The Protonated Water Trimer and Its Giant Fermi Resonances ⇑ Nagaprasad Reddy Samala, Noam Agmon

Chemical Physics 514 (2018) 164–175

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Chemical Physics

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The protonated water trimer and its giant Fermi resonances ⇑ Nagaprasad Reddy Samala, Noam Agmon

The Fritz Haber Research Center, Institute of Chemistry, The Hebrew University of Jerusalem, 91904, Israel article info abstract

Article history: The protonated water trimer is a prototype ‘‘proton wire” that can transport two protons nearly concert- Received 8 January 2018 edly. This ‘‘proton transfer mode” (PTM) is an important contributor to the infrared spectrum of the iso- In ﬁnal form 1 April 2018 lated gas-phase cluster. We have simulated its infrared spectrum for both the hydrated and deuterated Available online 3 April 2018 isotopologues, using vibrational 2nd order perturbation theory (VPT2) and ab initio molecular dynamics (AIMD) trajectories. VPT2 calculations explain quantitatively the experimental spectra at both high and Keywords: low frequencies, provided that high-level quantum chemistry is utilized. In the D2-tagged hydrated Cluster cluster, the PTM undergoes giant Fermi resonances (FR’s) with two combination bands. In the deuterated Fermi resonance analogue, one observes a single FR of ‘‘normal” intensity, manifested as the doublet recently Infrared Proton reported experimentally. We provide band assignment for both isotopologues, with and without the Water D2 tag applied in experiment. We discuss possible manifestations of the giant resonance on proton transfer through water wires. Ó 2018 Elsevier B.V. All rights reserved.

1. Introduction of occurrences involve longer-range concerted multi-proton hops [8,9]. Such events are prominent in trajectories of hydronium- Protons in water diffuse abnormally fast, which is attributed to hydroxide recombination [10], and are manifested experimentally the Grotthuss mechanism [1]. In bulk water, this is depicted as an in the inverse temperature (T) dependence of the rate coefficient þð Þ [11] for PT between photoacid and base in liquid water along ‘‘isomerization” reaction from a solvated hydronium, H3O H2O 3 (the ‘‘Eigen cation”), centered on the donor oxygen, via a shared- chains of hydrogen-bonded water molecules (‘‘water-wires”) þð Þ [12]. The shortest water wire beyond the Zundel cation is the pro- proton intermediate, H H2O 2 (the ‘‘Zundel cation”), to an Eigen cation localized on the acceptor oxygen [2]. While the rate limiting tonated water trimer. step involves hydrogen-bond rearrangement in the 2nd solvation Infrared (IR) studies demonstrated that traces of aqueous acids þ + ð Þ shell of the H3O , the actual proton transfer (PT) step in the Zundel in organic solvents consist predominantly of H H2O 3 cations cation can be as short as a vibrational period, or involve barrier [13], rather than the traditional Eigen or Zundel ions. In narrow recrossings during several vibrational periods. Thus, vibrational carbon nanotubes, which host a single-file of water molecules, PT dynamics plays a role in the kinetics of proton mobility. may involve isomerization between two Zundel cations via a tran- þð Þ Indeed, low temperature vibrational predissociation spectra of sient H H2O 3 structure [14]. In the green fluorescent protein þ (GFP) a water-trimer was found to connect the chromophore to the H5O2 Zundel cation show an intense ‘‘proton transfer mode” (PTM) at 1047 cm1 [3], arising from the proton shuttling between the bulk [15]. It may efficiently conduct the proton, generated the two oxygen centers, while at 928 cm1 appears a strong com- upon chromophore photoexcitation, outside the protein barrel bination band, gaining its unusual intensity from the PTM via a structure. Consequently, it is important to understand the dynam- ‘‘Fermi resonance” (FR) [4]. This combination, of OO stretch and a ics of the protonated water trimer and, particularly, its IR spectrum + wag overtone [5], tends to localize the proton as H3O on one side [16–23]. of the Zundel cation, at the expense of the shared proton mode. While it is easy to identify the IR absorption due to the flanking Here we propose that FRs play an important role also in larger pro- water free OH stretch, that of the hydrogen bonded OH (HB–OH) þ stretches is more elusive. Schwarz suggested that some small tonated water clusters, specifically the H ðH2OÞ cluster. 3 1 While the stepwise Eigen-Zundel-Eigen mechanism may absorption in the range 2000–2300 cm might be due to the þð Þ describe the majority of PT events in liquid water [6,7], a minority H H2O 3 cluster [16]. Subsequently, Okumura et al., who first applied vibrational predissociation spectroscopy to this cluster, identified a peak near 2300 cm1 as originating from the HB–OH ⇑ Corresponding author. stretch [17]. Headrick et al. [18] extended the accessible spectral E-mail address: [email protected] (N. Agmon). https://doi.org/10.1016/j.chemphys.2018.04.003 0301-0104/Ó 2018 Elsevier B.V. All rights reserved. N.R. Samala, N. Agmon / Chemical Physics 514 (2018) 164–175 165 range to below 2000 cm1, locating a very strong peak at 1880 cm1. Although they have also observed a broad weak feature near 2300 cm1, they have assigned the 1880 cm1 band to the HB–OH asymmetric stretch (as) mode [18] (denoted below HB–OHas). This concerted 2-proton motion is depicted in Fig. 1 (that also defines a numbering scheme for the atoms), and may be viewed as the PTM responsible for concerted PT in the liquid phase. From its high intensity one might expect facile concerted PT whenever a þð Þ H H2O 3 grouping is formed. As we argue below, due to the FRs this may not always be true. While the above assignment prevailed for a decade [19,20], theory has recently cast some doubt on this simplistic picture. Notice- ably, Duong et al. [22] have improved upon the analytical multidimensional potential energy surface (PES) and dipole surface of Yu & Bowman [21], for which a vibrational self-consistent field (VSCF) analysis combined with vibrational configuration interac- þð Þ tion (VCI) of the untagged (‘‘bare”) H H2O 3 cluster was performed. They observed that the dominant 1880 cm1 band arises ‘‘from a congested series of transitions to ...mixed levels” [22]. This is lamentable, because a major goal in performing vibrational analysis is to provide the physical insight gleaned from clear-cut assignments. Interestingly, in a recent M.Sc. thesis [24], Shtainmetz suggested a FR of the HB–OHas with a combination band involving the two normal modes (NM’s) depicted in Fig. 2. Even more recently, it became possible to measure this spectrum in the intermolecular, terahertz regime (below 1000 cm1) using a free electron laser [23]. The most intense low frequency bands were assigned to the water wag and hydrogen bond stretch. However, it was no longer possible to treat this regime with VSCF/ VCI, so quasiclassical molecular dynamics (QCMD) was used instead for the same analytical PES. The challenge then is to explain the whole spectrum from one theory.

