Using a recent, full-dimensional, ab initio potential energy surface [Y. Wang, X. Huang, B. C. Shepler, B. J. Braams, and J. M. Bowman, J. Chem. Phys.134, 094509 (2011)]

together with rigorous diffusion Monte Carlo calculations of the zero-point energy of the water trimer, we report dissociation energies, D0, to form one monomer plus the water dimer and three monomers. The calculations make use of essentially exact zero-point energies for the water trimer, dimer, and monomer, and benchmark values of the electronic dissociation energies, De, of the water trimer [J. A. Anderson, K. Crager, L. Fedoroff, and G. S. Tschumper, J. Chem. Phys.121, 11023 (2004)]. The D0 results are 3855 and 2726 cm−1 for the 3H2O and H2O + (H2O)2 dissociation channels, respectively, and 4206 and 2947 cm−1 for 3D2O and D2O + (D2O)2 dissociation channels, respectively. The results have estimated uncertainties of 20 and 30 cm−1 for the monomer plus dimer and three monomer of dissociation channels, respectively.

The dissociation energy of water clusters is an important topic that has received considerable attention theoretically.1–10 For a general water cluster consisting of N monomers, there are N-1 dissociation channels ranging from dissociation of a single monomer to complete “monomerization” of the cluster. The energy most often reported is the electronic dissociation energy De. The purposes of such calculations are manifold, ranging from establishing benchmark values of this important property to determining the trends of De with cluster size and the number of hydrogen bonds. (One interesting question is the extent to which the De scales linearly with the number of hydrogen bonds.)

However, to make contact with the experimental observable, the quantity of interest is D0. This is related to the De, as usual, by taking into account the zero-point energies (ZPEs) of a cluster and the fragments of interest. The harmonic approximation, which is often used to estimate ZPEs, is suspect for water clusters, which are fluxional and which contain numerous low-lying structural isomers.1–10 Thus, a rigorous calculation of the ZPE must take account of the vibrational anharmonicity, which, given the existence of low-lying isomers mentioned above, may be quite extreme and thus computationally highly demanding. Even for the smallest cluster, the water dimer, which has twelve vibrational degrees of freedom, this calculation is very demanding. Nevertheless, this was done recently, using an accurate ab initio full-dimensional potential energy surface, together with a diffusion Monte Carlo (DMC) calculation of the ZPE of the water dimer, in full dimensionality.11 The calculated D0 was reported to be 1103 cm−1. D0 was subsequently measured, with excellent precision, by Reisler and co-workers12 and reported to be 1105 ± 10 cm−1, in excellent agreement with the calculated value.

Here we report corresponding calculations of D0 for the water trimer, (H2O)3 and (D2O)3, dissociating to the water dimer plus a monomer and three monomers. These calculations make use of the latest ab initio water potential from our group, which is described in detail elsewhere.10 In brief, it a full dimensional, permutationally invariant fit to tens of thousands of electronic energies for the intrinsic 2- and 3-body potentials and a spectroscopically accurate 1-body (monomer) potential. The 3-body fitted potential is much more computer intensive to evaluate than the 2-body potential and so two 3-body fits were done. One is of maximum total polynomial order 5 and the other of order 6. These 1-2-3-body potentials combine to make the general PES for arbitrary number of monomers; they are denoted WHBB/3b5 or WHBB/3b6, depending on whether the more efficient 5th- or 6th-order 3-body fit is used. (Higher-body, long-range interactions have also been incorporated into this PES.10) Both versions of this PES have been shown to accurately reproduce stationary points and electronic dissociations energies, De, for clusters as large as 22 monomers, which are the largest clusters for which reliable benchmarks ab initio results are available.10 

Both WHBB/3b5 and WHBB/3b6 are used in the present trimer calculations to obtain the full-dimensional ZPE, using the DMC method13–15 that we used to obtain the ZPE of the water dimer.11 As noted above, water clusters are fluxional with multiple minima and saddle points. These have been investigated thoroughly for the water trimer.3,16–18 These stationary points are accurately predicted using WHBB/3b5 and WHBB/3b6. They are conventionally denoted by the orientations of the free OH-bonds relative to the plane defined by the three O atoms. The global minimum (GM), depicted in Fig. 1, is conventionally denoted “up-down-up,” with the shorthand notation [udu]. The lowest-lying saddle point (SP1), which separates two equivalent global minima, is denoted [udp], where “p” indicates planar, the local minimum (LM) is [uuu], and the saddle point (SP2) between a GM and LM is [uup].

