this case, the monomers are different to the ionic fragments used to compute the BSSE for the double ionic and transition state clusters. This can create some inconsistencies in the BSSE computed values, as have been already indicated in the literature (see ref 24, for example). (24) Turi, L.; Dannenberg, J. J. J. Phys. Chem. 1993, 97, 2488. (25) Rovira, M. C.; Novoa, J. J.; Whangbo. M. H.; Williams, J. M. Chem. Phys. 1995, 200, 319.
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