Similarly intriguing is the novel D2-tagged IR predissociation þð Þ spectrum for the deuterated analogue, D2 D3O D2O 2 [22]. Here 1 the 1878 cm PTM band (Fig. 3a) is replaced by a doublet, at 1680 Fig. 2. The hydronium (a) umbrella and (b) rocking modes (harmonic B3LYP-D3/6- and 1452 cm 1 (Fig. 3b). Unfortunately, a simple assignment was 311++G⁄⁄) are (a) symmetric and (b) antisymmetric to reﬂection. They combine and again not feasible, and the isotope effect was notably smaller than interact with the PTM of Fig. 1 to produce the unusually strong combination bands expected. It was consequently suggested that ‘‘calculations that are in the spectra shown in Fig. 3. simpler than the present VSCF/VCI ones (e.g. ...vibrational second order perturbation theory (VPT2)) could be performed to shed light on the unusual spectral features seen in these clusters” [22]. The problem with the VPT2 approach is that bands involving The present work lives up to this challenge by testing VPT2 the excess proton motion appear to be unstable to perturbations, from Gaussian 09 using 2nd-order Møller-Plesset perturbation the- varying sometimes wildly with method and bs. This has been ory (MP2) [with a frozen core (fc)] and density functional theory clearly demonstrated for the protonated water dimer and tetramer (DFT) with several density functionals and basis sets (bs’s) for [33]. Consequently, it has been postulated that for the trimer þ ‘‘VPT2 as implemented in Gaussian 09 will not even qualitatively the H ðH2OÞ cluster over the whole frequency range, as well as 3 reproduce the essential physics controlling the vibrational band for its deuterated analogue. These calculations have been per- pattern” [22]. It turns out that the recent availability of detailed formed with and without the D2 tag. experimental spectra [22,23] now allows to select, from a plethora of diverse spectra [24], those VPT2 calculations that reproduce the experimental results. These, in turn, provide the lacking understanding of the essential physics controlling the vibrational band pattern.

The calculations with the D2 tag are the closest to experiment, suggesting that the 1878 cm1 band is not due to the PTM fundamental (which remains where Schwarz has predicted), but rather + to a combination of the H3O umbrella and rocking modes (Fig. 2), which enters into an abnormally strong FR with the PTM [24]. As in previous studies of protonated water clusters [25–33],we have also performed ab initio molecular dynamics (AIMD) simulations, in which the forces are calculated from DFT (using the B3LYP-D3 functional), while the atoms are propagated classically, using Newton’s equations of motion. This method [34] thus lacks nuclear quantum effects (NQE), which we replace by frequency Fig. 1. The hydronium asymmetric stretch NM (termed herein HB–OHas or PTM) of þð Þ ⁄⁄ scaling, and specialized analysis is required to infer about mode the H H2O 3 cluster (harmonic B3LYP-D3/6-311++G ), with atom labels indicated. Note the antisymmetry to reﬂection. couplings [27,35]. With this, we obtain a consistent interpretation 166 N.R. Samala, N. Agmon / Chemical Physics 514 (2018) 164–175

(a) Hydrated isomer (with D tag)

2 3726 Fundamental 1883 MP2/aug-cc-pVTZ

Overtones 1878 3580 Combination bands 3640 2084 Expt 2284 1943 3540 2383 2109 1534 3726 1639 1059 2307 3814 2961 2410 2693 1992 2367 3628 2544 2664 2175 1553 1080 1621 1907 992 3039

1000 1500 2000 2500 3000 3500 4000 1680 Intensity (b) Deuterated isomer (with D tag) 1452 2 1740 1770 2631 2770 2660 800 1477 1202 1791 1143 1991 2631 2149 2314 2978 1540 1913 818 2063 2648 2767 1424 3038 1186 731 1145

1000 1500 2000 2500 3000 3500 4000

-1 Wavenumber (cm )

þð Þ þð Þ Fig. 3. The experimental D2-tagged IR predissociation spectrum of (a) the H H2O 3 cluster, and (b) the D D2O 3 cluster at 12 K (blue lines, digitized from Ref. [22]), compared with the corresponding (unscaled) VPT2 spectra calculated at the MP2(fc)/aug-cc-pVTZ level, with the tag at the H3 atom of the hydronium core (see Fig. 11b in Ref. [17]). Black, red and green sticks mark fundamental, overtone and combination bands, with their frequencies in cm1. Omitted from the ﬁgure are all combination bands involving a fundamental with a negative or near-zero frequency (modes 22 and 23 for both clusters). For visibility, the fundamental intensities in panel (a) were scaled by a factor of 2. (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version of this article.)

for both VPT2 and AIMD results that sheds new light on the infra- The harmonic/anharmonic frequencies and intensities, for red spectrum of this important cluster. fundamental and combination bands, are collected in the Excel file of the Supplementary Material (SM). The Excel sheet 2. Theoretical methods names identify the tagging, deuteration, method and bs. For example, mp2-avtz-D2-H indicates that the IR spectrum of the hydrated 2.1. VPT2 calculations tagged cluster was computed with MP2/aug-cc-pVTZ. Similarly, b3lyp-pople-D indicates that the IR spectrum of the deuterated The calculation of the anharmonic VPT2 spectra was performed bare cluster was computed with B3LYP-D3/6-311++G⁄⁄. On the using the Gaussian 09 (Revision D.01) suite of programs [36]. Two top of each sheet appear the accuracy conditions. methods were used: (a) MP2(fc), a quantum chemistry method The above mentioned calculations were initially performed only that accounts for electron correlations in the valence shell, and on the bare clusters (no tag), because experiment showed notice- (b) DFT with a variety of exchange–correlation functionals. The able effect of the tag only for the O3H3 vibration [22]: It appears functionals tested, B3LYP-D3, M06-2X, mPW2-PLYP and xB97, between the flanking water symmetric stretch (ss) and as modes 1 were chosen from a recent benchmarking of DFT functional perfor- of the bare cluster, moving some 90 cm to the red when a D2 mance in predicting harmonic frequencies of water clusters [37], molecule gets attached at this position. where D3 is Grimme’s third generation dispersion-correction In a second phase, we have performed additional calculations ð Þ [38]. For the H2O 3 cluster, the mean absolute deviation (MAD) with explicit inclusion of the D2 tag on the H3 (D3) atom of the + + with these functionals from the CCSD(T) near complete basis set H3O (D3O ) core [17]. Here we applied what appeared to us the limit was 31, 13, 14 and 20 cm 1, respectively, as compared to most promising of the above methods, MP2(fc) and B3LYP-D3. 13 cm 1 with MP2 [37]. The above methods were combined here These additional calculations were performed because, in-spite of ⁄⁄ with either Pople’s 6-311++G or Dunning’s aug-cc-pVTZ bs’s. All an apparently limited tag effect on the experimental spectrum, þð Þ these calculations were performed for both the H H2O 3 and the high sensitivity of perturbation expansion methods (such as þð Þ VPT2) to the environment might lead to more noticeable changes D D2O 3 clusters. þð Þ in the computed spectrum. The geometry of the H H2O 3 cluster was first optimized with the tight convergence criterion of Gaussian 09, confirming that the For the calculations showing the most promising agreement obtained structures are stable minima on the PES. Deuterated iso- with experiment (B3LYP-D3 and MP2 with the Dunning bs), we þð Þ have performed additional tests to check the numerical stability mers were reoptimized starting from the optimized H H2O 3 coordinates (which did not change). The harmonic NM spectra of the results. One set of calculations used the ‘‘VeryTight” instead 12 were then calculated at each of the optimized geometries with of ‘‘Tight” optimization condition, and 10 instead of the default the corresponding quantum method/bs. Anharmonic analysis was 10 10 integral accuracy. For B3LYP-D3 we have checked the effect subsequently performed with VPT2. of using a tighter, ‘‘Ultrafine” integration grid for the exchange N.R. Samala, N. Agmon / Chemical Physics 514 (2018) 164–175 167 functional. Finally, we have checked the effect of using a 3. Results ‘‘FourPoint” numerical differentiation formula instead of the default two-point differentiation. Results from these additional calcula- þð Þ We begin by presenting the results for the H H2O 3 cluster, tions are also collected in the SM Excel file. The results show that using VPT2 and AIMD, then proceed to its deuterated analogue. the very low frequency modes (below, say, 200 cm 1) are not con- The spectra are reported for both the high and low frequency vibra- sistently reproduced, while the variations in the higher frequency tional modes, without and with explicit inclusion of the D2 tag. This 1 1 modes is modest (under 15 cm with the D2 tag, and 5 cm with- exceeds the scope of other recent computations [22,23], which did out). Moreover the improvement is not systematic. Consequently, not include the tag, and reported VSCF results only at high frequen- we report below the results obtained with the Gaussian defaults. cies, and QCMD data only at low frequencies. All harmonic and anharmonic spectra are collected in the Excel file of the SM.

