Article pubs.acs.org/JPCB
Kinetics and Mechanism of Water Cluster Equilibria F. Weinhold Theoretical Chemistry Institute and Department of Chemistry, University of Wisconsin, Madison, Wisconsin 53706, United States S Supporting Information *
ABSTRACT: We combine hybrid density functional and transition state theory to investigate dynamical and mechanistic features of important aggregation, isomerization, and exchange pathways for cluster constituents of liquid water, building on the general concepts of quantum cluster equilibrium (QCE) theory. Such calculations confirm the extreme dynamical volatility of leading water cluster structural motifs, consistent with known ultrafast relaxation properties of liquid water and contrary to the superficial static imagery often associated with thermodynamic-level description. We identify low-barrier mechanistic pathways and associated donor−acceptor orbital interactions that lead to facile scrambling of covalent and H-bond cluster motifs with remarkably small energetic barriers, significantly less than required to break even a single covalent or H-bond in isolation.
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INTRODUCTION Quantum cluster equilibrium (QCE) theory has emerged as a clear alternative to conventional molecular dynamics (MD) simulation methods for a molecular-level description of liquid water and related H-bonded solution phenomena.1 In contrast to a MD description based on classical-type force fields,2 QCE theory is based on a full quantal description of the principal “building block” clusters of the liquid phase, while simpler mean-field and excluded-volume approximations are reserved for the long-range interactions between clusters of significant equilibrium population. Such thermodynamically viable cluster motifs are in turn obtained iteratively from the self-consistent QCE solutions for an initially chosen set of candidate clusters. The QCE formalism allows accurate quantum mechanical treatment of the resonance-type charge transfer interactions that can now be recognized as essential features of H-bonding phenomena3 but are essentially ignored (unless included in averaged form by empirical three-body correction terms4) in current classical-type force fields. On the other hand, the QCE thermodynamic description inherently lacks the richly visual dynamical detail that is an appealing feature of MD simulations. Although harmonic vibrational information is included in the QCE partition function, the physical consequences of such vibrations appear only indirectly in the entropic properties of QCE equilibrium cluster distributions. The leading QCE clusters correspond to the idealized “inherent structures”5 that underlie a full dynamical description, conventionally represented by the static imagery of the stationary minimum energy structures. As usual in a thermodynamic-level description,6 important dynamical and mechanistic details of the equilibrating cluster species are lost in the QCE framework, obscuring direct comparisons with modern experimental measurements of ultrafast dynamical properties of liquids.7 © XXXX American Chemical Society
Nevertheless, well-known methods of transition state theory (TST) can be enlisted to add significant kinetic and mechanistic detail to a thermodynamic-level description. Accordingly, in the present work, we explore the mechanistic pathways and activation barriers for some key isomerization, exchange, and aggregation processes in the QCE equilibrium distribution of liquid water clusters, leading to TST-based estimates of the associated rate constants and time scales. These calculations establish that many such cluster “reaction” processes must be extremely fast under near-ambient conditions, passing over surprisingly low activation barriers despite the apparent necessity to rupture multiple covalent or H-bonds of high dissociation energy. We employ natural bond orbital (NBO) analysis8 of orbital interactions to rationalize the activation barriers, showing their dependence on the powerful cooperative (Grotthus vs non-Grotthus) aspects of H-bond ordering patterns that are characteristic of exchange-type charge transfer interactions but absent in classical force field simulations.
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THEORY AND COMPUTATIONAL METHODS In QCE theory, the thermodynamic behavior of a liquid composed of molecules (W) is governed by simultaneous equilibrium conditions among its constituent clusters (Wn, n = 1, 2, 3, ...), as envisioned in the network of balanced cluster “reactions” of generic form W ⇄ 1 2 W2 ⇄ 1 3 W3 ⇄ 1 4 W4 ⇄ ...
(1)
Special Issue: James L. Skinner Festschrift Received: November 21, 2013 Revised: January 16, 2014
A
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Figure 1. Reactant (left), transition (center), and product (right) species (with parenthesized ΔGHB(0) values) for elementary cluster reactions (a−d) (see text). Principal H-bonding interactions are included to aid visualization.
The cluster reaction network may include various internal rearrangement (isomerization) and substitution-type (exchange) reactions Wn ⇄ Wn′
(isomerization)
W′ + Wn″ ⇄ Wn′ + W″
(exchange)
(3)
as well as stepwise aggregation reactions between clusters W + Wn ⇄ Wn + 1
(2) B
(aggregation)
(4)
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• Convoy 5H transfer (Figure 1b), to break five covalent OH bonds of the H-bonded chain and reattach each proton to the adjacent O atom (interchanging “covalent bonds” vs “H-bonds” in each OH···O linkage), denoted as
where W′, W″ represent distinct monomers substituted in or out of clusters Wn′, Wn″ of fixed size but distinguishable internal constitution. In analogy to ordinary chemical reactions, each cluster reaction 2−4 may be regarded as mechanistically “elementary” (single step) or “complex” (sequence of elementary steps), according to the number of intermediate transition state species (first-order saddle points) or minimum-energy species that connect the initial reactant and final product cluster species along a minimum energy pathway (IRC “intrinsic reaction coordinate”) on the quantum mechanical potential energy surface. As usual, the computational search for a transition state species Wn⧧ may be followed by searching along the IRC in each direction to verify direct connectivity to desired reactant and product species. Analytic harmonic vibrational frequencies establish the topological character [transition state (one imaginary frequency) or minimum (all positive frequencies)] and allow evaluation of corresponding temperature-dependent entropic corrections to obtain the standard-state Gibbs free energy (G(0)) at each stationary energy (E) value. Standard options in electronic structure programs such as Gaussian 099 allow transition species of cluster reactions to be determined rather straightforwardly for a chosen theory level, here taken as the B3LYP/6-311++G** level10 of hybrid density functional theory. Given the calculated forward (ΔGf⧧(0)) and reverse (ΔGr⧧(0)) activation free energy changes to surmount the transition state, one can also evaluate standard TST expressions for the rate constants (kf, kr) of each elementary reaction, which serve as input to coupled rate equations for the cluster reaction network.11 From such equations, one obtains the characteristic lifetimes for individual reactive species and full kinetics of the cluster distribution for desired experimental comparisons. In the present work, we focus theoretical attention on some elementary aggregation, isomerization, and exchange reactions of cluster species that are indicated to be of principal importance in the QCE distribution of liquid water. Under near-ambient conditions, the QCE distribution is found to be dominated by pentameric and hexameric cylic species W5c and W6c of a specific “Grotthus-like” proton ordering pattern,12 together with a smattering of water monomers, dimers, and other trace species. The electronic origins of the enthalpic and entropic preference for “near-ideal” pentameric and hexameric H-bond motifs are discussed elsewhere13 in terms of the underlying oxygen lone pair (nO, Lewis base electronic “donor”) and hydride antibond (σ*OH, Lewis acid electronic “acceptor”) NBOs that engage in the characteristic nO → σ*OH “charge transfer” (resonance-type donor−acceptor) interaction of H-bonding. Details of NBO orbital interactions will be examined in further detail below for specific cluster species, using the current NBO 6.0 program14 and associated NBOView 2.0 visualization software.15
