Theoretical studies of hydrogen bond network rearrangement (HBNR) dynamics in liquid water have indicated that librational motions initiate the hydrogen bond breaking/formation processes. We present the results of using a simple time evolution method to extract and compare the tunneling lifetimes for motions that break and reform the hydrogen bond for the water dimer, trimer, and pentamer from the experimentally measured tunneling splittings in the ground and excited intermolecular vibrational states. We find that the specific nature of the intermolecular vibrational excitation does not significantly influence the tunneling lifetime of the dimer, but that only excitations to a librational vibration affect the water trimer and pentamer lifetimes. The specific enhancement of bifurcation tunneling in larger clusters relative to the dimer also indicates that hydrogen bond cooperativity is a vital element of these dynamics.

It is well known that the remarkable properties of water result from the intricacies of the hydrogen bond network (HBN) connecting the individual water monomers.1–7 Understanding details of the processes by which the HBN rearranges is thus essential to understanding and modeling water. Many different experimental techniques have addressed these rearrangement dynamics, but these have generally required extensive interpretation to rationalize the measured properties.8–14 In contrast, molecular dynamics (MD) simulations have provided a detailed microscopic picture of many aqueous systems but are limited as they depend explicitly on the model potentials used in the simulation.1,2,5,15 More recent work has again indicated that nuclear quantum effects are important for correctly describing hydrogen bond behavior.16,17

Incoherent neutron scattering experiments of both H2O and D2O demonstrate evidence of relaxation processes with two distinct time scales.9 Similar experiments performed by Cabral et al. on mixtures of H2O with Dimethyl Sulfoxide (DMSO) showed good agreement with the theoretical results of Luzar and Chandler,9,18 viz., that the “residence” time, defined as the average time in which a molecule remains in a narrow geometric domain wherein a hydrogen bond could form, is 1.8 ± 0.4 ps for pure water and is 6 ± 2 ps in the mixture. These neutron scattering experiments supported the theoretical contention that librational motions (hindered rotations) are the primary means of breaking and forming hydrogen bonds in the liquid. Subsequent experiments using coherent quasielastic neutron scattering have provided further refinement of these deductions and also demonstrated that the temperature dependence of hydrogen bond dynamics is curiously weak compared with that of other transport properties of liquid water.9,19,20

More recently, mid-IR femtosecond pump-probe spectroscopy experiments and photon echo experiments have yielded additional insight into the intramolecular modes associated with hydrogen bond network rearrangement (HBNR) dynamics. Pump-probe spectroscopy experiments on the O–H stretch in HDO:D2O solutions demonstrate that a coupling between the intramolecular OH stretch and a translational intermolecular vibration is essential to understand the vibrational relaxation in the liquid.8,14,21–24 Further studies employing femtosecond mid-IR spectroscopy demonstrated coupling between intra- and inter-molecular modes and also identified two distinct time domains for relaxation processes, viz., sub-picosecond and picosecond.11,25–28 In photon echo experiments, Stenger et al. demonstrated that there may also be an even longer (5-15 ps) time scale process occurring that must be connected to the hydrogen bond rearrangement, indicating that these dynamics may be much slower than previously thought.29 Huse et al. then performed ultrafast pump probe experiments on the O–H bending mode and showed that explicit consideration of the librational excitations is necessary to rationalize ultrafast vibrational relaxation.23 

While the proceeding experiments have provided significant insight into aqueous hydrogen bond dynamics, some authors have argued that they possess a common weakness,1,6 viz., lack of a conclusive connection between the relaxation processes observed and the detailed hydrogen bond dynamics occurring in the bulk. While some measured properties, such as the “residence time” obtained in neutron scattering experiments, have been shown to agree well with the predicted hydrogen bond lifetimes, the connections of other properties to actual bulk phase dynamics are less transparent. Motivated by this history, and mindful of the ambiguities, our group has endeavored to explore the relation of our high-precision terahertz spectroscopy measurements of water cluster intermolecular vibrational modes to the HBNR dynamics of bulk water.