2.2. AIMD setup 3.1. The hydrated cluster Classical AIMD simulations were performed in the CP2K/Quick- step software package [39], see http://www.cp2k.org/quickstep. 3.1.1. Harmonic normal modes This code describes the electronic structure using DFT with a Fig. 4 shows the harmonic IR spectra of this cluster using MP2 mixed Gaussian and plane wave basis. We used the B3LYP-D3 and DFT/B3LYP-D3 with the Dunning bs. As expected from the functional with Grimme’s third generation dispersion-correction water cluster benchmarking [37], the methods indeed give similar (D3) [38] and the augmented triple-f valence polarization (aug- results. Around 3800 cm1 we have the free OH stretches, around TZVP) bs, both of which were not utilized in previous AIMD com- 2500 cm1 appear the HB–OH stretches, between 1600 and 1700 putations for this cluster [25–27]. The plane wave energy cutoff cm1 the 4 bending modes, and near 1250 cm1 – the umbrella was set to 350 Ry. Self-interaction correction was applied with mode. Animations of all the fundamental modes are shown in the Martyna-Tuckerman Poisson equation solver [40]. The orbital the Word file of the SM. transformation method was applied for faster convergence, with The normal mode visualization allows several observations. The the convergence criterion of 1 10 6 a.u. at every timestep (0.5 fs). isolated water asymmetric stretch (as) and symmetric stretch (ss) The trajectories generated in CP2K for this study are summa- modes each combine into two nearly degenerate modes, in which rized in Table 1. In the optimization stage, initial coordinates were the two water molecules vibrate either in phase or out of phase. + selected from the MP2 optimized geometries. In the equilibration The HB–OHas mode involves also some rocking of the H3O free stage, these structures were run in the canonical (NVT) ensemble, OH. The 4 bending modes are clearly divided into 2 symmetric with a target temperature of 50 K maintained by the Nosé-Hoover bends (bs) with respect to a perpendicular mirror plane, and 2 chain thermostat. From each NVT trajectory, we branched out three asymmetric bends (ba). The higher energy ba and bs pair is often + NVE production trajectories (at 9, 9.5 and 10 ps). These were con- ascribed to the H3O core whereas the lower energy pair arises tinued at constant energy for 20 ps each. While the measurements from the flanking water molecules [27]. However, these bending in Fig. 3 were performed at 12 K, equilibration at such a low tem- modes are rather thoroughly mixed and their localized origin is perature is prohibitively long, and not much change is expected hard to discern. The umbrella mode (U) involves also in-phase between 12 and 50 K or, more accurately, up to the average temper- rocking of the 2 terminal water molecules (hence this mode is ature, hTi, along the NVE trajectories. Coordinates and velocities symmetric to reflection), whereas the hydronium rocking (R, near from each production run were saved every timestep (0.5 fs) to 1060 cm1) and rotation (r, near 600 cm1) modes involve out of ensure that the spectra at high frequencies do not get distorted. phase water rocking (asymmetric to reflection), see Fig. 2.D2 trans- The dipole moment was calculated using Wannier localization, lations and rotations couple into several low frequency modes. and then the IR absorption coefficient was computed from the There are, of course, no overtones and combination bands on the dipole moment (time derivative) autocorrelation function (DACF), harmonic level. by taking its temporal Fourier transform [34]. A frequency-dependent nuclear ‘‘harmonic quantum correction” factor was applied to the computed IR intensities (see Eq. (1) in Ref. [33]). The spectrum 3.1.2. VPT2: High frequencies obtained contains anharmonic effects that are generated as the tra- In comparison with the experimental spectrum in Fig. 3a,itis jectory moves on the (anharmonic) multidimensional PES. Noise clear that all the indicated harmonic peaks in Fig. 4 should red- was reduced by averaging together the normalized spectra from shift with inclusion of anharmonicity. The different methods agree the three trajectories that were started from different initial condi- on the extent of the red-shift of the observed fundamental bands, tions. Missing from the picture, of course, are the NQE because of except for the two HB–OH modes. For example, the PTM frequency the use of Newton’s equations. varies within a frequency window of ca. 600 cm1 (Fig. 5). The In order to assign the vibrational frequencies, partial velocity accuracy of predicting the harmonic frequencies [37] seems to pro- autocorrelation functions (VACF) were calculated from each trajec- vide no guide on selecting the most accurate anharmonic method. tory, by restricting the Fourier transform to the velocity of the Another problem with a perturbation theory like VPT2 is that specified coordinate (when interatomic charge transfer can be for some modes it becomes so unstable that frequencies become neglected, the velocity is proportional to dipole moment negative or intensities become (nearly) infinite. When this occurs derivative). only for a few low frequency modes (e.g., modes 22 and 23 in

Table 1 AIMD (B3LYP-D3/aug-TZVP) trajectories run for this study.

Optimized cluster Equilibration Production type/name target T,K Dt; ps type # a trajectory name branching times, ps Dt,ps hTi,K þð Þ H H2O 3 NVT1 50 10 NVE 3 1, 2, 3 @ 50 K 9, 9.5, 10 20 68, 69, 89 þð Þ D D2O 3 NVT2 50 10 NVE 3 4, 5, 6 @ 50 K 9, 9.5, 10 20 60, 51, 48

a Number of trajectories of the given type. 168 N.R. Samala, N. Agmon / Chemical Physics 514 (2018) 164–175

+ and the VSCF/VCI calculation [22]. Clearly, the VPT2 results are in Harmonic IR spectra (a) H (H2O) (without D tag) 3 2 better agreement with experiment than any previous computation. For intensities, VSCF/VCI is possibly more reliable. MP2/aug-cc-pVTZ 2464 The assignment of most peaks becomes much clearer now. Par- B3LYP-D3/aug-cc-pVTZ 1 ticularly, band a8 at 1878 cm , long believed to be the PTM [18– 21], is actually a combination band, U þ R, of the hydronium umbrella (U) and rocking (R) modes. [For a non-linear H–O–H moi- ety, twist, wag and rock are deﬁned as the arrested rotations around the symmetry axis, the perpendicular in-plane axis, and the perpendicular out-of-plane axis, respectively]. The HB–OH fun- 2620 1 damentals contribute near 2300 cm (band a5), as suggested by 3897 3789 403 254 Schwarz [16], and they are comparatively weak (note that we have 369 1265 3812 1077 1702 scaled the fundamentals in Fig. 3 by 2, for visibility). Band a6 is

again a combination band, r þ bs, of hydronium rotation (r) around

0 500 1000 1500 2000 2500 3000 3500 4000 its main symmetry axis and its symmetric bending (bs) mode. Band Intensity + (b) H (H2O) (with D tag) a7 is the least certain, and could be the U overtone.