W5c ⇄ c5h⧧ ⇄ W5c″
• Monomer exchange (Figure 1c), to exchange a monomer W″ of of the reactant W5c″ with a free monomer W′ of the surrounding medium, denoted as W′ + W5c″ ⇄ ex⧧ ⇄ W″ + W5c′
(7)
• Additive insertion (Figure 1d), to insert a free monomer W into the pentamer W5c to give the corresponding Grotthus-ordered cyclic hexamer W6c, denoted as W + W5c ⇄ ins⧧ ⇄ W6c
(8)
Although eq 6 is presumably less important in the fast dynamics of liquid water, it forms a simple five-ring analogue of concerted Grotthus-type proton transfers that may be important in the six-rings of ice-I. For each reaction 5−8, we obtained the optimized reactant or reactant complex (left panel), transition species (middle panel), and product or product complex (right panel), as shown in Figure 1a−d. Calculated energies and standard-state free energies for each species are given in Table 1, each expressed Table 1. Energetic (ΔEHB) and Standard-State Free Energetic (ΔGHB(0)) Changes in H-Bond Dissociation Energy (kcal/mol; Relative to Isolated Monomers) for Elementary Cluster Reactions Considered in This Work (See Text, Figures 1 and 2), All at the B3LYP/6-311++G** Level reaction species W5c io⧧ c5h⧧ W6b ex⧧ W6b′ W6a ins⧧ W6c
ΔEHB (a) In−Out Twist −40.84 −37.52 (b) Convoy 5H Transfer −15.72 (c) Monomer Exchange −48.86 −48.43 −50.72 (d) Additive Insertion −48.68 −46.47 −49.13
ΔGHB(0) 3.23 6.16 23.01 6.96 7.30 4.05 8.45 9.65 7.50
with respect to the corresponding quantity for noninteracting water monomers at infinite separation. Graphical representations of the stationary points of the potential surface are presented in Figures 2 and 3 for the energy and free energy values, respectively. Comparison of Figures 2 and 3 shows the powerful influence of entropic corrections on the apparent “potential energy landscape” for water clusters. Thus, the E-diagram of Figure 2 exhibits steady descent in the sequence W → W2 → W5c → W6c → ..., whereas the corresponding G(0)-diagram of Figure 3 dramatically shifts the scale of numerical values and exhibits significant destabilization of W6c and higher clusters with respect to W5c. (In the full QCE description, the ΔGHB(0) values
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RESULTS We selected for study four specific dynamical cluster reactions 5−8 involving the cyclic pentameric water cluster W5c, as depicted in Figure 1a−d: • “In−out twist” isomerization (Figure 1a), to switch a particular OH bond in or out of the H-bond chain, denoted as W5c ⇄ io⧧ ⇄ W5c′
(6)
(5) C
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rather strained motif compared to W5c), whereas the latter exhibits Grotthus order in both four- and five-ring linkages. The transition state (middle panel of Figure 1c) therefore involves bridging shift to adjacent O(1), which restores full Grotthus parity to intruder O(16) and native O(5) monomers, giving each an equal chance to become the intruder bridge in subsequent in−out flip transition from W6b′ (“bag-1”4) to an isomeric equivalent of W6b. Other exchange mechnisms may be imagined (and tested), but that shown in Figure 1c appears optimal in preserving the Grotthus-type cooperative proton ordering that is uniquely characteristic of low-barrier pathways. The corresponding low-barrier pathway to additive insertion 8 also involves initial monomer attachment to W5c by formation of two new hydrogen bonds, in this case to create a noncooperative three-ring with a pentamer edge. From this starting complex (left panel of Figure 1d, “chaise-1”4), it then requires simple opening of the strained inner edge of the attachment triangle and minor torsional twists to achieve the cyclic Grotthus proton pattern of chairlike W6c (“ring-1”4). As shown in the calculated values of Table 1, the forward (ΔGf⧧(0)) and reverse (ΔGr⧧(0)) free activation energies for the cluster reactions 4−8 of Figure 1a−d are found to be
Figure 2. H-bond dissociation energy changes (ΔEHB, kcal/mol; relative to fully dissociated water monomers) for the elementary cluster reactions of Table 1 (see text). The binding energy (−5.63 kcal/mol) for the single H-bond of water dimer W2 is included for comparison.
ΔGf ⧧ (0) = ΔGr ⧧ (0) = 2.93 kcal/mol
(in−out twist) (9a)
ΔGf
⧧ (0)
= ΔGr
⧧ (0)
= 19.78 kcal/mol
(convoy 5H transfer)
ΔGf ⧧ (0) = 0.34 kcal/mol,
(9b)
ΔGr ⧧ (0) = 3.25 kcal/mol
(exchange)
(9c)
ΔGf ⧧ (0) = 1.20 kcal/mol,
Figure 3. Similar to Figure 2, for standard-state free energy changes (ΔGHB(0), kcal/mol). The value (+3.07 kcal/mol) for the single Hbond of W2 is included for comparison, and the location of c5h⧧ (above the figure boundary) is indicated schematically.
ΔGr ⧧ (0) = 2.15 kcal/mol
(additive insertion)
(9d)
Thus, the activation energies for H-bond rearrangements 9a, 9c, and 9d are all found to be in the range of 3 kcal/mol or less, significantly less than the nominal “bond energy” (ca. 5 kcal/ mol) of even a single H-bond in isolated W2. Similarly, the 5Htransfer reaction of Figure 1b apparently requires simultaneous breaking of five covalent OH bonds but is actually seen in eq 9b to proceed with less than 20% of the bond energy of a single OH bond in an isolated water molecule. These results point to the powerful cooperative properties of H-bond networks that allow nearly barrier-free pathways for a variety of protontransfer and monomer-exchange dynamical processes in the liquid phase, sustaining robust thermodynamic populations of W5c even as individual structural components of each cluster exchange rapidly with the surrounding environment. The TST rate constant (k) for a reaction with activation free energy ΔG⧧ is given by
are further altered by intercluster interactions to make W5c exoergic with respect to monomers, in accordance with the expected transition from gaseous to liquid cluster distribution under near-ambient conditions.) Nevertheless, within a reaction manifold of fixed monomer number, the energy and free energy transition barriers are reasonably similar (e.g., for the “in−out twist” reaction, the barriers are 3.32 and 2.93 kcal/mol, respectively), so we shall focus primarily on the ΔGHB(0) values in the ensuing discussion. As seen in Figure 1a and b, the intracluster isomerization reactions 5 and 6 proceed straightforwardly through “least motion” transition states whose general features could be intuitively anticipated. However, the intercluster exchange 7 and insertion 8 reactions are seen in Figure 1c and d to have more subtle mechanistic features. The low-barrier pathway to monomer exchange reaction 7 involves initial bridging of external monomer W′ [labeled as O(16)H(17)H(18) in Figure 1c] with monomers O(2) and O(4) of the precursor W5c″ cluster to give the “[1,1,2]propellane”-like structure (“bag-2” in the hexamer terminology of ref 4) that serves as the starting reactant complex. In this initial complex, the bridging O(16) “intruder” still appears disfavored compared to “native” O(3), because the former gains Grotthus-like proton ordering only in the four-ring (a
k = κ(kBT /h) exp( −ΔG⧧/RT )
(10)
where κ is a transmission coefficient (≤1), kB is the Boltzmann constant, h is the Planck constant, and T is the standard-state temperature. For κ = 1, the numerical rate constants and associated characteristic lifetimes (τ = 1/k) for reactions 9a−9d can therefore be estimated as
D
io⧧ : k f = k r = 4.44 × 1010 s−1, τf = τr = 22.5 ps
(10a)
c5h⧧: k f = k r = 2.03 × 10−2 s−1, τf = τr = 49.3 s
(10b)
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Figure 4. NBO overlap diagrams for leading nO → σ*OH donor−acceptor stabilizations in each transition species of reactions 5−8.
