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Complete Assignment of the Infrared Spectrum of the Gas-Phase Protonated Ammonia Dimer Huan Wang, and Noam Agmon J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.5b11062 • Publication Date (Web): 28 Apr 2016 Downloaded from http://pubs.acs.org on May 1, 2016

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The Journal of Physical Chemistry

Complete Assignment of the Infrared Spectrum of the Gas-Phase Protonated Ammonia Dimer 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]

∗

To whom correspondence should be addressed

1 ACS Paragon Plus Environment


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Abstract The infrared (IR) spectrum of the ammoniated ammonium dimer is more complex than those of the larger protonated ammonia clusters due to closelying fundamental and combination bands and possible Fermi resonances (FR). To-date, the only theoretical analysis involved partial dimensionality quantum nuclear dynamic simulations, assuming a symmetric structure (D3d ) with the proton midway between the two nitrogen atoms. Here we report an extensive study of the less symmetric (C3v ) dimer, utilizing both 2nd order variational perturbation theory (VPT2) and ab initio molecular dynamics (AIMD), from which we calculated the Fourier transform (FT) of the dipole-moment autocorrelation function (DACF). The resultant IR spectrum was assigned using FTed velocity autocorrelation functions (VACFs) of several interatomic distances and angles. At 50 K, we have been able to assign all 21 AIMD fundamentals, in reasonable agreement with MP2-based VPT2, about 30 AIMD combination bands and a difference band. The combinations involve a wag or the NN stretch as one of the components, and appear to follow symmetry selection rules. Based on this, we suggest possible assignments of the experimental spectrum. The VACF-analysis revealed two possible FR bands, one which is the strongest peak in the computed spectrum. Raising the temperature to 180 K eliminated the “proton transfer mode” (PTM) fundamental, and reduced the number of observed combination bands and FRs. With increasing temperature, fundamentals red-shift, and the doubly degenerate wags exhibit larger anharmonic splittings in their VACF bending spectra. We have repeated the analysis for the H3 ND+ NH3 isotopologue, finding that it has a simplified spectrum, with all the strong peaks being fundamentals. Experimental study of this isotopologue may thus provide a good starting point for disentangling the N2 H+ 7 spectrum.


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Keywords: ammonia, anharmonicity, combination bands, dimer, IR, proton



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Introduction 1 The protonated water dimer, H5 O+ 2 , sometimes known as the “Zundel cation”,

plays an important role in the Grotthuss mechanism of proton mobility in liquid water. 2 Therefore its gas-phase infrared (IR) spectrum has been well studied both experimentally 3–7 and theoretically. 8–10 This cation might be more important in small protonated water clusters than previously anticipated, if indeed it contributes appreciably to the IR signal from the gas-phase protonated water tetramer, 11 pentamer, 12 and the benzene-attached dimer. 13 It may also dominate the signal from protonated water in liquid acetonitrile, 14 and even in bulk water. 15 The spectral signatures of the excess (shared) proton, H∗ , in H2 O · · · H∗+ · · · OH2 , appear below 2000 cm−1 . The “pure” antisymmetric H∗ stretching mode, so-called “proton transfer mode” (PTM), absorbs at 1047 cm−1 , but the “water bend” at 1763 cm−1 is also strongly coupled to the H∗ stretch. 16 The water-water OO stretch (550 cm−1 ) is a totally symmetric mode that is IR inactive, but it lends intensity to a progression of overtones and combination bands. These bands have been predicted by full dimensional quantum calculations, in which the motions of all nuclei are treated quantum mechanically, using the time-dependent multi-configuration Hartree (MCTDH) method. 10 The protonated ammonia dimer, H3 N · · · H∗+ · · · NH3 , is isoelectronic with the Zundel cation. It may play a role in proton conduction along ammonia wires, such as observed in the excited-state of 7-hydroxyquinaline in the gas-phase. 17,18 Yet, there is a clear distinction between these two ions. In H5 O+ 2 , the OO distance is small (ca. 2.4 ˚ A), so that the excess proton resides in a single-minimum well ˚ between the two oxygen atoms. In N2 H+ 7 , the NN distance is larger (ca. 2.7 A), so that H∗ is located in a double-well potential. Thus, classically this cation may appear as composed of two moieties, ammonia (NH3 ) and ammonium (NH+ 4 ). This asymmetry leads to C3v symmetry. Quantum mechanically, if the zero point energy 4 ACS Paragon Plus Environment

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(ZPE) is above the central barrier, the proton may appear as located symmetrically between the two nitrogen atoms, resulting in D3d symmetry. In contrast to the Zundel cation, there has been relatively little work on its protonated ammonia analogue. The older experiments concentrated on the high frequency range, 19,20 showing no absorption in the 2000–3200 cm−1 spectral window, and then a broad absorption around 3400 cm−1 due to the various NH stretches of the peripheral H atoms. Previous theoretical work used harmonic normal modes (HNM’s), 21,22 dipolemoment autocorrelation function (DACF) from “ab initio” molecular dynamics (AIMD) trajectories, albeit with band assignments from HNM, 23 and a one dimensional (1D) quantum model. 24 The various methods vary appreciably in the location of the PTM. Its harmonic frequency at the MP2(full)/6-311++G(d,p) level is 1950 cm−1 , see e.g. Table S3 of the Supporting Information (SI). Ref. 23 identified it at 1610 cm−1 , whereas the 1D quantum model 24 gave a considerably red shifted PTM at 707 cm−1 . Significant progress on NH+ 4 (NH3 )n clusters was achieved recently by Tono et al. (n = 3 − 8) 25,26 and Asmis and coworkers (mainly n = 1, also n = 2 − 4). 27,28 Measurements were extended to lower frequencies, below 1700 cm−1 , using photodissociation with the free electron laser, 25,26 and IR multiphoton dissociation (MPD), a non-linear technique with enhanced signal from overtones and combination bands. 27,28 For n = 1, which is the focus of the present work, also IR vibrational predissociation (VPD) spectra were measured, involving linear absorption of the (Ar tagged) complex. 27 The measurements have been extended down to 330 cm−1 , revealing the PTM at a surprisingly low frequency (374 cm−1 ), 28 considerably lower than any previous estimate. In addition, restricted dimensionality (D) quantum calculations were performed in 4D 27 and 6D. 28,29 The PTM was found to shift all the way to 471 cm−1 by the



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6D model, 28 close to the experimental value. In addition, absorption bands for the umbrella mode and NH∗ N bending were assigned, while other fundamental modes remain unassigned. Due to the large anharmonicity, the coupling of the (asymmetric) PTM with the (symmetric) NN stretch is strong, so that a progression of 3 combination bands (PTM+NN, PTM+2NN and PTM+3NN) was identified, 28 similarly to the PTM+OO and PTM+2OO combinations identified for the Zundel cation. 10 It was maintained that quantum mechanically the protonated ammonia dimer has a symmetric D3d structure, 27,28 because the ZPE of the PTM is above the barrier separating the two minima. However, this was deduced from the earlier 1D theoretical model that found the PTM at 707 cm−1 , 24 in comparison to computations that found a central barrier between 1–3 kcal/mol. 21,24,30 Given the recent small experimental value for the PTM frequency, 28 of 374 cm−1 , its ZPE is actually around 0.5 kcal/mol, which is smaller than even the lowest theoretical estimate for the central barrier height. Thus there is place for recalculating the spectrum of the dimer in C3v symmetry. Several factors determine the accuracy of the computed IR spectra of floppy systems such as protonated water and ammonia clusters. • The accuracy of the potential energy surface. • Inclusion of its anharmonicity in the calculations. • Quantum nuclear effects. Generally speaking, empirical potential energy surfaces are not of “spectroscopic quality”, and one requires interatomic forces obtained from solutions of the Schrödinger equation. These can be computed on various levels of approximation. In particular, one may consider using either Density Functional Theory (DFT), with accuracy that greatly depends on an empirical functional, or a wavefunction-based approach, 6 ACS Paragon Plus Environment

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such as the 2nd-order Møller-Plesset perturbation theory, whose accuracy can be systematically enhanced by changing the quantum method and/or increasing the basis-set. Here we present results from (a) Classical AIMD simulations for “on the fly” DFT-based potential energy surfaces using the CP2K/Quickstep software package, 31 and (b) 2nd order vibrational perturbation theory (VPT2). 32 These two methods are, to an extent, complementary. AIMD is dynamic, VPT2 is static. AIMD trajectories sample the anharmonic regions of the potential more realistically, 33,34 up to an energy dictated by the chosen temperature. VPT2 introduces anharmonicities through a 4th-order expansion of the potential near its local minima at 0 K, which can be a good approximation for certain normal modes but poorer for others. MCTH seems to offer “the best of both worlds”, but scales stiffly with the 10 number of particles. Thus full dimensional MCTH is doable for H5 O+ but not 2,

for N2 H+ 7 . The approximation then introduced is in freezing some of the degrees of freedom and performing partial dimensional calculations. 27,28 A complete identification of all normal modes is consequently not possible. Some of the fundamentals and numerous combination bands are not included in the analysis. Here we strive to obtain a complete assignment of the N2 H+ 7 IR bands by a judicious combination of the AIMD and VPT2 methods. The analysis is “complete” in the sense of including all the molecular degrees of freedom. These give rise to all of the 21 fundamental modes. Particularly intriguing is the identification of the overtones and combination bands. VPT2 will compute over 200 frequencies and intensities, some more reliably than others. The intensities, in particular, can sometimes show large errors. The AIMD results can help in selecting the experimentally relevant combination bands.



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While the calculation of IR spectra from AIMD simulations has progressed in recent years, 33,34 it was noted that “mode coupling leading to combination bands and Fermi resonances are still not included” in this approach. 34 This has generally been our impression from applying this methodology to protonated water clusters. 11,12 It is thus surprising to find that, when implementing methods for noise-reduction in conjunction with partial velocity autocorrelation functions (VACFs), one can identify numerous combination bands (and even FRs) in the AIMD spectra. Comparison with the VPT2 results allows also for a complete, if tentative, assignment of fundamentals, overtones, combination bands and FRs in the experimental spectrum. We then proceed beyond the available experimental data, in investigating temperature and isotope effects on the spectrum. Interestingly, we find that elevating the temperature may eliminate the PTM fundamental, as previously suggested for the Zundel cation. 16 We also find that the AIMD spectrum for the H3 ND+ NH3 isotopologue has quite similar frequencies to the N2 H+ 7 cluster, but different intensities. Now all the strong peaks are fundamentals, and this could greatly assist in analyzing an experimental spectrum.