3 2 2558 Note that the VPT2 computations only report frequencies for transitions involving up to two quanta. For higher order transitions one may consult the VSCF/VCI results (Table S2 in Ref. [22]). The only 3-quanta transition reported there is the third harmonic of the hydronium arrested rotation mode, and it occurs within the

envelope of band a8. 2693 3708

240 364 3.1.3. VPT2: Low frequencies 1057 1244 3901 1705 Low frequency measurements (below 1000 cm1) are difficult, 3792 434 585 requiring the intensity of a free electron laser. Such measurements 0 500 1000 1500 2000 2500 3000 3500 4000 were just published [23]. ‘‘Unfortunately, in this range the quantum -1 approach becomes prohibitively difficult” so that VSCF/VCI results Wavenumber (cm ) could not be obtained, and classical dynamic results were reported þð Þ instead [23]. We find that also VPT2 has more problems at low fre- Fig. 4. Harmonic IR spectra for (a) the bare and (b) the tagged H H2O 3 cluster for two different quantum methods (indicated). Note that (a) with MP2 was previously quencies, such as negative anharmonic frequencies (while all the reported in Table 2 of Ref. [27], though with somewhat different assignments for modes have positive real harmonic frequencies, as expected for a the low frequency modes. minimum on the PES). Another problem is diverging intensities. However, not all is lost. From the many different computations, ⁄⁄ þð Þ we found that B3LYP-D3/6-311++G for the H H2O 3 cluster with 2600 aD2 tag has no negative anharmonic frequencies, and only two Dunning bs (without D tag) 2 anharmonic intensities which seem unphysically large (modes 22 Pople bs (without D2 tag) Dunning bs (with D tag) 2400 2 and 23). We replace these by the harmonic intensities and present ) Pople bs (with D tag) -1 2 the results in Fig. 6. The frequencies are listed in Table 3.

2200 With VPT2 one is able to assign these modes. We agree with the + assignment [23] of band a12 to frustrated H3O rotation. This mode is weak experimentally, and even weaker in the calculation. The 2000 exact intensity is difﬁcult to ascertain here. For example, in our Wavenumber (cm MP2(fc)/aug-cc-pVTZ calculation (SM Excel ﬁle) the intensity is 1800 50 times larger, but the frequency moves down to 618 cm1. The

previously unassigned band a13 is best attributed to a water wag 1600 overtone, although its intensity is not quite right.

P The most intense low frequency peak, a14, is assigned to the 97 -D3 2X B LY P MP2 OOO as, denoted herein OOO . The OOO is a weak peak predicted Y w as ss M06- 1 W2-P B3L to occur 67 cm to the red. The frequency of the OOOas band (347 Method mP cm 1) is in excellent agreement with experiment (344 cm 1). It is þ also reproducible between different methods, for example in the Fig. 5. Anharmonic PTM frequency of the H ðH2OÞ cluster for different quantum 3 1 methods, bs’s and tagging. above MP2 calculation it is observed at 338 cm , and then OOOss is at 311 cm1. Experimentally, there is a weak unmarked feature at nearly the same frequency (309 cm1), which is thus assigned the MP2 calculation in Fig. 3a), these modes and their combination to OOOss. bands can simply be excluded from the analysis. In agreement with Ref. [23], the other high-intensity low fre-

Considering the different computations for the hydrated cluster quency peak (a15) is due to the water wag. From the MP2 calcula- (Excel file of the SM), we find that a quantitative agreement with tion, we can only cite the harmonic frequency, 224 cm1 , which is experiment is obtained only with the highest level method (MP2) remarkably close to experiment (234 cm1). In fact, the B3LYP and largest bs (aug-cc-pVTZ) (Fig. 3a). The DFT PES’s appear to be (251 cm1) and MP2 results bracket this value, as they do for insufficiently accurate for quantitative IR spectroscopy of this clus- OOOas. For the D2-tagged cluster the wag is coupled to D2 transla- ter, even though they have performed well for neutral water clus- tion, either in- or out-of-phase with the two (in-phase) wagging ters [37]. The MP2/aug-cc-pVTZ frequencies are collected in water molecules. Only one of these is observed in the experimen- Table 2, where they are compared with those from experiment tally accessed spectral range. In contrast to the good agreement N.R. Samala, N. Agmon / Chemical Physics 514 (2018) 164–175 169

Table 2 1 þð Þ Calculated frequencies (in cm ) and assignments of the D2-tagged H H2O 3 IR spectrum.

Ref. [22] This work Label Experiment VSCF/VCIa modeb VPT2c AIMDd Assignment

a1 3726 3748 1/2 3726 3707 H2O as a2 3640 3658 3/4 3628 3617 H2O ss a3 3580 5 3540 O3H3 stretch D2 2961 6 3039 D2 stretch a4 2410 2554 7 2383 2450 HB–OHss a5 2307 2364 8 2284 2305 HB–OHas e a6 2109 2240 15 + 10 2084 FR (8) f a7 1943 2034 2 13 1992 FR (7) a8 1878 1950 14 + 13 1883 FR (8) a9 1639 1701 9 1621 1597 bend (ba) a10 1534 1477 12 1553 1527 bend (ba) + a11 1059 1202 13 1080 1157 H3O umbrella (U) + 14 992 1022 H3O rock (R) + 15 618 579 H3O rotation (r)

20 338 352 OOOas MADg 0 72 15.6 33

a Peak frequencies. This calculation does not include the D2 tag, hence we leave out the O3H3 frequency. b Fundamental normal modes for the D2 tagged cluster, numbered from the highest (harmonic) frequency to the lowest (see Excel ﬁle in the SM). c MP2/aug-cc-pVTZ, D2-tagged. d B3LYP-D3/aug-TZVP, not D2-tagged, VACF frequencies scaled by 0.966. e Combination band undergoing Fermi resonance with mode number 8. f Overtone undergoing FR with mode number 7. g 1 ; ; Mean absolute deviation from the experimental frequencies (in cm ), averaged over the fundamentals of the bare cluster and the FR (i.e., excluding the a3 D2 a6 and a7 modes).

+ in Refs. [18–20]. The U þ R combination has now inconspicuous B3LYP-D3/6-311++G** (anharmonic) D tagged H (H O) 2 2 3 intensity, marked by the dashed orange arrow in Fig. 7b. Fundamental 1 a 1875

8 1982 In both cases there is at least one HB–OH band near 2300 cm 344 Overtones 234 a Combination bands [16]. Thus the main change with the added D2 tag is in the reas- a 14 15 Expt 2149 signment of a to a combination band, rather than the PTM (both 2100 8 1942 a6 occurring at similar frequencies). Since the origin of an IR band a7 1533 508 (fundamental vs. combination) is not an experimental observable, 1640 a 1060 a 13 668 10 current experiments could not check this, whereas previous com- a11 a 9 2174 a12 putations did not explicitly include the tag. 2232 Intensity 251