We draw attention to studies suggesting that the behavior of other small water clusters can provide additional insight into the hydrogen bond dynamics of the bulk.30–32 In the aforementioned theoretical studies by Luzar and Chandler, the authors found that the dynamics of a particular hydrogen bond were not affected by the neighboring hydrogen bonds,1,15,22,24 i.e., the local environment of a hydrogen bond does not significantly affect the dynamics of the hydrogen bond itself. Furthermore, more recent studies show that even static correlations are insignificant, and this result holds regardless of temperature.33 Additionally, the spectra of clusters comprising 10-100 molecules resemble the spectra of bulk.12,34

In the past, we have also shown that the lifetimes associated with breaking of the hydrogen bonds via permutational tunneling in the water trimer librational band show surprisingly good agreement with the accepted hydrogen bond lifetime in liquid water.31,32,35,36 Recent studies of the water hexamer cluster have revealed tunneling pathways, which involve the concerted breaking of two hydrogen bonds. These motions were suggested to be important in interfacial and confined environments.7 While the water dimer, trimer, and pentamer do not possess the 3-dimensional hydrogen bond (HB) structures existing in liquid water, insights into the respective HBNR dynamics could similarly elucidate dynamics of water in bulk environments. We wish to emphasize that isolated cluster do not exist in the bulk and that the dynamics of the bulk are more complicated. The gas phase cluster systems presented herein provide an insight into the dynamics of small hydrogen bonded cluster systems and the effect of cluster size and vibrational motion. Here we present results from applying a simple time evolution method for the water dimer, trimer, and pentamer clusters to extract tunneling lifetimes associated with the different experimentally measured tunneling pathways that break and reform hydrogen bonds.

It is well known that the HBNR tunneling in water clusters can be described in terms of complete nuclear permutational inversion (CNPI) group theory, wherein the tunneling motions are represented by group operations that are termed “feasible.” For the purpose of this study, “feasible” will be taken to mean experimentally observed. The basis set becomes the set of permutational isomers that are connected by feasible operations. For the water dimer, it has been shown that there are 8 isomers connected by four tunneling pathways: acceptor switching (AS), geared interchange (G), antigeared interchange (I), and bifurcation (B), shown in Fig. 1(a). Three of these pathways break and reform the hydrogen bond: G, I, and B. The observed experimental splittings are accounted for by the eigenvalues of the tunneling Hamiltonian [shown in Fig. 2(a)]. The 8 permutational isomers (labeled 18) that make up the basis set for the dimer are shown in the supplementary material. The term υ represents the vibrational origin.

FIG. 1.

The feasible tunneling motions of the water dimer (a), trimer (b), and pentamer (c). In this study we limit our focus to those motions that break and reform a hydrogen bond. (a) Water dimer tunneling pathways: oxygens are colored red, protons 1 → 4 are labeled grey, blue, green, and purple. Acceptor switching does not involve breaking/reforming a hydrogen bond and is not explicitly considered in this study. The interchange motions results in the acceptor and donor monomer changing roles. Bifurcation results in the free proton of the donor monomer exchanging with the proton involved in the hydrogen bond. The bifurcation proceeds through a transition state in which both protons of the donor monomer are coordinated to the acceptor monomer. (b) Water trimer bifurcation tunneling: oxygens are labeled red, hydrogens unaffected by the tunneling are colored grey, and hydrogens participating in the tunneling are colored purple, green, and blue. The predicted transition state of the bifurcation pathway is also shown. (c) Water pentamer bifurcation tunneling: oxygens are labeled red, hydrogens unaffected by the tunneling are colored grey, and hydrogens participating in the tunneling are colored green and blue. The predicted transition state is also shown.

FIG. 1.

The feasible tunneling motions of the water dimer (a), trimer (b), and pentamer (c). In this study we limit our focus to those motions that break and reform a hydrogen bond. (a) Water dimer tunneling pathways: oxygens are colored red, protons 1 → 4 are labeled grey, blue, green, and purple. Acceptor switching does not involve breaking/reforming a hydrogen bond and is not explicitly considered in this study. The interchange motions results in the acceptor and donor monomer changing roles. Bifurcation results in the free proton of the donor monomer exchanging with the proton involved in the hydrogen bond. The bifurcation proceeds through a transition state in which both protons of the donor monomer are coordinated to the acceptor monomer. (b) Water trimer bifurcation tunneling: oxygens are labeled red, hydrogens unaffected by the tunneling are colored grey, and hydrogens participating in the tunneling are colored purple, green, and blue. The predicted transition state of the bifurcation pathway is also shown. (c) Water pentamer bifurcation tunneling: oxygens are labeled red, hydrogens unaffected by the tunneling are colored grey, and hydrogens participating in the tunneling are colored green and blue. The predicted transition state is also shown.

