Reinvestigation of the Infrared Spectrum of the Gas-Phase Protonated

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Reinvestigation of the Infrared Spectrum of the Gas-Phase Protonated Water Tetramer Huan Wang, and Noam Agmon J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.7b01856 • Publication Date (Web): 29 Mar 2017 Downloaded from http://pubs.acs.org on March 30, 2017

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The Journal of Physical Chemistry

Reinvestigation of the Infrared Spectrum of the Gas-Phase Protonated Water Tetramer Huan Wang and Noam Agmon∗ The Fritz Haber Research Center, Institute of Chemistry, The Hebrew University of Jerusalem, Jerusalem 91904, Israel E-mail: [email protected]

1 ACS Paragon Plus Environment


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Abstract + Gas-phase H9 O+ 4 has been considered an archetypal Eigen cation, H3 O (H2 O)3 .

Yet ab initio molecular dynamics (AIMD) suggested that its infrared spectrum is explained by a linear-chain Zundel isomer, alone or in a mixture with the Eigen cation. Recently, hole-burning experiments suggested a single isomer, with a 2nd order vibrational perturbation theory (VPT2) spectrum agreeing with the Eigen cation. To resolve this discrepancy, we have extended both calculations to more advanced DFT functionals, better basis sets and dispersion correction. For Zundel-isomers, we find VPT2 anharmonic frequencies for 4 low-frequency modes involving the excess proton motion unreliable, including the 1750 cm−1 band that is pivotal for differentiating between the two isomers. Because the analogous band of the H5 O+ 2 cation shows little effect of anharmonicity, we utilize the harmonic frequencies for these modes. With this caveat, both AIMD and VPT2 agree on the spectrum as originating from a Zundel isomer. VPT2 also shows that both isomers have the same spectrum in the high frequency region, so that the hole burning experiments should be extended to lower frequencies.

Keywords: cluster, IR spectrum, proton, water


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Introduction The structure of aqueous acid solutions has been a subject of intense debates 1–7 on whether the dominant protonated water structure is H3 O+ · (H2 O)3 (the “Eigen cation”) 3,6 or H2 O · · · H+ · · · OH2 (the “Zundel cation”). 2,4,5,7 It therefore seemed reassuring that for gas-phase clusters there was a consensus, 8–12 the protonated water tetramer being an Eigen cation (Scheme 1 left), which is the most stable isomer. 13–15 Its measured infrared (IR) spectrum 8–12 showed several characteristic bands, of which three were reproduced theoretically [the “dangling” water (dw) symmetric stretch (ss) and asymmetric stretch (as) modes near 3700 cm−1 , a strong hydrogen-bonded (HBed) OH band at 2665 cm−1 and the dw bend (b), 1615 cm−1 ]. This, however, left two prominent bands unassigned (at 1750 and 1050 cm−1 ), and a few smaller peaks (1847, 1904, 2307 and the “α band” at 2245 cm−1 ), which are thought to be combination bands. 16

Scheme 1: The optimized geometries of H9 O+ 4 Eigen (left), trans-Zundel (middle) and cis-Zundel (right) isomers, calculated in Gaussian 09 at the MP2(fc)/aug-ccpVTZ level. Coordinates given in Tables S2–S4 of the SI.

The 1750 cm−1 band was noted to be “markedly similar to that displayed by the isolated Zudnel ion”, 9 but was nevertheless interpreted as originating from an Eigen isomer. Subsequently it was (erroneously 17 ) assigned to the bending mode of the hydronium core. 16 Kulig and Agmon 18 then performed ab initio Molecular Dynamics (AIMD) simulations using Density Functional Theory (DFT) with 3 ACS Paragon Plus Environment


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the BLYP functional, Dunning’s double-ζ basis set, and no dispersion correction. [These simulations are quantum mechanical for the electrons, but classical for the nuclei.] The Fourier transform of the Dipole Moment Autocorrelation Function (DACF), as the computed IR spectrum, indeed showed the two low frequency features (at 1750 and 1050 cm−1 ) for the trans-Zundel isomer (Scheme 1, middle), but not for the Eigen isomer. [The cis-Zundel isomer could not be studied by AIMD because it rapidly converted to the trans form]. They consequently suggested that the observed cluster is a Zundel isomer, or at least a mixture of both isomers. This analysis has recently been contested by Fournier et al., 17 who have performed isomer selective photochemical hole burning IR-IR double resonance measurements on this cluster above 2000 cm−1 . Probing “either of the proposed Zundel specific transitions recovers the full suite of transitions, thus establishing that the observed spectrum is indeed homogeneous”. 17 Logically, if the Zundel assignment of the probed transitions is beyond doubt, such a result implies that the single isomer is actually the Zundel isomer (cf. ref 19), rather than the Eigen isomer advocated by Fournier et al. However, the frequencies of these “Zundel specific transitions” were taken from the AIMD simulations. 18 These are not quantitative, because classical trajectories do not include nuclear quantum effects (NQE), 20 such as zero-point energy (ZPE) and tunneling. To get a spectrum with NQE included, Fournier et al. have conducted a single anharmonic 2nd-order Vibrational Perturbation Theory (VPT2) 21 computation in Gaussian 09, 22 at the B3LYP/6-31+G(d) level of theory with no dispersion correction, as reported in the supporting information (SI) of ref 17. They concluded that the computed Eigen isomer spectrum agrees nicely with experiment, the 1750 cm−1 band being the combination of two H3 O+ rocking modes 17 and the 1050 cm−1 band assigned to the H3 O+ umbrella (U ) mode. In contrast, their trans-Zundel disagreed notably with experiment. It turns out that their “trans-


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Zundel isomer” was of C1 symmetry, differing from the C2 trans-Zundel isomer investigated by Kulig and Agmon (K.D. Jordan, private communication). Consequently, the VPT2 calculations should be repeated and extended to include the cis-Zundel isomer, which is also situated in a local minimum of the potential energy surface. 13 With these issues in mind, it seems that the strongest argument in favor of an Eigen isomer 8 is based on the relative cluster energetics. “The n = 4 complex is expected to be the classic ‘Eigen’ ion, having a central hydronium surrounded symmetrically by three water molecules each hydrogen bonded to a single OH. This structure is confirmed by calculations to be the most stable for this ion.” 10 While we also confirm this assertion (below), the electronic energy difference (ZPE not included) is only about 9.6 kJ/mol. The molecular beam is not an equilibrium system and thus the isomer distribution is not necessarily a Boltzmann distribution. In this respect, it is instructive to compare the H+ (H2 O)4 and Na+ (H2 O)4 clusters. The latter has been studied extensively by Lisy and coworkers. 23–25 Its lowest energy isomer has the four water molecules in the 1st solvation shell of the sodium cation (hence, it is denoted by 400). The next isomer on the electronic energy scale has three water ligands in the first solvation shell and the fourth in the 2nd shell, as a double-acceptor (AA) water molecule (hence, it is denoted 310). Similarly to the protonated water case, their electronic energy difference (ZPE not included) is only 9.6 kJ/mol. The spectra of this cluster is reproduced in Figure 1. Clearly, the 100 K spectrum (black line) cannot be accounted for solely by the lowest energy, 400 isomer. The band around 3550 cm−1 is characteristic of the weak hydrogen-bond of an AA water molecule, which can only be explained by the presence of the 310 isomer, as recently confirmed by the VPT2 calculations shown in the figure. 26 As this peak disappears in the high temperature IRMPD



1 .0

3 1 0

4 0 0

0 .8

In te n s ity

0 .6

2 2 0 -1

2 2 0 -2

0 .4

IR

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0 .0 3 3 5 0

3 4 0 0

3 4 5 0

3 5 0 0

3 5 5 0

3 6 0 0

3 6 5 0

3 7 0 0

3 7 5 0

3 8 0 0

F r e q u e n c ie s (c m -1 )

Figure 1: Comparison of experimental IR spectra 23–25 for the tetra-aqua-sodium ion cluster with anharmonic VPT2 calculations. 26 Black line depicts the Infrared Photodissociation (IRPD) spectrum of Na+ (H2 O)4 Ar at T ∼ 100 K, 23 whereas the red line is the Infrared Multiphoton Dissociation (IRMPD) spectrum of Na+ (H2 O)4 Ar at T ∼ 300 K. 24 Anharmonic spectra (colored sticks) were calculated at the full-MP2/6-31++G** level of theory. A scale factor of 0.9936 has been applied to all calculated frequencies. Reproduced from Figure 2 of ref 26 with permission from the American Chemical Society. spectrum at 300 K (red line), the lowest energy 400 isomer now becomes dominant. A possible explanation for the non-Boltzmann distribution at low temperatures comes from the kinetic pathways of cluster formation in the molecular beam, 26,27 although these pathways have not been precisely characterized. Here we revisit the protonated water tetramer, hoping to determine which of the structures in Scheme 1 corresponds to the measured IR spectrum. 11 We shall use two state-of-the-art methods with which one can nowadays tackle the problem, AIMD/DACF and VPT2. AIMD/DACF is the more mature and stable method, albeit lacking NQE. Use of Fourier transformed DACF to generate vibrational spectra of small molecules from classical trajectories has been practiced for a long time. 28,29 AIMD subse-