Methods 35 The geometry of the N2 H+ with 7 cluster was first optimized in Gaussian 09,

the MP2(full) method combined with the 6-311++G(d,p) or aug-cc-pVTZ basis sets. The static harmonic and anharmonic spectra were then calculated at each optimized geometry. Anharmonic frequency analysis was based on the VPT2 frequencies, which were used as is (not scaled). We have repeated this procedure using DFT with the B3LYP-D3, 36,37 B2PLYP-D3, 38 M06-2X 39 or PBE0-D3 40 functionals (where D3 implies dispersion correction), and either one of the above basis


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sets. Because we assess that MP2(full)/6-311++G(d,p) is superior over the other methods, we use this dataset in the sequel, but the other results can be found in Table S1 and the Excel file deposited as Supporting Information (SI). For comparison, the geometry of the optimized structure of N2 H+ 7 in the D 3d symmetry was determined by the same methods (Figure S1 and Table S2 of the SI). Since it was treated as a transition state, only anharmonic frequencies are available, but not anharmonic intensities (see Excel file in the SI.) In addition to the static anharmonic IR spectrum, we have also calculated the anharmonic spectrum from molecular dynamics using AIMD. In classical AIMD simulations one solves at each timestep the full Schrödinger equation for the electrons, usually using DFT with an empirical density functional. The interatomic forces are then computed from the electron density using the Hellmann-Feynman theorem, and the position and velocity of the atoms are propagated classically, by solving Newton’s equations in this force field. These AIMD computations were performed in the CP2K/Quickstep software package, 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 B3LYP functional with Grimme’s third generation dispersion correction (DFT-D3) was used. 41 This makes the computations considerably more costly than with, e.g., the BLYP functional. The augmented triple-ζ valence polarization (aug-TZV2P) basis set with the corresponding Goedecker-Teter-Hutter (GTH) pseudopotentials 42 were used for all atoms. The plane wave energy cutoff was set to 300 Ry. Self-interaction correction (SIC) was applied with the Martyna-Tuckerman Poisson equation solver. 43 The orbital transformation (OT) method 44 was applied for faster convergence, with the tight convergence criterion of 1 × 10−7 a.u. at every MD step. The initial coordinates for the simulations were taken from the Gaussian 09



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optimizations. Two independent trajectories, with optimized geometries from the MP2(full)/6-311++G(d,p) and B3LYP-D3/aug-cc-pVTZ calculations, were equilibrated in CP2K for 7 to 15 ps in the canonical (NVT) ensemble. The target temperature was maintained by the Nosé-Hoover chain thermostat at 50 K, which is the temperature in the experiments. More precisely, the clusters in the experiment are in contact with a 20 K helium cryostat, so that the temperature may be as low as 20 K. 28 Experience from DFT simulations suggests that they should be conducted at temperatures 30–40 degrees higher, which better correspond to the experimental ones. A similar approach was taken in a recent simulation of the protonated water hexamer, 45 where experiments were at 15 K and the simulations were conducted at 50 K. To better sample the initial conditions, three production runs were performed. Two started from the coordinates and velocities obtained from the first NVT equilibration trajectory at 7.5 and 14.5 ps, respectively. The third started from the second equilibration trajectory at 7.75 ps. Each of the three trajectories was continued for 20 ps in the microcanonical (NVE) ensemble. To check the temperature effects on the spectrum, we have conducted a fourth 20 ps NVE trajectory at 180 K. A fifth, 20 ps trajectory was run for the H3 ND+ NH3 isotopologue at 50 K. The time step for our simulations was ∆t = 0.5 fs, which is standard in AIMD simulations. 45 For too large timesteps the trajectory can show instabilities and drift in energy/temperature. Figure S8 in the SI shows that such instabilities do not appear in our computations. In addition, large ∆t may cause blue shift in vibrational frequencies, which for ∆t ≤ 0.5 fs are well converged (see Fig. 2 in Ref. 46). Occasionally, BLYP simulations are conducted with smaller time-steps. 13 But B3LYP simulations are about an order of magnitude more costly in computer time, so that smaller than necessary timesteps will unavoidably result in shorter trajectories and poorer statistics. Coordinates and velocities from each production


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run were saved every time step to ensure that the spectra at high frequencies do not get distorted. The (angular) frequency (ω) dependence of the IR absorption coefficient, αµ (ω), was computed from the autocorrelation function (ACF) of the system’s total dipole moment (µ): 34,47–49 n ω ω[1 − exp(−β~ω)] · αµ (ω) ∝ 1 − exp(−β~ω) Z ∞ 2 dt exp(−iωt)hµ(0) · µ(t)i = ω

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, 50 and β is 1/(kB T ). The time (t) dependent dipole moment ACF, hµ(0) · µ(t)i, is a convolution whose FT is therefore a products of FT’s, giving rise to the third line, in which µj is the j’th Cartesian component of the dipole moment. Integration by parts leads to the 4’th line, which is the FT of the ACF of the time derivative of the dipole moment. For calculating αµ (ω) one can thus use either the dipole moment itself or (more commonly) its derivative. ˙ In the procedure applied here, µ(t) was first transformed into the frequency domain using discrete fast Fourier transform (FFT), then multiplied by its complex conjugate, and transformed back into the time domain using inverse FFT (iFFT) to obtain the DACF:

˙ F(ω) = FFT[µ(t)] DACF(t) = iFFT[F(ω) · F ∗ (ω)]



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Next, the Blackman-Harris window function was applied to suppress noise in the DACF, and the result was again Fourier transformed to produce the IR spectrum. DACF spectra obtained with different window functions are demonstrated in Figure S2 of the SI. The vibrational frequencies from the AIMD runs were corrected by a scaling factor of 0.968 that is recommended for the B3LYP functional. 51 Normally, the scaling factor is used to correct HNMs for anharmonicity. Here we use the same factor to correct for the lack of nuclear quantum effects in AIMD trajectories, finding that this brings the AIMD/DACF frequencies to close agreement with the VPT2 results. At T = 50 K we have utilized 3 trajectories starting from three different initial states. These finite trajectories sample different regions of the multi-dimensional potential energy surface, and as a result the generated spectrum for each trajectory is somewhat different (see Figure S16 in the SI). Their comparison reveals intensity differences (so that an IR band can appear in one trajectory but not in another) and likely also frequency shifts. This limits the accuracy of the method. The final spectrum was obtained by averaging over the three different trajectories. In order to identify the AIMD absorption bands, partial VACFs were calculated e.g., for some of the stretching and bending modes. 13 Thus if ζ(t) is a coordinate of interest (atomic position, bond distance, bond angle, etc.), its contribution to the spectrum can be assessed from the FT of its VACF: 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. Unlike the DACF spectrum, which is subject to the IR selection rules, there are no selection rules for VACF spectra. For example, the NN stretch in N2 H+ 7 is not


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IR active: due to symmetry it does not involve a change in µ(t). Its frequency can, nevertheless, be determined from the NN-stretch VACF.

Results Ammonia and Ammonium + Unlike H5 O+ 2 , the N2 H7 cluster (with classical nuclei) is asymmetric, with a shared

proton that is not midway between the two N atoms. While larger protonated ammonia clusters can definitely be viewed as composed of ammonia plus ammonium, 25,26 the dimer, though closer to being symmetric, may still be described this way. 20 Therefore, we begin from these two molecules. The optimized structures in Gaussian 09 using either MP2 or DFT with the B3LYP-D3 functional are shown in Figure 1. It can be seen that the two methods yield essentially identical results. As expected, ammonium is tetrahedral, while ammonia is pyramidal, with HNH angles that are slightly smaller than the tetrahedral angle of 109.5◦ . In the Valence-Shell Electron-Pair Repulsion (VSEPR) model, 52 this is attributed to the extra repulsion between the lone- and bondingelectron pairs (compared with the smaller repulsion between two bonding pairs).

Figure 1: The optimized geometries of ammonia (left) and the ammonium ion (right), calculated in Gaussian 09 at the B3LYP-D3/aug-cc-pVTZ (MP2(full)/6311++G(d,p)) levels. Diatomic distance (in angstroms) and triatomic angles (in degrees) are marked on the structures. 13 ACS Paragon Plus Environment


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Table 1 compares the experimental IR frequencies 20 of ammonia and ammonium with our two anharmonic calculations, using VPT2 and AIMD, both with the B3LYP-D3 functional. The complete VPT2 output, using both DFT/B3LYPD3 and MP2, is given in the Excel file of the SI, while the computed AIMD/VACF spectra are shown in Figure S3 of the SI. a) Ammonia has N = 4 atoms and hence 3N − 6 = 6 vibrational modes. In the high frequency end there are 3 NH stretching modes (same as the number of NH bonds). These are divided into one (IR active) symmetric stretch (ss) and two degenerate asymmetric stretches (as). In the middle of the frequency scale there are two degenerate HNH bending modes (b), at a similar frequency to the HOH bending mode of water. At the low end of the spectrum we find the celebrated umbrella mode (U ). Table 1 shows that the VPT2 and AIMD spectra are in good agreement with experiment, with a mean absolute deviation (MAD) of 23 and 32 cm−1 , respectively. These results are substantially better than the harmonic frequencies (MAD = 96 cm−1 ). b) Ammonium has N = 5 atoms and hence 3N − 6 = 9 vibrational modes. In the high frequency end there are 4 NH stretching modes (same as the number of NH bonds). These are divided into one (non-IR active) ss and three degenerate as modes, at lower frequencies than the corresponding NH3 bands. (Apparently the 4’th NH bond utilizes not only the nitrogen lone-pair electron density, but also borrows some electron density from the other three NH bonds). In the middle of the frequency scale there are two degenerate HNH bending modes (not-IR active), at a higher frequency than in ammonia. Somewhat lower in frequency we find three degenerate umbrella modes. They absorb at a frequency that is substantially higher than the U (NH3 ) mode. Interestingly, the harmonic frequencies (MAD = 70 cm−1 ) and AIMD frequencies (MAD = 15 cm−1 ) are in even better agreement with experiment than for