Close modal
FIG. 2.

Tunneling Hamiltonians used for the water dimer (a) and trimer (b). The labels represent the permutational isomers connected by the feasible tunneling motions. Descriptions of the elements in the matrices can be found in the text. The tunneling matrix for the pentamer can be found in the supplementary material for this article. (a) Dimer Hamiltonian. (b) Trimer Hamiltonian.

FIG. 2.

Tunneling Hamiltonians used for the water dimer (a) and trimer (b). The labels represent the permutational isomers connected by the feasible tunneling motions. Descriptions of the elements in the matrices can be found in the text. The tunneling matrix for the pentamer can be found in the supplementary material for this article. (a) Dimer Hamiltonian. (b) Trimer Hamiltonian.

Close modal

The water trimer exhibits two feasible tunneling motions: flipping and bifurcation. The flipping motion has been shown to be essentially barrierless and can thus be neglected within the context of this study. However, the bifurcation pathway, shown in Fig. 1(b), results in 8 degenerate minima. We can construct the tunneling Hamiltonian by inspection of the group operation that represents bifurcation. As shown by Keutsch et al.,31 the tunneling Hamiltonian for the trimer is given in Fig. 2(b), where A represents a single bifurcation event, B represents two consecutive bifurcation events, and C represents three consecutive bifurcation events. Experimentally, we have only observed single bifurcation events, so we set B = C = 0. The term υ represents the vibrational origin. We show the permutational isomers corresponding to the labels 18 in the supplementary material.

The water pentamer also exhibits two feasible tunneling pathways: flipping and bifurcation. Again, the flipping pathway is essentially barrierless and also can be ignored here. The bifurcation pathway, shown in Fig. 1(c), results in 32 permutational isomers. A tunneling Hamiltonian can be constructed to account for the experimental splitting pattern and is shown in the supplementary material. From these simple tunneling Hamiltonians, we can extract a measurement of the tunneling lifetime for each water cluster in turn, using the typical time evolution operator. As a note, the tunneling magnitudes and relative signs were taken from experimental studies31,35–44 and correspond to the eigenvalues observed. In more sophisticated dynamical tunneling analysis, the relative signs of matrix elements can play an important role in the observed dynamics. Due to the simple approach applied here, a change in relative signs would change the lifetime for an individual pathway, but overall the same lifetimes would be observed. As each pathway represents an essentially identical motion (breaking and reforming the hydrogen bond), further analysis of this change would provide insight into how the different tunneling pathways might interfere with each other. We only use experimentally observed values for the tunneling; though we acknowledge that theoretical values exist for some missing modes.

The results of the time-dependent calculations are shown numerically in Table I and pictorially in Fig. 3. We define the tunneling lifetime in these clusters as the time required to break and reform a hydrogen bond, which corresponds to the period of the oscillating cosines in the probability equation. This is similar to the approach employed by Keutsch et al. in their earlier study of dynamics in the water trimer.31 

TABLE I.

Tunneling magnitudes and lifetimes for the water dimer. Values are taken from Refs. 35–45.

