JOURNAL OF CHEMICAL PHYSICS
VOLUME 111, NUMBER 17
1 NOVEMBER 1999
High symmetry effects on hydrogen bond rearrangement: The 4.1 THz vibrational band of „D2O…4 Mac G. Brown, Frank N. Keutsch, Linda B. Braly, and Richard J. Saykallya) Department of Chemistry, University of California Berkeley, Berkeley, California 94720
共Received 19 May 1999; accepted 4 August 1999兲 Vibration–rotation–tunneling 共VRT兲 spectroscopy has been extended to the 4 THz spectral region through the observation of a second intermolecular vibration of (D2O) 4 . Analysis of the precisely measured perpendicular transition confirms the previously reported cyclic homodromic structure and reveals a dramatically increased 共30⫻兲 hydrogen bond rearrangement rate in the excited state. © 1999 American Institute of Physics. 关S0021-9606共99兲00441-9兴
framework to the other yields a higher energy structure.10 The two degenerate minimum energy structures of the water tetramer interconvertable through torsion of the free hydrogens are shown in Fig. 1. All four free hydrogens must ‘‘flip’’ across the ring in order to produce a degenerate structure, a reasonably high barrier and long path length tunneling process, very different from the flipping pathway in the trimer. Theoretical predictions have been unable to agree on a specific mechanism that interconverts the two minimum energy structures in Fig. 1, and several different tunneling pathways have been proposed.10–12 It is possible that several of the proposed mechanisms are active simultaneously with relative contributions depending on the vibration excited. As described below, it is observed that the doublet tunneling splitting changes dramatically from vibration to vibration and with isotopic substitution.
INTRODUCTION
Much progress has been realized in the last few years in understanding the vibrational and hydrogen bond tunneling dynamics of small water clusters. Laser spectroscopy data for the water dimer has now been rigorously fit to a sixdimensional potential energy surface, achieving the best agreement between spectroscopic observations and a model water potential to date.1 The myriad perturbations observed in the spectra of both the (H2O) 23 and (D2O) 33 isotopic forms of the water trimer have also been precisely analyzed through a global fit of the data based on a model Hamiltonian derived by van der Avoird.2–5 The nature of vibration– rotation–tunneling 共VRT兲 dynamics in the water tetramer, however, remain an interesting, unsolved problem wherein the effects of high symmetry on the hydrogen bond tunneling dynamics become the central issue.
EXPERIMENT
HYDROGEN BOND TUNNELING DYNAMICS
The vibrational band of (D2O) 4 reported here was observed with the Berkeley terahertz spectrometers. These instruments have been described in previous papers,13,14 and therefore only details specific to the work at hand are reported here. Briefly, terahertz laser light is generated by using a line-tunable CO2 laser to pump a molecular gas terahertz laser. The terahertz laser lines used in this work were the 70 m 共4.251 673 6 THz兲 CH3OH laser and 72 m 共4.158 915 8 THz兲 13CH3OH laser. The resulting fixed frequency light is coupled to an antenna contacted to a Schottky barrier diode while at the same time tunable microwave radiation is coupled via the antenna to the diode. The nonlinear properties of the Schottky barrier diode mix the microwave and terahertz light, creating tunable sidebands with frequencies of 共laser兲⫾ 共microwave兲, which gives us an effective tuning range of ⫾2 cm⫺1 共60 GHz兲 around each laser. These tunable sidebands are multipassed in the throat of a pulsed supersonic expansion of argon and H2O and detected on an unstressed germanium/gallium photoconductive detector developed by Haller and co-workers at the University of California Berkeley. A large number of terahertz gas lasers have been discovered, a minimal reference search finding lasers covering the range of frequencies from 4 to 350 cm⫺1. We have been limited in the past to frequencies below 3.3 THz
The equilibrium structure of the water tetramer 共Fig. 1兲 is a nearly planar homodromic ring, with each water monomer in the cluster acting as both a single donor and single acceptor of hydrogen bonds.6 The tetramer equilibrium structure belongs to the S 4 point group, making it the only water cluster smaller than the octamer with an equilibrium structure with higher than C s symmetry.7 This high symmetry imposes very different dynamical constraints from those observed in the water trimer and pentamer. The free hydrogens in the tetramer alternate above and below the quasiplane defined by the oxygen framework. In odd-numbered water clusters viz. the trimer8 and the pentamer,9 a pair of free hydrogens are forced to be on the same side of the oxygen plane, resulting in a frustrated equilibrium structure. In such frustrated structures a simple ‘‘flip’’ of the hydrogen from one side of the plane to the other becomes a low barrier degenerate rearrangement process which dominates the low frequency 共far-infrared兲 spectrum of these clusters, creating manifolds of low energy torsional levels. Such a low barrier rearrangement is not accessible to the water tetramer, as a simple flip of a single hydrogen from one side of the oxygen a兲
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E 共 J,K 兲 ⫽ 0 ⫹B 共 J 共 J⫹1 兲兲 ⫹ 共 C⫺B 兲 K 2 ⫺D J 共 J 共 J⫹1 兲兲 2 ⫺D JK 共 J 共 J⫹1 兲 K 2 兲 ⫺D K K 4 .
