Terahertz VRT spectroscopy of the water hexamer-d₁₂ prism: Dramatic enhancement of bifurcation tunneling upon librational excitation Free

The challenge in developing a general predictive molecular-scale description of water is essentially that of correctly describing its extended and dynamic hydrogen-bonded network.^1–9 High-resolution spectroscopy of water clusters provides a means of systematically untangling its complexities. In this context, the water hexamer is of particular interest, as it is the smallest cluster exhibiting a 3-D hydrogen bond network.^4,10–23 Previous studies have revealed the existence of low energy cage, prism, and book hexamer isomers within a supersonic expansion.^12,24–27 It has been established that the cage isomeric structure is the global minimum energy structure for the all-H₂O isomer,^12,28 although the prism isomer lies in close energetic proximity, and is actually the lowest energy isomer for the all-D₂O isomer.^4,25,28–36 Additionally, cyclic hexamer clusters have been observed in frozen rare gas matrices.^37,38

The first detailed experimental study of a water hexamer cluster was reported by Liu et al., who observed and characterized terahertz spectra of the cage isomer in a supersonic expansion.^12,24 More recently, Perez et al. observed the cage, prism, and book isomers (in addition to heptamer and nonamer clusters) in a supersonic expansion using broadband microwave spectroscopy.^25,39 This study characterized the ground state structures of these isomers at high resolution and provided critical insights into the associated hydrogen bond network tunneling motions. A subsequent study by Richardson et al. provided an in-depth description of the tunneling motions of the prism, revealing a pathway which simultaneously breaks two hydrogen bonds.²⁶ The structure of the water prism is shown in Fig. 1.

Here we report the measurement of a parallel vibration-rotation-tunneling (VRT) librational band of a (D₂O)₆ cluster centered in the 15 THz (510 cm⁻¹) region. The experimental rotational constants describing the mass distribution in the cluster via the principal moments of inertia extracted from the present VRT spectra agree most closely with those predicted for the prism structure. We find a dramatic librational enhancement in the bifurcation tunneling splittings of the D₂O cluster relative to the ground state, in analogy with other recent cluster studies in this spectral region.^40–44 

Our previous investigations of water dimer, trimer, and pentamer librational motions in the 15 THz region prompted us to search for similar transitions in larger clusters.^43,45 The Berkeley diode laser/supersonic beam spectrometer used in this study has been described in detail elsewhere and only a short description is provided here.^46,47

A helium-cooled spectrometer (Spectra Physics) using lead-salt diodes (Laser Photonics) was used to produce infrared radiation from 509 to 514 cm⁻¹. The beam was multipassed 18-22 times through a pulsed planar supersonic expansion of a mixture of D₂O and He using a Herriott cell and detected using a helium-cooled (Si:B) photoconductive detector (IR Labs). The supersonic expansion was produced by bubbling pure He gas, with a backing pressure of 1-2 atm through liquid D₂O (Cambridge Labs, 99.96% purity), and then expanding through a 101.6 mm long slit at a repetition rate of 35 Hz into a vacuum chamber maintained at ∼200 mTorr by a Roots blower (Edwards 4200) backed by two rotary pumps (E2M 275).⁴⁸ Simultaneously, the fringe spacing of a vacuum-spaced etalon and carbonyl sulfide (OCS) reference gas spectra was detected with a liquid He cooled (Cu:Ge) detector (Santa Barbara Research Center) and recorded to enable precise frequency calibration. The observed linewidths of ∼30-40 MHz full-width half maximum (FWHM) are somewhat larger than the Doppler-limited linewidths extrapolated from earlier experiments using argon expansions. Typical frequency measurement accuracy is 10-20 MHz, limited by both linewidths of the cluster absorptions and laser drift. Spectra were detected in direct absorption using a time-gated phase sensitive signal processing approach.⁴⁸ 

Accessing the 15 THz (500 cm⁻¹) region of the electromagnetic spectrum has generally been notoriously difficult. The spectra reported here required the use of 10 separate laser diodes, each scanned across several modes to cover the specified spectral range. Moreover, large laser gaps are present in the spectra, which causes considerable difficulty in the assignment. Additionally, the spectrum reflects several distinct laser intensity fluctuations across different devices that are apparent in the complete spectra shown in Fig. 2. Specifically, between 512.4 cm⁻¹ and 513.2 cm⁻¹ the laser intensity is enhanced (as was found in a previous study⁴¹) and between 509.5 cm⁻¹ and 510 cm⁻¹ the laser intensity is depressed.

We have assigned 142 of the 188 observed transitions in the studied region using the PGOPHER platform.⁴⁹ The transitions belong to three distinct a-type (ΔK_a = 0) subbands, which are assigned to different tunneling levels of the water hexamer-d₁₂ prism isomer. The observed transitions were fit to an S-reduced Watson Hamiltonian. For the assignment, we fixed the ground state constants to those obtained in Ref. 44. Correlation matrices of the fit and a list of all assigned transitions are given in the supplementary material.