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Table 1: Vibrational modes and frequencies of gas-phase ammonia and ammonium ion, experiment 20 vs. calculations (this work). Static harmonic and VPT2 anharmonic infrared frequencies are obtained at the B3LYP-D3/aug-cc-pVTZ level. Dynamic infrared frequencies are obtained from AIMD/VACF with the B3LYP-D3 functional combined with aug-TZV2P basis set. NH+ 4

NH3 mode ν1 ν2 ν3 ν4 MAD

Exp. Har. 3336 3468 950 1024 3444 3588 1627 1663 0 96

freq. assn. VPT2a AIMDb 3322 3368 ss 923 997 U 3411 3482 as 1611 1616 b 23 32

deg. 1 1 2 2

Exp. 3270 1699 3343 1447 0

Har. 3365 1722 3466 1484 70

freq. VPT2c – – – –

AIMDb 3266 1675 3364 1437 15

freq. = frequency (cm−1 ), assn. = assignment, deg. = degeneracy, Exp. = Experimental, Har. = Harmonic, MAD = Mean Absolute Deviation (cm−1 ). ss = symmetric stretch, as = asymmetric stretch, b = bend, U = umbrella. a as and b frequencies averaged over the 2 modes. b VACF spectra, in which all modes are active, with frequencies scaled by a factor of 0.968. c No anharmonic analysis for spherical tops.

ammonia. The maximal deviation between AIMD and experiment (24 cm−1 ) occurs for the b(NH+ 4 ) mode. If we make the simplified assumption that the maximal error in frequency of any dimer mode can be as large as the sum of the maximal errors in the monomers, we expect a maximal absolute deviation of ca. 70 cm−1 , which may be nearly sufficiently accurate for identifying the dimer modes. Consider forming N2 H+ 7 by hydrogen bonding one of the ammonium hydrogens (subsequently denoted by H∗ ) to the ammonia nitrogen: H3 NH∗+ · · · NH3 . In this process, the abovementioned modes are conserved, with the following changes: • One of the 3 as(NH+ 4 ) modes becomes the PTM. • One of the 3 NH+ 4 umbrella modes, whose axis is along the NN line, becomes U (NH+ 4 ), while the other two that have their main axis along ammonium NH bonds become “perpendicular umbrella modes” (U⊥ ). They include a 15 ACS Paragon Plus Environment

assn.

deg.

ss b as U

1 2 3 3


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considerable component of H∗ motion perpendicular to the NN axis. There are 6 additional low-frequency vibrational modes replacing the 3 translational and 3 rotational modes of one of the two fragments. These include two degenerate gerade wags (wg ), two degenerate ungerade wags (wu ), the NN symmetric stretch and an internal rotation mode (i-rot). These expected fundamental modes of the protonated ammonia dimer are listed in Table 2. In the experimentally investigated spectral range of, say, 300 − 1700 cm−1 , one would not observe the 6 NH stretches (that occur at higher frequencies), the IR-inactive NN stretch or the i-rot mode. This leaves 13 modes with 8 different fundamental frequencies that should be manifested in the experimental spectrum. Out of these, Asmis and collaborators have identified one U mode in their first paper. 27 In a subsequent work, they have identified the PTM and the (doubly degenerate) NH∗ N bend, 28 which corresponds to U⊥ (NH+ 4 ) in our notation. They have also suggested that wu might be identified with a feature at 409 cm−1 . This leaves the two bending modes, wg , and possibly one U mode unidentified.

Dimer Geometry The optimized structure of the dimer is depicted in Figure 2, with the indicated bond lengths and bond angles from 10 different quantum chemistry methods collected in Table S1 of the SI. These include MP2(full) and DFT with 4 different functionals, for 2 different basis sets each. It can be seen that in the dimer the shared proton, H∗ , is withdrawn (by 0.1 ˚ A) from the ammonium towards the ammonia moiety, though it is still far from being midway between the two N atoms. With it, one expects the NH bond distances and HNH bond angles in the ammonium moiety to shift closer to their values in the ammonia moiety, and vice versa. A comparison with Figure 1 shows that this is generally so, expect for the ammonia HNH angles that instead of increasing from around 107◦ back toward 16 ACS Paragon Plus Environment

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Table 2: Expected peaks in the IR spectrum of protonated ammonia dimer assigned to fundamental vibrational modes of the two monomeric fragments. Modes between the dashed lines are expected to be observed in the measurements of Asmis and collaborators. mode as(NH3 ) as(NH+ 4) ss(NH3 ) ss(NH+ 4) b(NH+ 4) b(NH3 ) U⊥ (NH+ 4) aU ≡ U (NH+ 4) sU ≡ U (NH3 ) wg wu PTM NN i-rot Total

deg. 2 2 1 1 2 2 2 1 1 2 2 1 1 1 21

109.5◦ , further shrink in the dimer to around 106◦ . Table S1 shows that the optimized bond distances and angles are quite similar for the different quantum chemistry calculations. For example, the NN distance (in C3v symmetry) is slightly under 2.700 ˚ A for all these methods. For comparison with AIMD, we have averaged the NN distances in 3 trajectories (Figure S4), finding a value of 2.698 ˚ A. Thus our dynamic and static geometries agree quite well. For D3d symmetry we find an even smaller value, of 2.606 ˚ A (see Table S2 of the SI). In contrast, the NN distance in the 6D model of Ref. 28 (D3d symmetry) is notably larger, 2.83 ˚ A, possibly because some of the coordinates were frozen there, whereas we have performed a full optimization. + The NN distance in N2 H+ 7 is substantially larger than the OO distance in H5 O2 ,

which is under 2.4 ˚ A. This distance between the two heavy atoms in the dimer correlates with the barrier for proton transfer between them, which is zero for the Zundel cation and a few kcal/mol for the protonated ammonia dimer. 30 17 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

Figure 2: The geometry of the protonated ammonia dimer cluster in C3v symmetry. The marked distances and angles, for the various quantum chemistry methods and basis sets, are collected in Table S1 of the SI. H-bond distances are depicted by dashed gray lines, and the shared proton denoted by H∗ .

Dimer Fundamental Bands The HNM and anharmonic (VPT2) frequencies and intensities (computed at the MP2 level, see Methods) for the protonated ammonia dimer are listed in Table S3 of the SI, and depicted graphically in Figure S5 there. An animation of the 21 normal modes is provided in Figure S6 of the SI. By consulting these animations, one gets a feeling as to the degree of coupling between the two monomers within the dimer. The high frequency modes correspond to fast timescales during which the two parts of the cluster cannot communicate. These modes are localized on the two monomers. In contrast, the low frequency modes allow ample time for inter-monomer energy flow, so these modes are delocalized leading, e.g. to the gerade and ungerade wags. The borderline is roughly at the umbrella modes, hence Table 2 retains both notations for these modes. The “static” spectra will be now compared with the dynamic (AIMD) DACF and VACF spectra.


Page 18 of 61

Page 19 of 61

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High frequency stretching modes. Let us begin with the high frequency end (not probed by the Asmis experiments), which corresponds to the free NH stretching modes. Figure 3 shows the averaged DACF from the three AIMD trajectories, decomposed into contributions from the ammonia and ammonium sides using partial VACFs. In this methodology, one computes separately the averaged FT of the VACFs for the NH distances on each of the nitrogen atoms. Just as for the isolated fragments (Table 1), the as and ss frequencies on the ammonia side are higher than the corresponding ones on the ammonium side. However, this difference is much lower than in the free fragments (about 20 cm−1 ). Upon forming the dimer, the ammonia frequencies go down, and the ammonium ones go up, until they become nearly equal. This behavior is summarized in Table 3. It is also seen in Table 3 that the VPT2 frequencies are within ca. 10 cm−1 of the (scaled) AIMD frequencies. Such a close agreement, if maintained in the other regions of the spectrum, will help out in the assignments. Concerning comparison −1 with experiment, we note that the as(NH+ 4 ) band, at 3417 cm , is in excellent

agreement with the older measurements of Schwarz (3420 cm−1 ), 19 though somewhat newer results from Y. T. Lee’s lab 20 find its center at 3397 cm−1 . Table 3: Ammonia and ammonium stretching vibration frequencies (in cm−1 ) as monomers (top) and in the dimer (bottom), determined from our AIMD/B3LYP-D3 simulations. Values in parentheses are from a VPT2 analysis of the MP2 quantum chemistry calculations. mode ss as ss as

NH3 NH+ 4 Monomers 3368 3266 3482 3364 Dimer 3345 3327 (3334) (3316) 3437 3417 (3443) (3427)



3 4 2 0

D A C F a v e r a g e o f tr a j. 1 , 2 , 3

4

) 3 4 0 3

3 3 2 7

0 .5

+

3 4 5 7

2 b (N H

3 4 3 3

(A ) 1 .0

0 .0

(B )

V A C F a v e r a g e o f tr a j. 1 , 2 , 3 N 1 H s tr e tc h ( N H 3 m o ie ty )

1 .0

ss(N H 3

3 4 3 7

3 3 4 5

)

a s(N H 3

)

0 .5

0 .0

(C )

V A C F a v e r a g e o f tr a j. 1 , 2 , 3

3 3 2 7

1 .0

3 4 1 7

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

Page 20 of 61

+

ss(N H 4

)

s t r e t c h ( N H +4 m o i e t y )

N 6 H +

a s(N H 4

)

0 .5

0 .0 3 2 0 0

3 2 5 0

3 3 0 0

3 3 5 0

3 4 0 0

3 4 5 0

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

-1

3 5 0 0

3 5 5 0

3 6 0 0

)

Figure 3: The simulated AIMD/B3LYP-D3 DACF IR spectrum of the N2 H+ 7 cluster at 50 K in the range above 3200 cm−1 (panel A) is decomposed into the contributions from the NH stretch of the NH3 (panel B) and NH+ 4 (panel C) moieties, respectively, using partial VACF analysis. AIMD frequencies here (and in all subsequent figures) were scaled by a factor of 0.968. Intermediate frequencies. We continue to the frequency range below 2000 cm−1 that was probed by the Asmis experiments. 27,28 Figure 4 compares the experimental IRMPD spectrum 27,28 (panel A) of the N2 H+ 7 cluster with the DACF/AIMD spectrum, calculated using the CP2K software with the B3LYP-D3 functional and averaged over three 20 ps trajectories (panel B). While the calculated spectrum resembles the experimental one, it is not a priori clear which calculated band corresponds to which measured band, particularly because the measured bands are quite numerous, closely spaced, and some of them are combination bands rather than fundamentals.