347 3.1.5. Indications for Fermi resonances 2020 1075 171

603 FR’s are omnipresent in vibrational spectroscopy [41], for exam- 63 321 127 388 1601 1057 2047 1537 529 ple in CO2 [4], methanol [42] and the Zundel cation [5]. A FR arises 646 when a combination band (or an overtone) becomes nearly degen- 0 500 1000 1500 2000 erate (i.e. of equal frequency) with a fundamental mode having the -1 Wavenumber (cm ) same molecular symmetry. As a result, the combination gains intensity at the expense of the fundamental, and the two bands þð Þ Fig. 6. The experimental IR predissociation spectrum of the D2-tagged H H2O 3 move away from each other along the frequency axis. cluster at low frequencies [23] compared with the (unscaled) VPT2 spectrum In the present case, U is symmetric while R and the PTM are calculated at the B3LYP-D3/6-311++G⁄⁄ level. Black, red and green sticks mark 1 antisymmetric to reflection in a mirror perpendicular to the fundamental, overtone and combination bands, with their frequencies in cm . þ Experimental data (blue line with mode labels) courtesy of Tim Esser and Knut H3’O3H3” plane (Cs point group). Consequently, U R is also Asmis. (For interpretation of the references to colour in this figure legend, the antisymmetric, like the PTM, allowing a FR to occur between them. reader is referred to the web version of this article.) Consider the B3LYP-D3 and MP2/aug-cc-pVTZ calculations that were found to agree best with experiment. In the bare cluster calculation, the U þ R combination must be too remote from the PTM, just described, the main band a8 is not accurately predicted by the so that no such FR occurs and the PTM is near 1800 cm1 (Fig. 5). B3LYP calculation, which we attribute to the FR discussed below. The small perturbation induced by the D2 tag appears to be suffi- cient to bring the PTM into FR with U þ R, pushing the PTM to 3.1.4. The bare cluster the blue and U þ R to the red. This shift is only partially developed Fig. 7 compares the computed spectra of the bare and tagged in the B3LYP-D3/6-311++G⁄⁄ calculation in Fig. 6. clusters using the same MP2 method for both. The bare cluster fre- An indication that this is indeed the case is obtained by plotting quencies are collected in Table 4. According to Ref. [22],D2 binding the intensity of the U þ R combination band as a function of the affects predominantly the O3H3 bond to which it binds. Upon fundamental frequency difference, PTM U R, for all the compu- removing the D2 tag, the O3H3 stretch blue-shifts by approximately tations performed in this study (Fig. 8). For the bare cluster, it is 90 cm1 (in excellent agreement with 92 cm1 in VPT2). Here we seen that when this difference (in absolute value) is small, the

ﬁnd that, in addition, the two HB–OH bands red shift: HB–OHss intensity of the U þ R combination becomes very strong, dropping 1 1 by 90 cm , while HB–OHas (PTM) jumps nearly 500 cm . Conse- abruptly when this frequency gap widens. The latter occurs for the quently, it identiﬁes in the bare cluster with band a8, as suggested B3LYP and MP2 methods that reproduce experiment, and thus we 170 N.R. Samala, N. Agmon / Chemical Physics 514 (2018) 164–175

Table 3 1 þð Þ Measured and calculated band positions in the low frequency IR spectrum (in cm ) of the D2-tagged H H2O 3 cluster with band assignments.

Ref. [23] This work Label Experimenta QCMDb mode # VPT2c AIMDd Assignment

e + a11 1060 1176 13 1075 1157 H3O umbrella (U) + ? 1015 14 1057 1022 H3O rock (R) + a12 668 15 646 579 H3O rotation (r) a13 508 456 2 23 529 wag overtone a14 344 355 19 347 352 OOOas

309? 21 321 322 OOOss e a15 234 241 23 251 228 water wag 22 171 195 water wag MADf 0471543

a Free electron laser spectrum. b Quasiclassical molecular dynamics. This calculation does not include the D2 tag. c DFT based spectrum using the B3LYP functional with the D3 correction and Pople’s 6-311++G⁄⁄ bs (see Excel ﬁle in the SM). d B3LYP-D3/aug-TZVP, not D2-tagged, VACF frequencies scaled by 0.966. e Looks more like noise than a real peak, see Fig. 2b in Ref. [23]. f Mean absolute deviation from the experimental frequencies (in cm1), averaged over all the available frequencies.

a ^ 1 (a) Hydrated isomer (with D2 tag) Fundamental a MP2/aug-cc-pVTZ Overtones 8 1883 a3 Combination bands a 2084 2 Expt 2284 a7

a a 3540 10 6 2383 3726 a9 a11 a5 a D 4 v 2 2693 1992 2367 3628 2544 2664 2175 1553 1907 1080 1621 992 3039

1000 1500 2000 2500 3000 3500 4000 Intensity 2263 3393 1613 2563 1943

2157 ^ 1539 1018 3632 1985 1113 2454 3623 3721 2293 2373

(b) Hydrated isomer (without D2 tag) 1790 -1 Wavenumber (cm )

þð Þ Fig. 7. Comparison of the computed VPT2 IR spectra for (a) the tagged and (b) the bare H H2O 3 cluster (with an upside-down intensity scale) using MP2(fc)/aug-cc-pVTZ [24]. Black, red and green bars indicate the calculated fundamental, overtone and combination bands, respectively. Orange arrows mark the location of the U þ R combination band. In panel (a) the experimental spectrum is reproduced (blue) with the peak labels according to Ref. [22], and the VPT2 fundamental intensities are scaled by a factor of 2. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) assume that it is the correct result. Curiously enough, for the com- These effects are connected by a theory [43] of FR intensity putations with a D2 tag the U þ R intensity is large (as expected ratios (q), as a function of the (hypothetical) frequency split (d0) q when the FR is operative) but nearly independent of the frequency and intensity ratio ( 0) in the absence of the resonance. Assuming gap. q 3 that 0 0 (e.g., for the bare cluster it is 3 10 ), one obtains Eq. What is unusual here is the large intensity of the combination (3) in Ref. [42], namely: band (Ic), which is larger than the intensity of the fundamental d d0 (If ) with which it is in resonance. Defining the intensity ratio as q ¼ : ð1Þ d þ d0 q Ic=If , we find for the D2-tagged cluster that q ¼ 4:05 (see Excel file in SM), which is rather exceptional [43]. Another curious This can be rearranged to read observation is the crossover of the two bands as a function of the 1 q interaction at the O3H3 position. Denoting the frequency differ- d ¼ d; ð2Þ 0 1 þ q ence by d mc mf , we find that it changes sign between the tagged and bare clusters: For the tagged cluster, d ¼402 cm1, from which one immediately sees that d0=d changes sign when d ¼ 1 whereas for the bare cluster 472 cm . q ¼ 1. For the tagged cluster, d ¼402 cm 1 and q ¼ 4:05, so that N.R. Samala, N. Agmon / Chemical Physics 514 (2018) 164–175 171

1 Table 4 d0ðtagÞ¼d0ðbareÞþDd0 ¼ 339 cm ; ð3Þ 1 þð Þ Calculated frequencies (in cm ) and assignments of the bare H H2O 3 IR spectrum. as compared with 243 cm1 that we obtained from Eq. (2). Thus, an Ref. [22] This work (MP2/aug-cc-pVTZ) additional shift of 100 cm1 is due either to the anharmonic effect a b Label Exper. VSCF/VCI VPT2 Assignment and/or reﬂects the accuracy limits of this approach.

a1 3725 3748 3721 H2O as a2 3640 3658 3623 H2O ss 3.1.6. More Fermi resonances a 3668 3664 3632 O3H3 stretch 3 The unusually large q value observed for the tagged cluster can a4 2484 2554 2454 HB–OHss + OOOas be understood if the PTM enters simultaneously into a second FR a5 2320 2364 2373 HB–OHas + OOOas 2293 HB–OHss that reduces its intensity independently of the first FR. Indeed, 1 2263 U þ R (v.w.) the bs þ r combination (2084 cm in Table 2) is asymmetric to a 2115 2240 b þ r 6 2157 s reflection and even closer to the PTM than U þ R. Thus bs þ r also 2153 b þ r a gets enhanced by a FR with the PTM. This explains band a6 in both a7 1950 2034 1943 2R the tagged and bare clusters, contrasting with U þ R that loses all of a8 1860 1950 1790 HB–OHas its unusual intensity in the bare cluster. Band a is explained as an a9 1636 1701 1613 ba 7 a10 1510 1477 1539 ba overtone both in the tagged (2U) and bare (2R) clusters. Because a a11 1202 1113 U first overtone is symmetric, it might also gain in intensity from a FR 1018 R with the HB–OHss band. 585 r