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quently replaced the empirical potential energy surfaces by on-the-fly solutions for the electronic problem, opening the road for more accurate calculations of vibrational spectra. 30,31 Since then, AIMD/DACF has been applied extensively to small protonated water clusters. 14,15,18,32–38 Here we present additional AIMD/DACF computations using the BLYP and B3LYP functionals with Grimme’s third-generation dispersion correction (DFTD3), 39 and Dunning’s triple-ζ basis set. In particular, AIMD simulations with the B3LYP functional have not been previously conducted for this cluster. These simulations are considerably more time consuming than with the BLYP functional, but expected to exhibit better accuracy than BLYP simulations: For example, recent benchmarking of harmonic OH frequencies in water clusters 40 and halidewater dimers 41 confirm that B3LYP-D3 results are considerably more accurate than BLYP. Application of VPT2 to protonated water 17,42,43 and ammonia clusters 44 is more recent. The method includes NQE, is valid up to quartic anharmonicities and suffers from the instabilities of perturbation theories. Unlike AIMD, it is a static 0 K theory with no account for linewidths and temperature effects. Because we assess that this method should be tested in more detail, our analysis is more extensive than previous VPT2 studies of protonated water clusters. We report 30 anharmonic VTP2 calculations using DFT (B3LYP-D3, HCTH470 and PBE0-D3 functionals with ultrafine grids) and MP2(fc), all with either Dunning’s tripleζ (aug-cc-pVTZ) or Pople’s 6-311+G(d,p) basis sets (abbreviated bs1 and bs2, respectively), and (as an extra test) also B3LYP-D3/6-311++G(3df,2p) with a superfine grid. With the large data set of spectra generated herein, one can appreciate the merits of VPT2 and identify some pitfalls. On the one hand, the VPT2 frequencies in the high frequency regime are very accurate, and can be used to correct the



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AIMD frequencies for NQE. On the other hand, a few modes in the low frequency range are highly unstable, and should preferably be replaced with the harmonic results. With these issues properly addressed, all theoretical methods (AIMD and VPT2) generate the same spectra (at least for the fundamental bands), agreeing with experiment for the Zundel isomer, but not for the Eigen isomer as the sole contributor to the spectrum.

Computational Methods 22 The geometry of the H9 O+ with the 4 cluster was optimized in Gaussian 09,

second-order Møller-Plesset perturbation theory [MP2(fc)] method and DFT with either the B3LYP-D3, 45,46 PBE0-D3 47 or HCTH470 48 functional (where D3 implies dispersion correction 39 ), combined with either the aug-cc-pVTZ or 6-311+G(d,p) basis sets (denoted herein bs1 and bs2, respectively), under the “tight” geometry convergence criteria. VPT2. The static harmonic and anharmonic spectra were then calculated at each optimized geometry from the “standard orientation” as suggested by Gaussian technical support team (see the communication in the SI), especially for symmetric molecules/clusters. All DFT calculations were performed using the “ultrafine” grid as recommended, 21 except the B3LYP-D3/6-311++G(3df,2p) level of theory was performed with “superfine” grid and higher accuracy for the two-electron integrals (10−13 ). Anharmonic frequency analysis was based on 2nd order Vibrational Perturbation Theory (VPT2), 21 which utilizes an expansion of the interaction potential up to 4th order. Sometimes (e.g., very anharmonic systems) this description is inadequate and then the method can produce errors. However, when it is applicable it generates quite accurate frequencies because NQE (important for light


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atoms such as hydrogen) are included here, 20 particularly the zero-point energy (ZPE). Therefore the computed frequencies were used as is (not scaled). Surprisingly, for the highly symmetric Eigen isomer (C3 symmetry) only MP2 could generate the anharmonic frequencies, whereas all DFT-based VPT2 calculations ended in errors. Similar phenomena were reported in the literature, in which the authors excluded the calculations of the molecules with 3-fold or higher symmetry axes, “due to the limitations of the Gaussian 09”. 49 As a result, in this work the DFT-based anharmonic frequencies of the Eigen isomer were computed at the lower C1 symmetry, in which the dihedral angles between the dangling water molecules and the hydronium core are slightly different from each other. The MP2 anharmonic spectra were obtained for both C1 and C3 symmetries. In contrast, for both trans- and cis-Zundel isomers it was possible to perform VPT2 anharmonic frequency analysis with C2 symmetry. For comparison with the Zundel isomers, geometry optimization and VPT2 anharmonic frequency analysis were performed also on the H5 O+ 2 cation (with C2 symmetry) at both MP2(full)/bs1 and B3LYP-D3/bs1 theory levels. All the anharmonic results can be found in the Excel file accompanying the SI. AIMD. We have also calculated the anharmonic spectra from AIMD. In AIMD simulations, one solves the full Schrödinger equation for the electrons “on the fly”, usually based on DFT. The interatomic forces are then generated from the electron density using the Hellmann-Feynman theorem, and the positions and velocities of the atoms are propagated classically, by solving Newton’s equations in this force field. These simulations show greater stability than VPT2, because they do not involve a perturbation expansion. They include any kind of anharmonic effect (not only the quartic one). On the other hand, NQEs are not included in these classical simulations, which would typically cause a blue shift of some of the IR bands. Overtones and combination bands are usually not observed in AIMD/DACF



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spectra. 50 There is no fundamental reason for this, because such bands were seen in DACF spectra from classical trajectories of small molecules, such as OCS, 28 D2 O, 31 Ba(NO3 )2 , 51 methanol, 52 and small clusters (e.g., the protonated ammonia dimer 44 ). With increasing size of the strongly coupled molecular system the noise in the computation increases, and this could hinder the observation of weak features in the spectrum. The AIMD computations were performed in the CP2K/Quickstep software package, 53 see http://www.cp2k.org/quickstep. This code describes the electronic structure using DFT with a basis of mixed Gaussian and plain wave functions. The BLYP and B3LYP functionals with Grimme’s third generation dispersion correction (DFT-D3) were used. 39 The augmented triple-ζ valence polarization basis set with the corresponding Goedecker-Teter-Hutter (GTH) pseudopotentials 54 (aug-TZVP-GTH) were used for all atoms. The plane wave energy cutoff was set to 350 Ry. Self-interaction correction (SIC) was applied with the Martyna-Tuckerman Poisson equation solver. 55 The orbital transformation (OT) method 56 was applied for faster convergence, with the tight convergence criterion of 1 × 10−6 a.u. at every MD step. The time step for our simulations was 0.5 fs, which is the standard time step for protonated water clusters simulations. 35,37 The masses of the hydrogen atoms were 1 a.m.u. The initial coordinates for the simulations were taken from the Gaussian 09 MP2(fc)/aug-cc-pVTZ optimizations. Three independent trajectories (Eigen, trans-Zundel, cis-Zundel isomers), were equilibrated for 9 to 50 ps in the canonical (NVT) ensemble to a target temperature of 50 K maintained by the Nosé-Hoover chain thermostat. Subsequently, several NVE AIMD trajectories were initiated for each isomer from different time points along the NVT equilibration trajectory, as detailed in Table S1 of the SI. The spectra from these NVE trajectories were averaged together, so as to better sample the initial conditions. Similar to the previous report, 18 cis-to-trans isomerization


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occurred during the equilibration of the cis-Zundel isomer. Consequently, only the trans-Zundel isomer survived in the AIMD simulations. Determination of the IR intensities depend on the total dipole moment of the system, µ(t), but its computation is not as straightforward in AIMD as in empirical force fields of classical MD, which assign a partial charge to each atom. As discussed in the literature, for AIMD there are two main methods for calculating the total dipole moment, “the Berry phase approach to polarization, and the maximally localized Wannier function scheme. The former provides the total dipole moment of the simulation box and this is perfectly suited as long as only a single molecule is simulated”. 50 Thus we have used the Berry phase to calculate µ(t). The IR absorption coefficient, αµ (ω), was computed from the autocorrelation function (ACF) of (µ) by: 29,50,57–59 n ω ω[1 − exp(−β~ω)] · αµ (ω) ∝ 1 − exp(−β~ω) Z ∞ 2 = ω dt exp(−iωt)hµ(0) · µ(t)i

Z

∞

dt exp(−iωt)hµ(0) · µ(t)i

o

0

0

2 2 3 Z ∞ 3 Z ∞ X X 2 dt exp(−iωt)µ˙ j (t) = ω dt exp(−iωt)µj (t) = 0 0 j=1 j=1 Z ∞ ˙ ˙ dt exp(−iωt)hµ(0) · µ(t)i . =

(1)

0

The prefactor ω/[1 − exp(−β~ω)] is the quantum correction factor for a harmonic oscillator, 29 and β is 1/(kB T ). The dipole moment ACF, hµ(0) · µ(t)i , is a convolution whose FT is therefore a product of FT’s, giving rise to the third line. Integrating by parts, one obtains the 4’th line namely, the FT of the ACF of the ˙ time (t) derivative of the dipole moment, µ(t). One can thus use either the dipole moment itself or (more commonly) its time derivative for calculating αµ (ω). ˙ As described previously, 44 µ(t) was first transformed into the frequency domain using discrete fast Fourier transform (FFT), then multiplied by its complex



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conjugate, and transformed back into the time domain using inverse FFT (iFFT) to obtain the DACF. Next, a Gaussian window function was applied to suppress noise in the DACF, and the result was again Fourier transformed to produce the power spectrum. The line width of each spectrum was controlled within 10 cm−1 . The final spectrum was obtained by averaging over the four trajectories generated for each functional (BLYP-D3 and B3LYP-D3). In order to identify the contribution of the excess proton (H∗ ) in the Zundel clusters to the AIMD IR bands, its partial velocity autocorrelation functions (pVACFs) were calculated e.g., for the two OH∗ distances, ζ(t). 42,44 Then: Z

∞

Iζ (ω) =

˙ . ˙ ζ(t)i dt exp(−iωt)hζ(0)

(2)

0

This local mode approach reveals all the frequencies to which ζ(t) contributes. Other local modes were similarly probed using pVACF (Figures S1-S5 in the SI). Unlike the DACF spectrum, which is subject to IR selection rules, there are no selection rules for pVACF spectra.