FIG. 1. The equilibrium structure of the water tetramer has free hydrogens 共deuteriums兲 alternating above and below the OOOO ring. The two equivalent structures interconvertable through degenerate tunneling are shown as structure 共a兲 and structure 共b兲. The pathway for this process is not known, although several proposed tunneling pathways are discussed in the text.
共100 cm⫺1兲 due to limitations in the mixing response of Schottky barrier diodes at high frequencies. Recent improvements in Schottky barrier technology by Crowe and co-workers15 have allowed us to expand the frequency coverage of the spectrometer to 4.5 THz 共150 cm⫺1兲. This extension of our experimental frequency coverage is crucial because it allows us to access an important new class of vibrational modes in water clusters, the hydrogen bond stretch motions. An earlier version of the high frequency Schottky barrier diode, the 1T15 Schottky barrier diode, was used for initial experiments on the 70 and 72 m lasers. This diode was found to be overly sensitive to microwave and terahertz laser power fluctuations for effective day-to-day use, small power surges resulting in a loss of contact between the antenna and the diode. The 1T24 Schottky barrier diode has proven to be a vast improvement in this area. In fact, in practical use the 1T24 diode was found to be nearly as easy to use as the 1T12 Schottky barrier diode used in previous, lower frequency experiments.
共1兲
The results of the fit are summarized in Table I. The ground ⬙ , and D K⬙ were fixed state vibrational constants B ⬙ , D J⬙ , D JK at the values determined for the ground state by the analysis of the (D2O) 4 tetramer band7,16,17 at 67.8 cm⫺1. The poorer quality of the fit for the new band is caused by the Coriolis interactions which are still present even for K⬎4, although they are much smaller than for the lower K values. As the vibrational band reported here is a perpendicular band, it therefore represents the first absolute determination of the ground state C rotational constant. The parallel band at 67.8 cm⫺1 only allowed the difference in C rotational constants (C ⬘ ⫺C ⬙ ) to be fit. Surprisingly, it was found that the inertial defect for the tetramer had a positive value in both the ground and excited states, a strong indication of planarity. This indicates that tunneling between the two structures in Fig. 1 creates an expectation value equivalent to a planar structure without significant out-of-plane vibrational motion. The vibration, in fact, may even be driving the deuterons into the plane.12 The water trimer, in contrast, was found to have a large negative inertial defect, indicating large-scale out-ofplane torsional motion by the free hydrogens. Note that the nonplanar individual structures in Fig. 1 would both give symmetric top spectra with a negative inertial defect. Until the Coriolis perturbations are explained, however, these results should be interpreted with caution. It is possible that the rotational constant C is contaminated through the Coriolis interactions. DISCUSSION
ANALYSIS