Page 21 of 61

(A )

1 3 2 5 D '

E x p e r im e n t

3 7 4 a

1 .0

1 4 1 5 E ' 1 4 5 1 1 5 4 5 F ' G '

4 0 9 a ' 9 0 4 A 4

7 4 3 A '

0 .5 4 4 4 a "

1 0 9 7 C ' 1 1 3 3 1 2 6 9 C 2 C 3

9 3 8 B ' 9 9 8 B 2

8 2 4 8 7 3 A 2 A 3

1 5 9 6 H '

0 .0 +

H

tr a j. 1 , 2 , 3 7

(1 8 0 K ) 7

7 0 6

a U

6 0 0

7 0 0

2 0 0

4 0 0

6 0 0

9 0 0

8 0 0

1 5 6 5 1 5 9 7

1 6 8 0 1 7 0 0

1 4 9 5

3

1 1 0 0

8 0 0

0 .0 1 0 0

g

sU

1 2 0 0

×5

× 50

1 2 3 2

5 8 1

5 0 6 5 0 0

2 w g

1 1 1 6 1 1 4 2

P T M 4 0 0

)

9 9 5 1 3 1

1 5 2 1 8 1

0 .5

× 100

(N N )

w

1 6 8 7 b ( N H +4 )

+

1 4 0 0

1 3 0 0

1 6 6 5

H

1 6 8 9

1 5 3 1 2

1 3 0 0

1 5 9 5 b (N H

D A C F o f N

1 2 0 0

4

1 1 0 0

1 0 0 0

× 50

)

9 0 0

+

8 0 0

1 4 8 5 1 4 9 8

1 1 1 7 1 1 5 9 1 1 9 3

9 7 0 9 4 0

8 6 8 80 0

75 0

5 7 1 5 9 2

70 0

7 0 0

u

3 5 5

1 .0

6 0 0

1 2 5 5

w

5 0 0

×5

1 5 2 1 U ⊥( N H

25 0

4 0 0

(C )

1 2 6 2

8 9 6

3 7 4

×105

3 8 1

×104 × 40

20 0

0 .0

(5 0 K )

× 300

2 9 6

0 .5

3 0 1

2 3 7

7 4 5

2

1 6 4 2

D A C F a v e ra g e o f N

(B ) 1 .0

15 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


7 0 0

8 0 0

1 4 0 0

1 0 0 0

1 0 0 0


1 2 0 0 -1

1 4 0 0

1 6 0 0

1 8 0 0

)

Figure 4: IR spectrum of the protonated ammonia dimer. (A) Experimental IR multiphoton dissociation spectrum (top panel of Figure 5 in Ref. 28, data courtesy of Knut R. Asmis). (B) Theoretical IR spectrum of the N2 H+ 7 cluster at 50 K obtained by averaging the Fourier transforms of the DACFs from three AIMD/DFT 20 ps production runs and scaling the frequencies by a factor of 0.968. (C) Theoretical IR spectrum of the N2 H+ 7 cluster at 180 K, obtained from the DACF of a single 20 ps AIMD trajectory under the same conditions as for the 50 K case. Insets are blow-ups of regions with relatively weak peaks. a) Bending modes. We begin by searching for the ammonia- and ammoniumfragment bending modes, that were not probed by the quantal 4D and 6D calculations of Asmis and collaborators. In these (as well as the umbrella modes), the main change is in the HNH angles (where H is not the shared proton, which is denoted here by H∗ ). Therefore, we present in Figure 5 the partial VACF spectrum of these angles for the two parts of the dimer (panels B and C), to be compared with the DACF spectrum in panel A. One might have expected the strongest band in the DACF spectrum, at 1642 cm−1 , to represent one (or both) of these bending modes, because it occurs midway between the NH3 and NH+ 4 bending NMs (Table 1). Surprisingly, perhaps, we find 21 ACS Paragon Plus Environment


two strong VACF peaks in this region that do not correspond to any DACF peak. The bending of the NH3 fragment is located at 1613 cm−1 , very close to the AIMD frequency of isolated NH3 (1616 cm−1 ), but slightly red- rather than blue-shifted from it. At that location there is a minimum in the DACF spectrum. The bend−1 ing of the NH+ 4 fragment is located at 1699 cm , close to the AIMD frequency of −1 the isolated NH+ 4 (1675 cm ), but blue- rather than red-shifted from it. At that

location there is no observable feature in the DACF. As a check on this result, consider the VPT2 frequencies, which are 1600 and 1708 cm−1 for b(NH3 ) and b(NH+ 4 ), respectively. This nice agreement (to within about 10 cm−1 ) suggests that this partial-VACF result is not an artefact, and

1 0 0 0

1 4 0 0

3 4 2 0

1 5 3 1

1 2 0 0

×6 3 2 0 0

3 3 2 7 3 4 0 3 3 4 3 3

6 0 0

1 3 5 0 1 4 0 0

1 5 6 5 1 5 9 7

5 0 0

×5

× 500

1 2 6 2 5 7 1 5 9 2

3 5 0 4 0 0

× 40

1 1 1 7 1 1 5 9 1 1 9 3

3 7 4 3 0 1

3 8 1

2 9 6

0 .5

1 3 4 5

D A C F (5 0 K ) a v e ra g e o f tra j 1 ,2 ,3

1 4 8 5

(A ) 1 .0

1 6 4 2

hence the strongest peak in the spectrum, at 1642 cm−1 , is not a fundamental.

3 3 0 0

3 4 0 0

3 5 0 0

(C )

1 1 9 7

× 104

× 15

5 0 0

6 0 0

1 0 0 0

1 2 0 0

V A C F ( 5 0 K ) a v e r a g e o f tr a j. 1 , 2 , 3 ( N H +4 m o i e t y )

4 0 0

5 0 0

4 )

× 10

6 0 0

1 0 0 0

6 0 0

1 0 0 0

1 2 6 3

× 10

3 3 0 0

3 4 0 0

3 5 0 0

+

a U g

1 1 9 7

2 9 5 3 0 1 3 7 1 3 0 0

5

5 7 0

w 0 .5

U ⊥( N H 1 5 1 2

5 8 4

b e n d

1 3 4 0

H N 6 H

1 .0

3 2 0 0

1 4 0 0

b (N H

+

a s (N H +

4 )

4 )

3 4 1 7

4 0 0

a s (N H 3 )

1 6 9 9

3 0 0

0 .0

× 107

1 5 2 4

3 7 1

2 9 5 3 0 1

0 .5

b (N H 3 )

a U

sU

5 8 2

5 7 3

g

1 2 6 3

w

3 4 3 8

V A C F ( 5 0 K ) a v e r a g e o f tr a j. 1 , 2 , 3 H N 1 H b e n d ( N H 3 m o ie ty )

(B ) 1 .0

1 6 1 3

0 .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

Page 22 of 61

× 103

1 2 0 0

1 4 0 0

1 2 0 0

1 4 0 0

3 2 0 0

3 3 0 0

3 4 0 0

3 5 0 0

0 .0 2 0 0

4 0 0


1 6 0 0 -1

3 2 0 0

3 4 0 0

3 6 0 0

)

Figure 5: Simulated AIMD/DACF IR spectrum of the N2 H+ 7 cluster at 50 K in the −1 range 200–3600 cm (panel A) is decomposed into the contributions from the HNH bend for the NH3 moiety (panel B) and the NH+ 4 moiety (panel C), respectively, using partial VACF analysis. Insets depict the details of the spectrum in regions exhibiting relatively weak bands. The VPT2 calculation also suggests that b(NH3 ) is about factor 4 more intense 22 ACS Paragon Plus Environment

Page 23 of 61

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


than b(NH+ 4 ). Hence, b(NH3 ) may be the only bending mode seen experimentally, and it could correspond to band H’ at 1596 cm−1 . This conclusion agrees with Tono + et al., 26 who did not observe a b(NH+ 4 ) band in the spectra of the NH4 (NH3 )3,4

clusters as well. b) Umbrella modes. The remaining peaks in the partial-VACF spectrum for the HNH bending coordinates in Figure 5 correspond to umbrella modes. Firstly, −1 we find the U⊥ (NH+ 4 ) band as a doublet at 1512/1524 cm , identifying it as a

doubly degenerate mode. We assume it corresponds to the 1531 cm−1 band in the DACF. Its identification relies on (i) The matching MP2/VPT2 assignment of modes 12/13 at 1537 cm−1 ; (ii) The 6D MCTDH result of Yang et al., 28 who have found a “doubly degenerate proton bending mode” at 1542 cm−1 , corresponding to the IRMPD band G’ at 1545 cm−1 . Thus all methods agree on the assignment of this mode within a frequency window of only 14 cm−1 . Although both bending and umbrella motions involve both the HNH and HNN angles, the HNH angles are dominant for bending whereas the HNN angle is the coordinate of the umbrella motion. Figure 6 shows the partial VACF for the HNN angles. Unlike Figure 5, the U modes are now more intense than the b modes, though both figures show peaks at 1202, 1263 and 1343 cm−1 . Only the first two are manifest in DACF peaks (1193 and 1262 cm−1 , respectively), so we assign them to the two U -modes. Because sU and aU correspond to the sum and difference of the HNN angles on the two sides, we present their partial VACFs in Figure 7. It clearly identifies the 1202 and 1263 cm−1 peaks as the sU and aU modes, respectively. In comparison, the VPT2 frequencies are at 1224 and 1246 cm−1 , respectively (Table S3), differing from the AIMD by only ∼ 20 cm−1 . Finally, partial VACFs for the HNH∗ angles are shown in Figure S7. The features are similar to those of Figure 6, further confirming our assignments of sU and aU . In contrast, Asmis and coworkers find the sU and aU bands from their 6D



model at 1330 and 1348 cm−1 , respectively. 28 The higher values for these modes are in accord with the VPT2 vibrational analysis for the D3d transition-state (at the MP2(full)/6-311++G(d,p) level), which produces frequencies of 1339 and 1383 cm−1 for the two U -modes (see the Excel file in the SI). Possibly also, the band at 1343 cm−1 seen in our DACF spectrum and in the U -mode VACF spectra is due to trajectory segments that sample conformations close to the D3h geometry.