Results Relative Energies of Isomers Table 1 lists the relative electronic energies, (∆Eelec ), of the three isomers and their ZPE corrected values (∆E0 ). Consistent with the previous reports, 13–15 the Eigen isomer is the global minimum energy structure at all theory levels tested in this work. Both the cis- and trans-Zundel clusters have similar electronic energies, 12–16 kJ mol−1 higher than the Eigen isomer. With ZPE included, both isomers stabilize, but the trans isomer stabilizes more, becoming 1–2 kJ mol−1 more stable than the cis-Zundel isomer, ca. 8 kJ mol−1 above the Eigen isomer.


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Table 1: Relative Electronic Energies (in kJ mol−1 ) of the Three H9 O+ 4 isomers without ZPE (∆Eelec ) and with ZPE Correction (∆E0 ) Obtained from the MP2(fc), B3LYP-D3, PBE0-D3 and HCTH470 Methods with Either the aug-cc-pVTZ (bs1) or 6-311+G(d,p) (bs2) Basis Sets. MP2(fc) Species

bs1 ∆Eelec

Eigen

B3LYP-D3 bs2

∆E0

∆Eelec

bs1 ∆E0

∆Eelec

PBE0-D3 bs2

∆E0

bs1

∆Eelec

∆E0

∆Eelec

HCTH470 bs2

∆E0

∆Eelec

bs1 ∆E0

∆Eelec

bs2 ∆E0

∆Eelec

∆E0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

trans-Zundel

15.10

8.04

16.24

8.15

14.19

7.31

14.32

8.88

12.52

6.48

12.57

8.09

12.82

7.39

13.06

7.62

cis-Zundel

14.75

9.63

15.90 10.57

14.05

9.13

14.39 10.13

12.26

8.14

12.62

9.50

12.39

8.57

12.73

9.40

diff(cis − trans)

-0.35

1.59

-0.34

-0.13

1.81

-0.26

1.66

0.05

1.42

-0.43

1.19

-0.33

1.78

2.41

0.07

1.25

Table 2: Relative Gibbs Free Energies (in kJ mol−1 ) of the Three H9 O+ 4 isomers (at 298.15 K, 1 atm) with ZPE Correction Obtained from the MP2(fc), B3LYP-D3, PBE0-D3 and HCTH470 Methods with Either the aug-cc-pVTZ (bs1) or 6-311+G(d,p) (bs2) Basis Sets.

Species

MP2(fc)

B3LYP-D3

PBE0-D3

HCTH470

bs1

bs2

bs1

bs2

bs1

bs2

bs1

bs2

0

0

0

0

0

0

0

0

trans-Zundel

5.19

3.23

7.39

12.82

6.59

12.67

7.70

8.13

cis-Zundel

9.08

9.63

11.63

14.90

10.55

15.23

10.78

12.56

diff(cis − trans)

3.89

6.40

4.24

2.08

3.96

2.56

3.09

4.43

Eigen

Entropy further favors the trans isomer (likely because of more facile internal rotations 26 ). As evidenced from the Gibbs free energy in Table 2, at roomtemperature the trans-Zundel isomer becomes 2–6 kJ mol−1 more stable than the cis-Zundel isomer. This explains why cis-Zundel is short lived in the AIMD trajectories, converting in a few ps to the trans form. 18 With MP2/bs1 the Gibbs energy of the trans-Zundel isomer is only 5.2 kJ/mol higher than that of the Eigen isomer. This means that when the isomers are fully equilibrated at 298 K there will be over 10% of the Zundel isomer in the mixture.



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AIMD Spectra The previous AIMD/DACF spectra of the Eigen and trans-Zundel isomers by Kulig and Agmon 18 utilized the BLYP functional, with no dispersion correction, and with the double-ζ basis set. These computations have been extended here in several directions: using both BLYP and B3LYP functionals, both with dispersion correction (DFT-D3) and with the triple-ζ basis set. Nevertheless, the computed spectra shown in Figure 2, are quite similar to theirs. 18 In the high-frequency end there are 6 free OH stretching modes. For the C3 Eigen isomer they are grouped into 3 symmetric and 3 antisymmetric stretches of the 3 water molecules (denoted ss and as, respectively). Each motion combines to produce 1 in-phase and 2 out-of-phase modes. The large dipole moment variations for the in-phase asymmetric and out-of-phase symmetric modes are responsible for the two observable peaks, in agreement with experiment. For the Zundel isomer, there are 2 acceptor water molecules, each with a symmetric and antisymmetric stretching mode, whose in-phase superposition contributes two bands to the transZundel spectrum, with a third band originating from the in-phase stretch of the dangling OH bonds of the two acceptor-donor water molecules. This appears to be in less good agreement with experiment, which shows two pronounced peaks in this region. For both isomers, the B3LYP-D3 functional (Figure 2A) gives blue shifted high frequency peaks (dangling and HBed OH). This can be ascribed to the omission of NQE, because VPT2 with the same functional and basis set reproduces the experimental frequencies (see below). The BLYP-D3 calculations (Figure 2B) do not show such large blue shifts, which is likely due to cancelation of errors: 60 when NQE are properly accounted for, BLYP underestimates OH, NH and CH stretching frequencies. This error largely offsets the blue-shift due to lack of NQE. The important differences between the two isomers occur below 2000 cm−1 . 14 ACS Paragon Plus Environment

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10 00

0 .6

4

3 8 0 1 3 8 5 1

×10

1 7 5 9

0 .8

2 0 p s ( E ig e n ) 2 0 p s ( Z u n d e l) 2 8 3 9

1 1 3 7

1 .0

1 6 3 2

(A )

1 .2

0 .0

5 0 0

1 0 0 0

2 2 0 0

3 6 0 0

1 5 0 0

2 4 0 0

2 0 0 0

2 5 0 0

W a v e n u m b e r (c m

×1 0

2

3 8 0 0

3 6 7 0

2 0 0 0

3 7 1 4 3 6 2 5

2 3 7 5

2 8 2 7 2 8 7 1

×1 0

4

-1

3 0 0 0

3 5 0 0

3 7 2 6

1 8 0 0

×10 4

3 6 2 8

0 .0

1 6 0 0

α?

2 0 p s ( E ig e n ) 2 0 p s ( Z u n d e l)

1 5 9 1

10 00

1 0 6 5

0 .2

×1 0 3 12 00

9 4 7

0 .6 0 .4

1 7 0 7

0 .8

1 7 2 0

1 1 3 2

1 .0

1 5 8 4

E x p e r im e n t B L Y P - D 3 a v e r a g e tr a j. 1 , 2 , 3 , 4 , e a c h o f th e m B L Y P - D 3 a v e r a g e tr a j. 1 , 2 , 3 , 4 , e a c h o f th e m

2 6 9 7

(B )

1 .2

80 0

N o r m a liz e d In te n s ity ( a .u .)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


2 9 9 7

Page 15 of 46

4 0 0 0

)

Figure 2: Comparison of the photo predissociation experimental spectrum (gray) 11 of the H9 O+ 4 cluster with the computed AIMD/DACF spectra of the Eigen (blue and green lines) and trans-Zundel (red and orange lines) isomers, using DFT with (A) the B3LYP-D3 and (B) the BLYP-D3 functionals at 50 K. Intensities are normalized based on the 2665 cm−1 band. Insets depict enlargements of the weaker bands.

Firstly, the Zundel isomer has reasonable intensity in the water bend mode (1615 cm−1 ), whereas for the Eigen isomer it is very weak. More importantly, the Eigen isomer shows no IR bands near 1750 cm−1 , whereas at 1137 cm−1 we find the U mode that is a 1000 times too weak, and it red shifts due to NQE to below 1000 cm−1 (see below). In contrast, the Zundel isomer reproduces both 1750 and 1050 cm−1 bands quite accurately. Both bands have a contribution from the excess proton rattling between the two oxygen atoms (a so-called “proton transfer mode”, PTM), 18 as verified in partial VACF of the OH* stretch in Figure S1 of the SI. Interestingly, in spite of the engagement of the PTM in these two modes, they seem to be less 15 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

influenced by NQE, as judged from the AIMD/B3LYP-D3 peaks being blue shifted to less than 10 cm−1 from the experimental bands. Based on AIMD, one would conclude that if a single isomer contributes to the measured spectrum, 17 it is the Zundel rather than the Eigen isomer. The only drawback is that AIMD tends not to show overtones and combination bands. 50 Hence if the Eigen isomer has strong combination bands in these regions, as suggested by Fournier et al., 17 they might be missed by AIMD.

