J. Phys. Chem. B 1997, 101, 6251-6253
6251
Quantum Mechanical Investigation of the Energetics of Proton Transfer along Hydrogen Bonds Patricia L. Moore Plummer Departments of Physics and Chemistry, UniVersity of MissourisColumbia, Columbia, Missouri 65211 ReceiVed: October 18, 1997X
Proton transfer along hydrogen bonds is examined using medium and large basis set ab initio quantum mechanical calculations. Post Hartree-Fock analysis is applied to the results. The model systems for these studies are clusters containing three or four water molecules. The proton in the hydrogen bond chosen for study belongs to a molecule which is involved in at least two hydrogen bonds and serves as both a donor and an acceptor in those bonds. The transfer of the proton along the bond between adjacent oxygen atoms thus creates an ion pair in the model system. For comparison similar calculations were undertaken in which the proton in question was associated with a H3O+ ion. These latter calculations model the energy of proton migration without the initial formation of an ion pair.
Introduction Proton transfer reactions play important roles in many natural phenomena, ranging from the environmental to the biological. In ice systems, it is the motion of the protons that is responsible for a variety of mechanical and electrical properties. Despite this importance, details of a realistic mechanism of charge transfer via hydrogen bonds in water and ice systems are largely absent. The situation for both experimental and theoretical understanding is complicated by the inherent disorder of the protons in many phases of ice and the presence of both ionic and configurational defects. The cooperativity of hydrogen bonds, the polarizability of the the water molecule, and water’s permanent dipole moment all add additional complexity. What is needed is detailed information about the potential energy surfaces of charge transfer. Only in the past few years has the development of advanced computer architecture and the evolution of numerical procedures to take advantage of this architecture made the potential energy surface for even a few molecules accessible to quantum mechanical calculations. In this report the potential surfaces for proton transfer for several small aggregates of water are examined. The use of small clusters as models for more extensive systems to investigate processes on the atomic level has proven very useful. Properties which primarily depend on the immediate environment can be well-characterized and realistically reproduced in the small clusters. Such model calculations also permit the separation of these properties from those which depend on longer range interactions. The ability to employ a high level of theory together with the flexibility to realistically describe the system gained with the use of extensive basis sets can clarify the importance of the roles of electron correlation and of the presence of diffuse and polarization functions in the process being simulated. Description of the Calculations The water trimer is the primary cluster chosen for these studies. A limited number of calculations were made on the cyclic pentamer. The trimer system is large enough to have several stable isomers and at least one molecule participating as both a donor and an acceptor of a hydrogen in a hydrogen bond. Reference calculations were carried out to obtain a X
Abstract published in AdVance ACS Abstracts, June 15, 1997.
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minimum energy structure for the water monomer, hydronimum ion, and the hydrated hydronimum ion or dihydated proton, H2O2+. For each of the energy minimization calculations, all geometric parameters were unconstrained. Each stationary point obtained was identified as a minimum (stable conformer) or a transition state on the potential surface by examination of the corresponding Hessian matrix. For the estimation of energy barriers associated with proton migration, the proton was moved from its position in the minimum energy surface along the O‚‚‚O axis in small discrete steps. At each shift in the position of the proton, the energy was reminimized allowing all other geometric parameters to relax. In this way the energy required to initiate the formation of an ion pair can be studied without confusion from strain energies that would be present if the atoms in the cluster were “clamped” in a predetermined geometry. A second series of calculations introduced an additional proton into the neutral cluster, creating an H3O+ at one of the molecular positions. Here all geometric parameters were allowed to change in the search for a stable structure for the cationic cluster. When a proton is transferred from one oxygen to an adjacent one in these calculations, the energy differences calculated can be attributed solely to the movement of the proton and the response of the other atoms in the cluster to this motion. Gaussian 94 Rev.C.31 installed on a SGI Power Challenge was used for these calculations. The basis sets employed were 6-31G(d) and aug-cc-pVDZ, the augmented correlationconsistent polarized valence double-zeta basis set of Dunning, et al.2 Correlation effects were included through second-level Møller-Plesset3 perturbation theory (MP2). The energies, geometries, and properties reported were all calculated at this level of theory. Results and Discussion Energy and geometric data for the structures calculated to have the lowest energy for the reference systems of H2O, the hydrated proton H3O+, the dihydrated proton H5O2+, and the cyclic water trimer are given in Table 1. A linear neutral trimer was also predicted to be stable but less so than the cyclic system. In both trimers, the orientation of the hydrogen bonds is sequential (see Figure 1), i.e. each molecule donating or accepting no more than one hydrogen in forming the hydrogen bonds. In contrast to previously reported results,4 when electron © 1997 American Chemical Society
6252 J. Phys. Chem. B, Vol. 101, No. 32, 1997
Plummer TABLE 1: Structural Parameters and Energiesa,b H2O R(OH)c
Figure 1. The cyclic water trimer illustrating the sequential hydrogenbonding pattern preferred in small clusters.