To elucidate the nature of the observed vibration, anharmonic frequencies and rotational constants were calculated by applying generalized second-order vibrational perturbation theory (GVPT2)^50,51 to the second-order Møller–Plesset (MP2) perturbation theory potential energy surface using the Gaussian 09 D.01 package.⁵² The geometry of the lowest energy D₂O prism was optimized at the MP2/aug-cc-pVDZ level of theory enforcing “very tight” convergence criteria, following which a GVPT2 calculation was performed at the same level of theory using the parameters recommended by Temelso and Shields.²³ The Gaussian input file, the full list of calculated harmonic and anharmonic frequencies, and the rotational constants are listed in the supplementary material.

We obtained a good quality fit of the observed transitions; the average RMS of the fit is ∼37 MHz as a result of the wavelength accuracy of <20 MHz and the observed linewidths of 30-40 MHz (Table I). We note that the A, B, and C constants of all three bands show good agreement with one another, providing evidence that these bands originate from tunneling sublevels of a common excited state. Additionally, a fit with floating ground state constants yielded good agreement with the observed values from Ref. 44, although we have chosen to keep those values fixed in the reported fit.

We refer the reader to the recent work of Richardson et al. for a detailed treatment of the water hexamer prism tunneling dynamics and instead focus only on the relevant considerations for the fully deuterated cluster.²⁶ The feasible prism tunneling motions can be described by the complete nuclear permutation inversion (CNPI) subgroup isomorphic to point group D_2d. Previous work has shown that the water hexamer prism exhibits two feasible tunneling motions (here we define “feasible” as “experimentally observed”⁵³), which are referred to as P_a and P_g. In the CNPI notation, P_a = (A D)(B F)(C E)(1 7)(2 8)(3 11)(4 12)(5 9)(6 10), where the labels correspond to the structure in Fig. 1. The motion associated with this element involves a double flip of the free hydrogens, resulting in breaking a single hydrogen bond. Likewise, in the CNPI notation, P_g = (A D)(B F)(C E)(1 8 2 7)(3 11)(4 12)(5 9)(6 10), and the motion is described as a double flip accompanied by a bifurcation, which breaks two hydrogen bonds. The character table for this group is given in Table II.

The character representation of the nuclear spin wavefunction (Γ_ns) and the electric dipole moment (Γ_dip) is given at the bottom of Table II. We can reduce Γ_ns to its irreducible representation: $118341A1⊕29403A2⊕29646B1⊕117855B2⊕118098 E$⁠.

For the fully deuterated hexamer prism, the total spin wavefunction must transform as A₁ which leads to the spin statistical weights: A₁:A₂:B₁:B₂:E of 487:121:122:485:486. The irreducible representation of the electric dipole moment is $A1⊕2B2$ and for a-type (ΔK_a = 0) transitions, the dipole transforms as B₂. Thus, the expected selection rules are A₁ <-> B₂, A₂ <-> B₁, and E <-> E. These selection rules lead to a “doublet of triplets” pattern with a central doublet, as shown in Fig. 3. Richardson et al. established that the energy level pattern of the hexamer prism is represented by the eigenvalue of the tunneling matrix shown in Fig. 3(a).²⁶ 

Based on these considerations, we assign the B₂ → A₁, E₂ → E₁, and A₂ → B₁ transitions to the subbands 1, 2, and 3, respectively. This assignment is based on the equal spacing between three subbands and the intensity ratio of 1:1:0.85. While this intensity ratio does not agree quantitatively with the expected ratio (based on spin statistics above), given the large intensity fluctuations of the lasers used, we consider this assignment as the most likely of those possible under parallel selection rules. Based on this assignment, we can estimate the tunneling matrix element, h_g. The separation between subband 1 and subband 2 corresponds to the quantity 2h_g′ + 2h_g″, wherein the prime and double prime represent the values in the excited and ground state, respectively. The separation between subbands 2 and 3 corresponds to the same quantity. Given the measured values in the ground state (which are <1 MHz), we can explicitly calculate the value for the excited state (assuming these transitions originate in the ground vibrational state). Based on our fitted values of the band origins, we find the average value of h_g′ to be ∼1720 MHz, which corresponds to a ca. 1000× enhancement of that tunneling splitting relative to the value for the ground state. This enhancement would be consistent with the enhancement of tunneling splitting observed for the water dimer, trimer, and pentamer upon excitation of a single quantum of 15 Thz librational vibrations. We note that using the intensity ratio to assign the observed subbands to the lower triplet is in some ways arbitrary due to the power fluctuations of the diode laser. It can be argued that the subbands could be assigned to the upper triplet just as easily; however, that assignment does not change the analysis of the tunneling enhancement, as both triplets are symmetric. Further experiments are needed to definitively establish which triplet these subbands represent.