The vibrational band of (D2O) 4 reported here centered at 137.7 cm⫺1 共4.132 885 THz兲 appears as a simple perpendicular band of a symmetric top until a closer analysis is performed. Severe Coriolis perturbations were observed for low K transitions. Examples of these perturbations are shown in the inset of Fig. 2. Notice that the K ⬘ ⫽0←K ⬙ ⫽1 and K ⬘ ⫽1←K⬙⫽2 transitions are ‘‘pushed together’’ by Coriolis perturbation while the K ⬘ ⫽1←K ⬙ ⫽0 and K ⬘ ⫽2←K ⬙ ⫽1 transitions are ‘‘pulled apart.’’ The K ⬘ ⫽1←K ⬙ ⫽2 Q branch is dramatically split, probably through Coriolis interaction. The K ⬘ ⫽3←K ⬙ ⫽4 and K ⬘ ⫽3←K ⬙ ⫽2 Q branches are split to a smaller degree. These effects, along with the splitting in K⫽2 of the previously observed 67.9 cm⫺1 band,7,16,17 remain unexplained. Similar perturbations observed in the water trimer have now been rationalized by van der Avoird in terms of Coriolis interactions of the torsional motion of the hydrogens and the overall rotation of the cluster.2–5 It is hoped that a similar treatment of Coriolis effects in the water tetramer will eventually allow us to include all of the transitions observed in the tetramer in one coherent fit, as accomplished for the water trimer. Due to these complications, the fit reported here only includes transitions with values of K larger than four. These high K transitions were fit to the standard symmetric top rotational energy level expression:
Analysis of the vibrational assignments and quantum mechanical tunneling in these highly nonrigid water clusters requires the use of permutation-inversion group theory.18 The effects of hydrogen bond rearrangement tunneling are present in the spectrum of (D2O) 4 reported here, splitting each transition by 192 MHz into a doublet. This spacing does not change with J and K, indicative of degenerate tunneling. The equilibrium structure of the tetramer 共Fig. 1兲 is a highly symmetric structure belonging to the S 4 point group. Tunneling between the two degenerate structures shown in Fig. 1 doubles the size of group, giving rise to a character table isomorphic to C 4h 共Table II兲. It should be noted that the symmetry labels in Table II are slightly different than those used in our previous papers.7,16,17 We have changed our notation in order to obtain consistency with standard Schoenflies notation. The new labeling scheme uses A, B, and E symmetry labels to indicate correlation with the symmetry labels of the S 4 rotational subgroup. The ⫾ superscripts are used to indicate symmetry with respect to the E * operation, while the g and u subscripts indicate symmetry with respect to the tunneling pathway between the two versions of the equilibrium structure in Fig. 1, the (AC)(BD) * operation. Previously we used g and u subscripts to indicate symmetry with respect to E * . The symmetry labels reported here are now consistent with those used by Leutwyler et al. in his treatment of the tetramer.11,12
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Vibrational band of (D2O) 4
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FIG. 2. A stick spectrum representation of the 4.1 THz (D2O) 4 tetramer band is shown. A fit of the observed perpendicular 共delta K⫽⫾1兲 rovibrational transitions to the Watson S-reduced Hamiltonian revealed several intense perturbations. Inset 共a兲 details some of the perturbations observed. These perturbations are likely caused by Coriolis interactions similar to those recently described for the water trimer 共Refs. 2 and 3兲. Inset 共b兲 gives a sense of the signal-to-noise recorded.
The rotational subgroup (S 4 ) is contained in the upper left quadrant of Table II. The irreducible representation共irrep兲 of each 兩J,K典 rotational level is defined by R Z 兩 J,K 典 ⫽e iK  兩 J,K 典 ,
共2兲
from which the symmetry assignments in Table III are derived. Table III and simple direct products give the information required to assign the symmetry of the vibration. All transition integrals must follow the standard relation for nonvanishing integrals, ⌫ vib⬘ 丢 ⌫ rot⬘ 丢 ⌫ 丢 ⌫ vib⬙ 丢 ⌫ rot⬙ ⫽A ⫹ g .