1 6 4 2

D A C F ( 5 0 K ) a v e r a g e o f tr a j. 1 , 2 , 3

3 2 0 0

3 3 2 7 3 4 0 3 3 4 3 3

×6

1 4 8 5

7 0 0

1 2 6 2

× 40

6 0 0

5 0 0

1 1 9 3

4 0 0

5 7 1 5 9 2

2 9 6

0 .5

3 8 1

3 0 1

1 5 3 1

3 7 4

3 4 2 0

(A ) 1 .0

3 3 0 0

3 4 0 0

3 5 0 0

0 .0 V A C F ( 5 0 K ) a v e r a g e o f tr a j. 1 , 2 , 3 H N N b e n d ( N H 3 m o ie ty )

0 .5

a s (N H 3 ) 3 4 3 8

1 2 6 3

5 7 4

u

g

1 2 0 2

w

3 7 2

w

3 8 5

1 .0

5 8 2

(B )

b (N H 3 )

a U

sU 5 9 5

1 6 1 3

3 5 8

× 10 3 2 0 0

3 3 0 0

3 4 0 0

3 5 0 0

0 .0

w g

a U

sU

3 5 7

3 8 7

0 .5

1 2 0 2

3 7 1

+

4 )

3 4 1 7

u

a s (N H

+

4 )

1 3 4 3

5 8 5

b (N H

5 9 0

w

( N H +4 m o i e t y )

H N N b e n d

5 7 0

1 .0

1 6 9 7

V A C F ( 5 0 K ) a v e r a g e o f tr a j. 1 , 2 , 3

(C )

× 60

1 2 7 3


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 61

3 2 0 0

3 3 0 0

3 4 0 0

3 5 0 0

0 .0 2 0 0

4 0 0

6 0 0

1 0 0 0

1 2 0 0

1 4 0 0


1 6 0 0 -1

3 2 0 0

3 4 0 0

3 6 0 0

)

Figure 6: Simulated AIMD/DACF IR spectrum of the N2 H+ 7 cluster at 50 K (panel A) is decomposed into the contributions from the HNN bend for the NH3 moiety (panel B) and the NH+ 4 moiety (panel C), respectively, using partial VACF analysis. Insets depict the details of the spectrum in regions exhibiting relatively weak bands.


a U

1 3 4 5

1 1 9 3

1 1 5 9

0 .5

× 50 1 2 8 1 1 2 8 7 1 2 9 5

sU

1 3 0 3

1 .0

1 3 7 5

D A C F a v e ra g e tra j 1 ,2 ,3

1 2 6 2 1 2 6 7

(A )

1 3 2 0

1 3 4 0

1 3 6 0

1 3 8 0

a U

1 2 8 1

V A C F a v e r a g e tr a j. 1 , 2 , 3 a n ti- s y m m e tr ic u m b r e lla

1 2 9 2

(B ) 1 .0

1 2 6 3 1 2 7 3

0 .0

1 3 4 3

1 3 0 1

1 2 0 2

0 .5

0 .0 (C )

V A C F a v e r a g e tr a j. 1 , 2 , 3 s y m m e tr ic u m b r e lla

1 2 0 2

1 .0

sU 1 2 1 9

0 .5

1 3 4 3


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 2 7 5

Page 25 of 61

0 .0 1 1 0 0

1 1 5 0

1 2 0 0

1 2 5 0


1 3 0 0 -1

1 3 5 0

1 4 0 0

)

Figure 7: Simulated AIMD/DACF IR spectrum of the N2 H+ 7 cluster at 50 K in the range 1100–1400 cm−1 (panel A) is decomposed into the contributions from the difference (B) and sum (C) of the HNN angles on the two sides of the dimer, using partial VACF analysis. Insets depict the details of the spectrum in regions exhibiting relatively weak bands.

Based on the 6D model, Asmis and coworkers have assigned the D’ band at 1325 cm−1 to the aU mode. The sU mode has not been previously assigned. We suggest to identify it with the previously unidentified C3 band at 1269 cm−1 . Although the absolute value of the AIMD aU and sU frequencies is smaller than in the experiment, their difference of ca. 60 cm−1 is similar to the difference between the D’ and C3 frequencies. We also note that the VPT2 anharmonic intensities suggest that aU is about 5 times more intense than sU , in qualitative agreement with the relative intensities of the D’ and C3 bands. Finally, we note an interesting progression just blue of the DACF aU band, see Figure 7. The frequency difference between two consecutive peaks in the progression is about 7 cm−1 . Possibly, the U -mode couples to the cluster (rigid-body) rotation to produce this progression. High resolution measurements of vibrationalrotational spectra could test this prediction. 25 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

Table 4: Fundamental vibrational frequencies (in cm−1 ) for gas-phase N2 H+ 7 at 50 K from VPT2 (at the MP2(full)/6-311++G(d,p) level), DACF and partial-VACF spectra from AIMD with the B3LYP-D3 functional, and IRMPD experiment. We exclude from this table the high frequency NH stretches (modes 1–6) that were already summarized in Table 3. mode Table S3 8/9 10/11 12/13 14 15 16/17 18/19 20 7 21

assn. VPT2 AIMD Table 2 freq. freq. + b(NH4 ) 1708 1699a b(NH3 ) 1600 1612a U⊥ (NH+ 1537 1531b 4) aU 1246 1263a sU 1224 1202a wg 624 571b wu 408 374b NN 271 301b PTM 485 296b i-rot 10i 27

IRMPD 28 freq. band 1596 1545 1325 1269

H’ G’ D’ C3

409 a’ 369c 374 a

a

Value from VACF. Value from DACF. c This mode is not IR active. The experimental frequency is based on the assumption that mode A’ at 743 cm−1 is the combination PTM+NN. b

All the assignments made thus far are collected in Table 4. Listed are DACF frequencies, when available, and partial-VACF frequencies when their corresponding DACF bands are not observed.

Low frequencies. a) Wags. We begin the discussion of the low frequency regime with the wagging modes (w). These also involve oscillations of the HNN angles, whose spectrum is depicted in Figure 6. To the red of the U modes, one observes two prominent peaks, at 571 and 374 cm−1 , which are candidates for the wagging modes. Indeed, their AIMD frequencies are close to the VPT2 values of 624 and 408 cm−1 , assigned to wg and wu , respectively. As reported by Yang et al., 28 the gerade wag is higher in frequency than the ungerade. In both the DACF and VPT2 spectra wg has a 26 ACS Paragon Plus Environment

Page 26 of 61

Page 27 of 61

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considerably smaller intensity than wu , which is understandable from the NM vector representation in Figure S6. For example, mode 17 (a wg mode) exhibits pairs of vectors of nearly identical magnitude and opposing directions. Consequently, the wu mode is observed experimentally 28 at 409 cm−1 , in good agreement with our computations, whereas the wg band is absent there. b) PTM. We now proceed to identify the PTM and NN stretching modes. For this end, we present in Figure 8(B) and (C) the partial-VACF for the two NH∗ stretching motions. They both show a strong peak around 300 cm−1 . In comparison, VPT2/MP2 has the PTM at 475 cm−1 . This relatively large difference indicates that the PTM is a very anharmonic mode. Indeed, the corresponding harmonic frequency is at 1950 cm−1 (see Table S3 of the SI), which undergoes a huge red-shift due to anharmonicity. Table 5: PTM (anharmonic) frequency (in cm−1 ) calculated from different quantum chemistry methods/basis-sets using VPT2 analysis. methoda /basis MP2(full) B3LYP-D3 B2PLY-D3 PBE0-D3

6-311++G(d,p) 485 75 394 287

aug-cc-pVTZ 571 191 472 118ı

a

Anharmonic results for the M06-2X functional are not listed here, because they are implausible (see the Excel file in the SI). Poor performance of the M06-2X functional was reported in the benchmark studies of Ref. 53, suggesting that it is not suitable for anharmonic analysis.