0.966 0.968
H3O+ 0.983 0.969
H5O2+
0.972 0.978, R(O‚‚‚H) 1.202 1.210, 1.209 ∠HOH 103.9 111.0 108.6 104.0 106.0 108.6 ∠OH‚‚‚O 175.1 173.0 dipole 2.016 1.554 1.497 2.20 1.812 1.644 energy -76.260 91 -76.530 06 -152.845 233 -76.196 85 -76.475 11 -152.732 93 ZPVEd 13.387 21.651 35.584 13.48 21.740 36.007
(H2O)3 0.965, 0.0978 0.969, 0.984 1.905, 1.906, 1.924 1.874, 1.876, 1.896 105.0, 105.2, 105.3 104.5, 104.6, 104.9 148.4-151.1(2) 149.5-151.6 1.124 1.457 -228.808 80 -228.628 23 43.91e 46.917
a Values in boldface are MP2(FC)/aug-cc-pVDZ results, while those in standard font are MP2(FC)/6-31G(d) results. b Bond distances are in angstroms, angles in degrees, dipole moments in Debye. c The first value given is for the hydrogen not participating in a hydrogen bond, while the second is for the hydrogen-bonded hydrogen. d The zeropoint vibrational energy (ZPVE) is quoted in kcal/mol. e The MP2/ aug-cc-pVDZ frequencies and ZPVE did not converge to the same accuracy as the other results reported, and the value reported is the ZPVE from a modified G2 calculation.
Figure 2. The water cations: (a) hydronium ion, H3O+, (b) dihydrated proton, H5O2+, (c) dihydrated hydronium ion, (H2O)2H3O+.
correlation was included at the MP2 level, trimer structures which contained molecules which served as a double donor or a double acceptor of a proton in a hydrogen bond were not found to be stable. In all attempts to introduce such a molecule into either the cyclic or the linear trimer, the system eventually returned to a sequentially bonded structure. No intermediate stationary points were found on the potential surface. To begin the examination of proton transfer in these model systems, a proton in the cyclic trimer was shifted in a series of 0.05 Å steps from its equilibrium position. At each step, the other atoms in the cluster were allowed to relax. When the proton had been shifted by 0.6 Å, the hydrogens involved in
the other two hydrogen bonds began to shift also. Closer examination of the structures as the proton continued to be shifted away from its equilibrium oxygen showed an apparent cooperative motion of the other protons. The result was not a structure containing an ion pair but was a cyclic and seemingly concerted permutation of all the protons in the structure to the neighboring molecule. The energy reflected this structural change; after reaching a maximum of 52.2 kcal/mol, the energy begins to fall, returning to its original value as the final structure is an isoenergetic isomer. This result appears to contradict earlier work of Scheiner5 in whose system cooperative motion of protons increased the energy required when compared with a “hop and wait” mechanism. To see if an ion pair could be stabilized at the terminal positions in the linear trimer, an initial zwitterion structure was constructed. The initial structure had the ions separated by 4.5 Å. However, when all constraints were removed and a
a
b
d
c
Figure 3. Illustration of proton transfer between the central and terminal molecules in the trimer cation. Structure a is the most stable. The energy added to produce structure b is ca. 3 kcal/mol. Structure c requires an additional ca. 4.5 kcal/mol of energy. Nearly 13 kcal/mol additional energy is needed to move the proton to the final structure, d.