Reliability of the VPT2 Spectra In comparison, VPT2 routinely calculates all overtones and combination bands, and includes NQE and anharmonicities up to quartic order. In these respects it is complementary to AIMD. However, because it is based on a perturbation expansion, when the quartic expansion is not valid for a given vibrational mode, or near-degeneracies occur, it may break down for that mode. It can be difficult to decide which modes are described well in VPT2 and which ones are not. To overcome this difficulty we have computed, for each of the three isomers in Scheme 1, the VPT2 spectra using DFT with the B3LYP-D3, HCTH470 and PBE0-D3 functionals (dispersion correction not available for HCTH470), and also for MP2(fc). From the two basis sets used, bs1 appears to be more reliable, as deduced below from the MP2(fc) spectra, so that results for bs2 are shown only in the SI. For the Eigen cluster, the DFT methods in Gaussian 09 could not compute VPT2 spectra for the high (C3 ) symmetry, only for the low (C1 ) symmetry [MP2(fc) could calculate the spectra for both]. For the Zundel isomer there are four modes involving the PTM (modes 22, 25, 26 and 27, see Excel file in the SI). As seen in Figure 3, VPT2 does not succeed in calculating the first three of these accurately. In comparison with the HBed OH stretching modes 4 and 21 (panels E and F), modes 22, 25 and 26 are characterized 16 ACS Paragon Plus Environment

Page 16 of 46

Page 17 of 46

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by: 1. Blue shift of the anharmonic frequencies compared to the harmonic ones, whereas usually the opposite holds true. 2. The shifts are sometimes excessive. 3. While the harmonic frequencies do not differ by more than ca. 100 cm−1 between different methods, the anharmonic ones are irreproducible and vary erratically between methods. Of the different methods, MP2(fc) that might have been expected to produce the most accurate frequencies, actually produced the most erratic results, particularly in conjunction with bs2. The anharmonic trans-Zundel frequency is blue shifted (compared to the harmonic result) by about 600 cm−1 for mode 25 and 1300 cm−1 for mode 26. Such huge blue shifts are not physically viable, and contrast with some of the other quantum chemistry methods that show only minor shifts (e.g., B3LYP-D3/bs1 for mode 25 and HCTH470/bs1 for mode 26). The huge discrepancy between methods and basis sets suggests that the anharmonic frequencies of these three modes are affected by the inherent instability of perturbation methods, and should better be replaced by their harmonic counterparts.



2 4 0 0 (A )

1 6 0 0

M o d e 2 tra c is tra c is

P T -Z u n -Z u n

M u n d e u n d e

(B ) d e l h d e l a

l h a r l a n h

a rm m o n n h a a rm

o n ic ic r m o n ic o n ic

tra c is tra c is

2 0 0 0 -1

1 4 0 0

F re q u e n c y (c m

-1


Z -c o re (H B e d O H )

c o u p le d w ith P T M

u n d e u n d e

d e l h d e l a

l h a r l a n h


c k (Z u n d d e l u n d u n d e l

-c e l h a e l a n

o re h a rm a n h a

M o d e 2 6 : w a g

2 2 0 0

n s -Z -Z u n n s -Z -Z u n


)

5 : n s -Z n s -Z

)

1 5 0 0

1 3 0 0 1 2 0 0

1 8 0 0 1 6 0 0 1 4 0 0 1 2 0 0

1 1 0 0 1 0 0 0 1 0 0 0

8 0 0

2 0 0 0

7 0 0 l h a r l a n h



(D ) 6 5 0 6 0 0 -1

1 9 0 0


d e l h d e l a



-1

)

1 9 5 0

re H )

)

M o d e 2 2 : b Z( H - c O o tra n s -Z u n c is - Z u n d e tra n s -Z u n c is - Z u n d e

(C )

1 8 5 0

1 8 0 0

5 5 0 5 0 0 M o d e 2 tra c is tra c is

4 5 0 1 7 5 0

4 0 0

1 7 0 0

3 5 0

7 : n s -Z n s -Z

ro -Z u n -Z

) c o u p le d w ith P T M r m o n ic o n ic h a r m o n ic r m o n ic

3 6 0 0

3 2 0 0


3 0 0 0

)

h a rm rm o n a n h a h a rm


-1

)

3 4 0 0

(F )

3 2 0 0

O H )

e l h a e l a n

-1

M o d e 4 : s s Z( H - c B o e r d e tra n s -Z u n d c is - Z u n d e l tra n s -Z u n d c is - Z u n d e l

(E )


3 0 0 0

2 8 0 0

2 8 0 0 2 6 0 0 2 4 0 0

M o d e 2 1 : a s Z( H - c B o e r d e tra n s -Z u n d e c is - Z u n d e l h tra n s -Z u n d e c is - Z u n d e l a

2 2 0 0 2 0 0 0

2 6 0 0

O H )

l h a r l a n h


o n ic ic r m o n ic o n ic s2

s1

0/ b

s2 3/ b

0/ b

3/ bs 1

s1

s2

3/ bs 2

3/ b

s1

)/b

)/b P 2 (fc

P 2 (fc M

M

P 2 (fc

M

C o m p u ta tio n a l L e v e ls

B 3 LY P D B 3 LY P D P B E 0 -D P B E 0 -D H C T H 47 H C T H 47

s2

s1

s2

s1

s2

s1

s2

)/b P 2 (fc B 3 )/b LY P D B 3 3/ b LY P D 3 P B /b E 0 -D 3 P B /b E 0 -D 3/ H C b T H 47 0/ H b C T H 47 0/ b

s1

1 8 0 0

M

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 18 of 46

C o m p u ta tio n a l L e v e ls

Figure 3: Frequencies of some major Zundel-core vibrational modes from our VPT2 calculations at different quantum chemistry levels. (A) mode 25: the PTM; (B) mode 26: Zundel-core HBed OH wag coupled with the PTM; (C) Mode 22: Zundel core water bend coupled with the PTM; (D) Mode 27: Zundel-core rock coupled with the PTM; (E) Mode 4: Symmetric stretch ss of the two HBed OH groups; (F) Mode 21: Asymmetric stretch as of the two HBed OH groups. Data from the Excel file in the SI. bs1 and bs2 are abbreviations for the aug-cc-pVTZ and 6-311+G(d,p) basis sets, respectively. The harmonic frequencies are shown as solid triangles (trans-Zundel) and circles (cis-Zundel). The anharmonic frequencies are shown as hollow triangles (trans-Zundel) and open circles (cis-Zundel).


Page 19 of 46

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The problem now is that the modes that were replaced, particularly 22 and 25, are the ones that are crucial for differentiating between the Zundel and Eigen forms. They are also the two shared proton modes with counterparts in the spectrum of the bare Zundel cluster, H5 O+ 2 , so that one might take advantage of the more quantitative results available in the latter case. Table 3 compares the frequencies of these two modes for the trans-Zundel isomer with those of the bare-Zundel cluster using VPT2, and also multi-configuration time-dependent Hartree (MCTDH) 61 and diffusion Monte-Carlo (DMC) calculations 62 on a fitted multi-dimensional potential energy surface. 63 These two dynamic methods include anharmonicity and NQE, but without the inherent instabilities of VPT2. For the PTM (mode 25), DMC and MCTDH indeed show an anharmonic blueshift (compared to the harmonic frequency), amounting to 172 cm−1 for MCTDH contra 670 cm−1 for VPT2. This proves that VPT2 indeed produces an excessive blue-shift for the PTM of the Zundel cation. Thus for the trans-Zundel isomer, we expect a PTM frequency that is smaller than that of VPT2 namely, smaller than 1200 cm−1 . For the PTM-coupled water bend of the bare-Zundel isomer, VPT2 again shows a blue anharmonic shift (71 cm−1 ) whereas MCTDH and DMC show a modest red -shift (of 30–40 cm−1 ). Amusingly, the harmonic value is in better agreement with experiment than either of these sophisticated methods, corroborating the observation in ref 62 that “the harmonic HOH bend frequencies are actually in surprisingly good agreement with experiment”. For mode 22 of the trans-Zundel isomer this conclusion is also valid, although the VPT2 blue-shift is modest (of order 10 cm−1 ) for all our calculations with bs1. With this caveat we present the VPT2 spectra for the different methods, in which harmonic frequencies/intensities are used for modes 22, 25, 26 and 27, and anharmonic ones for all other modes. Overtones and combination bands in which



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 20 of 46

Table 3: Comparison of harmonic and anharmonic vibrational frequencies (in cm−1 ) of modes 22 and 25 for the trans-Zundel isomer of H+ (H2 O)4 and the bare Zundel cation, H5 O+ 2. Cluster

trans-Zundel harmonic

Mode

bare Zundel

anharmonic

harmonic

anharmonic

MP2/bs1a VPT2a AIMDa exper. 11 MP2/bs1a VPT2a DMC 62 MCTDH 61 exper. 64

25 (PTM)

1017

1210

1063

1050

884

1556

1037

1033

1047

22 (bend)

1763

1770

1758

1750

1765

1836

1728

1741

1763

a

This work. AIMD used the B3LYP-D3 functional, VPT2 with MP2(fc)/bs1.

these four modes are involved are displayed only when their fundamental anharmonic frequencies are blue shifted by less than 150 cm−1 from their harmonic values, and provided they do not have unreasonably large intensities. Similar spectra for bs2 are displayed in Figures S6-S10 of the SI. For completeness, the pure VPT2 spectra are shown in Figures S11-S18 of the SI.