Table 5 gives the PTM frequency for VPT2 with different quantum chemistry methods and basis sets. There is indeed a large variation in frequency here, but all values are below 570 cm−1 , and they average (real frequencies only) to 354 cm−1 . The experimental value, 28 374 cm−1 , is quite close to this average value. The remaining peaks in the NH∗ VACF are thus not fundamental modes, and will be discussed with the combination bands below. c) NN symmetric stretch. Figure 8(D) shows the partial VACF for the NN 27 ACS Paragon Plus Environment


6 0 0

8 0 0

1 0 0 0

1 1 0 0

1 2 0 0

1 4 0 0

1 9 0 0

2 0 0 0

2 1 0 0

2 2 0 0

2 3 0 0

3 2 1 7

× 105

3 0 0 0

2 5 0 0 2 6 0 0

2 4 0 0

3 3 2 7 3 4 0 3 3 4 3 3

2 4 0 4

× 104

2 3 8 4

2 0 9 9

2 3 1 8

1 9 3 9

1 8 0 0

× 15

3 4 2 0

3 0 4 1

1 7 9 2

2 2 7 4

1 7 8 2

1 6 4 2

1 3 0 0

× 5000

1 8 6 3 1 8 8 3 1 9 0 1

×5

13 50 14 00

0

1 1 5 9 1 1 9 3

10 50 11 0 11 0 4

9 4 0

9 0 0

1 5 3 1

× 500

× 260

7 0 0

8 6 8

× 300

1 5 6 5 1 5 9 7

5 0 0

6 5 0 7 0 0 7 5 0 8 0 0

1 4 8 5 1 4 9 8

4 0 0

9 7 0

6 7 1

× 40

5 7 1 5 9 2

2 9 6

3 8 1

3 0 1

× 104

0 .5

1 3 4 5

8 9 6

3 7 4

1 .0

D A C F ( 5 0 K ) a v e r a g e o f tr a j. 1 , 2 , 3

1 2 6 2

1 1 1 7

7 4 5

(A )

3 3 0 0

3 2 0 0

3 4 0 0

×6

3 5 0 0

1 9 0 0

4

+

U ⊥( N H

2 0 0 0

3

ss(N H 3 3 4 5

× 104 × 20

2 1 0 0

2 2 0 0

× 105

2 5 7 0

2 1 0 5

1 8 0 0

3 2 1 8

g

) + w

u

1 7 9 3 1 8 6 4 + 1 9 0 2 U ⊥( N H 4 ) + w 1 9 4 1

u

1 7 7 9

+ w a U + N N

1 6 3 8

a U

× 200 0

10 50 11 0

1 0 0 0

× 5000

2 4 6 3

1 1 1 5

9 3 7 w 9 0 0

1 5 6 0

8 0 0

g

g

2 2 6 9

u

7 0 0

1 4 9 1 sU + N N

6 0 0

2 w

1 2 6 3 1 3 0 1

5 0 0

8 9 8

g

× 900

× 5000

a U 1 1 6 1 1 1 9 7

4 0 0

6 5 0 7 0 0 7 5 0 8 0 0

9 6 7

u

3 7 6

0 .5

g

7 4 2

× 104

w

V A C F ( 5 0 K ) a v e r a g e o f tr a j. 1 , 2 , 3 N 1 H * s tr e tc h ( N H 3 m o ie ty ) b ( N H +4 ) + w

+ w

u

8 6 2 N N + w

2 9 4

1 .0

2 w

5 9 7 P T M

3 0 1

+ N N

(B )

)

0 .0

2 3 0 0

2 4 0 0

2 5 0 0

× 5300

3 0 0 0

2 6 0 0

3 3 0 0

3 2 0 0

3 4 0 0

3 5 0 0

0 .0 V A C F ( 5 0 K ) a v e r a g e o f tr a j. 1 , 2 , 3

7 0 0

8 0 0

1 0 0 0

1 8 0 0

4

3 2 1 9

)

)

2 1 0 0

2 3 0 0

2 4 0 0

2 5 0 0

2 6 0 0

)

3 3 4 5

× 104

2 5 7 0 2 2 0 0

4

ss(N H

× 104

× 400

+

ss(N H 3 3 2 7

N H

3

3 0 3 8

) + w

2 0 0 0

⊥(

+

4

+

U ⊥( N H × 10

1 9 0 0

2 U 2 4 6 3

g

+ 2 w

u

2 2 6 9

2 b (N H

2 1 0 5

1 4 9 1 1 5 6 0 1 5 9 7

u

0

10 50 11 0

9 0 0

1 7 7 9 1 7 9 3 sU + 2 N N 1 8 6 4 a U + 2 N N

g

1 1 9 7 P T M

+ 6 0 0

1 2 6 3 1 3 0 1 w

5 0 0

× 400

1 1 6 1

4 0 0

1 6 3 8

+ 3 N N

1 1 1 5 u

× 4000

3 0 0

× 100

9 6 7 w 6 5 0 7 0 0 7 5 0 8 0 0

8 6 2

× 35

0 .5

( N H +4 m o i e t y )

N 6 H * s tre tc h

2 w

+ 2 N N

× 10

8 9 8 P T M 9 3 7

6 7 3 5 9 6

u

+ 2 N N

7 4 2

P T M

6

1 .0

3 w

1 9 4 1 s U

u

+ w

(C ) 2 9 4 3 0 1


3 0 0 0

3

)

× 1300

3 2 0 0

3 3 0 0

3 4 0 0

3 5 0 0

0 .0

5 0 0

6 0 0

7 0 0

8 0 0

9 0 0

1 0 0 0

1 1 0 0

1 2 0 0

1 3 0 0

1 4 0 0

1 6 0 0

1 7 0 0

1 8 0 0

2 1 0 0

2 2 0 0

2 3 0 0

2 4 0 0

2 5 0 0

2 6 0 0

) 3

3 3 4 5 s s (N H

3 2 1 9

4

+

3 3 2 7 s s (N H

× 1010 2 5 7 0

2 4 6 3

+

U ⊥( N H

2 0 0 0

)

3 0 3 8

2 2 6 9

g

) + w 1 9 0 0

× 109

× 107

2 1 0 5

1 8 6 4 1 8 9 2 1 9 4 1

1 4 9 1 1 5 0 0

4

1 6 3 8

1 7 7 9 1 7 9 3

× 104

+

2 N N

× 103

1 5 6 0 1 5 9 7

8 0 0

× 104

1 3 0 1 1 3 4 0 2 w

7 0 0

× 140

0

4 0 0

6 0 0

8 9 8

× 8000

10 50 11 0

u

× 106

8 6 2

0 .5

1 1 9 7

× 104

a U 1 2 6 3

6 7 3

9 3 7

3 0 1

1 .0

3 7 6

N N

V A C F ( 5 0 K ) a v e r a g e o f tr a j. 1 , 2 , 3 N 1 N 6 s tre tc h

1 1 1 5

(D )

2 9 5 P T 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 28 of 61

3 0 0 0

3 2 0 0

3 0 0 0

3 2 0 0

× 125 3 3 0 0

3 4 0 0

3 5 0 0

0 .0 2 0 0

4 0 0

6 0 0

8 0 0

1 0 0 0

1 2 0 0

1 4 0 0

1 6 0 0

1 8 0 0

2 0 0 0


-1

2 2 0 0

2 4 0 0

3 4 0 0

3 6 0 0

)

Figure 8: Theoretical AIMD/DACF IR spectrum of the N2 H+ 7 cluster at 50 K (panel A) is decomposed into the contributions from the NH∗ distance for the NH3 moiety (panel B) and the NH+ 4 moiety (panel C), and from the NN distance (panel D), using partial-VACF analysis. Assignments of overtones and combination bands are shown in blue (2nd order) and red (higher order). Insets show blow-ups of the spectrum in regions containing relatively weak peaks. stretch, with a doublet at 296 and 301 cm−1 . Because the latter has the larger amplitude we assign it to the NN stretch, with the smaller amplitude peak assigned to the PTM. The NN stretch appears at 271 cm−1 in the VPT2 calculation, as compared with 434 cm−1 in the 6D calculation of Yang et al. 28 This mode is not IR active and thus manifested experimentally only in combination bands. Assuming 28 ACS Paragon Plus Environment

0 .8

+

H

V A C F 7

1

N

H 6

b (N H 3 )

( 5 0 K ) a v e r a g e o f tr a j. 1 , 2 , 3

in te r n a l r o ta tio n

w u

1 1 4 6

3 7 2

2 w g

1 4 9 9

2 7

H N

i-r o t

2

1 1 6 4 2w

+ u

4 0 1 w

0 .4

1 5 1 1

g

i-r

+

ot

i-r

0 .6

ot

N 1 .0


w g 0 .2

×4 0 3 0 0

0 .0 0

2 0 0

×1 0

5 7 4 5 8 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


4 0 0

4 0 0

6 0 0

+ 1 5 3 1 U ⊥( N + H 4 ) 1 5 5 1 U ⊥( N H 4 ) + i - r o t 1 5 9 1 1 6 1 2 1 6 3 4 b (N H 3 ) + i-r o t 1 6 4 5

Page 29 of 61

3

1 0 0 0

1 2 0 0

1 0 0 0

1 2 0 0


1 4 0 0

1 4 0 0 -1

1 6 0 0

1 8 0 0

)

Figure 9: Theoretical VACF spectrum of the HNNH dihedral angle of the N2 H+ 7 cluster at 50 K using the three AIMD 20 ps trajectories. (following Asmis and coworkers) that the A’ band at 743 cm−1 (Figure 4A) is the combination PTM+NN, we subtract 374, obtaining 369 cm−1 for the NN stretch. Thus, experimentally, the PTM and NN stretch occur at nearly the same frequency. This is nicely reproduced by our AIMD simulations, but not by VPT2. d) i-rot. In order to identify the internal rotation (i-rot) mode (for rotation around the N–N axis), we have computed the HNNH dihedral angle VACF spectrum as shown in Figure 9. A strong peak is seen at 27 cm−1 , which we identify with i-rot. Because i-rot belongs to the A2 irrep of the C3v symmetry group, it is not IR active and is therefore not seen in the DACF spectrum. For the same reason, it can combine with modes possessing E but not A1 symmetry (see discussion of combination bands below). Thus we find in this VACF also the combinations wu +i-rot, 2wg +i-rot, U⊥ +i-rot and b(NH3 )+i-rot, all involving E-symmetry modes.