nO → σ*OH interactions available to the oxygen at the right, one strengthening and one weakening as the monomer at the left twists OH bonds in or out of OH···O alignment.) The diagrams emphasize that H-bond formation is essentially a continuous gain or loss of nO → σ*OH stabilization, vestiges of which can guide low-barrier reactive pathways even in highly distorted nonequilibrium geometries.
ex⧧: k f = 3.50 × 1012 s−1, τf = 0.3 ps; k r = 3.50 × 1012 s−1, τr = 38.6 ps
(10c)
ins⧧: k f = 8.21 × 1011 s−1, τf = 1.2 ps; k r = 1.65 × 1011 s−1, τr = 6.1 ps
(10d)
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All but process 10b fall into the picosecond range, and will hence lead to rapid scrambling of structure-specific features of H-bonding that can only be frozen out with ultrafast spectroscopic methods such as IR7 or X-ray16 spectroscopy.
SUMMARY AND CONCLUSIONS The present results add a degree of mechanistic and kinetic detail to thermodynamic QCE description of the cluster composition of liquid water, consistent with experimental results of X-ray spectroscopy16 and other theoretical and experimental evidence3,4 for the importance of cooperative “three-body interactions” of nO → σ*OH type. Low-barrier pathways for important aggregation, exchange, and isomerization reactions confirm the dynamical importance of cooperative Grotthus-type proton ordering patterns that are known to govern QCE equilibrium cluster populations. NBO orbital interaction diagrams further illuminate the electronic origins of the surprising pliability of H-bond networks, which seemingly allow wholesale disruptions of multiple elements of the collective motif with less energy than required for even single H-bond dissociation in isolation. Specifically, numerical TST rate constants for calculated B3LYP/6-311++G** transition pathways indicate the remark-
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NBO ANALYSIS Additional electronic insights into the low-barrier cluster reaction pathways can be gained by visualizing the key nO → σ*OH donor−acceptor interactions of transition species in (preorthogonal) NBO overlap diagrams. Figure 4 presents such interaction diagrams for the transition states of reactions 5−8, showing the leading oxygen lone pair (nO) and hydride antibond (σ*OH) interaction that stabilizes the strained transition region of each reaction. From the orbital shapes in Figure 4, one can see that significant nO → σ*OH interaction is maintained even in transition geometries with interatomic bending angles or distances far from nominal near-equilibrium H-bond values. (Note that Figure 4a illustrates one of the two near-equivalent E
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of the Hydrogen Bond (IUPAC Recommendations 2011). Pure Appl. Chem. 2011, 83, 1637−1641. Weinhold, F.; Klein, R. A. What is a Hydrogen Bond? Mutually Consistent Theoretical and Experimental Criteria for Characterizing H-Bonding Interactions. Mol. Phys. 2012, 110, 565−579. (4) Tainter, C. J.; Skinner, J. L. The Water Hexamer: Three-Body Interactions, Structures, Energetics, and OH-Stretch Spectroscopy at Finite Temperature. J. Chem. Phys. 2012, 137, 104304−104319. Tainter, C. J.; Ni, T.; Shi, L.; Skinner, J. L. Hydrogen Bonding and OH-Stretch Spectroscopy in Water: Hexamer (Cage), Liquid Surface, Liquid, and Ice. J. Phys. Chem. Lett. 2013, 4, 12−17. (5) Stillinger, F. Inherent Structure in Liquids. J. Phys. Chem. 1983, 87, 2833−2840. (6) For the contrary example of a volume-based thermodynamic description of dynamical properties, see: Glasser, L.; Jenkins, H. D. B. Volume-Based Thermodynamics: A Prescription for its Application and Usage in Approximation and Prediction of Thermodynamic Data. J. Chem. Eng. Data 2011, 56, 874−880. (7) Fecko, C. J.; Eaves, J. D.; Loparo, J. J.; Tokmakoff, A.; Geissler, P. L. Ultrafast Hydrogen-Bond Dynamics in the Infrared Spectroscopy of Water. Science 2003, 301, 1698−1702. Khalil, M.; Demirdöven, N.; Tokmakoff, A. Coherent 2D IR Spectroscopy: Molecular Structure and Dynamics in Solution. J. Phys. Chem. A 2003, 107, 5258−5279. Zheng, J.; Kwak, K.; Roberts, S. T.; Ramasesha, K.; Tokmakoff, A. Structural Rearrangements in Water Viewed Through Two-Dimensional Infrared Spectroscopy. Acc. Chem. Res. 2009, 42, 1239−1249. Skinner, J. L. Following the Motions of water molecules in aqueous solutions. Science 2010, 328, 985−986. Nicodemus, R. A.; Corcelli, S. A.; Skinner, J. L.; Tokmakoff, A. Collective Hydrogen Bond Reorganization in Water Studied with Temperature-Dependent Ultrafast Infrared Spectroscopy. J. Phys. Chem. B 2011, 115, 5604− 5616. Fayer, M. D., Ed. Ultrafast Infrared Vibrational Spectroscopy; CRC Press: Boca Raton, FL, 2013. (8) Glendening, E. D.; Landis, C. R.; Weinhold, F. Natural Bond Orbital Methods. Wiley Interdiscip. Rev.: Comput. Mol. Sci. 2012, 2, 1− 42. (9) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Scalmani, G.; Barone, V.; Mennucci, B.; Petersson, G. A.; Nakatsuji, H.; Caricato, M.; Li, X.; Hratchian, H. P.; Izmaylov, A. F.; Bloino, J.; Zheng, G.; Sonnenberg, J. L.; Hada, M.; Ehara, M.; Toyota, K.; Fukuda, R.; Hasegawa, J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.; Nakai, H.; Vreven, T.; Montgomery, J. A., Jr.; Peralta, J. E.; Ogliaro, F.; Bearpark, M.; Heyd, J. J.; Brothers, E.; Kudin, K. N.; Staroverov, V. N.; Kobayashi, R.; Normand, J.; Raghavachari, K.; Rendell, A.; Burant, J. C.; Iyengar, S. S.; Tomasi, J.; Cossi, M.; Rega, N.; Millam, N. J.; Klene, M.; Knox, J. E.; Cross, J. B.; Bakken, V.; Adamo, C.; Jaramillo, J.; Gomperts, R.; Stratmann, R. E.; Yazyev, O.; Austin, A. J.; Cammi, R.; Pomelli, C.; Ochterski, J. W.; Martin, R. L.; Morokuma, K.; Zakrzewski, V. G.; Voth, G. A.; Salvador, P.; Dannenberg, J. J.; Dapprich, S.; Daniels, A. D.; Farkas, Ö .; Foresman, J. B.; Ortiz, J. V.; Cioslowski, J.; Fox, D. J. Gaussian 09, revision D.01; Gaussian, Inc.: Wallingford, CT, 2009. (10) Foresman, J. B.; Frisch, A. E. Exploring Chemistry With Electronic Structure Methods: A Guide To Using Gaussian, 2nd ed.; Gaussian, Inc.: Pittsburgh, PA, 1995. (11) Laidler, K. J. Theories of Chemical Reaction Rates; McGraw-Hill: New York, 1969. (12) Weinhold, F. Quantum Cluster Equilibrium Theory of Liquids: Illustrative Application to Water. J. Chem. Phys. 1998, 109, 373−384. (13) Weinhold, F. Nature of H-Bonding in Clusters, Liquids, and Enzymes: An Ab Initio, Natural Bond Orbital Perspective. J. Mol. Struct.: THEOCHEM 1997, 398−399, 181−197. Weinhold, F. Resonance Character of Hydrogen-Bonding Interactions in Water and Other H-Bonded Species. In Peptide Solvation and H-Bonds: Advances in Protein Chemistry; Baldwin, R. L., Baker, D., Eds.; Elsevier: New York, 2006; Vol. 72, pp 121−155. (14) Glendening, E. D.; Badenhoop, J. K.; Reed, A. E.; Carpenter, J. E.; Bohmann, J. A.; Morales, C. M.; Landis, C. R.; Weinhold, F. NBO 6.0; Theoretical Chemistry Institute, University of Wisconsin:
ably short time scales for intracluster isomerizations (ca. 22 ps), intercluster exchange (ca. 39 ps), or additive insertion (ca. 6 ps), corresponding to activation free energies in the 1−3 kcal/ mol range. Even the concerted “convoy” break-up of f ive covalent OH bonds (if preserving the Grotthus order of proton transfer in bicoordinate cyclic topology) can occur with far lower activation energy than required to rupture a single OH bond of an isolated water molecule. The present results for W5c illustrate the general methodology but cannot yet provide a full QCE-based picture of the temperature (T) and pressure (P) dependence of water dynamics for detailed experimental comparisons. For a more complete description, the calculated dynamical properties for other possible cluster constituents should be combined with calculated QCE cluster populations at chosen T and P to obtain population-weighted kinetics of the composite cluster mixture, analogous to previous QCE treatment of (T, P)-dependent vibrational or other spectroscopic properties.17 Nevertheless, the expected dominance of W5c, W6c, and related small cluster motifs18 under near-ambient conditions suggests that the W5cbased dynamical time scales quoted above should be broadly representative of those measurable in liquid water. All these results challenge conventional assumptions of pairwise additive potential forms in MD force fields, as well as the “dipole−dipole” rationalizations of H-bonding that are commonly advocated in freshman chemistry textbooks. They also provide emphatic support for recent initiatives to fundamentally revise the IUPAC Gold Book definition of Hbonding.3 While the present results fall far short of the vivid detail of classical-type MD simulations, they serve to anticipate how dynamical details of such simulations may differ significantly from corresponding quantum mechanical descriptions that properly incorporate the important exchange-type charge transfer aspect of H-bonding.
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ASSOCIATED CONTENT
S Supporting Information *
Gaussian input files containing energies, free energies, and optimized coordinates for all species. This material is available free of charge via the Internet at http://pubs.acs.org.
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AUTHOR INFORMATION
Notes
The authors declare no competing financial interest.
ACKNOWLEDGMENTS I am grateful to Jim Skinner for many discussions of the structure and dynamics of water. REFERENCES
(1) Weinhold, F. Quantum Cluster Equilibrium Theory of Liquids: General Theory and Computer Implementation. J. Chem. Phys. 1998, 109, 367−372. Kirchner, B. Theory of Complicated Liquids; Investigation of Liquids, Solvents and Solvent Effects with Modern Theoretical Methods. Phys. Rep. 2007, 440, 1−111. Matisz, G.; Fabian, W. M. F.; Kelterer, A.-M.; Kunsagi-Mae, S. Weinhold’s QCE Model A Modified Parameter Fit. Model Study of Liquid Methanol Based on MP2 Cluster Geometries. J. Mol. Struct.: THEOCHEM 2010, 956, 103−109. (2) Leach, A. Molecular Modelling: Principles and Applications, 2nd ed.; Prentice Hall: Upper Saddle River, NJ, 2001. (3) Arunan, E.; Desiraju, G. R.; Klein, R. A.; Sadlej, J.; Scheiner, S.; Alkorta, I.; Clary, D. C.; Crabtree, R. H.; Dannenberg, J. J.; Hobza, P.; Kjaergaard, H. G.; Legon, A. C.; Mennucci, B.; Nesbitt, D. J. Definition F
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Madison, WI, 2013. Glendening, E. D.; Landis, C. R.; Weinhold, F. NBO 6.0: Natural Bond Orbital Analysis Program. J. Comput. Chem. 2013, 34, 1429−1437. (15) Wendt, M.; Weinhold, F. NBOView 2.0; Theoretical Chemistry Institute, University of Wisconsin: Madison, WI, 2013 (http://nbo6. chem.wisc.edu). (16) Wernet, P.; Nordlund, D.; Bergmann, U.; Ogasawara, H.; Cavalleri, M.; Näslund, L.-Å.; Hirsch, T. K.; Ojamäe, L.; Glatzel, P.; Odelius, M.; Pettersson, L. G. M.; Nilsson, A. Science 2004, 304, 995− 999. (17) Ludwig, R.; Reis, O.; Winter, R.; Weinhold, F.; Farrar, T. C. Quantum Cluster Equilibrium Theory of Liquids: Temperature Dependence of Hydrogen Bonding in Liquid N-Methylacetamide Studied by IR Spectra. J. Phys. Chem. B 1998, 102, 9312−9318. (18) As shown by QCE (ref 1) and other modelling methods (ref 4) at various theory levels, small cyclic clusters rapidly gain majority free energy precedence over noncyclic (bridged, caged, or other threedimensional) clusters as T increases. Although W5c serves adequately as a prototype of the dominant equilibrium cyclic motif, species of lesser equilibrium population might nevertheless contribute significantly to the full spectrum of dynamical rates. For example, the calculated B3LYP/6-311++G** free energy activation barriers for convoy proton transfer processes (cf. eq 6) in the sequence of small cyclic clusters W3c (24.69 kcal/mol), W4c (18.70 kcal/mol), W5c (19.78 kcal/mol), and W6c (21.29 kcal/mol) indicate that the most rapid diffusional proton transfers occur through the minority W4c species. Further work along these lines (also at other method and basis levels) is required for a more complete picture of liquid water dynamics.