The VPT2/DFT Spectra Let us start with results for the B3LYP-D3 functional (with bs1). For the water tetramer (also with bs1), B3LYP-D3 was shown to produce an average absolute deviation of only 35 cm−1 of the harmonic frequencies from values considered close to the CCSD(T) complete basis set limit. 40 Thus, on the harmonic level, this functional performs satisfactorily. Figure 4A1 shows the VPT2 spectrum of the Eigen cation with C1 symmetry, whose bands appear slightly red-shifted compared to experiment. In agreement with the AIMD/DACF spectra in Figure 2, there are no fundamental bands that could explain the experimental peaks at 1750 and 1050 cm−1 . There are indeed a few combination bands (not seen in the AIMD/DACF spectrum) that might explain the 1904 and 2245 cm−1 peaks, as well as the red shoulder of the 2665 cm−1 peak (or else they are an artefact of 20 ACS Paragon Plus Environment

Page 21 of 46

the C1 symmetry, see below). Because all of the peaks appearing in the computed spectrum exist in the experimental spectrum, this calculation does not rule out participation of an Eigen isomer. It shows, however, that the Eigen isomer alone cannot explain the observed spectrum because of the missing peaks at 1750 and 1050 cm−1 .

m o d e 2 7 m o d e 2 6

0 .8 0 .6 0 .4

(D 1 )

H 9 O 4 tr a n s - Z u n d e l is o m e r ( C 2 ) B 3 L Y P -D 3 /a u g -c c -p V T Z

1 .0 0 .8

m o d e 2 2

(C 1 )

1 .0

0 .0 +

m o d e 2 5

0 .0

(E 1 )

6 4 2 0 0

0 .1 0

0 .4 0 .0 5

0 .2 0 .0 +

H 9 O 4 c is - Z u n d e l is o m e r ( C 2 ) B 3 L Y P -D 3 /a u g -c c -p V T Z

1 .0

8

0 .1 5

0 .6

0 .2 0 .0

0 .8

(F 1 )

1 .0

0 .8

0 .0 0 0 .1 5

m o d e 2 2

0 .4

m o d e 2 5

m o d e 2 7 m o d e 2 6

0 .8

0 .6

0 .1 0

0 .6 0 .4

0 .2

0 .0 5

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0

2 5 0 0

3 0 0 0

0 .0 0 .0 0 3 5 0 0 3 5 0 0 3 6 0 0 3 7 0 0 3 8 0 0 3 9 0 0

W a v e n u m b e r (c m

0 .6 0 .4

1 .2

e r (C 1 ) -p V T Z fu n d a m o v e rto n c o m b in n d a m e n

e n ta ls e s a tio n s ta ls

0 .2 0 .0 (C 2 )

1 .0 0 .8 0 .6 0 .4 0 .2 0 .0

0 .4

-1

0 .2 5 0 .2 0

0 .6

0 .1 5

0 .4

0 .1 0

0 .2

0 .0 5 0 .0 0 (D 2 )

H 9 O 4 tr a n s - Z u n d e l is o m e r ( C 2 ) H C T H 4 7 0 /a u g -c c -p V T Z

1 .0

0 .1 5

0 .8 0 .1 0

0 .6

0 .0 5

0 .2 0 .0 +

(E 2 )

0 .6

0 .3 0

0 .8

0 .4

H 9 O 4 c is - Z u n d e l is o m e r ( C 2 ) H C T H 4 7 0 /a u g -c c -p V T Z

1 .0 0 .8

(B 2 )

1 .0

0 .0 +

(F 2 )

0 .0 0 0 .1 5

1 .0 0 .8

0 .1 0

0 .6 0 .4

0 .2

0 .2

0 .0

H 9 O 4 E ig e n is o m E x p e r im e n t H C T H 4 7 0 /a u g -c c a n h a r m o n ic a n h a r m o n ic a n h a r m o n ic h a r m o n ic fu

0 .0 5

0 .2

0 .0 5 0 0

1 0 0 0

1 5 0 0

2 0 0 0

2 5 0 0

3 0 0 0

0 .0 0 .0 0 3 5 0 0 3 5 0 0 3 6 0 0 3 7 0 0 3 8 0 0 3 9 0 0

W a v e n u m b e r (c m )

-1

)

Figure 4: Anharmonic VPT2 spectra (unscaled) at the B3LYP-D3/aug-cc-pVTZ (A1, B1, C1, D1, E1 and F1) and HCTH470/aug-cc-pVTZ (A2, B2, C2, D2, E2 and F2) levels for (A1, B1, A2, B2) Eigen isomer, (C1, D1, C2, D2) transZundel, (E1, F1, E2, F2) cis-Zundel. The anharmonic fundamental, overtone and combination bands are represented by blue, green and red bars, respectively. Intensities are normalized based to the experimental 2665 cm−1 peak. In the region above 3500 cm−1 , the calculated anharmonic intensities are zoomed in (see the blue axes) for easier comparison with experiment. The blue dotted lines represent the fundamental harmonic frequencies and intensities of modes 22, 25, 26 and 27, whose anharmonic VPT2 frequencies are suspect of errors, as discussed in the text. The experimental IR predissociation spectrum 11 is depicted by the gray shaded area.

In contrast, for the two Zundel isomers all the high-intensity fundamentals have frequencies close to experiment. Both Zundel isomers predict pronounced harmonic fundamental bands around 1050 and 1750 cm−1 (modes 25 and 22, respectively), agreeing well with the experimental spectrum. These frequencies 21 ACS Paragon Plus Environment

C a lc u la te d In te n s ity ( a .u .)

0 .2

0

+

(A 2 )

m o d e 2 2

0 .2

1 .0

m o d e 2 2

0 .4

1 .2 2

m o d e 2 6

0 .6

4

m o d e 2 5

0 .8


0 .1 0 .1 0 .1 0 .0 0 .0 0 .0 0 .0 0 .0 0 .2

m o d e 2 5

0 .4

1 .0

m o d e 2 7

0 .6

(B 1 )

1 .2

m o d e 2 6

0 .8

ig e n is o m e r ( C 1 ) r im e n t /a u g -c c -p V T Z r m o n ic fu n d a m r m o n ic o v e r to n r m o n ic c o m b in o n ic fu n d a m e n

x p e -D 3 n h a n h a n h a a rm

m o d e 2 7

H 9 O 4 E E B 3 L Y P a a a h


+

(A 1 )

1 .0

E x p e r im e n ta l a n d C a lc u la te d In te n s itie s ( a .u .)

1 .2


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60



(A )

1 .0

H 9

O 4

+

E ig e n is o m e r ( C 1

P B E 0 a a a

)

E x p e r im e n t

D 3 n h n h n h

/a u a rm a rm a rm

g o o o

c c n ic n ic n ic

p V T fu n o v e c o m

Z d a m e n ta ls rto n e s b in a tio n s

1 .0

0 .5

0 .5

0 .0

0 .0 (B ) m o d e 2 5

9

O

+

tr a n s - Z u n d e l is o m e r ( C 4

H m o d e 2 6

9

O 4

+

D 3 n h n h n h a r

/a u g a rm a rm a rm m o n

-c c -p V T o n ic fu n o n ic o v e o n ic c o m ic fu n d a

Z d a m rto n b in m e n

1 .0


0 .5

c is - Z u n d e l is o m e r ( C 2

P B E 0 a a a h

)

m o d e 2 2

E x p e r im e n t

m o d e 2 5

m o d e 2 7

P B E 0 a a a h

0 .0

(C )

1 .0

)

E x p e r im e n t

m o d e 2 7

0 .0

2

m o d e 2 2

m o d e 2 6

1 .0

0 .5

H

0 .5

D 3 n h n h n h a r

/a u g a rm a rm a rm m o n

-c c -p V T o n ic fu n o n ic o v e o n ic c o m ic fu n d a

Z d a m rto n b in m e n




1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 22 of 46

1 .0

0 .5

0 .0

0 .0 5 0 0

1 0 0 0

1 5 0 0

2 0 0 0 2 5 0 0 W a v e n u m b e r ( c m -1 )

3 0 0 0

3 5 0 0

4 0 0 0

Figure 5: Anharmonic VPT2 spectra (unscaled) at the PBE0-D3/aug-cc-pVTZ level for (A) Eigen, (B) trans-Zundel, and (C) cis-Zundel isomers. The anharmonic fundamental, overtone and combination bands are represented by blue, green and red bars, respectively. The experimental IR predissociation spectrum 11 is depicted by the gray shaded area. Intensities are normalized based on the experimental peak around 2665 cm−1 . The blue dotted lines represent the fundamental harmonic frequencies and intensities of modes 22, 25, 26 and 27, whose anharmonic VPT2 frequencies are suspect of errors, as discussed in the text. agree also with those of the bare-Zundel cation (Table 3). The most notable improvement of the VPT2 spectrum compared with AIMD is that the HBed OH modes agree quantitatively with the main peak at 2665 cm−1 . This holds for all three isomers. In the AIMD spectrum (Figure 2A), the peak for the trans-Zundel isomer appears more blue-shifted than that of the Eigen isomer (because its ss mode has very small intensity). With VPT2 there is a larger NQE correction for the Zundel isomer, leading to similar anharmonic frequencies for both isomers. This similarity implies that it may not be possible to probe selectively one of the isomers in this frequency regime. 17 In the high-frequency end one again notes three rather than two OH stretching


Page 23 of 46

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bands. These arise from the as and ss modes of the terminal water molecules, with an intermediate frequency due to the in-phase vibration of the dangling hydrogens of the two acceptor-donor (AD) water molecules in the Zundel core. The latter is closer to the ss mode so as to merge into a single experimental band. To check this, we present in Table 4 these two frequencies and their differences for all methods and basis sets used in our VPT2 calculations. For the trans-Zundel isomer we note that all bs1 frequencies differ by less than 30 cm−1 . Given that this difference is smaller than the error in the calculation and that the VPT2 bands have no widths, it is possible that these two bands will merge. Indeed, Figure 4 of Douberly et al. 10 shows a 3643 cm−1 band with a red shoulder that might conceal an extra band. In addition, the Zundel isomers have stronger combination bands than the Eigen isomer. In particular, both Zundel isomers exhibit several combination bands that may correspond with the 1904 and 2245 cm−1 peaks, as well as the blue-shoulder of the 2665 cm−1 band around 3000 cm−1 . Thus the Zundel isomer accounts for all of the main peaks in the measured spectrum. It is therefore either present alone in the molecular beam, or in combination with the Eigen isomer. To check whether these results are maintained for other functionals, we present in Figure 4A2 to F2 the spectra for the HCTH470 functional. That of the Eigen isomer is similar to the B3LYP-D3 spectrum, only of somewhat lesser quality. It also shows the two combination bands near 1904 and 2245 cm−1 . Likewise, the 1050 and 1750 cm−1 bands that remain unassigned for the Eigen isomer, are again assigned to modes 25 and 22 (respectively) of the Zundel isomers. PBE0-D3 gives similar results, as seen in Figure 5 and Figure S5 of the SI. In addition to the 1050 and 1750 cm−1 bands predicted for both Zundel isomers, their HBed OH stretches agree even better with experiment than those of the Eigen isomer (Figure 5B and C). The symmetric and asymmetric stretches split, and this accounts for the width of the 2665 cm−1 band (whereas its red and blue



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 24 of 46

Table 4: The VPT2 Frequencies (in cm−1 ) of the Symmetric OH Stretch (ss) of the Flanking Water Molecules and the Free OH Stretch of the AD Water Molecule in the Zundel-core of the trans- and cis-Zundel Isomers, as Obtained with Different Functionals and Basis Sets. trans-Zundel

Methods/basis sets

cis-Zundel

AD

ss

Diff.