332 OOOas 3.1.7. AIMD MADc 05628 The DACF spectrum from the AIMD (B3LYP-D3/aug-TZVP) tra- a From Table S1 in Ref. [22]. þ jectories for the bare H ðH2OÞ cluster is shown in Fig. 9a. By far b 3 Main contributor underlined. the most intense peak here is the PTM: Due to the inherent weak- c Averaged over all the listed experimental bands. ness of combination bands and FRs in the AIMD approach [34], the PTM did not lose intensity to any overtone or combination band. In d ¼ 1 0 243 cm . Thus, ‘‘before” switching on the interaction, the PTM comparison to the PTM, all other fundamentals appear to be rather was lower in energy than the U þ R combination. In comparison, for weak and need to be magnified as indicated in the figure. d d ¼ 1 the bare cluster 0 472 cm . This larger gap precluded FR. All fundamentals are blue shifted from their experimental (and Addition of the tag weakens the HBs, strengthening the O3H3’ VPT2) values because classical trajectories lack NQE. Inclusion of, and O3H3” covalent bonds, and blue shifting their absorption e.g. zero point energies, will reduce the 0 ! 1 gap and hence 1 already on the harmonic level (by 94 cm , Fig. 4). At the red-shift these bands. Assuming that this can be corrected for by same time, the U and R fundamentals red-shift by ca. 20 cm1 applying a single frequency-independent scaling factor, sf ,we each. This diminishes the unperturbed frequency separation, determine sf that gives the best fit to experiment in the least- Dd ¼ 1 0 133 cm , so we expect squares sense from [44]

4 Hydrated isomer

D2 tagged Hydrated isomer 3

2 wB97/6-311++G** MP2/6-311++G** MP2/aug-cc-pVTZ wB97/aug-cc-pVTZ mPW2-PLYP/6-311++G** B3LYP-D3/aug-cc-pVTZ B3LYP-D3/6-311++G** B3LYP-D3/6-311++G** M06-2X/aug-cc-pVTZ MP2/aug-cc-pVTZ B3LYP-D3/aug-cc-pVTZ

1 MP2/6-311++G**

0 -400 -200PTM-(U+R) 0 200 400 0

log (intensity of combination(U+R)) 1 MP2/6-311++G** MP2/aug-cc-pVTZ M06-2X/aug-cc-pVTZ B3LYP-D3/6-311++G** 2 B3LYP-D3/aug-cc-pVTZ

Deuterated isomer MP2/6-311++G** mPW2-PLYP/6-311++G** wB97/6-311++G** D tagged Deuterated isomer wB97/aug-cc-pVTZ

2 B3LYP-D3/aug-cc-pVTZ B3LYP-D3/6-311++G** MP2/aug-cc-pVTZ 3

Fig. 8. The dependence of the intensity of the U þ R combination band (logarithmic scale) on the separation between the frequency sum of the U and R fundamentals and the PTM. Top: Hydrated cluster, and bottom: deuterated cluster. Data from the VPT2 calculations collected in the Excel ﬁle of the SM. 172 N.R. Samala, N. Agmon / Chemical Physics 514 (2018) 164–175

DACF @ 50 K (average of Traj 1, 2 and 3) (a) distances and bond angles. Their Fourier transforms then show 2738 2365 2540 1200 where these motions absorb in the IR, helping to assign the AIMD 202 236 2897

1652 peaks (independently from the NM analysis reported above). 3826 2469 364 2009 3741 2150 Fig. 9b shows such results for various bond distances. For example,

2235 + 2637

2975 it shows that the H3O free OH, O3H3, appears for the bare cluster 3895 1 1060 420

1581 inbetween the water ss and as free OH bands (3764 cm there). 132 3521

3314 This agrees with the experimental and VPT2 results for the bare 94 278 cluster, discussed above, lending credibility to the AIMD results. 0 500 1000 1500 2000 2500 3000 3500 4000 2

2 Then, of course, it unequivocally identifies the dominant HB–OH 3 3 2 3 3 3 3 x5 x10 x30 x10 x10 x20 x10 2x10 as and ss bands as arising from the O3H3’ and O3H3” vibrations. 3x10 x10 x10 2x10 5x10 2x10 In addition, it identifies the OO stretch (actually, the OOO ), which 0 500 1000 1500 2000 2500 3000 3500 4000 as (after scaling) is within 8 cm1 from the experimental result in Ref. VACF of stretch @ 50 K (average of Traj 1, 2 and 3) O1H1 (b) O1H1 - [23] (Table 3). Similarly, panel (c) identifies two major HOH bends. 3744 2536 3764 2386 364 O3H3 Previous work showed that combination bands are revealed by - O3H3 3837 333 O1O3 amplification of VACF spectra [35]. Mostly, the frequency of the 383 combination band is just slightly lower than the sum of its two fundamental components. We have already identified some impor- 0 500 1000 1500 2000 2500 3000 3500 4000 tant combination bands of the bare cluster in the VPT2 results of 1198 2149

333 + (c) Table 4. These bands are likely identiﬁed also in the VACF spectra 2535 Intensity 1652 1058 VACF of H O bend 74 3 2240 435 377 2386 e.g., that of the HBed O3H3’ stretch in Fig. 9d. Table 5 shows that 1274 599 946 521 115 3764 1446 2010 the assignments of the major combinations from VPT2 and AIMD 704 are consistent, although those involving the HB–OH bands differ 3 3 3 2 2 4 4 3 1582 4 3 in frequencies. x50 x50 x10 x10 5x10 x10 x10 x10 5x10 5x10 x10 x10

0 500 1000 1500 2000 2500 3000 3500 4000 - 2239

2150 (d)