G
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Kinetics and Mechanism of Water Cluster Equilibria F. Weinhold Theoretical Chemistry Institute and Department of Chemistry, University of Wisconsin, Madison, Wisconsin 53706, United States S Supporting Information *
ABSTRACT: We combine hybrid density functional and transition state theory to investigate dynamical and mechanistic features of important aggregation, isomerization, and exchange pathways for cluster constituents of liquid water, building on the general concepts of quantum cluster equilibrium (QCE) theory. Such calculations confirm the extreme dynamical volatility of leading water cluster structural motifs, consistent with known ultrafast relaxation properties of liquid water and contrary to the superficial static imagery often associated with thermodynamic-level description. We identify low-barrier mechanistic pathways and associated donor−acceptor orbital interactions that lead to facile scrambling of covalent and H-bond cluster motifs with remarkably small energetic barriers, significantly less than required to break even a single covalent or H-bond in isolation.
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INTRODUCTION Quantum cluster equilibrium (QCE) theory has emerged as a clear alternative to conventional molecular dynamics (MD) simulation methods for a molecular-level description of liquid water and related H-bonded solution phenomena.1 In contrast to a MD description based on classical-type force fields,2 QCE theory is based on a full quantal description of the principal “building block” clusters of the liquid phase, while simpler mean-field and excluded-volume approximations are reserved for the long-range interactions between clusters of significant equilibrium population. Such thermodynamically viable cluster motifs are in turn obtained iteratively from the self-consistent QCE solutions for an initially chosen set of candidate clusters. The QCE formalism allows accurate quantum mechanical treatment of the resonance-type charge transfer interactions that can now be recognized as essential features of H-bonding phenomena3 but are essentially ignored (unless included in averaged form by empirical three-body correction terms4) in current classical-type force fields. On the other hand, the QCE thermodynamic description inherently lacks the richly visual dynamical detail that is an appealing feature of MD simulations. Although harmonic vibrational information is included in the QCE partition function, the physical consequences of such vibrations appear only indirectly in the entropic properties of QCE equilibrium cluster distributions. The leading QCE clusters correspond to the idealized “inherent structures”5 that underlie a full dynamical description, conventionally represented by the static imagery of the stationary minimum energy structures. As usual in a thermodynamic-level description,6 important dynamical and mechanistic details of the equilibrating cluster species are lost in the QCE framework, obscuring direct comparisons with modern experimental measurements of ultrafast dynamical properties of liquids.7 © XXXX American Chemical Society
Nevertheless, well-known methods of transition state theory (TST) can be enlisted to add significant kinetic and mechanistic detail to a thermodynamic-level description. Accordingly, in the present work, we explore the mechanistic pathways and activation barriers for some key isomerization, exchange, and aggregation processes in the QCE equilibrium distribution of liquid water clusters, leading to TST-based estimates of the associated rate constants and time scales. These calculations establish that many such cluster “reaction” processes must be extremely fast under near-ambient conditions, passing over surprisingly low activation barriers despite the apparent necessity to rupture multiple covalent or H-bonds of high dissociation energy. We employ natural bond orbital (NBO) analysis8 of orbital interactions to rationalize the activation barriers, showing their dependence on the powerful cooperative (Grotthus vs non-Grotthus) aspects of H-bond ordering patterns that are characteristic of exchange-type charge transfer interactions but absent in classical force field simulations.
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THEORY AND COMPUTATIONAL METHODS In QCE theory, the thermodynamic behavior of a liquid composed of molecules (W) is governed by simultaneous equilibrium conditions among its constituent clusters (Wn, n = 1, 2, 3, ...), as envisioned in the network of balanced cluster “reactions” of generic form W ⇄ 1 2 W2 ⇄ 1 3 W3 ⇄ 1 4 W4 ⇄ ...
(1)
Special Issue: James L. Skinner Festschrift Received: November 21, 2013 Revised: January 16, 2014
A
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Figure 1. Reactant (left), transition (center), and product (right) species (with parenthesized ΔGHB(0) values) for elementary cluster reactions (a−d) (see text). Principal H-bonding interactions are included to aid visualization.
The cluster reaction network may include various internal rearrangement (isomerization) and substitution-type (exchange) reactions Wn ⇄ Wn′
(isomerization)
W′ + Wn″ ⇄ Wn′ + W″
(exchange)
(3)
as well as stepwise aggregation reactions between clusters W + Wn ⇄ Wn + 1
(2) B
(aggregation)
(4)
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• Convoy 5H transfer (Figure 1b), to break five covalent OH bonds of the H-bonded chain and reattach each proton to the adjacent O atom (interchanging “covalent bonds” vs “H-bonds” in each OH···O linkage), denoted as
where W′, W″ represent distinct monomers substituted in or out of clusters Wn′, Wn″ of fixed size but distinguishable internal constitution. In analogy to ordinary chemical reactions, each cluster reaction 2−4 may be regarded as mechanistically “elementary” (single step) or “complex” (sequence of elementary steps), according to the number of intermediate transition state species (first-order saddle points) or minimum-energy species that connect the initial reactant and final product cluster species along a minimum energy pathway (IRC “intrinsic reaction coordinate”) on the quantum mechanical potential energy surface. As usual, the computational search for a transition state species Wn⧧ may be followed by searching along the IRC in each direction to verify direct connectivity to desired reactant and product species. Analytic harmonic vibrational frequencies establish the topological character [transition state (one imaginary frequency) or minimum (all positive frequencies)] and allow evaluation of corresponding temperature-dependent entropic corrections to obtain the standard-state Gibbs free energy (G(0)) at each stationary energy (E) value. Standard options in electronic structure programs such as Gaussian 099 allow transition species of cluster reactions to be determined rather straightforwardly for a chosen theory level, here taken as the B3LYP/6-311++G** level10 of hybrid density functional theory. Given the calculated forward (ΔGf⧧(0)) and reverse (ΔGr⧧(0)) activation free energy changes to surmount the transition state, one can also evaluate standard TST expressions for the rate constants (kf, kr) of each elementary reaction, which serve as input to coupled rate equations for the cluster reaction network.11 From such equations, one obtains the characteristic lifetimes for individual reactive species and full kinetics of the cluster distribution for desired experimental comparisons. In the present work, we focus theoretical attention on some elementary aggregation, isomerization, and exchange reactions of cluster species that are indicated to be of principal importance in the QCE distribution of liquid water. Under near-ambient conditions, the QCE distribution is found to be dominated by pentameric and hexameric cylic species W5c and W6c of a specific “Grotthus-like” proton ordering pattern,12 together with a smattering of water monomers, dimers, and other trace species. The electronic origins of the enthalpic and entropic preference for “near-ideal” pentameric and hexameric H-bond motifs are discussed elsewhere13 in terms of the underlying oxygen lone pair (nO, Lewis base electronic “donor”) and hydride antibond (σ*OH, Lewis acid electronic “acceptor”) NBOs that engage in the characteristic nO → σ*OH “charge transfer” (resonance-type donor−acceptor) interaction of H-bonding. Details of NBO orbital interactions will be examined in further detail below for specific cluster species, using the current NBO 6.0 program14 and associated NBOView 2.0 visualization software.15