FIG. 1.

Depiction of the up-down-up [udu] global minimum of the water trimer. The local minimum is the [uuu] configuration, the lowest energy saddle point separating two global minima is up-down-planar [udp], and the higher energy saddle point separating the global and local minima is [uup].

FIG. 1.

Depiction of the up-down-up [udu] global minimum of the water trimer. The local minimum is the [uuu] configuration, the lowest energy saddle point separating two global minima is up-down-planar [udp], and the higher energy saddle point separating the global and local minima is [uup].

Close modal

The energies of these stationary points from WHBB/3b6 and WHBB/3b5, relative to the global minimum, are given in Table I, along with benchmark values obtained from complete basis extrapolation CCSD(T) calculations.3 First, note the very good agreement of the PES results with the benchmark values, with WHBB/3b5 being in slightly better agreement. The Des for the two dissociation channels are also in very good agreement with benchmark values; however, differences of roughly 100 cm−1 (0.28 kcal/mol) are seen. These differences between the PES and benchmark ab initio values for De are larger than for water dimer, which does not have a 3-body interaction. Thus, it seems reasonable to assign this increased difference for the trimer to the 3-body interaction, which is significant and quite difficult to describe accurately with ab initio methods owing to the relatively large basis set superposition error.3 Thus, even the benchmark calculations, which employed several extrapolation methods to the complete basis set limit, have estimated uncertainties in the De for the monomer plus dimer channel of ±14 cm−1 and ±21 cm−1 for the three monomer channel.3 The 3-body component of the WHHB PES is a fit to roughly 40 000 MP2/aug-cc-pVTZ 3-body energies, and this level of theory/basis is certainly subject to a basis set superposition error that could be several times the above estimated benchmark uncertainties.

Table I.

Energies in cm−1 of indicated structures relative to the udu global minimum of the water trimer and indicated dissociation energies.

StructureWHHB/3b5WHHB/3b6Benchmark
SP1 [udp] 83.2 100.8 81.5a 
LM [uuu] 269.6 273.8 269.6a 
SP2 [uup] 278.0 290.6 275.5a 
De [H2O + (H2O)23701.0 3686.8 3788.4a 
De [3H2O] 5441.5 5427.2 5558.3a 
D0 [H2O + (H2O)22642.1 2620.9 2729.1(5th) 
      2722.3(6th) 
D0 [3H2O] 3742.2 3721.0 3858.2(5th) 
      3851.4(6th) 
D0 [D2O + (D2O)22863.6 2842.1 2950.6 (5th) 
      2943.6 (6th) 
D0 [3D2O] 4094.0 4072.6 4210.0 (5th) 
      4202.9 (6th) 
StructureWHHB/3b5WHHB/3b6Benchmark
SP1 [udp] 83.2 100.8 81.5a 
LM [uuu] 269.6 273.8 269.6a 
SP2 [uup] 278.0 290.6 275.5a 
De [H2O + (H2O)23701.0 3686.8 3788.4a 
De [3H2O] 5441.5 5427.2 5558.3a 
D0 [H2O + (H2O)22642.1 2620.9 2729.1(5th) 
      2722.3(6th) 
D0 [3H2O] 3742.2 3721.0 3858.2(5th) 
      3851.4(6th) 
D0 [D2O + (D2O)22863.6 2842.1 2950.6 (5th) 
      2943.6 (6th) 
D0 [3D2O] 4094.0 4072.6 4210.0 (5th) 
      4202.9 (6th) 
a

Reference 3.