VACF of O3H3 2020 2386 2535 2602

363 3.2. The deuterated cluster 3742 1581 2882 2736 1654 3.2.1. VPT2 3764 þð Þ

3 Fig. 3b shows the IR spectrum of the D -tagged D D O clus- 3 2 2 3 3 x10 x10 x30 x50 x50

x10 ter, using the same MP2(fc)/aug-cc-pVTZ level of theory as pre- 5x10 þð Þ 0 500 1000 1500 2000 2500 3000 3500 4000 sented in panel (a) for the D2-tagged H H2O 3 cluster. A list of -1 Wavenumber (cm ) the experimentally relevant frequencies is reproduced in Table 6. The agreement of the fundamental frequencies with experiment þ Fig. 9. IR spectra of the H ðH OÞ cluster computed from AIMD trajectories with 1 2 3 is excellent, their MAD being only 17 cm . The U þ R combination the B3LYP-D3/aug-TZVP functional/bs (unscaled), with some weak features (in color) magnified as indicated. (a) The DACF spectrum. (b) Partial VACF spectra for explains now the ‘‘extra” band b8b, but its intensity ratio relative to the indicated OH and OO distances. (c) Partial VACF spectra for the hydronium HOH the PTM, q ¼ 0:49, is notably smaller than the value of 4 obtained angles. (d) Partial VACF spectra for the hydrogen-bonded OH distance. Atom for the hydrated cluster. This is mainly because the PTM, which is numbers defined in Fig. 1. A somewhat similar calculation was reported by Kaledin not involved in the second, bs þ r resonance, is much stronger for and Wood [27], but with a smaller bs and no dispersion correction. (For the deuterated cluster. Our assignment of band b agrees with interpretation of the references to colour in this figure legend, the reader is 8b referred to the web version of this article.) Duong et al. [22], who have described it as a mixture of the HB–OHas with ‘‘the perturbed hydronium wag and umbrella modes” (apparently their ‘‘wag” is a ‘‘rock” in our nomenclature). R f g d ¼ 1 s ¼ i i i ; ð4Þ Following a frequency gap, 263 cm (computed value), f R 2 d ¼ i gi appears the PTM (band b8a). Thus, according to Eq. (2), 0 90 cm1. Table 6 also presents the frequencies from a similar VPT2 1 where f i are the experimental frequencies of the fundamental calculation for the bare cluster. The gap now is 265 cm , almost modes identified in Table 2, and gi the corresponding AIMD fre- identical to its value for the tagged case. In particular, there is no quencies from Fig. 9a. The scaled AIMD fundamental frequencies crossover of the PTM and U þ R combination upon tagging, as seen q (sf gi) are listed in Table 2, and they agree rather well with the for the hydrated case, possibly because here is smaller than 1. experimental and VPT2 frequencies, even though the calculation The agreement of the VPT2 fundamental and FR frequencies þð Þ did not include the tag and utilized B3LYP-D3 rather than MP2. with the D2 D D2O 3 experiment is unprecedented, with In the partial-VACF approach, we compute the autocorrelation MAD of only 18 cm1 (Table 6). The bare cluster VPT2 calculation of velocities of specific coordinates (‘‘local modes”), such as bond gives similar results, with a better PTM frequency though a larger

Table 5 þð Þ Main combination bands of the bare H H2O 3 cluster identiﬁed from AIMD.

Assignment AIMDa VPT2b Fundamentals Freq. sum VACFc Comb.

bs þ r 1582, 599 2181 2150 2157 U þ R 1198, 1058 2256 2239 2263

HB—OHasþ OOOas 2386, 363 2749 2736 2373

HB—OHssþ OOOas 2536, 363 2899 2882 2454

a Unscaled, B3LYP-D3/aug-TZVP. b From Table 4. c See Fig. 9d. N.R. Samala, N. Agmon / Chemical Physics 514 (2018) 164–175 173

Table 6 1 þð Þ Calculated frequencies (cm ) and assignments of the D2-tagged D D2O 3 IR spectrum.

Ref. [22] This work Label Exper. VSCFa modeb VPT2c VPT2d AIMDe Assignment

D2 2978 1 3038 D2 stretch b1 2770 2778 2/3 2767 2742 2749 D2O as b2 2660 2670 4/5 2648 2625 2633 D2O ss b3 2631 6 2631 O3D3 stretch b4 2314 2383 comb. b5 2149 2181 7 + 19 2063 comb. ( ) b6 1991 1980 11 + 14 1913 1993 OOOas + HB–ODss b7 1770 1802 7 1791 1758 1786 HB–ODss b8a 1680 1755 8 1740 1704 1736 HB–ODas

b8b 1452 1541 13 + 14 1477 1439 FR(8) + b9 1202 1230 9/11 1186 1191 1182 D3O &D2O bends + b10 1143 1191 10/12 1145 1132 1122 D3O &D2O bends + b11 800 869 13 819 853 849 D3O umbrella + 14 731 733 749 D3O rock + 15 355 420 422 D3O rotation

19 315 319 335 OOOas MADf 045 182330

a VSCF/VCI without the D2 tag. b Fundamental NM’s for the D2 tagged cluster, numbered from the highest (harmonic) frequency to the lowest. c MP2/aug-cc-pVTZ, D2-tagged. d MP2/aug-cc-pVTZ, no D2 tag. e B3LYP-D3/aug-TZVP, no D2 tag, using VACF frequencies scaled by 0.975. f 1 ; ; Mean absolute deviation from the experimental frequencies (in cm ) excepting the D2 b4 b5 and b6 modes.

Table 7 as in Fig. 7b, the isotope effect becomes 1.05. This may cast doubts H/D isotope effect on the experimental [22] fundamental frequencies (in cm1) of the on the validity of this assignment. þð Þ D2-tagged H H2O 3 cluster given the assignments of the present study.

Assignmenta H D H/D 3.2.3. AIMD The DACF spectrum from the AIMD (B3LYP-D3/aug-TZVP) tra- H O as 3726 2770 1.345 2 þð Þ H2O ss 3640 2660 1.368 jectories for the bare D D2O 3 cluster is shown in Fig. 10a. Again, O3H3 3580 2631 1.361 the most intense peak is the PTM, with an isotope effect of HB–OHss 2410 1770 1.362 2386/1781 = 1.34. All peaks are blue shifted from the experimental HB–OHas 2307 1680 1.373 due to lack of NQE. However, for the heavier isotope one expects b 1639 1202 1.364 a smaller NQE and indeed, the scaling factor calculated from Eq. ba 1534 1143 1.342 U 1059 800 1.324 (4) is 0.975, as compared with 0.966 for the hydrated cluster. The scaled AIMD fundamental frequencies are listed in Table 6, a Table 2 and 6. and they are quite close to the VPT2 frequencies, thus corroborat- ing the results. overall MAD of 23 cm1 (the bare cluster measurement is too noisy The VACF spectrum in panel (b) again shows that for the bare so that the comparison here is with the tagged cluster). Both calcu- cluster the O3H3 stretch is inbetween the flanking D2O as and ss lations are in better agreement with experiment than the VSCF/VCI bands. Weak features in the VACF spectra attest to mode couplings calculation [22] (MAD of 45 cm1). However, the combination and resultant combination bands. For example, both the U and R 1 + ; modes (871 and 768 cm ) couple to the D3O bend in panel (c). bands b4 b5 and b6 are uncertain in the VPT2 calculations, and here the VSCF/VCI results may be more informative. They can thus combine, with an expected frequency of 871 + 768=1639 cm1. U þ R is indeed manifested by the sharp peak (1645 cm1) in the O3D3 VACF (panel (d)), in which the HB–OD 3.2.2. Isotope effect modes are the dominant peaks. Hence, one can expect that U þ R Of interest is the isotope effect on the PTM frequency. Under the couples with one of them (to give the FR). Similarly, the presence assignment of band a8 to the PTM of the hydrated cluster [18–21], 1 of the OOOas band (343 cm ) in panel (d) suggests that it may b8a being the PTM for the deuterated analog [22], the isotope effect add to the HB–OH as and ss frequencies (1781 and 1832 cm1)to on the experimental PTM frequency is 1878/1680 = 1.12. This is give the weak features at 2132 and 2179 cm1, respectively. considerably smaller than the expected deuterium isotope effect Indeed, OOO þ HB—OD was identified in the VPT2 spectrum of for OH oscillators of ca. 1.36 [41], smaller even from the value of as ss the bare cluster (Table 6). The OOOas mode also interacts with 1.23 suggested for the lower frequency bands of the OH ðH2OÞ 2 the bend, 343 + 1151 = 1494 cm1, manifested in panel (d) as a cluster [45]. With the current VPT2 assignment of the PTM to band peak at 1491 cm1. a5 (Fig. 3), its isotope effect becomes 2307=1680 ¼ 1:37. Table 7 shows the experimental isotope effects on the frequencies of all the measured fundamental bands according to our present assign- 4. Conclusion ment, all of which lie in the range 1.32–1.37. Thus the isotope effect lends support to the Schwarz assignment of the PTM of the In this work we set to explain the ‘‘extra” IR band (b8b) recently 1 þð Þ hydrated cluster near 2300 cm , and consequently a8 must be a observed for the D D2O 3 cluster, finding that the really surprising þð Þ combination band undergoing a strong FR. system is the H H2O 3 cluster. We started by running VPT2 using The computed VPT2 spectra give similar isotope effects, except DFT with 4 different functionals and 2 different bs’s, finding that 1 for the HB–OHas band of the bare cluster. When it is assigned to a8 above 1000 cm DFT is not sufficiently accurate for quantitative 174 N.R. Samala, N. Agmon / Chemical Physics 514 (2018) 164–175