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

Combination Bands The large anharminicity for the protonated ammonia dimer appears to produce an unprecedented number of overtones and combination bands, both experimentally and theoretically. Let us first settle-down the theoretical side using the VPT2 and AIMD outputs. When an anharmonic calculation is instigated in Gaussian 09, the output (e.g., Table S3 in the SI) includes first all 3N − 6 fundamentals in the order of decreasing HNM frequency, and their harmonic and anharmonic intensities (the latter may occasionally diverge). Then come the (first) overtones in the same order, and then the combinations i, j, where for each mode i all combinations involving mode j < i are listed. The program does not go beyond binary interactions, except in listing possible Fermi resonances (FR). Determination of combination bands from AIMD trajectories is non-standard. For assignment, we rely on patrial-VACFs and frequency summation. Consider the partial-VACF for the NH∗ stretch (Figure 8 above) and the NH∗ N bend (Figure 10 below). The first is sensitive to modes in which H∗ moves parallel to the NN axis. This includes all the A1 modes of the C3v symmetry group: The PTM, NN stretch, the two U -modes and the ss. In contrast, the NH∗ N bending VACF is predominantly sensitive to H∗ motion perpendicular to the NN axis. All the modes belonging to the E-type irreducible representation (irrep) have such a component: The wags, bends and U⊥ modes. In addition, we have used the frequency-sum property namely, that the frequency sum (of two fundamentals) equals to their combination band frequency. Often, the latter is slightly smaller than the frequency-sum due to anharmonicity. However, in our AIMD results this red-shift is possibly within the computational error, and we find that the frequency-sum rule is obeyed rather closely in nearly all the AIMD-derived combinations. Between the PTM and NN modes the difference in frequencies is only 5 cm−1 , so we consulted the VPT2 intensities to determine 30 ACS Paragon Plus Environment

Page 30 of 61

8 3 0 9 0 0

7 0 0

1 0 0 0

2 0 9 9

× 15 2 0 0 0

2 3 0 0

2 2 0 0

2 4 0 0

3 4 2 0 3 2 1 7

× 105

2 5 0 0

2 6 0 0

3 0 0 0

3 1 0 0

3 3 2 7 3 4 0 3 3 4 3 3

2 2 7 4 2 3 1 8 2 3 8 4

22 00

00 0

1 8 0 0

21 00

1 923 9

1 8 6 3

× 600

2 0 1 4 2 0 6 9

1 7 9 2 1 8 8 3

× 5000

1 9 0 1

1 7 2 3

1 5 6 5 1 5 9 7

1 3 0 0 1 4 0 0

3 0 4 1

1 6 4 2

1 3 7 5 1 3 4 5 1 4 1 6

1 4 8 5 1 4 9 8

0

13 5 14 0 0

1 1 1 7 1 1 5 9 1 1 9 3 1 1 0 0 1 2 0 0

× 106

26 00

6 0 0

9 4 0

× 300

5 7 1 5 9 2

5 0 0

D A C F a v e r a g e o f tr a j. 1 , 2 , 3

25 00

4 0 0

9 7 0

75 0 80 0

65 0 70 0

3 8 1

×104

× 40

2 0 0

8 9 6

×5

2 3 7 2 9 6

0 .5

1 5 3 1

3 0 1

6 7 1

3 7 4

×104

1 2 6 2

1 .0

1 4 3 1

× 500

7 4 5

(A )

3 3 0 0

3 2 0 0

3 4 0 0

×6

3 5 0 0

4 0 0

6 0 0

1 0 0 0

1 6 0 0 1 8 0 0 2 0 0 0 W a v e n u m b e r (c m -1 )

)

4

2 b (N H 3 3 9 8

3 2 1 9

) 4

+

)

3

2 2 0 0

w

2 U

25 00

g

3 4 2 1

4

24 00

+

⊥(

N H

+

2 sU g

2 3 6 9

w

×103

3 0 3 8

2 0 7 4

×200

b (N H

+

1 8 9 2 2

0

8 0 0

2 b (N H

+ 2 g

sU

1 7 9 3 s U 1 8 0 0

2 2 6 9 b (2N H 4 ) + 30 2 2 8 3 0

1 4 0 0

1 6 0 0

20 50

1 2 0 0

1 5 0 5

1 4 1 3 3

u 1 1 1 4 0

1 4 0 0

×103

0 .0 1 0 0 2 0 0

+ 2 N N

+ N N

1 4 8 7

+

1 6 9 7 b ( N H 4 )

w

u

w

w

u + N N

g + 2 w u 1 3 2 8

w 1 1 4 5 2 1 1 5 8

7 0 0

w

19 50 20 0

6 0 0

w 3

u

w

w + g

5 0 0

×8 12 00

4 0 0

9 3 7

3 0 0

4

u

+ 2 N N

5

w

3 0 0

u

4 1 4

×10

w

g

11 00 11 5

w

9 5 4

2 0 0

u

g

u

7 4 3

- w

5 0 8 5 3 4 5 6 5 5 9 7 P T M + N N 6 5 5 P T M + w u

g

3 8 6

0 .5

w

2 0 8

1 .0

w 2

w

V A C F a v e r a g e o f tr a j. 1 , 2 , 3 N 1 H *N 6 b e n d

(B )

1 9 0 2 U ⊥( N H ) + w 4 u 1 9 9 5 P T M + b ( N H + ) 4 + ) + w u 4 2 1 0 5 U ⊥( N H + ) + w 4 g

u

0 .0

×103


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 7 8 2

Page 31 of 61

× 10

3 3 0 0 3 4 0 0 3 5 0 0

2 4 0 0

3 0 0 0

3 2 0 0

3 4 0 0

3 6 0 0

Figure 10: Theoretical AIMD/DACF IR spectrum of the N2 H+ 7 cluster at 50 K ∗ (panel A) is compared with the VACF spectrum for the NH N angle (panel B). Assignments of overtones and combination bands are given in blue, with difference bands in red. Insets show blow-ups of the spectrum in regions containing relatively weak peaks. which of them participates in a given combination. The results of this analysis are collected in Table 6.



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

Table 6: Overtone and combination band frequencies (in cm−1 ) of the gas-phase N2 H+ 7 cluster from AIMD/VACF analysis at 50 K, see Figures 8 and 10, as compared with the VPT2 results. assignment 2b(NH+ 4) 2b(NH3 ) 2U⊥ (NH+ 4) 2sU 2wg 2wu b(NH+ 4 ) + wg U⊥ (NH+ 4 ) + wg b(NH+ ) 4 + wu U⊥ (NH+ 4 ) + wu wg + wu aU +NN sU +NN PTM+NN PTM+b(NH+ 4) aU + wu (FR)e NN+wg PTM+wu PTM+2NN wu +2NN PTM+3NN wg + 2wu 2wu +2NN 3wu +NN sU +2NN 2wg + 2wu aU +2NN sU + 2wu

VACF Suma DACF VPT2b Ic E × E overtones 3398 3398 3403 3406 0.42 3219 3226 3217 3221 0.61 3038 3062 3041 3062 3.56 2368 2404 2384 2424 2.96 1145 1142 1159 1234 9.86 743 748 745 809 7.70 E × E combinations 2269 2270 2274 2324 97 2105 2102 2099 2158 126 2074 2073 2069 2120 265 1902 1905 1901 1941 1.79 937 945 940 1025 0.15 A1 × A1 combinations 1560 1564 1565 1476 357d 1505 1503 1498 1472 140d 597 597 592 787 939 A1 × E combinations 1995 1995 2014 2166 3.1 1638 1637 1642 1662 0.08 862 872 868 889 0.79f 655 670 671 939 0.57g Higher order 898 898 896 – – 967 976 970 – – 1197 1199 1193 – – 1301 1319 1303 – – 1340 1350 1345 – – 1413 1423 1416 – – 1793 1804 1792 – – 1892 1890 1883 – – 1864 1865 1863 – – 1941 1950 1939 – –

a

Sum of AIMD fundamental frequencies from Table 4. Averaged VPT2 frequencies (MP2(full)/6-311++G(d,p) level in Gaussian 09) for E-modes (Table S3). c Anharmonic intensity from VPT2 analysis (Table S3). d Ca. 10 fold more intense than with PTM replacing NN (Table S3). e Combination enhanced by a Fermi resonance with the ammonia HNH bending mode. f Intensity 0.06 with PTM replacing NN (Table S3). g Intensity 0.00 with NN replacing PTM (Table S3). b


Page 32 of 61

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Interestingly, we find that binary combinations are observed predominantly when the two modes both belong to the E-irrep or both to the A1 -irrep (of the C3v point group). This is evident also from the VPT2 intensities (the I column in Table 6). One may understand this from symmetry considerations. In the multiplication table of C3v , the direct products A1 × A1 = A1 , A2 × A2 = A1 , and E × E = A1 + A2 + E, contain the totally symmetric irrep, A1 , but A1 × E = E, A2 × E = E, and A1 × A2 = A2 do not. The z coordinate (NN axis) also belongs to A1 , but the perpendicular axes (x and y) belong to the E-irrep. This means that the A1 × A1 and E × E combinations get excited by the z component of the dipole moment, whose time-variations must be large due to motion of the excess charge parallel to the NN axis. The smaller variation of the dipole moment in the x and y directions gives rise to the weaker A1 × E combination bands that belong to the E-irrep. We also note that the E × E combinations always involve a wag, whereas the A1 × A1 combinations always involve the NN stretch. In the higher (third and fourth) order combinations, we also find that one of the wags or the NN stretch always participates.

Difference Bands Also of interest is the identification of difference bands. These involve de-excitation of a hot band together with the excitation of a fundamental. These are rather rare events because both anharmonic coupling and a hot band are involved. The highest probability for observing a hot band is for the lowest frequency mode, wu . Indeed, wg − wu is identified with a clear peak at 208 cm−1 in Figure 10(B), which corresponds with the 197 cm−1 difference between the two wag frequencies in Table 4.



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

Fermi resonances 1. The bending FR. With all the detailed analysis presented thus far, the strongest peak in the DACF/AIMD spectrum, at 1642 cm−1 , remains unassigned. This peak falls inbetween the VACF peaks for the two bending modes. It is closest to the b(NH3 ) VACF peak at 1613 cm−1 , yet too far from it (ca. 30 cm−1 ) to allow its identification with this fundamental. It is even closer to the wu + aU combination band, whose DACF frequency sum is 1637 cm−1 . Because wu and aU have different symmetries, E and A1 , respectively, we expect it to be weak (indeed, its VPT2 intensity is only 0.08, see Table 6). However, the E symmetry of wu + aU is the same as that of b(NH3 ). These two bands are nearly degenerate, separated by only 20 cm−1 . These are the two conditions required for a FR. As a consequence of the FR, the combination band gains intensity from the fundamental and moves up in frequency (to 1642 cm−1 ). The fundamental loses intensity and moves down in frequency (to 1597 cm−1 ). This can explain these two peaks in the DACF spectrum (Figure 4B). The Gaussian 09 output predicts a similar FR, but because it is based on harmonic frequencies it is slightly different. The strongest FR is expected between −1 wu + aU and b(NH+ 4 ), whose harmonic frequencies differ by only 1 cm .