共3兲
If we take the example of a transition arising from the 兩 J ⫽1,K⫽0 典 rotational level of the totally symmetric (A ⫹ g ) ground vibrational state being promoted to the 兩 J⫽1,K⫽1 典 rotational level of an unknown excited vibrational state (⌫ vib) by the dipole operator (A ⫺ u ) we find that the symmetry ⫹ of the excited vibrational state is either E ⫺ g or E u . This is in agreement with the standard spectroscopist’s rule that perpendicular vibrational bands involve degenerate vibrational states. If we assume that the ground vibrational state tunnel⫺ ing splitting (A→A ⫹ g 丣 B u ) gives a doublet of energy levels ⫹ with the A g symmetry component at lower energy, it follows that the observation of an E ⫺ g symmetry vibration would give rise to a doublet that is a sum of the tunneling splittings in the ground and excited states while an E ⫹ u vibration would
give rise to a difference of tunneling splittings 共Fig. 3兲. Energetically the most likely vibrational assignment is to the ⫺1 E⫺ by g vibrational mode of (D2O) 4 predicted at 174.7 cm 12 Schu¨tz et al. using ab initio methods and at 184.7 cm⫺1 by Sabo et al.11 through DVR calculations on a torsional potential19 fit to ab initio points. All other degenerate vibrational levels were predicted to arise at considerably higher frequencies. It is understandable that the harmonic frequencies obtained through ab initio calculations would give higher frequencies than experimentally observed and a similar effect was observed for the torsional vibration in the water trimer. It is less clear why fully coupled calculations on a torsional potential should give results that are further from the experimentally observed values than the harmonic results. This suggests that the torsional potential surface used, a fit to ab initio calculated points, does not describe this particular vibration well, which is somewhat surprising, as the 67.9 cm⫺1 band was found to be fairly well described by the initial ab initio calculations.12 Tunneling splittings: The observation of large tunneling splittings in the water tetramer has created an interesting problem as there are a number of tunneling pathways between the two permutational isomers of the equilibrium structure shown in Figs. 1共a兲 and 1共b兲, all of which involve long tunneling pathways, high potential barriers, and large effective masses. Finding a tunneling splitting in (D2O) 4 at
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TABLE I. The parameters for the 67.76 and 137.76 cm⫺1 VRT bands of (D2O) 4 and the 67.84 cm⫺1 VRT band of (H2O) 4 are shown. All values are in megahertz. The ground state parameters for the 137.76 cm⫺1 band are fixed at the values determined for the 67.76 cm⫺1 band of (D4O) 4 . (H2O) 4
(D2O) 4
(D2O) 4 共this work兲
0
2 035 397.57共40兲
2 032 688.38共33兲
4 132 885共1兲
B⬙ C⬙ D J⬙ ⬙ D JK D ⬙K
3509.994共79兲
3079.512共36兲
0.0123共13兲 ⫺0.0287共31兲 0a
0.0089共04兲 ⫺0.0177共07兲 0a
3079.512a 1497.0共3兲 0.0089a ⫺0.0177a 0.04共1兲
B⬘ C⬘ D J⬘ ⬘ D JK D K⬘
3526.817共83兲
3091.726共31兲
(⫽D J⬙ ) ⬙ ) (⫽D JK 0a
0.0092共03兲 ⫺0.0181共06兲 0a
⌬C
⫺4.1465共53兲
⫺3.4771共67兲
Splitting
2260.83
5.60
Parameter
a
3070.7共1兲 1494.5共3兲 0.007共1兲 ⫺0.018共3兲 0.035共8兲
192
Fixed.