(H2O)2
Vibrational origin (cm−1)Geared interchange tunneling (cm−1)Anti-geared interchange tunneling (cm−1)Bifurcation tunneling (cm−1)Lifetime range (ps)Shortest bifurcation lifetime (ps)Shortest interchange lifetime (ps)
7.6 1.75 × 10−1 −1.26 × 10−2 2.20 × 10−2 44.3-667 758 44.3 
87.75 1.02 × 10+00 3.88 × 10−1 5.40 × 10−3 5.93-26.7 3080 5.93 
102.8 1.48 × 10+00 1.50 × 10+00 1.38 × 10−1 2.80-476 120 2.80 
124.255 1.30 × 10+00 −1.24 × 10+00 8.16 × 10−1 3.29-139 20.4 3.29 
151.875 5.48 × 10−1 5.50 × 10−2 6.55 × 10−1 13.3-318 25.4 13.8 
531.67 8.46 × 10−1 −1.73 × 10−1 5.19 × 10−1 8.18-109 32.2 8.18 
(H2O)2
Vibrational origin (cm−1)Geared interchange tunneling (cm−1)Anti-geared interchange tunneling (cm−1)Bifurcation tunneling (cm−1)Lifetime range (ps)Shortest bifurcation lifetime (ps)Shortest interchange lifetime (ps)
7.6 1.75 × 10−1 −1.26 × 10−2 2.20 × 10−2 44.3-667 758 44.3 
87.75 1.02 × 10+00 3.88 × 10−1 5.40 × 10−3 5.93-26.7 3080 5.93 
102.8 1.48 × 10+00 1.50 × 10+00 1.38 × 10−1 2.80-476 120 2.80 
124.255 1.30 × 10+00 −1.24 × 10+00 8.16 × 10−1 3.29-139 20.4 3.29 
151.875 5.48 × 10−1 5.50 × 10−2 6.55 × 10−1 13.3-318 25.4 13.8 
531.67 8.46 × 10−1 −1.73 × 10−1 5.19 × 10−1 8.18-109 32.2 8.18 
(D2O)2 
Vibrational origin (cm−1)Geared interchange tunneling (cm−1)Anti-geared interchange tunneling (cm−1)Bifurcation tunneling (cm−1)Lifetime range (ns)Shortest bifurcation lifetime (ns)Shortest interchange lifetime (ns)
4.16 9.38 × 10−3 −3.75 × 10−4 2.40 × 10−4 0.855-22.2 69.4 0.855 
65.726 2.66 × 10−1 6.24 × 10−2 −1.23 × 10−3 0.0254-0.134 13.5 0.0254 
83.5 1.21 × 10−1 9.71 × 10−3 3.24 × 10−3 0.0637-0.858 5.15 0.0637 
89.873 1.09 × 10−1 −1.39 × 10−3 −2.54 × 10−3 0.0753-6.06 6.58 0.0753 
(D2O)2 
Vibrational origin (cm−1)Geared interchange tunneling (cm−1)Anti-geared interchange tunneling (cm−1)Bifurcation tunneling (cm−1)Lifetime range (ns)Shortest bifurcation lifetime (ns)Shortest interchange lifetime (ns)
4.16 9.38 × 10−3 −3.75 × 10−4 2.40 × 10−4 0.855-22.2 69.4 0.855 
65.726 2.66 × 10−1 6.24 × 10−2 −1.23 × 10−3 0.0254-0.134 13.5 0.0254 
83.5 1.21 × 10−1 9.71 × 10−3 3.24 × 10−3 0.0637-0.858 5.15 0.0637 
89.873 1.09 × 10−1 −1.39 × 10−3 −2.54 × 10−3 0.0753-6.06 6.58 0.0753 
Lifetimes for trimer tunneling
(H2O)3
Vibrational origin (cm−1)Bifurcation tunneling (cm−1)Shortest bifurcation lifetime (ps)
42.9 6.50 × 10−4 8550 
65.6 −4.25 × 10−3 1310 
87.1 −4.82 × 10−3 1150 
520 6.67 × 10−1 8.33 
Lifetimes for trimer tunneling
(H2O)3
Vibrational origin (cm−1)Bifurcation tunneling (cm−1)Shortest bifurcation lifetime (ps)
42.9 6.50 × 10−4 8550 
65.6 −4.25 × 10−3 1310 
87.1 −4.82 × 10−3 1150 
520 6.67 × 10−1 8.33 
(D2O)3 
Vibrational origin (cm−1)Bifurcation tunneling (cm−1)Shortest bifurcation lifetime (ns)
19.5 1.67 × 10−5 333 
28 −1.50 × 10−5 371 
41.1 −2.50 × 10−5 222 
81.8 4.50 × 10−5 124 
89.6 1.00 × 10−4 55.6 
98 8.34 × 10−5 66.7 
142.8 1.67 × 10−5 333 
(D2O)3 
Vibrational origin (cm−1)Bifurcation tunneling (cm−1)Shortest bifurcation lifetime (ns)
19.5 1.67 × 10−5 333 
28 −1.50 × 10−5 371 
41.1 −2.50 × 10−5 222 
81.8 4.50 × 10−5 124 
89.6 1.00 × 10−4 55.6 
98 8.34 × 10−5 66.7 
142.8 1.67 × 10−5 333 
Lifetimes for pentamer tunneling
(H2O)5
Vibrational origin (cm−1)Bifurcation tunneling (cm−1)Shortest bifurcation lifetime (ps)
89 8.01 × 10−5 41 600 
Lifetimes for pentamer tunneling
(H2O)5
Vibrational origin (cm−1)Bifurcation tunneling (cm−1)Shortest bifurcation lifetime (ps)
89 8.01 × 10−5 41 600 
(D2O)5
Vibrational origin (cm−1)Bifurcation tunneling (cm−1)Shortest bifurcation lifetime (ns)
27.3a 1.33 × 10−6 2510 
512 3.14 × 10−2 0.106 
(D2O)5
Vibrational origin (cm−1)Bifurcation tunneling (cm−1)Shortest bifurcation lifetime (ns)
27.3a 1.33 × 10−6 2510 
512 3.14 × 10−2 0.106 
a