Proton Transfer along Hydrogen Bonds stationary point was finally located, the structure had cyclized and a proton had been transferred, thus recombining the ion pair. Limited calculations were also carried out for the cyclic pentamer. This cluster also optimized with each molecule participating as both a donor and an acceptor in a sequentially hydrogen-bonded conformation. Attempts to move the proton from one end of the bond to the other met with the same fate as in the cyclic trimer. Introduction of an ion pair separated by a neutral molecule could not be stabilized; it relaxed when the constraints were removed to the ion free sequentially hydrogen-bonded ring structure. When combined with the cyclization of the trimer described above, this result appears to support the conclusion that stabilization of an ion pair requires at least one molecule in the cluster to participate in three or more hydrogen bonds.6 Next a series of cations were constructed by adding a proton to one of the molecules in the cyclic cluster. Again when the constraints were removed and the geometry of the cation optimized, the resulting structure had a linear chain structure with the H3O+ group located in the central position. See Figure 2c. It appears that the cyclic structure cannot withstand the strain of an added proton and the hydrogen bond opposite the proton’s position breaks. In a final attempt to examine the energy associated with proton transfer, the stable trimer cation was used. Again the proton was stepped along the line connecting the central and terminal molecules in the cluster. The initial and final structures together with two intermediate structures along the path are illustrated in Figure 3. In this case, no energy maximum was detected along the proton’s path. The energy calculated to move the proton and to have the cation in a terminal location was 20.28 kcal/mol, a value less than half that observed when elongating an OH bond in the direction of producing an ion pair. However the structure having a terminal cation is not a minimum on the potential surface. It is, in fact, a saddle point if the proton is constrained to remain on the terminal molecule. But since there is no barrier along the proton’s path, when the constraint is removed the proton returns to its central location and the structure relaxes to its initial conformation. Conclusions It is of interest to compare the structures of the cations considered here. The cation systems are shown in Figure 2. It has long been known that the reaction of a single proton with a molecule of water produces a cation in which the OH bonds are identical, and hence the resulting H3O+ can not be regarded as a proton with a water of hydration. This stands in sharp contrast to the situation when an additional molecule of water is added to form H5O2+ whose structure is clearly consistent with a proton associated with two waters of hydration. When a third molecule of water is introduced, the cluster assumes a structure resembling a hydronium ion with two waters of hydration. But here the two OH bonds which donate protons to the hydrogen bonds are lengthened relative to the nonbonded OH bond. Here too there is a preference for the H3O+ to occupy the central position and to serve as a double-donor molecule. Attempts to rotate the hydronium ion so as to introduce a sequential bonding pattern into the trimer cation always relaxed to the double-donor conformation as soon as the constraints were removed. Thus we see, even in these very small clusters, the diversity of environments favored by the proton. It is suggestive that the presence of the other partner of the ion pair provides the “driving force” for proton migration and restoration of sequential
J. Phys. Chem. B, Vol. 101, No. 32, 1997 6253 hydrogen bonds between neutral molecules. This apparently barrier-free migration can be interrupted, at least in small clusters of water, by the presence of a molecule with “saturated” or nearly saturated bonding (three or four hydrogen bonds). While we have been able to learn a good deal about a model local environment for proton transfer, it appears that with the flexibility of the hydrogen bond, the cooperative nature of hydrogen-bonded networks, and the ability of protons to migrate in a concerted fashion the size of clusters employed in this study are not large enough to provide all of the desired realistic information about the potential energy surface for proton transfer in ice. To improve the model, more of the molecules should participate in three or four hydrogen bonds and inclusion of second-neighbor interactions may also be needed. These extensions are planned. On the other hand, comparison of the results for these small clusters with different basis sets and different levels of theory illustrate the importance of electron correlation in the description of proton transfer reactions. The presence of additional polarization or diffuse function does not appear to change the results in any significant fashion. The general structural parameters, whether a system has a high degree of symmetry or not, and relative energies of different conformers are more dependent on electron correlation than on expanded basis sets. In addition, these simulations have shown the ease of migration of the proton in the cation system. In a semiempirical AM1 study of H+(H2O)n, n ) 2 to 11, Davidson et al.7 found that the barrier to proton transfer in the linear chain cation initially decreases with increasing numbers of waters, becoming constant at ca. 1 kcal/mol. He found the frequency associated with the proton transfer motion to be 589 cm-1 for H5O2+, which can be compared with 603 cm-1 from the MP2/6-31G(d) calculation. For (H2O)3H+, the associated frequency is 622 cm-1. The lack of any sizable barrier to proton migration in our trimer clusters is in agreement with the observations8 that while there is an activation energy associated with the formation of an ion pair and perhaps also a barrier for the injection of protons into the ice system, once a proton is present there is no apparent barrier to its diffusion. The easy rearrangement of protons in the octamer aggregates suggests that this property will not be significantly altered in larger systems with more “saturated” bonding. Acknowledgment. The author expresses appreciation to the Computer Center, University of MissourisColumbia, the SHIVA support staff, and the Special Projects Group for the computer time needed for these species and for support of the software systems used in the analysis. References and Notes (1) Gaussian 94 (Rev. C-3); Gaussian, Inc.: Pittsburgh, PA, 1995. (2) Dunning, T. H. J. Chem. Phys. 1989, 90, 1007. Kendall, R. A.; Dunning, T. H.; Harrison, R. J. J. Chem. Phys. 1992, 96, 6796. (3) Møller, C.; Plesset, M. S. Phys. ReV. 1934, 46, 618. (4) Hankins, D.; Moskowitz, J. W.; Stillinger, F. H. J. Chem. Phys. 1970, 53, 4544; Chem. Phys. Lett. 1970, 4, 527. (5) Scheiner, S. In Proton Transfer in Hydrogen Bonded Systems; Bountis, T., Ed.; Plenum Press: New York, 1992; pp 29-47. (6) Plummer, P. L. M. J. Phys. Chem., preceding paper in this issue. (7) Choi, J. Y.; Cave, R. J.; Davidson, E. R. Department of Chemistry, Indiana University, preprint, 1992. (8) Eisenberg, D.; Kausman, W. Structure and Properties of Water; Oxford University Press: Oxford, 1969; p 118.