all was something of a surprise, to say nothing of the very large 共192 MHz兲 splitting observed. Several papers have been written discussing the possible tunneling pathways between the two versions of the equilibrium structure.10–12,16,19–21 With the notable exception of a recent paper by Loerting, Liedl, and Rode,20 the pathways involve either sequential flips of the free hydrogens from one side of the ring to the other or bifurcated hydrogen bonds. The most complete treatment of the tunneling pathways is given in the paper by Wales and Walsh10 共WW兲 in which four distinct pathways, abbreviated as 共pppp兲, 共updp兲, (uupd)/(uu pd), and 共udbd兲, connect the version of the equilibrium structure in Fig. 1共a兲 and the version in Fig. 1共b兲. Three of these pathways, the 共pppp兲, 共updp兲, and (uupd)/(uudp) pathways, are torsional pathways, while 共udbd兲 involves a bifurcated hydrogen bond as a transition state. The torsional pathways interconvert the structures in Fig. 1 by four, two, and one simultaneous flips of the free hydrogens from one side of the oxygen framework to the other, respectively. The important features to consider in weighing the importance of these vari-
TABLE III. The symmetry of the rotational energy levels depend only on the rotational projection quantum number K, as explained in the text. The symmetry of the two tunneling components for different values of K is shown. Rotational symmetry K⫽0
⫺ A⫹ g 丣 Bu
K⫽⫾1
⫹ E⫺ g 丣 Eu
K⫽⫾2
A 2g 丣 B ⫺ u
K⫽⫾3
⫹ E⫺ g 丣 Ed
K⫽⫾4
⫺ A⫹ g 丣 Bu
ous pathways are the length of the tunneling pathway, the height of the potential barrier, and the effective mass that must tunnel through the barrier. The pathway suggested by us in our first paper on the water tetramer was 共pppp兲, a simultaneous flip of all four hydrogens. This pathway has a short tunneling path length but a high barrier and a large effective mass. WW predict this pathway to be orders of magnitude less important than the alternatives with longer path length but lower barrier. The 共updp兲 pathway involves the simultaneous flip of two hydrogens and an intermediate nondegenerate local minimum, a longer pathway but one involving less mass than the 共pppp兲 pathway. The (uu pd)/(uud p) pathway may be described as a sequential flipping of the four free hydrogens, a long but low mass pathway involving three separate intermediate nondegenerate minima. The bifurcation pathway 共udbd兲 is a relatively low mass and short path length pathway that involves a high barrier. WW suggest that the sequential flipping or (uu pd)/(uud p) pathway is likely to be the most important contributor to the tunneling splitting experimentally observed, although they note that several pathways may simultaneously contribute. The level of calculation required to determine which of these pathways is the ‘‘correct’’ one is prohibitive at present, and in fact they may all contribute to the splittings. They all involve longer path lengths and higher barriers than any of the tunneling pathways observed in water clusters studied thus far. It is hoped, therefore, that a suitably large body of
TABLE II. The permutation-inversion character table for the water tetramer, C 4h , is shown. The rotational subgroup is shown as a subset of the full table.
C 4h
R (0) E
R z( /2) (ABCD) *
R z( ) (AC)(BD)
R z(3 /2) (ABCD) *
共ABCD兲
共ABCD兲
E*
(AC)(BD) *
A⫹ g B⫹ g E⫺ g E⫺ g A⫺ u B⫺ u E⫹ u E⫹ u
1 1 1 1 1 1 1 1
1 ⫺1 ⫺i i ⫺1 1 i ⫺i
1 1 ⫺1 ⫺1 1 1 ⫺1 ⫺1
1 ⫺1 i ⫺i ⫺1 1 ⫺i i
1 ⫺1 i ⫺i 1 ⫺1 i ⫺i
1 ⫺1 ⫺i i 1 ⫺1 ⫺i i
1 1 ⫺1 ⫺1 ⫺1 ⫺1 1 1
1 1 1 1 ⫺1 ⫺1 ⫺1 ⫺1
28 38
22 32
24 34
22 32
(H2O) 4 (D2O) 4
⌫ nspin ⌫ nspin
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Vibrational band of (D2O) 4
J. Chem. Phys., Vol. 111, No. 17, 1 November 1999
FIG. 3. Quantum tunneling between degenerate minima splits each rovibrational energy level of the water tetramer into a doublet. Transitions between such split energy levels give rise to the observed doublet splittings reported in Table I and Table III. Depending on the relative ordering of the doublet symmetry labels, each observed doublet splitting may arise from either a sum or a difference of doublet splittings in the ground and excited states. The transition shown in 共a兲 would give rise to a observed doublet splitting equal to a difference between the doublet splitting in the ground and excited states. 共b兲 A transition where a sum of doublet splittings would be observed.