These experiments were conducted in a He supersonic expansion, which cools the rotational temperature to approximately 10 K and the vibrational temperature to ca. 100 K. The most likely vibrational origin of the transitions is the ground state. Based on the calculated anharmonic vibrational frequencies and assuming that the transitions observed originate from the absolute ground state, we attempted to assign a vibrational mode to the experimental transitions; however, we note several interesting observations. We find two likely candidates for anharmonic vibrational modes with transition frequencies from the absolute ground state of 505.693 and 539.359 cm⁻¹, both of which lie close to the observed average band origin of 509.7183 cm⁻¹. However, we stress that this experimental band origin corresponds to the average origin of individual tunneling bands and is not the true vibrational band origin. In Table III, we show a comparison between the observed rotational constants and the predicted values for the 4 closest vibrational modes.

From the table we can see that the observed values are all larger in magnitude than the predicted values. The most striking observation in the calculations is that the calculated values of ΔM (where M is A, B, or C) are opposite in sign and about an order of magnitude smaller than what is observed experimentally. Some of that disagreement can be attributed to the fact that there are large perturbations to the excited state, shown by the large higher order terms in the fit results, indicating a very “floppy” excited state. We have previously observed dramatic fluctuations of the centrifugal distortion constants in this experimental region.^41,43,45 Another possibility would be that the ground state of these transitions is not actually the absolute ground state of this cluster, but rather a “hot band.” While this is unlikely, given the strong cooling in a supersonic expansion, there are several vibrational states located within 10 cm⁻¹ of the absolute ground state. However, further experimental studies and calculations will be needed to determine a definitive resolution of this disagreement. We also stress that these values represent only the lower triplet of the tunneling pattern, which should be fairly representative of the excited state.

We have previously studied the water dimer, trimer, and pentamer in the 15 Thz librational region, with the most salient observation being the dramatic enhancement of the tunneling splittings in these clusters.^40,41,43 We observe a similar effect here; however, we can only determine the enhancement for the h_g pathway. As described by Richardson et al.,²⁶ the h_g pathway motions are characterized by a double flip accompanied by a bifurcation, which involves the breaking of two hydrogen bonds. Studies of the water dimer, trimer, and pentamer tunneling in the 500 cm⁻¹ region involve motions which only break a single hydrogen bond.^40,41,43 We observe a 1000× enhancement relative to the observed ground state tunneling for the (D₂O)₆ prism from microwave experiments.²⁶ This observation indicates that the observed enhancement in the librational region does not seem to be influenced by the number of hydrogen bonds broken in the tunneling pathway. Additionally, the reported enhancement is with respect to the observed tunneling in the (H₂O)₆ prism reported by Richardson et al.²⁶ As observations of the tunneling in the fully deuterated ground state do not presently exist, an accurate measure of the enhancement cannot be obtained; however, due to the larger mass-weighted path for the deuterated species, we can consider the observed 1000× enhancement as a lower bound.

Due to the absence of the higher energy triplet [Fig. 3(c)] in our measured spectra, we cannot presently characterize the h_a tunneling pathway in this excited librational state. The absence of this triplet is most likely a result of the large laser gaps present in the experiment, coupled with the fact that the observed transitions occur near the edge of the available laser coverage.

The results reported here are significant, as the water hexamer represents a transition of the minimum energy structure of water clusters from ring-like forms to a fully 3-D structure. The water prism tunneling motion has been predicted to potentially describe the motions of water in interfacial and confined environments; hence, the results presented here indicate that excitation of librational vibrations has a significant impact on the hydrogen bond dynamics in these macroscopic environments.

We have measured 3 a-type subbands belonging to a common librational vibration of the (D₂O)₆ prism cluster. Assigning the transitions to tunneling sublevels allows us to extract a value of ∼1720 MHz for the h_g tunneling motion, representing a ca. 1000× enhancement of the tunneling splitting, relative to the ground state splitting observed for the (H₂O)₆ cluster. This enhancement is consistent with the dramatic tunneling enhancement observed previously for the water dimer, trimer, and pentamer in the same (15 THz) experimental region.

From comparison to theoretical calculations, we find the observed change in the all-rotational constants to be dramatically larger than predicted. This large change indicates a very “floppy” excited state, which would be consistent with the large tunneling splittings observed and the breaking of one or two hydrogen bonds in the tunneling pathway.

See supplementary material for a list of all the assigned transitions in Table S1 and correlation matrices for the fits in Table S2. Calculated anharmonic vibrational frequencies are given in Table S3. The Gaussian input file for calculations is given in Fig. S1.

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. This project is currently supported by the CALSOLV collaboration, an affiliate program of RESOLV (Ruhr-Unversitaet Bochum). D.J.W. gratefully acknowledges financial support from the EPSRC.