Table 8 DACF @ 50 K (average of Traj 4, 5 and 6) (a) þ ð Þ 1780 Summary of assignments of the problematic H H2O bands in its bare and D2-

2697 3 866 tagged states.a 340 239 72

1212 Label Bare Tagged 2827 1632 172

2748 a4 HB–OHss + OOOas HB–OHss 1494 a5 HB–OHss HB–OHas 2152 321 150 2177 a6 bs þ rbs þ r 763 1153 a7 2R 2U a8 HB–OHas U þ R

03 5002 1000 1500 2000 2500 3000 2 3 3 3

3 a 1832 From Table 2 and 4. x50 2x10 3x10 x10 x20 x10 2x10 x10 5x10

0 500 1000 1500 2000 2500 3000 2743

VACF of stretch @ 50 K (average of Traj 4, 5 and 6) 2819 (b) 2701 O1D1 For the bare cluster, the PTM corresponds to the highest-inten- - 1781 1832

344 O1D1 O3D3 sity, lowest-energy band within this group, a8.D2 attachment -

327 O3D3 weakens the HBs, blue-shifting the HB–OH bands. Consequently O1O3 HB–OHss moves from position a5 to a4. The behavior of the HB–

Intensity OHas band is more complex. Moving up the frequency axis, it 0 500 1000 1500 2000 2500 3000 326 251 encounters two otherwise weak combination bands of the same 1777 871 + 2741 939 VACF of D O bend (c) þ þ 768 3 symmetry, ﬁrst bs r then U R (Fig. 7b). This results in a giant 1490 906 85 433 1212

1151 FR. Both combination bands become stronger than the fundamental 2176 1022 67 835 387 1837 2700 1408 (Fig. 7a). Instead of ‘‘repelling” each other they cross, so that the 1645 2151 order of these bands reverses (Table 8). 3 2 3 3 4 x5 x50 x5 This remarkable scenario (particularly the assignment of a8 to 2x10 x10 x10 x10 x10 the PTM of the bare cluster) is questioned by the isotope effect 0 500 1000 1500 2000 2500 3000

1832 and AIMD results. These would suggest that the PTM remains in 1781 343

- (d) 1645 VACF of O3D3 1151 the same position (a5) as in the tagged cluster, in contrast with 2741 2132

1491 assignments of the last decade. Even if this mechanism is not rele- 2179 2700

1437 vant at the precise conditions of the bare vs. tagged measurements, 2

2 the computation indicates that it might occur under conditions 3 3 x10 x10 5x10 x10 x10 where the interaction strength at the H3 site is varied.

0 500 1000 1500 2000 2500 3000 In the deuterated analogue, quantum effects are smaller and the -1 Wavenumber (cm ) FR is ‘‘normal”, with q ¼ 0:5. As a result, the bare and tagged spe- cies exhibit similar spectra (Table 6). Most noticeably, the splitting þð Þ Fig. 10. IR spectra of the D D2O 3 cluster (no tag) computed from AIMD in the FR is of conventional magnitude, leading to the more cus- trajectories with the B3LYP-D3/aug-TZVP functional/bs (unscaled), with some weak tomary FR doublet. The two partners in the FR, PTM and U þ R, features (in color) magnified as indicated. (a) The DACF spectrum. (b) Partial VACF spectra for the indicated OH and OO distances. (c) Partial VACF spectra for the appear here as the b8a and b8b bands, respectively. hydronium HOH angles. (d) Partial VACF spectra for the hydrogen-bonded OD An interesting question to conclude with is whether the giant distance. Atom numbers as defined in Fig. 1. (For interpretation of the references to þð Þ FR in the H H2O 3 cluster is merely a curiosity, or could it mani- colour in this figure legend, the reader is referred to the web version of this article.) fest itself in proton transfer/mobility in water and proteins? It has long been recognized that fast ‘‘promoting” vibrations play þð Þ an important role in H-transfer reactions [46]. Proton mobility in analysis of H H2O 3, but VPT2 using MP2 with the larger bs (aug- cc-pVTZ) gave surprisingly good spectra in comparison with exper- liquid water is aided, to some extent, by near-concerted proton iment. At lower frequencies, it exhibited some negative anhar- hopping along water-wires, among which the trimer plays a cen- monic frequencies, so we replaced the method by B3LYP-D3/6- tral role. Simulations noted that protons move intermittently, with 311++G⁄⁄ that showed unprecedented agreement with experiment bursts of activity followed by quiescent periods [8,9]. One partici- + in the low-frequency range. pant in the burst-to-rest transitions was suggested to be H3O Cautioned that ‘‘VPT2...will not even qualitatively reproduce inversion via the U mode [47]. The U þ R FR could play an impor- the essential physics controlling the vibrational band pattern” tant role here. Depending on the magnitude of the interaction felt + [22], we found that the spectra from the higher level VPT2 calcula- at the H3O free-OH site (O3H3, Fig. 1), the trimer may go in and tions fitted experiment with a MAD of ca. 15 cm1, while the error out of resonance. When out of resonance, the PTM is the dominant using the best-scaled DACF from AIMD trajectories with the B3LYP- mode, which facilitates PT. When in-resonance, the PTM loses its D3 functional are at least twice as large, (4.5 times as large for intensity, as it mixes with the R and U modes (Fig. 2). Their vector VSCF/VCI on a fitted PES [22]). It thus seems that difficulties in sum is a motion in which one proton (e.g., H3”) is nearly static, obtaining a trustworthy spectrum [24] lie more with the accuracy while the other (H3’) vibrates perpendicular to the O1O3 axis. This of the PES than with the vibrational methodology. should impede PT through the wire, possibly explaining the liquid- Even more important than the close match with experiment is phase observations. Thus the FR becomes the gatekeeper of the the emergence of a much clearer band assignment scheme. The protonated water trimer. þð Þ only problematic bands in the H H2O 3 spectrum are a4—a8, which appear to be a complex mixture of fundamentals and combination bands [22]. These 5 bands are due to 2 fundamentals: HB– Acknowledgements

OHas (the PTM) and HB–OHss, and 3 combination bands. While experimentally nearly identical spectra are observed with and SNR acknowledges support from the Lady Davis Foundation. without the D2 tag, theory suggests two drastically different The Fritz Haber Research Center is supported by the Minerva assignment schemes (Table 8), both reproducing the observed Gesellschaft für die Forschung, München, FRG. We thank Tim Esser spectra (Fig. 7). and Knut Asmis for the experimental data shown in Fig. 6. N.R. Samala, N. Agmon / Chemical Physics 514 (2018) 164–175 175

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