data can be accumulated on the doublet splitting to enable an assignment of the tunneling mechanism through examining the changes in the tunneling splitting when different vibrational modes are excited. Spectroscopic data for the bands observed thus far are collected in Table IV. The (H2O) 4 and (D2O) 4 bands at 68 cm⫺1 have been classified as resulting from an in-plane ring deformation mode. The tunneling splitting for this vibration changes by more than a factor of 400 from (D2O) 4 to (H2O) 4 , a dramatic isotope effect, far greater than that observed in any other water cluster. It was noted in the paper by WW that while the four proposed tunneling pathways exhibit a large tunneling isotope effect, the magnitude of the tunneling isotope effect is almost the same for the different pathways. Unfortunately, this prevents us from distinguishing between the different mechanisms based on the isotope effect. The type of vibration, however, is perhaps able to give us more information. It seems unlikely that a ring deformation mode would contribute a significant amount of energy toward a torsional pathway, perhaps explaining the small tunneling splitting for the 68 cm⫺1 vibration, whereas a ring deformation mode might seem likely to assist a pathway with a bifurcated transition state, where hydrogen bonds are broken and reformed. In comparison, the E g symmetry torsional mode to which we have assigned the 137.7 cm⫺1 vibration of (D2O) 4 can be directly compared with the sequential flip tunneling mechanism (uu pd)/ (uudp). This vibrational mode appears to lead directly along
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this pathway,12 potentially explaining the large increase 共35 times兲 in tunneling splitting for this torsional vibration compared to that for the 68 cm⫺1 ring deformation vibration. This analysis fits the data but is not, of course, conclusive. Further experimental and theoretical studies are still required to characterize the dynamics of the water tetramer. It is interesting to compare the effects of symmetry on the rearrangement processes of water clusters and ice. In the water tetramer the hydrogen bond rearrangement dynamics are very different from those dynamics observed for the water trimer and pentamer. In the water trimer and pentamer, low symmetry equilibrium structures are vibrationally averaged to higher symmetry through a facile local rearrangement mechanism, the flipping of single hydrogen bonds. In the water tetramer the increased symmetry of the equilibrium structure inhibits this mechanism rather completely, allowing the observation of different, less facile, rearrangement pathways. Similarly, it has been experimentally observed that the dielectric constant in ice, a measurement of the rearrangement dynamics, is considerably greater in the proton disordered forms 共Ice In , etc.兲 when compared to proton ordered ice 共Ice II and Ice IX兲.22 Symmetry apparently plays an important role even in the dynamics of the bulk phases of water.
CONCLUSIONS
The extension of tunable terahertz laser technology to 4 THz has made it possible to obtain and fit a new perpendicular vibrational band of (D2O) 4 . It is anticipated that the new capability of the Berkeley terahertz spectrometers will allow us to obtain spectra that will test new and interesting regions of water cluster potential energy surfaces than have been obtained previously. Of particular interest is the hydrogen bond stretching region of the spectrum 共the ‘‘translational band’’ of liquid water兲, a region that our instrument has previously been unable to study. The effects of tunneling in the water tetramer give rise to a doublet splitting that varies dramatically with vibrational type and isotopic substitution. A vibrationally averaged planar structure results from tunneling between the two degenerate structures depicted in Fig. 1. The most likely tunneling pathway between these two structures involves sequential flips of the free hydrogens from one side of the oxygen framework to the other. The large increase in the tunneling splitting when the 137.7 cm⫺1 E ⫺ g vibration reported here is excited is an indication that the vibration assists the tunneling. It will be interesting to see the results obtained when vibrations that assist other pathways are excited. In summary, the tetramer is perhaps the least understood water cluster observed to date. A considerable amount of experimental
TABLE IV. Tunneling splittings for the experimentally observed vibrational modes of the water tetramer. Cluster type (H2O) 4 (D2O) 4 (D3O) 4
Band frequency cm⫺1
Splitting 共MHz兲
Description of vibration
Reference
67.9 67.8 137.8
2260 5.6 192
In-plane ring deformation In-plane ring deformation Torsional
16 7, 16 This work
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and theoretical effort will still have to be expended before an understanding of the energetics and dynamics of the water tetramer comparable to that achieved for the trimer is accomplished. ACKNOWLEDGMENT
This work was supported by the Experimental Physical Chemistry program of the National Science Foundation. 1
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J. Chem. Phys., Vol. 111, No. 17, 1 November 1999
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