This entry represents all of the values of (D2O)5 where bifurcation tunneling is below experimental resolution: the vibrational origins are 27.3, 30.2, 45, 45.4, 47.7, 50.7, and 81.2 cm−1.

FIG. 3.

Typical plots of probability to be in specific minima versus time for the three cluster systems. (a) Probability of being in minima 1 of the water dimer when I, G, and B are active for the out-of-plane vibrational state. (b) Probability of the 8 degenerate minima of the trimer versus time for the 520 cm−1 vibration. The 4 plotted sets correspond to: P(1) → probability of being in minima 1, P(2,3,4) → combined probability of being in minima 2, 3, or 4, P(5,6,7) → combined probability of being in minima 5,6, or 7, and P(8) is the probability of being in minima 8. (c) Probability of the 32 degenerate minima of the pentamer versus time for the 512 cm−1 vibration. The probability of being in a specific minima, i, is indicated by the sets labeled P(i). Likewise, the combined probability of being in a specific set of minima, i through j, is represented by the sets labeled P(i:j).

FIG. 3.

Typical plots of probability to be in specific minima versus time for the three cluster systems. (a) Probability of being in minima 1 of the water dimer when I, G, and B are active for the out-of-plane vibrational state. (b) Probability of the 8 degenerate minima of the trimer versus time for the 520 cm−1 vibration. The 4 plotted sets correspond to: P(1) → probability of being in minima 1, P(2,3,4) → combined probability of being in minima 2, 3, or 4, P(5,6,7) → combined probability of being in minima 5,6, or 7, and P(8) is the probability of being in minima 8. (c) Probability of the 32 degenerate minima of the pentamer versus time for the 512 cm−1 vibration. The probability of being in a specific minima, i, is indicated by the sets labeled P(i). Likewise, the combined probability of being in a specific set of minima, i through j, is represented by the sets labeled P(i:j).

Close modal

For the dimer, the presence of multiple tunneling pathways complicates the analysis; in order to isolate interchange and bifurcation, we sequentially set the corresponding matrix elements to 0 in order to isolate the pathway of interest. All the calculated lifetimes are shown with respect to vibrational origin in Fig. 4. As a note, we initialized the total wavefunction in the minima labeled 1 for all systems, but this, of course, was an arbitrary choice. Additionally, for the water dimer we report both the range of tunneling lifetimes in addition to the shortest lifetimes for bifurcation and interchange. The shortest lifetimes are reported specifically for comparison with the lifetimes of the trimer and pentamer.

FIG. 4.

Comparison of the shortest tunneling lifetimes for each cluster studied with respect to vibrational origin. The values are given in Table I.

FIG. 4.

Comparison of the shortest tunneling lifetimes for each cluster studied with respect to vibrational origin. The values are given in Table I.

Close modal

Substituting hydrogen by deuterium results in a large decrease in tunneling rates for all studied clusters, as would be expected due to larger mass-weighted pathways. This effect is essentially the same for both the bifurcation and interchange pathways. However, perdeuteration does not seem to negate the tunneling rate enhancement that has been observed for excitations to librational states—particularly for the trimer and pentamer systems.