AD

ss

Diff.

MP2(fc)/bs1

3666.18

3639.42

26.76

3622.86 3583.18

39.68

MP2(fc)/bs2

3706.60

3691.65

14.94

3717.96 3681.26

36.70

B3LYP-D3/bs1

3645.10

3619.18

25.92

3639.99

3621.72

18.27

B3LYP-D3/bs2

3682.41

3633.26

49.15

3673.55

3629.42

44.13

PBE0-D3/bs1

3703.68

3685.54

18.13

3702.11

3683.76

18.35

PBE0-D3/bs2

3743.82

3690.58

53.24

3734.11

3684.40

49.71

HCTH470/bs1

3632.83

3604.67

28.16

3630.69

3584.26

46.43

HCTH470/bs2

3651.07

3601.12

49.95

3663.63

3606.37

57.27

shoulders are likely due to combination bands). The high frequency free OH bands are blue shifted as compared with experiment, in contrast to the red shift seen for the B3LYP and HCTH470 functionals.

VPT2/MP2 Spectra MP2(fc) is possibly of higher accuracy than the DFT methods. MP2(fc)/bs1 (Figure 6) gives similar VPT2 spectra for the three isomers as the DFT calculations, with some differences. On the one hand, on the high frequency end, the transZundel is in excellent agreement with experiment, whereas the intensities of the Eigen isomer are too small (note the scale on the right of panel A). On the other hand, the in-phase and out-of-phase HBed OH modes of the main peak exhibit a rather large split. Because the magnitude of this split is method/basis-set dependent (reaching a maximum for MP2(fc)/bs2, see Figure S10 in the SI), this might be yet another manifestation of VPT2 instability. Indeed, while the AIMD results 24 ACS Paragon Plus Environment

Page 25 of 46

also exhibit a split in the main Zundel band (Figure 2), it is rather modest (within 50 cm−1 for both BLYP and B3LYP functionals. In contrast to the narrow Eigen peak (the hydronium ss mode is very weak), the split Zundel peak might account for the substantial width of the 2665 cm−1 band. We also note that the trans-Zundel combination bands are much stronger than those of the Eigen isomer, so that if some of the experimental bands originate from combination bands, they are likely to be due to the trans-Zundel isomer. (A )

1 .0

H 9

+

O

E ig e n is o m e r 4

M P 2 (fc a a a

(C 3)

E x p e r im e n t

)/a n h n h n h

u g a rm a rm a rm

-c c o n o n o n

-p ic ic ic

V T Z fu n d a m e n ta ls o v e rto n e s c o m b in a tio n s

1 .0 0 .5

0 .0

tr a n s - Z u n d e l is o m e r ( C

H 9 O 4

m o d e 2 7

0 .0

9

O

+ 4

g -c c rm o n rm o n rm o n o n ic

-p ic ic ic fu

V T Z fu n o v e c o m n d a

d a m rto n b in m e n

e n ta s e s a tio n s ta s

1 .0

0 .5

c is - Z u n d e l is o m e r ( C 2

M P 2 (fc a a a h

)

E x p e r im e n t

m o d e 2 5

m o d e 2 6

H

m o d e 2 7

0 .5

)/a u n h a n h a n h a a rm

0 .0

(C )

1 .0

M P 2 (fc a a a h

)

m o d e 2 2

0 .5

2

E x p e r im e n t

m o d e 2 2

m o d e 2 6

m o d e 2 5

+

(B )

1 .0

0 .0 2 0 .0 1 0 .0 0

)/a u n h a n h a n h a a rm

g rm rm rm o n

c c o n o n o n ic

-p ic ic ic fu

V T Z fu n o v e c o m n d a




0 .5


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1 .0

0 .5

0 .0

0 .0 5 0 0

1 0 0 0

1 5 0 0

2 0 0 0 2 5 0 0 W a v e n u m b e r (c m

3 0 0 0 -1

3 5 0 0

4 0 0 0

)

Figure 6: Anharmonic VPT2 spectra (unscaled) at the MP2(fc)/aug-cc-pVTZ level for (A) Eigen, (B) trans-Zundel, and (C) cis-Zundel isomers. The anharmonic fundamental, overtone and combination bands are represented by blue, green and red bars, respectively. The experimental IR predissociation spectrum 11 is depicted by the gray shaded area. Intensities are normalized based on the experimental peak around 2665 cm−1 . The blue dotted lines represent the fundamental harmonic frequencies and intensities of modes 22, 25, 26 and 27, whose anharmonic VPT2 frequencies are suspect of errors, as discussed in the text. The blue intensity scale on the right refers to the computed intensities. Note in panel A, the axis breaks between 0.03 and 0.1 for magnifying the weak overtone and combination bands. Of interest are the MP2(fc)/aug-cc-pVTZ results for the Eigen isomer in Figure 7, with overtones and combinations bands magnified by a factor of about 25 ACS Paragon Plus Environment


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100. For the C3 -Eigen cluster, overtones (e.g., of the hydronium core wag) are excited in the vicinity of the 1750 cm−1 band, while the combinations remain mute. For the C1 -Eigen cluster, combinations are excited near the 1904 and 2245 cm−1 bands, and in the red and blue shoulders of the 2665 cm−1 band. Both symmetries would occur in low temperature clusters, interconverting by dangling water rotation around the HB axis. This might seem like a promising alternative scenario for assigning the spectrum, 17 provided there were mechanisms for increasing the intensities of overtones and combination bands a hundred fold. However, our C1 and C3 Eigen structures are actually identical (both in structure and energy, up to 6 significant digits). Thus Gaussian 09 appears to have an error in symmetry assignment (e.g., C1 instead of C3 ). Normally, this is not a problem, but here we note that identical structures produce different VPT2 spectra, with vastly different combination bands (Figure 7). It turns out that “an incorrect symmetry assignment would translate into incorrect results in different other places in the anharmonic analysis because they depend on how the normal modes are grouped” (see “Communication from the Gaussian Support Team” in the SI). Therefore we tentatively conclude that the combination bands seen in the C1 Eigen isomer spectrum are artefacts of the incorrect symmetry assignment.


Page 26 of 46

1 .0

(A )

E x p e r im e n t M P 2 (fc )/a u g -c c -p V T Z

0 .8

E ig e n ( C 3

)

a n h a r m o n ic fu n d a m e n ta ls a n h a r m o n ic o v e r to n e s a n h a r m o n ic c o m b in a tio n s

0 .6 0 .4 0 .2 0 .0 1 .0

(C )

E x p e r im e n t M P 2 (fc )/a u g -c c -p V T Z

0 .8

E ig e n ( C 1 )

a n h a r m o n ic fu n d a m e n ta ls a n h a r m o n ic o v e r to n e s a n h a r m o n ic c o m b in a tio n s

0 .6 0 .4

1 .0 0 .8 0 .6 0 .4 0 .2

1 .0

0 .0 2 0 .0 1 0 .0 0 1 .0 0 .8 0 .6 0 .4 0 .2

0 .2

0 .0 0 .0 0 .0 0 .0

0 .2 0 .0 5 0 0

1 0 0 0

1 5 0 0

2 0 0 0

2 5 0 0

(B )

0 .8

0 .1 5

0 .6

0 .1 0

0 .4

0 .0 1 .0

0 .0 5 0 .0 0 (D )

0 .8 0 .6

1 .5 1 .0


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60



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0 .4

0 .5 1 0 0 .2 0 5 0 .0 0 0 0 .0 3 5 0 0 3 6 0 0 3 7 0 0 3 8 0 0 3 9 0 0

1 5

3 0 0 0 W a v e n u m b e r(c m -1 )

Figure 7: Anharmonic VPT2 spectra (unscaled) at the MP2(fc)/aug-cc-pVTZ level for nearly identical Eigen isomers that were assigned different symmetries: (panels A and B) the correct high symmetry (C3 ) and (panel C and D) incorrect low (C1 ) symmetry. The anharmonic fundamental, overtone and combination bands are represented by blue, green and red bars, respectively. Intensities are normalized to the experimental 2665 cm−1 band. In the region below 3500 cm−1 , breaks are marked on the blue axes of the calculated anharmonic intensities in range of 0.03 to 0.1 for panel A and 0.02 to 0.1 for panel C, in order to show the weak overtones and combination bands. In the region above 3500 cm−1 , the calculated anharmonic intensities are zoomed in or out for easier comparison with the experiment (blue axes of panels C and D). The experimental IR predissociation spectrum 11 is depicted by the gray shaded area. The dashed line in panel D represents “infinite” intensity for that OH stretch.