It is already known that the Des of the water trimer are not accurately given by simple additivity of the water dimer De, owing the to 3-body interaction. However, it is worth quantifying that here because below we will examine this issue using the newly reported values of the trimer D0s. From the benchmark De for the water dimer of 1764 cm−1, simple additivity yields 3528 cm−1 for the trimer De for the H2O + (H2O)2 channel and 5292 cm−1 for the 3H2O channel. These values underestimate the benchmark Des by 260 and 266 cm−1, respectively.

DMC calculations were performed to obtain the rigorous ZPE of (H2O)3 and (D2O)3. To ensure the robustness of the results of these calculations, independent DMC runs were initiated at the global minimum and the three low-lying stationary points listed in Table I, using both WHBB/3b5 and WHBB/3b6. Each DMC trajectory was propagated using 20 000 walkers for 80 000 steps with a time-step of 10 atomic units, and an α-parameter of 0.1. The definitions of these quantities are given elsewhere,13–15 and their values were determined by running exploratory trajectories, with standard error analyses.13–15 The resulting energies, using WHBB/3b6 for illustration purposes, are given in Fig. 2 as a function of the imaginary time variable. As seen, the results are essentially identical for all four trajectories. The ZPE from each of these runs for (H2O)3 is 15 587 ± 2 and 15 594 ± 2 cm−1 using WHBB/3b5 and WHBB/3b6, respectively. For (D2O)3 the corresponding ZPEs are 11490.4 and 11497.4 cm−1. From standard statistical analyses, e.g., uncorrelated rolling averages, of these fluctuating zero-point energies,13–15 we estimate a statistical uncertainty of ±2 cm−1. The difference in the ZPEs for the two PESs is (gratifyingly) quite small, i.e., 7 cm−1, and that difference will contribute to the final reported uncertainties in the D0s. To get those D0 we use the benchmark De values for the two channels3 together with the above ZPEs for the trimer, the previously reported ZPE for the water dimer11 and the rigorous ZPE for the monomer.11 The results are given in Table I, along with the D0 using the De values from WHBB/3b5 and WHBB/3b6, for reference. Thus, the reported benchmark values of D0 are 3855 and 2726 cm−1, respectively, for 3H2O and H2O + (H2O)2 and 4206 and 2947 cm−1, respectively, for 3D2O and D2O + (D2O)2. We estimate the total uncertainties to be ±20 and ±30 cm−1, for the two channels. The major source of these estimates is the uncertainty in the De already mentioned. The additional uncertainty is from the trimer ZPE of roughly ±6 cm−1 from the two PESs and statistical uncertainty of the DMC calculations. The stated total uncertainties are obtained from adding these three uncertainties.

FIG. 2.

Zero-point energies of (H2O)3 from diffusion Monte Carlo trajectories initiated at four stationary points indicated, the global minimum (GM), a local minimum (LM), and two saddle points (SP1 and SP2) described in detail in the text.

FIG. 2.

Zero-point energies of (H2O)3 from diffusion Monte Carlo trajectories initiated at four stationary points indicated, the global minimum (GM), a local minimum (LM), and two saddle points (SP1 and SP2) described in detail in the text.

Close modal

With these benchmark D0s we can re-visit the issue of additivity of the water dimer dissociation energy, now using the dimer D0. As noted already this D0 is 1104 cm−1 and so additivity would predict 3312 and 2208 cm−1 for 3H2O and H2O + (H2O)2. These are even larger underestimates (roughly 500 cm−1) of the calculated D0s than the underestimates of the Des (roughly 260 cm−1).

Finally, we have begun examining the distribution of the DMC walkers as a function of the imaginary time with the goal of characterizing the zero-point wavefunction. We see significant population at numerous equivalent global minima (and of course significant population of walkers transitioning between them) and much less population at the local minima. This work and recent full-dimensional “instanton” calculations19 of splittings in the water trimer, using an earlier version of the WHHB PES (Ref. 20) and other PESs, are beginning to elucidate the complex, full-dimensional quantum fluxionality of this water cluster.

We thank the National Science Foundation (NSF) (CHE-0848233) for financial support. We also thank Stuart Althorpe for sending a preprint of the work by Richardson, Althorpe, and Wales.19 

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