W5c ⇄ c5h⧧ ⇄ W5c″
• Monomer exchange (Figure 1c), to exchange a monomer W″ of of the reactant W5c″ with a free monomer W′ of the surrounding medium, denoted as W′ + W5c″ ⇄ ex⧧ ⇄ W″ + W5c′
(7)
• Additive insertion (Figure 1d), to insert a free monomer W into the pentamer W5c to give the corresponding Grotthus-ordered cyclic hexamer W6c, denoted as W + W5c ⇄ ins⧧ ⇄ W6c
(8)
Although eq 6 is presumably less important in the fast dynamics of liquid water, it forms a simple five-ring analogue of concerted Grotthus-type proton transfers that may be important in the six-rings of ice-I. For each reaction 5−8, we obtained the optimized reactant or reactant complex (left panel), transition species (middle panel), and product or product complex (right panel), as shown in Figure 1a−d. Calculated energies and standard-state free energies for each species are given in Table 1, each expressed Table 1. Energetic (ΔEHB) and Standard-State Free Energetic (ΔGHB(0)) Changes in H-Bond Dissociation Energy (kcal/mol; Relative to Isolated Monomers) for Elementary Cluster Reactions Considered in This Work (See Text, Figures 1 and 2), All at the B3LYP/6-311++G** Level reaction species W5c io⧧ c5h⧧ W6b ex⧧ W6b′ W6a ins⧧ W6c
ΔEHB (a) In−Out Twist −40.84 −37.52 (b) Convoy 5H Transfer −15.72 (c) Monomer Exchange −48.86 −48.43 −50.72 (d) Additive Insertion −48.68 −46.47 −49.13
ΔGHB(0) 3.23 6.16 23.01 6.96 7.30 4.05 8.45 9.65 7.50
with respect to the corresponding quantity for noninteracting water monomers at infinite separation. Graphical representations of the stationary points of the potential surface are presented in Figures 2 and 3 for the energy and free energy values, respectively. Comparison of Figures 2 and 3 shows the powerful influence of entropic corrections on the apparent “potential energy landscape” for water clusters. Thus, the E-diagram of Figure 2 exhibits steady descent in the sequence W → W2 → W5c → W6c → ..., whereas the corresponding G(0)-diagram of Figure 3 dramatically shifts the scale of numerical values and exhibits significant destabilization of W6c and higher clusters with respect to W5c. (In the full QCE description, the ΔGHB(0) values
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RESULTS We selected for study four specific dynamical cluster reactions 5−8 involving the cyclic pentameric water cluster W5c, as depicted in Figure 1a−d: • “In−out twist” isomerization (Figure 1a), to switch a particular OH bond in or out of the H-bond chain, denoted as W5c ⇄ io⧧ ⇄ W5c′
(6)
(5) C
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rather strained motif compared to W5c), whereas the latter exhibits Grotthus order in both four- and five-ring linkages. The transition state (middle panel of Figure 1c) therefore involves bridging shift to adjacent O(1), which restores full Grotthus parity to intruder O(16) and native O(5) monomers, giving each an equal chance to become the intruder bridge in subsequent in−out flip transition from W6b′ (“bag-1”4) to an isomeric equivalent of W6b. Other exchange mechnisms may be imagined (and tested), but that shown in Figure 1c appears optimal in preserving the Grotthus-type cooperative proton ordering that is uniquely characteristic of low-barrier pathways. The corresponding low-barrier pathway to additive insertion 8 also involves initial monomer attachment to W5c by formation of two new hydrogen bonds, in this case to create a noncooperative three-ring with a pentamer edge. From this starting complex (left panel of Figure 1d, “chaise-1”4), it then requires simple opening of the strained inner edge of the attachment triangle and minor torsional twists to achieve the cyclic Grotthus proton pattern of chairlike W6c (“ring-1”4). As shown in the calculated values of Table 1, the forward (ΔGf⧧(0)) and reverse (ΔGr⧧(0)) free activation energies for the cluster reactions 4−8 of Figure 1a−d are found to be
Figure 2. H-bond dissociation energy changes (ΔEHB, kcal/mol; relative to fully dissociated water monomers) for the elementary cluster reactions of Table 1 (see text). The binding energy (−5.63 kcal/mol) for the single H-bond of water dimer W2 is included for comparison.
ΔGf ⧧ (0) = ΔGr ⧧ (0) = 2.93 kcal/mol
(in−out twist) (9a)
ΔGf
⧧ (0)
= ΔGr
⧧ (0)
= 19.78 kcal/mol
(convoy 5H transfer)
ΔGf ⧧ (0) = 0.34 kcal/mol,
(9b)
ΔGr ⧧ (0) = 3.25 kcal/mol
(exchange)
(9c)
ΔGf ⧧ (0) = 1.20 kcal/mol,
Figure 3. Similar to Figure 2, for standard-state free energy changes (ΔGHB(0), kcal/mol). The value (+3.07 kcal/mol) for the single Hbond of W2 is included for comparison, and the location of c5h⧧ (above the figure boundary) is indicated schematically.
ΔGr ⧧ (0) = 2.15 kcal/mol
(additive insertion)
(9d)
Thus, the activation energies for H-bond rearrangements 9a, 9c, and 9d are all found to be in the range of 3 kcal/mol or less, significantly less than the nominal “bond energy” (ca. 5 kcal/ mol) of even a single H-bond in isolated W2. Similarly, the 5Htransfer reaction of Figure 1b apparently requires simultaneous breaking of five covalent OH bonds but is actually seen in eq 9b to proceed with less than 20% of the bond energy of a single OH bond in an isolated water molecule. These results point to the powerful cooperative properties of H-bond networks that allow nearly barrier-free pathways for a variety of protontransfer and monomer-exchange dynamical processes in the liquid phase, sustaining robust thermodynamic populations of W5c even as individual structural components of each cluster exchange rapidly with the surrounding environment. The TST rate constant (k) for a reaction with activation free energy ΔG⧧ is given by
are further altered by intercluster interactions to make W5c exoergic with respect to monomers, in accordance with the expected transition from gaseous to liquid cluster distribution under near-ambient conditions.) Nevertheless, within a reaction manifold of fixed monomer number, the energy and free energy transition barriers are reasonably similar (e.g., for the “in−out twist” reaction, the barriers are 3.32 and 2.93 kcal/mol, respectively), so we shall focus primarily on the ΔGHB(0) values in the ensuing discussion. As seen in Figure 1a and b, the intracluster isomerization reactions 5 and 6 proceed straightforwardly through “least motion” transition states whose general features could be intuitively anticipated. However, the intercluster exchange 7 and insertion 8 reactions are seen in Figure 1c and d to have more subtle mechanistic features. The low-barrier pathway to monomer exchange reaction 7 involves initial bridging of external monomer W′ [labeled as O(16)H(17)H(18) in Figure 1c] with monomers O(2) and O(4) of the precursor W5c″ cluster to give the “[1,1,2]propellane”-like structure (“bag-2” in the hexamer terminology of ref 4) that serves as the starting reactant complex. In this initial complex, the bridging O(16) “intruder” still appears disfavored compared to “native” O(3), because the former gains Grotthus-like proton ordering only in the four-ring (a
k = κ(kBT /h) exp( −ΔG⧧/RT )
(10)
where κ is a transmission coefficient (≤1), kB is the Boltzmann constant, h is the Planck constant, and T is the standard-state temperature. For κ = 1, the numerical rate constants and associated characteristic lifetimes (τ = 1/k) for reactions 9a−9d can therefore be estimated as
D
io⧧ : k f = k r = 4.44 × 1010 s−1, τf = τr = 22.5 ps
(10a)
c5h⧧: k f = k r = 2.03 × 10−2 s−1, τf = τr = 49.3 s
(10b)
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Figure 4. NBO overlap diagrams for leading nO → σ*OH donor−acceptor stabilizations in each transition species of reactions 5−8.