Comparison with the VPT2 Spectra of Ref 17 Given our detailed VPT2 study (above), one wonders how Fournier et al. 17 concluded that “the observed spectral pattern is well reproduced by the anharmonic VPT2 calculated spectrum for the Eigen structure”, and why their Zundel-isomer spectrum looks so inappropriate (Figure S1 in their SI). We find three reasons for this. a) Use of a small basis-set, B3LYP/6-31+G(d), without dispersion correction. In our higher level computations with D3 correction, the IR spectrum of Eigen isomer (e.g., Figure 4) differs significantly from that of Fournier et al. The hydronium umbrella mode previously 17 seen around 1000 cm−1 is now the weak band 27 ACS Paragon Plus Environment


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at 977 cm−1 , which can no longer explain the strong 1050 cm−1 band. The water bend fundamental gained tremendously in intensity (it is now 100 times stronger than the 2665 cm−1 band). b) Use of C1 symmetry for the Eigen complex. As noted above, in Gaussian 09 it is not possible to calculate the DFT spectrum of the Eigen isomer with C3 symmetry. Lowering the symmetry to C1 results in significant enhancement of combination bands (Figure 7C). The combination bands previously assigned to the 1750 cm−1 and α bands, 17 supporting the Eigen assignment, might therefore be artefacts of the wrong (C1 ) symmetry assignment. c) Use of a different (C1 instead of C2 ) Zundel isomer. The Zundel isomer spectrum from Figure S1 of Fournier et al., 17 reproduced in Figure 8A, is very different from our trans-Zundel spectra reported above. For comparison, we have calculated it with the same B3LYP/6-31+G(d) method/bs as Fournier et al. have used, without and with dispersion correction (Figures 8B and C, respectively). Surprisingly, we obtained significantly different spectra from theirs, much closer to the trans-Zundel spectra reported above. Checking the Gaussian 09 outputs (Table 5), we find that geometry minimization in the hands of Fournier et al. produced C1 symmetry (i.e., no symmetry), with a distorted Zundel core (slightly elongated O–O distance, and unequal OH* distances). This contrasts with the C2 symmetry (with perfectly equal OH* distances) obtained in all of our trans-Zundel computations that are summarized in Table 5. Use of the C1 trans-Zundel isomer was justified (K.D. Jordan, personal communication) because it has slightly lower electronic energy. However, when ZPE is included, the C2 trans-Zundel isomer becomes about 1.3 kJ mol−1 more stable than the C1 isomer (see Table S19 of the SI, which also shows that inclusion of dispersion correction has a significant stabilization effect on the cluster). The structure seen


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Table 5: Comparison of the O-H∗ and O-O Distances (in ˚ A) in the Optimized Geometry of the C1 trans-Zundel Isomer of Ref 17 (Second Column) with Our C2 trans-Zundel Isomer Calculated with Different Methods. See Scheme 1 for Atom Labels. B3LYP 17

B3LYP

B3LYP-D3

bs3

bs3

bs3

bs1

bs2

bs1

bs2

bs1

bs2

bs1

bs2

C1

C2

C2

C2

C2

C2

C2

C2

C2

C2

C2

*

1.17401

1.20678

1.20627

1.19483

1.18950

1.19994

1.19523

1.19388

1.18899

1.20146

1.19630

O2 -H*

1.24459

1.20678

1.20627

1.19483

1.18950

1.19994

1.19523

1.19388

1.18899

1.20146

1.19630

O1 -O2

2.41728

2.41118

2.41007

2.38655

2.37647

2.39752

2.38869

2.38529

2.37612

2.39986

2.39009

trans-Zundel

O1 -H

MP2(fc)

B3LYP-D3

PBE0-D3

HCTH470

in the AIMD simulations averages to C2 symmetry: For the C1 isomer, one might expect the two OH* bonds to oscillate around 1.17 and 1.24 ˚ A (Table 5), whereas in the AIMD trajectory both bonds are on average equal, 1.21 ˚ A (Figure 6C of ref 18).

Discussion and Conclusions We start the discussion with an assessment of the quality of the computed results, then the likelihood of either isomer to dominate the spectrum. A) AIMD. The results from the AIMD calculations here and in ref 18 are consistent, and reproducible with respect to different functionals and basis sets. The commonly aired criticism, that AIMD does not include NQEs, is not an obstacle in the spectral assignment here. Most strongly affected are the OH stretching fundamentals at the higher frequencies (large ZPE). Their NQE-corrected frequencies can be gleaned from a VPT2 calculation with the same DFT functional and basis set. Thus the trans-Zundel HBed OH peak at 2997 cm−1 (DACF, Figure 2) red-shifts by 300 cm−1 (or more) with inclusion of NQE (VPT2, Figure 4C1). The HBed OH band of the Eigen isomer red-shifts less, so that both overlap with the 2665 cm−1 experimental peak. Consequently, the high frequency part of the spectrum (above 2000 cm−1 ) cannot be used to distinguish between the two isomers, e.g. by hole-burning experiments (Figure 6 of ref 17). 29 ACS Paragon Plus Environment



)

d a ta in F ig B 3 L Y P /6 -3 a n h a a n h a a n h a h a rm

. S 1 + rm rm rm o n

1 o f G (d o n ic o n ic o n ic ic fu

th e F o u r in e r e t a l. p a p e r ) fu n o v e c o m n d a



-3 a a a

G (d o n ic o n ic o n ic ic fu

)

1 .0

0 .5

0 .0

0 .0

d w (H O H )

m o d e 2 6

m o d e 2 5

E x p r im e n t

)

B 3 L Y P /6 a n h a n h a n h h a r

1 + rm rm rm m o n

fu n o v e c o m n d a



1 .0

0 .5

b

b

0 .5

2



ss

1 .0

4


+

O

a s

9


H

Z -c o re (H O H )

(B )

0 .0 )

B 3 L Y P a n a n a n h a

D 3 h a h a h a rm

/6 rm rm rm o n

3 1 o n o n o n ic

+ G (d ) ic fu n d a m ic o v e r to n ic c o m b in fu n d a m e n


1 .0

0 .5

b

Z -c o re (H O H )

b

0 .5

2



ss

4

+


O

d w (H O H )

m o d e 2 6

9

E x p r im e n t m o d e 2 5

1 .0

H

a s

(C )


0 .0


s

b

Z -c o re (H O H )

m o d e 1 9

0 .5

1


E x p r im e n t


4


+

O 9

m o d e 2 0

1 .0

H

s

(A )


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 30 of 46

0 .0

0 .0 5 0 0

1 0 0 0

1 5 0 0

2 0 0 0 2 5 0 0 W a v e n u m b e r ( c m -1 )

3 0 0 0

3 5 0 0

4 0 0 0

Figure 8: Anharmonic VPT2 spectra at the B3LYP/6-31+G(d) level for the transZundel isomer with (A) C1 symmetry and no dispersion correction, 17 and (B) C2 symmetry without dispersion correction (our work). (C) C2 symmetry with dispersion correction (our work). The experimental IR predissociation spectrum 11 is depicted by the gray shaded area. The anharmonic fundamental, overtone and combination bands are represented by blue, green and red bars, respectively, whereas harmonic fundamentals are depicted by dotted blue lines. Modes 19 and 20 in panel A are the rock mode of water molecules of the trans-Zundel-core and the OO stretch of the trans-Zundel isomer, respectively. We thank Kenneth D. Jordan for the Gaussian 09 output producing the spectrum in panel (A). Another criticism of the AIMD results is that the Zundel isomer shows three dangling OH peaks rather than the two observed experimentally. 17 Theoretically, there can be as many OH modes as OH bonds (6 for each isomer). How many are actually observed depends on their splitting (e.g., Table 4), which is below the computational accuracy, and the relative intensities, which are even more difficult to calculate accurately. Thus with the present level of theory it seems imprudent to base the isomer identification on such detail. The most serious drawback of the AIMD/DACF approach is the difficulty of obtaining overtone or combination bands. 50 While this explains why the AIMD/DACF


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calculation does not find the α band, it may also result in bias against the assignments of Fournier et al., which is based more extensively on combination bands. [However, this is not universally true, because our recent AIMD/DACF calculations for the protonated ammonia dimer revealed over two-dozen combination bands. 44 ] We thus turn to the VPT2 results for assessment of combination bands. B) VPT2. The VPT2 spectra are a cause of confusion for the protonated water tetramer. We have found that this is due to a combination of computational errors. 1. The difficulty for VPT2 to predict consistently the 4 fundamental modes of the Zundel isomer that involve the PTM (Figure 3). 2. Problems in symmetry assignment of the Eigen isomer by Gaussian 09, which possibly leads to artefacts in the anharmonic spectrum, particularly for combination bands (Figure 7). The suggested resolution of these problems is as follows: 1. Use the harmonic (rather than the anharmonic) output for the PTM-involving modes (22, 25, 26 and 27). 2. Rely preferably on the C3 Eigen isomer, which could be computed here only with MP2/bs1 (Figure 7A, B). With these precautions, the VPT2 fundamentals agree with the AIMD spectrum, in which the bands need to be red-shifted to account for the missing NQE. We are thus in position to reassess the likelihood of each isomer to dominate the experimental spectrum. C) Why not Eigen. For the Eigen isomer, VPT2 does not run into particular problems as far as the fundamental bands are considered, and we consider the predicted frequencies and intensities as rather reliable. In order to determine 31 ACS Paragon Plus Environment