We observe that the tunneling lifetimes decrease with increasing cluster size, which is not surprising, as the hydrogen bond strength also increases with cluster size. For the water dimer, it is important to point out that for all observed vibrational states, the interchange lifetimes are shorter than the bifurcation lifetimes, which is again unsurprising, since the lifetimes are inversely related to the tunneling magnitudes. We also find that cluster size is a much more reliable predictor of tunneling lifetimes as compared with isotopic identity, underscoring the importance of cooperativity in the hydrogen bond dynamics of clusters.

The effect of intermolecular vibrational excitation depends explicitly on the cluster species. The water dimer is relatively unaffected by the nature and frequency of the excited vibration, which indicates that the tunneling pathways are relatively uncoupled to the vibrational motions. For both dimer isotopomers, excitation of any intermolecular vibration reduces the tunneling lifetime to a common time scale: picoseconds for (H2O)2 and 10s of picoseconds for (D2O)2.

There is a drastic difference between the dynamics of the dimer and those of the trimer and pentamer. For the latter, only excitations of the librational motions reduce the tunneling lifetime, viz., the trimer lifetime reduces to a picosecond time scale (a ∼ 100× decrease relative to the next shortest time scale) and the pentamer reduces to a 100 ps time scale (a ∼ 380× decrease). These enhancements in the tunneling rates indicate that librational vibrations are indeed very effective in inducing hydrogen bond breaking in water clusters, similar to predictions for the liquid.1,2

The most interesting results extracted from these calculations are that the specific nature of the vibrational excitation does not significantly influence the tunneling lifetime of the water dimer, and that only excitations of a librational vibration affect the water trimer or pentamer tunneling lifetimes significantly, relative to those in the ground state. Previous studies of the hydrogen bond breaking dynamics in the bulk liquid have indicated that such librational motions initiate the HB breaking process.1,2 Thus, observing and quantifying dynamics related to hydrogen bond breaking/reformation events for these types of vibrations in water clusters supports this prediction. The specific enhancement of tunneling rates in larger clusters relative to the dimer indicates that hydrogen bond cooperativity is a vital element of these cluster dynamics.

The above theoretical studies also indicated that hydrogen bond dynamics in the liquid are essentially uncorrelated with the number of additional hydrogen bonds the two relevant monomers are participating in, i.e., a very local process. Therefore, we might have expected the water dimer to most closely represent the behavior of the bulk with respect to hydrogen bond dynamics. Indeed, we find that the tunneling lifetimes of the hydrogen bond in the dimer agree quite well with the accepted bulk liquid water hydrogen bond lifetime, which is on the order of a picosecond. However, we also find that the tunneling lifetime for the librationally excited state of the water trimer has a similar time scale. If we compare just the bifurcation pathways, we instead find that the dimer lifetimes are longer compared with those in the excited trimer librational mode. The significance of this being that the bifurcation pathways for all three systems involve breaking and reforming the same hydrogen bond (acceptor and donor monomer roles are not exchanged), whereas the interchange pathway of the dimer exchanges the monomer roles of acceptor and donor. We wish to emphasize that isolated clusters do not exist in the bulk, and that the dynamics of the bulk are much more complicated than for these gas phase cluster system. The possibly coincidental agreement between the tunneling lifetimes observed here with the accepted hydrogen bond lifetime of the bulk warrants further investigation. Specifically, the observation that only excitations to librational vibrations significantly enhance the hydrogen bond breaking tunneling motions in larger water clusters. Recent microwave measurements of the water hexamer and octamer indicate that tunneling is greatly quenched in the ground state of those systems,7,46 probably as a result of their high symmetry, which requires more concerted motions of constituent monomers.3 

See supplementary material for permutational isomers representing the degenerate minima of the water dimer, trimer, and pentamer in Figs. S1, S2, and S5. The tunneling Hamiltonian for the water trimer and pentamer is given in Figs. S3 and S4. The tunneling Hamiltonians with zeroed out elements, as described in the text, are shown in Fig. S6.

This work was supported by the CALSOLV project, an affiliate of the RESOLV collaboration from Bochum, Germany. The authors are grateful to Dr. W. H. Miller and Dr. A. Luzar for useful discussion. The authors thank Mr. M. I. Jacobs for help implementing the calculation. The Berkeley Terahertz project was previously supported by the Chemical Structure, Dynamics, and Mechanisms—a Division of the National Science Foundation under Grant No. 1300723.

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