nO → σ*OH interactions available to the oxygen at the right, one strengthening and one weakening as the monomer at the left twists OH bonds in or out of OH···O alignment.) The diagrams emphasize that H-bond formation is essentially a continuous gain or loss of nO → σ*OH stabilization, vestiges of which can guide low-barrier reactive pathways even in highly distorted nonequilibrium geometries.
ex⧧: k f = 3.50 × 1012 s−1, τf = 0.3 ps; k r = 3.50 × 1012 s−1, τr = 38.6 ps
(10c)
ins⧧: k f = 8.21 × 1011 s−1, τf = 1.2 ps; k r = 1.65 × 1011 s−1, τr = 6.1 ps
(10d)
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All but process 10b fall into the picosecond range, and will hence lead to rapid scrambling of structure-specific features of H-bonding that can only be frozen out with ultrafast spectroscopic methods such as IR7 or X-ray16 spectroscopy.
SUMMARY AND CONCLUSIONS The present results add a degree of mechanistic and kinetic detail to thermodynamic QCE description of the cluster composition of liquid water, consistent with experimental results of X-ray spectroscopy16 and other theoretical and experimental evidence3,4 for the importance of cooperative “three-body interactions” of nO → σ*OH type. Low-barrier pathways for important aggregation, exchange, and isomerization reactions confirm the dynamical importance of cooperative Grotthus-type proton ordering patterns that are known to govern QCE equilibrium cluster populations. NBO orbital interaction diagrams further illuminate the electronic origins of the surprising pliability of H-bond networks, which seemingly allow wholesale disruptions of multiple elements of the collective motif with less energy than required for even single H-bond dissociation in isolation. Specifically, numerical TST rate constants for calculated B3LYP/6-311++G** transition pathways indicate the remark-
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NBO ANALYSIS Additional electronic insights into the low-barrier cluster reaction pathways can be gained by visualizing the key nO → σ*OH donor−acceptor interactions of transition species in (preorthogonal) NBO overlap diagrams. Figure 4 presents such interaction diagrams for the transition states of reactions 5−8, showing the leading oxygen lone pair (nO) and hydride antibond (σ*OH) interaction that stabilizes the strained transition region of each reaction. From the orbital shapes in Figure 4, one can see that significant nO → σ*OH interaction is maintained even in transition geometries with interatomic bending angles or distances far from nominal near-equilibrium H-bond values. (Note that Figure 4a illustrates one of the two near-equivalent E
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of the Hydrogen Bond (IUPAC Recommendations 2011). Pure Appl. Chem. 2011, 83, 1637−1641. Weinhold, F.; Klein, R. A. What is a Hydrogen Bond? Mutually Consistent Theoretical and Experimental Criteria for Characterizing H-Bonding Interactions. Mol. Phys. 2012, 110, 565−579. (4) Tainter, C. J.; Skinner, J. L. The Water Hexamer: Three-Body Interactions, Structures, Energetics, and OH-Stretch Spectroscopy at Finite Temperature. J. Chem. Phys. 2012, 137, 104304−104319. Tainter, C. J.; Ni, T.; Shi, L.; Skinner, J. L. Hydrogen Bonding and OH-Stretch Spectroscopy in Water: Hexamer (Cage), Liquid Surface, Liquid, and Ice. J. Phys. Chem. Lett. 2013, 4, 12−17. (5) Stillinger, F. Inherent Structure in Liquids. J. Phys. Chem. 1983, 87, 2833−2840. (6) For the contrary example of a volume-based thermodynamic description of dynamical properties, see: Glasser, L.; Jenkins, H. D. B. Volume-Based Thermodynamics: A Prescription for its Application and Usage in Approximation and Prediction of Thermodynamic Data. J. Chem. Eng. Data 2011, 56, 874−880. (7) Fecko, C. J.; Eaves, J. D.; Loparo, J. J.; Tokmakoff, A.; Geissler, P. L. Ultrafast Hydrogen-Bond Dynamics in the Infrared Spectroscopy of Water. Science 2003, 301, 1698−1702. Khalil, M.; Demirdöven, N.; Tokmakoff, A. Coherent 2D IR Spectroscopy: Molecular Structure and Dynamics in Solution. J. Phys. Chem. A 2003, 107, 5258−5279. Zheng, J.; Kwak, K.; Roberts, S. T.; Ramasesha, K.; Tokmakoff, A. Structural Rearrangements in Water Viewed Through Two-Dimensional Infrared Spectroscopy. Acc. Chem. Res. 2009, 42, 1239−1249. Skinner, J. L. Following the Motions of water molecules in aqueous solutions. Science 2010, 328, 985−986. Nicodemus, R. A.; Corcelli, S. A.; Skinner, J. L.; Tokmakoff, A. Collective Hydrogen Bond Reorganization in Water Studied with Temperature-Dependent Ultrafast Infrared Spectroscopy. J. Phys. Chem. B 2011, 115, 5604− 5616. Fayer, M. D., Ed. Ultrafast Infrared Vibrational Spectroscopy; CRC Press: Boca Raton, FL, 2013. (8) Glendening, E. D.; Landis, C. R.; Weinhold, F. Natural Bond Orbital Methods. Wiley Interdiscip. Rev.: Comput. Mol. Sci. 2012, 2, 1− 42. (9) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Scalmani, G.; Barone, V.; Mennucci, B.; Petersson, G. A.; Nakatsuji, H.; Caricato, M.; Li, X.; Hratchian, H. P.; Izmaylov, A. F.; Bloino, J.; Zheng, G.; Sonnenberg, J. L.; Hada, M.; Ehara, M.; Toyota, K.; Fukuda, R.; Hasegawa, J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.; Nakai, H.; Vreven, T.; Montgomery, J. A., Jr.; Peralta, J. E.; Ogliaro, F.; Bearpark, M.; Heyd, J. J.; Brothers, E.; Kudin, K. N.; Staroverov, V. N.; Kobayashi, R.; Normand, J.; Raghavachari, K.; Rendell, A.; Burant, J. C.; Iyengar, S. S.; Tomasi, J.; Cossi, M.; Rega, N.; Millam, N. J.; Klene, M.; Knox, J. E.; Cross, J. B.; Bakken, V.; Adamo, C.; Jaramillo, J.; Gomperts, R.; Stratmann, R. E.; Yazyev, O.; Austin, A. J.; Cammi, R.; Pomelli, C.; Ochterski, J. W.; Martin, R. L.; Morokuma, K.; Zakrzewski, V. G.; Voth, G. A.; Salvador, P.; Dannenberg, J. J.; Dapprich, S.; Daniels, A. D.; Farkas, Ö .; Foresman, J. B.; Ortiz, J. V.; Cioslowski, J.; Fox, D. J. Gaussian 09, revision D.01; Gaussian, Inc.: Wallingford, CT, 2009. (10) Foresman, J. B.; Frisch, A. E. Exploring Chemistry With Electronic Structure Methods: A Guide To Using Gaussian, 2nd ed.; Gaussian, Inc.: Pittsburgh, PA, 1995. (11) Laidler, K. J. Theories of Chemical Reaction Rates; McGraw-Hill: New York, 1969. (12) Weinhold, F. Quantum Cluster Equilibrium Theory of Liquids: Illustrative Application to Water. J. Chem. Phys. 1998, 109, 373−384. (13) Weinhold, F. Nature of H-Bonding in Clusters, Liquids, and Enzymes: An Ab Initio, Natural Bond Orbital Perspective. J. Mol. Struct.: THEOCHEM 1997, 398−399, 181−197. Weinhold, F. Resonance Character of Hydrogen-Bonding Interactions in Water and Other H-Bonded Species. In Peptide Solvation and H-Bonds: Advances in Protein Chemistry; Baldwin, R. L., Baker, D., Eds.; Elsevier: New York, 2006; Vol. 72, pp 121−155. (14) Glendening, E. D.; Badenhoop, J. K.; Reed, A. E.; Carpenter, J. E.; Bohmann, J. A.; Morales, C. M.; Landis, C. R.; Weinhold, F. NBO 6.0; Theoretical Chemistry Institute, University of Wisconsin:
ably short time scales for intracluster isomerizations (ca. 22 ps), intercluster exchange (ca. 39 ps), or additive insertion (ca. 6 ps), corresponding to activation free energies in the 1−3 kcal/ mol range. Even the concerted “convoy” break-up of f ive covalent OH bonds (if preserving the Grotthus order of proton transfer in bicoordinate cyclic topology) can occur with far lower activation energy than required to rupture a single OH bond of an isolated water molecule. The present results for W5c illustrate the general methodology but cannot yet provide a full QCE-based picture of the temperature (T) and pressure (P) dependence of water dynamics for detailed experimental comparisons. For a more complete description, the calculated dynamical properties for other possible cluster constituents should be combined with calculated QCE cluster populations at chosen T and P to obtain population-weighted kinetics of the composite cluster mixture, analogous to previous QCE treatment of (T, P)-dependent vibrational or other spectroscopic properties.17 Nevertheless, the expected dominance of W5c, W6c, and related small cluster motifs18 under near-ambient conditions suggests that the W5cbased dynamical time scales quoted above should be broadly representative of those measurable in liquid water. All these results challenge conventional assumptions of pairwise additive potential forms in MD force fields, as well as the “dipole−dipole” rationalizations of H-bonding that are commonly advocated in freshman chemistry textbooks. They also provide emphatic support for recent initiatives to fundamentally revise the IUPAC Gold Book definition of Hbonding.3 While the present results fall far short of the vivid detail of classical-type MD simulations, they serve to anticipate how dynamical details of such simulations may differ significantly from corresponding quantum mechanical descriptions that properly incorporate the important exchange-type charge transfer aspect of H-bonding.
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ASSOCIATED CONTENT
S Supporting Information *
Gaussian input files containing energies, free energies, and optimized coordinates for all species. This material is available free of charge via the Internet at http://pubs.acs.org.
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AUTHOR INFORMATION
Notes
The authors declare no competing financial interest.
ACKNOWLEDGMENTS I am grateful to Jim Skinner for many discussions of the structure and dynamics of water. REFERENCES
(1) Weinhold, F. Quantum Cluster Equilibrium Theory of Liquids: General Theory and Computer Implementation. J. Chem. Phys. 1998, 109, 367−372. Kirchner, B. Theory of Complicated Liquids; Investigation of Liquids, Solvents and Solvent Effects with Modern Theoretical Methods. Phys. Rep. 2007, 440, 1−111. Matisz, G.; Fabian, W. M. F.; Kelterer, A.-M.; Kunsagi-Mae, S. Weinhold’s QCE Model A Modified Parameter Fit. Model Study of Liquid Methanol Based on MP2 Cluster Geometries. J. Mol. Struct.: THEOCHEM 2010, 956, 103−109. (2) Leach, A. Molecular Modelling: Principles and Applications, 2nd ed.; Prentice Hall: Upper Saddle River, NJ, 2001. (3) Arunan, E.; Desiraju, G. R.; Klein, R. A.; Sadlej, J.; Scheiner, S.; Alkorta, I.; Clary, D. C.; Crabtree, R. H.; Dannenberg, J. J.; Hobza, P.; Kjaergaard, H. G.; Legon, A. C.; Mennucci, B.; Nesbitt, D. J. Definition F
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Madison, WI, 2013. Glendening, E. D.; Landis, C. R.; Weinhold, F. NBO 6.0: Natural Bond Orbital Analysis Program. J. Comput. Chem. 2013, 34, 1429−1437. (15) Wendt, M.; Weinhold, F. NBOView 2.0; Theoretical Chemistry Institute, University of Wisconsin: Madison, WI, 2013 (http://nbo6. chem.wisc.edu). (16) Wernet, P.; Nordlund, D.; Bergmann, U.; Ogasawara, H.; Cavalleri, M.; Näslund, L.-Å.; Hirsch, T. K.; Ojamäe, L.; Glatzel, P.; Odelius, M.; Pettersson, L. G. M.; Nilsson, A. Science 2004, 304, 995− 999. (17) Ludwig, R.; Reis, O.; Winter, R.; Weinhold, F.; Farrar, T. C. Quantum Cluster Equilibrium Theory of Liquids: Temperature Dependence of Hydrogen Bonding in Liquid N-Methylacetamide Studied by IR Spectra. J. Phys. Chem. B 1998, 102, 9312−9318. (18) As shown by QCE (ref 1) and other modelling methods (ref 4) at various theory levels, small cyclic clusters rapidly gain majority free energy precedence over noncyclic (bridged, caged, or other threedimensional) clusters as T increases. Although W5c serves adequately as a prototype of the dominant equilibrium cyclic motif, species of lesser equilibrium population might nevertheless contribute significantly to the full spectrum of dynamical rates. For example, the calculated B3LYP/6-311++G** free energy activation barriers for convoy proton transfer processes (cf. eq 6) in the sequence of small cyclic clusters W3c (24.69 kcal/mol), W4c (18.70 kcal/mol), W5c (19.78 kcal/mol), and W6c (21.29 kcal/mol) indicate that the most rapid diffusional proton transfers occur through the minority W4c species. Further work along these lines (also at other method and basis levels) is required for a more complete picture of liquid water dynamics.
G
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