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whether the H+ (H2 O)4 cluster is built around an Eigen or a Zundel core, one should firstly consider the IR modes of the core, whereas those of the solvating water molecules provide secondary evidence. The Eigen core is the hydronium, H3 O+ , that consists of 4 atoms namely, 12 modes. Three are arrested translations and three arrested rotations, all well below 1000 cm−1 , leaving 6 fundamental modes to focus upon. These are: the ss mode, the doubly degenerate as and b modes, and the U mode. The frequencies of these 6 modes for some of the quantum methods are reproduced in Table 6 (intensities and other DFT methods appear in the SI Excel file). The only intense band is the doubly degenerate as mode, in which the dipole moment changes perpendicular to the C3 axis. The other modes are at least a factor of 10 weaker (for all methods/bs). This agrees with the AIMD spectra, in which the ss and b modes are seen only in the partial VACF spectra (Figures S2A and S3B, respectively), but not in the DACF. With the absence of the ss band, the single sharp peak of the as mode is left to account for the rather wide experimental band. The U mode is seen in the DACF spectrum after a 103 magnification (Figure 2). Experimentally, the intensity of the 1050 and 1750 cm−1 bands is 40–60% of the main peak. Previous attempts were made to explain these two bands as originating from the Eigen isomer. The 1750 cm−1 band has been assigned to the Eigen b mode. 16 The VPT2 results are unanimous in showing that the b mode is not only 100 fold weaker than the main peak, it also red-shifts by anharmonicity close to the solvating water bending modes, at 1600 cm−1 or below (Table 6). This is nowhere near 1750 cm−1 . Subsequently this band was attributed to a combination band. 17 Figure 7C shows that seemingly promising combination bands arise when the cluster is assigned C1 symmetry, although these are still a factor of 100 too weak. Figure 7A shows that once the correct C3 symmetry is assigned to the Eigen cluster, all these combination


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Table 6: VPT2 Frequencies (in cm−1 ) of the Six Fundamentals of the Hydronium Core of the Eigen cation.a MP2(fc)/bs1b MP2(fc)/bs2b B3LYP-D3/bs1c B3LYP-D3/bs2c B3LYP-D3d ss 1

2712

2811

2727

2660

2771

as 2

2680/2633

2833/2751

2621/2603

2590/2578

2654/2653

b 2

1601/1599

1597/1580

1541/1533

1593/1592

1561/1561

U 1

928

957

977

991

979

a

From Excel file in SI; b C3 symmetry assignment; bs= 6-311++G(3df,2p), superfine grid.

c

C1 symmetry assignment;

d

bands disappear. The experimental 1050 cm−1 band has been assigned to the Eigen U mode, based on lower level VPT2 calculations. 17 Table 6 shows that with the larger bs utilized herein, its anharmonic frequency further red-shifts to below 1000 cm−1 , and when MP2/bs1 replaces B3LYP, it further downshifts to below 950 cm−1 . This brings the U mode close to its experimental value for gas-phase H3 O+ , 954.4 cm−1 (see NIST WebBook, http://webbook.nist.gov/cgi/cbook.cgi?ID=C13968086&Mask=800). D) Why Zundel. With the abovementioned precautions, VPT2 and AIMD agree on the spectrum of the trans-Zundel isomer that exhibits fundamentals explaining the experimental 1050 and 1750 cm−1 bands, and also quite a number of combination bands (near 1847, 1904, 2245 and 3000 cm−1 ) that might explain some weaker features in the spectrum. The bands at 1050 and 1750 cm−1 can now be assigned. They represent a (nearly pure) PTM (mode 25) and PTM-coupled water bend (mode 22), respectively. Indeed, these two characteristic peaks appear in the spectra of the H5 O+ 2 and H+ (H2 O)6 clusters, 37,65,66 and could only be explained by the shared proton of Zundel-like structures both in experiment and theory. 9,18,37 Interestingly, they seem to be only minorly affected by NQE and anharmonicity, as verified by com-



parison to the analogous peaks of the bare-Zundel cation (Table 3). Most telling is the 1750 cm−1 band (mode 22), justly branded “a unique Zundel peak”. 14 Figure 9 highlights the similarities between the trans- and bare-Zundel isomers, both having the same frequency to within 15 cm−1 , with the harmonic value slightly higher than the experimental, yet somewhat closer to it than the predictions of VPT2 or high level quantum nuclear dynamics simulations. The robustness of this peak is further confirmed from its identification in the computed IR spectrum of acidic liquid water. 36

1 8 3 6

1 8 4 0

V P T h a rm e x p e M C T D M C

1 8 0 0

-1

)

1 8 2 0


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

2 o n ic r im e n ta l D H

1 7 8 0

1 7 7 0

1 7 7 0 1 7 6 3

1 7 6 3

1 7 6 0

1 7 5 0 1 7 4 1

1 7 4 0

1 7 2 8 1 7 2 0 1 7 0 0

tra n s -Z u n d e l

b a re -Z u n d e l

Figure 9: Comparison of the frequencies of the PTM-coupled water bending mode in the trans-Zundel isomer of H+ (H2 O)4 and in H5 O+ 2 . Data from Table 3. This robustness might be traced to the involvement of water bending in this mode. In gas-phase H2 O, the harmonic bending frequency at the MP2/aug-ccpVTZ level, 1628 cm−1 , downshifts by 50 cm−1 (to 1578 cm−1 ) upon inclusion of anharmonicity. 67 Compared with the experimental frequency of 1595 cm−1 , the error in using the harmonic frequency would be only 33 cm−1 . For the flanking water molecules in the trans-Zundel isomer, these frequencies at the same quantum34 ACS Paragon Plus Environment

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chemistry level are 1643 cm−1 (harmonic) and 1598 cm−1 (anharmonic), see the Excel file in the SI. Although these water molecules are HBed now, their bending frequencies are similar to those of gas-phase water, attesting to the bending mode insensitivity to the environment. Compared with an experimental frequency of 1615 cm−1 , the error in using the harmonic frequency would be only 28 cm−1 . The small anharmonic corrections to the water bending mode may explain its robustness. It stands to reason that the 1750 cm−1 Zundel band has inherited its robustness from the water bending mode, and its strong intensity from the PTM. This unique peak cannot be overlooked when assigning the spectrum. In conclusion, we have used state-of-the-art methods with which it is currently possible to compute the IR spectra of the clusters under consideration while taking into account anharminicity and NQE. Yet, these methods are not free of errors, some well-known and others exposed in the present work. This makes an unequivocal IR band assignment a challenging task. Certainly, conclusions in this case cannot be drawn based on a single computation. The detailed understanding of the various error sources gleaned from multiple classical and quantum mechanical computations allows one to reach a tentative conclusion. The theoretical analysis points to the involvement of the Zundel isomer in the observed spectrum, either alone or in a mixture with the Eigen isomer. The latter possibility should be further clarified by extending the hole-burning IR-IR double resonance experiments to frequencies below 2000 cm−1 , because we have shown that above 2000 cm−1 the spectra of the Eigen and Zundel isomers are nearly indistinguishable. The possibility that the second most stable isomer makes the dominant contribution to the low-temperature spectrum is, in retrospect, not that surprising because (see Introduction) similar behavior was also seen for the Na+ (H2 O)4 cluster, implying that the isomeric distribution in the low-temperature molecular beam is not at



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equilibrium.

Supporting Information Available PDF file with full citation of Gaussian 09; Communication from the Gaussian Support Team; Starting times of AIMD trajectories; Optimized geometries of the three H9 O+ 4 isomers; AIMD and VPT2 fundamental frequencies and assignments; Energies of C1 vs. C2 trans-Zundel isomers; Figures of partial VACF spectra of the Eigen and trans-Zundel isomers stretching and bending modes; VPT2 spectra of H9 O+ 4 isomers and their frequencies for different quantum chemistry methods; Excel file with harmonic and anharmonic frequencies of the three H9 O+ 4 isomers from 30 different VPT2 calculations, and 2 additional ones for H5 O+ 2.

This

material is available free of charge via the Internet at http://pubs.acs.org/. Acknowledgments. We thank Mark A. Johnson for the data displayed in Figure 5 of ref 11, Kenneth D. Jordan for the Gaussian 09 output producing Figure S1 in ref 17, and Fernando R. Clemente and Douglas J. Fox (Gaussian 09 technical support) for helpful discussions of anharmonic frequency calculations for symmetric molecules in VPT2. This research was supported by the Israel Science Foundation (grant number 766/12). The Fritz Haber Center is supported by the Minerva Gesellschaft f¨ ur die Forschung, M¨ unchen, FRG.

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Graphical TOC Entry E x p e r im e n ta l a n d C a lc u la te d In te n s itie s ( a .u .)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

E x p e r i m e n t a l , H 9 O +4 a n h a r m o n ic fu n d a m a n h a r m o n ic o v e r to n a n h a r m o n ic c o m b in h a r m o n ic fu n d a m e n

1 .8 1 .6 1 .4 1 .2


1 .0 0 .8 0 .6 0 .4 0 .2 0 .0 5 0 0

1 0 0 0 W

1 5 0 0 2 0 0 0 2 5 0 0 3 0 0 0 a v e n u m b e r ( c m -1 )

3 5 0 0


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Reinvestigation of the Infrared Spectrum of the Gas-Phase Protonated

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