We note that a previous theoretical work assigned the PTM to a band at 1610 cm−1 , the strongest peak found from those AIMD simulations. 23 Evidently, the newer experimental observation of the PTM at a significantly lower frequency invalidated this assignment. But now, with different theoretical methods, we find that the strongest peak in the spectrum, at 1642 cm−1 , is close to the previously observed intense band at 1610 cm−1 . While in our analysis this band is not the PTM fundamental, it does have a strong contribution from the excess proton due to the participation of the aU mode in the FR. The intriguing question, then, is where is this band experimentally? To check 34 ACS Paragon Plus Environment

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Page 35 of 61

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this, we take the experimental frequencies from Table 4. The frequency sums of wu + sU and wu + aU are 1678 and 1734 cm−1 , respectively. In the isolated −1 ammonium spectrum (Table 1), b(NH+ 4 ) occurs at 1699 cm , which is in-between

these two frequencies, so it may interact with either one of these combination bands to produce a FR. In both cases it is expected to appear above 1650 cm−1 , in a spectral region not yet probed by experiment. 2. The umbrella FR. A small shift in the sU band is also evident in our data. While in the VACF spectrum (Figure 7) its frequency is 1202 cm−1 , the peak in the DACF spectrum occurs at 1193 cm−1 . We explain this by another FR. The (4’th order) combination PTM+3NN is expected (by frequency addition) at 1199 cm−1 , red-shifted by only 3 cm−1 from the VACF sU peak. Both have A1 symmetry, so their FR is symmetry allowed. As a result of the FR, the PTM+3NN band moves to the red and gains intensity from the umbrella mode, so that a single peak is seen in the DACF at 1193 cm−1 . Because the sU mode is symmetric, and the FR further reduced its intensity, it becomes silent in the DACF spectrum.

Temperature Effects To check the temperature effect on the DACF and VACF spectra, we have propagated a single 20 ps trajectory at 180 K. Figure 4(C) shows its DACF spectrum compared to that at 50 K (panel B there). The partial VACF spectra are collected in Figures S9-S15 of the SI. Figure 4(C) shows that the activity now spreads out to more frequencies, particularly in the 1400–1800 cm−1 range. The highest peak here is at 1687 cm−1 , which at 50 K appeared as a tiny feature at 1689 cm−1 , assigned (based on the bending VACF spectra) to the b(NH+ 4 ) vibration. Therefore, the “bending FR” observed at 1642 cm−1 at 50 K, now possesses diminished intensity. This FR involved the wu + aU combination band, seen in the stretching VACF (Figure S14 in 35 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

the SI) at 1634 cm−1 , essentially the same frequency as at 50 K. Concomitantly, the bending VACF spectra (Figures S10(B) and S11(B) in the SI) suggest that this FR has moved to 1653 cm−1 , which could correspond to the 1665 cm−1 band in the 180 K DACF. −1 The other strong fundamentals at 180 K are U⊥ (NH+ 4 ) (at 1521 cm ), aU

(at 1255 cm−1 ), and wu (at 355 cm−1 ). These frequencies are red shifted by up to 20 cm−1 in comparison to the 50 K DACF spectrum, possibly because of anharmonicity (the hotter trajectory samples higher regions of the potential well). In particular, wu gained considerable intensity while the PTM became quite inconspicuous. The loss of PTM intensity at high temperatures might parallel the observation that the 1047 cm−1 band of H5 O+ 2 loses its intensity when embedded in liquid water near room temperature. 16 Commensurate with the above observation, we now see overtones and combinations that involve mainly the wags, and rarely the PTM. The most extensive set of combinations is again seen for the NH∗ N bend VACF, Figure 11. One would hope to assign these using the VACF spectra (Figures S9–S15 of the SI). However, the HNH and HNN bending VACFs (Figures S10 and S11) look at first confusing, as the wags seem to occur at an assortment of frequencies. Figure S11(C) suggest that the wags actually appear as doublets separated by over 35 cm−1 . Consulting the HNN VACF at 50 K (Figure 6) suggests that the wags were split already at 50 K, but to a lesser extent (< 15 cm−1 ). This suggests the following scenario. In the harmonic limit, each wag, which is doubly degenerate (E irrep), is composed of two modes having exactly the same frequency. This is equivalent to a particle moving in a 2D well whose energy contours are perfectly circular. Anharmonicity may lift this degeneracy, so that the contours resemble an ellipse, leading to different frequencies for motion along its two axes. With increasing temperature, the motion samples regions further


Page 36 of 61

1 4 0 0

1 6 0 0

) +

2 b (N H

4

)

3

2 b (N H

3 3 9 8

3 2 1 9

2 sU

+

3 0 3 8 2 U ⊥( N H 4 )

25 00

24 00

g

g

b e n d (1 8 0 K ) 6

1 0 0 2 0 0

4 0 0

6 0 0

6 0 0

1 0 0 0

1 1 0 0

1 2 0 0

1 3 0 0

1 0 0 0

1 2 0 0

1 4 0 0

1 6 0 0

1 8 0 0


4

+

) 2 0 0 0

2 2 0 0

2 4 0 0

3 2 0 8

a s(N H

+

3 3 8 6 2 b (N H 4 ) 3 4 0 7

3

2 b (N H

2 U 3 0 0 9

× 150

2 2 0 0 2 3 0 0 2 4 0 0 2 5 0 0 2 6 0 0

-1

4

+

N H ⊥(

4

3

b (N H

b (N H

2 0 0 0 + 2 0 7 1 b ( N H 4 ) + w u 2 1 7 8

2 2 7 4

+

1 4 0 0

7 0 0

8 0 0

) +

) +

) + 4

+

N H ⊥(

1 8 0 0

× 120

× 10

1 7 0 0

2 3 9 4 2 s U

1 6 0 0

20 00

9 0 0

0 .0

5 0 0

1 5 0 0

× 40

w

1 4 0 0

× 103 4 0 0

1 6 6 5

× 103

6 0 0

1 2 2 9

5 0 0

U

w 1 3 1 0

g

w

+ 2 N N

sU

+ u

w 4 0 0

9 4 9 9 6 2 × 10

3 0 0

7 4 4

2 0 0

1 7 7 7 1 8 8 5

1 7 0 3

1 1 3 7

g + 2

w

g

4

b (N H

u

P T M

2 w

u

×104

1 6 9 1 4 3

0 .5

2 w

g

u

w

7 0 6

u

3 6 0

w

5 6 8 5 9 7 P T M

1 .0

+ N N

6 4 9

+

)

)

)

w

H *N 1

× 10 3 3 0 0 3 4 0 0 3 5 0 0

w

V A C F N 7

g

u

H 2

w

w

+

u

+ w

N

2 3 6 9

g

w ) +

0 .0

(B )

×103

4

+

1 8 0 0 0

6 0 0

20 50

5 0 0

19 50 20 0

4 0 0

9 3 7

3 0 0

23 00

+ 2 N N

+ 1 6 9 7 b ( N H 4 )

1 4 8 7

sU + N N 1 5 0 5

1 4 1 3 3

b (N H

4

0

×8

11 00 11 50 12 0

u

×103

w

w

u + N N

g + 2 w u 1 3 2 8

w 1 1 4 5 2 1 1 5 8

u

w 1 1 1 4 3

u

+ 2 N N u

w g + 9 5 4 w

3 0 0

w

1 7 9 3 s U

4

2 2 6 9 2 2 8 3

b e n d (5 0 K )

2 0 7 4

6

u

3 8 6

4 1 4

u

×105

w

g

w

w

2 0 8 2 0 0

H *N 1

g

u

×104

0 .5

- w

V A C F N 7

7 4 3

g

5 6 5 5 9 7 P T M + N N 6 5 5 P T M + w u

w 1 .0

w 2

+

H 2

a v e r a g e o f tr a j. 1 , 2 , 3

1 8 9 2 2 w g + 2 w u + 1 9 0 2 U ⊥( N H 4 ) + w u 1 9 9 5 P T M + b ( N H + ) 4 + b (N H 4) + w u 2 1 0 5 U ⊥( N H + ) + w 4 g

N

(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


×200

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3 4 0 0

3 5 0 0

3 0 0 0 3 1 0 0 3 2 0 0 3 3 0 0

3 0 0 0

3 2 0 0

3 4 0 0

3 6 0 0

)

Figure 11: The VACF spectrum for the NH∗ N angle at (A) 50 K and (B) 180 K. Insets show blow-ups of the spectrum in regions containing relatively weak peaks.

elevated from the bottom of the well, so that anharmonicity effects increase and so does this “anharmonic splitting”. Taking the HNN bending VACF from Figure S11(C) as an example, we observe wu at frequencies 351 and 386 cm−1 . The first peak is more intense and it dictates the frequency observed in the DACF (356 cm−1 ). Similarly, we observe wg at frequencies 557 and 598 cm−1 . The latter is more intense and it dictates the frequency observed in the DACF (581 cm−1 ). However, in combinations we assume that the two modes contribute equally, so we assign to each wag a frequency that is the arithmetic mean of the two modes. For wu , (351 + 386)/2 = 368 cm−1 . For wg , (557 + 598)/2 = 577 cm−1 . With this, we obtain a consistent assignment of the combination bands, as summarized in Table 7. The most complex combinations appear in the NH∗ N bending VACF (perpendicular proton motion). Figure 11 compares this VACF spectrum at 180 and 50 K. In spite of the temperature effects discussed above, one observes a remarkable 37 ACS Paragon Plus Environment


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correspondence between these two spectra, with somewhat fewer combinations at the higher temperature. This suggests that our assignments are indeed consistent.

Experimental Combination Bands and FRs Table 6 lists combination bands observed in our AIMD simulations, which are natural candidates for being manifested as experimental combination bands. However, trying to match the frequencies of the DACF combination bands with the experimental ones (e.g., dashed vertical lines in Figure 4) may be misleading, because the calculated AIMD fundamentals deviate from experiment. Without quantitative match of the fundamental frequencies it is not likely that such a match exists for the combinations. One alternative is to use the sum of the corresponding experimental fundamental frequencies. The problem, however, is that the low frequency fundamentals are not all experimentally identified. The PTM band at 374 cm−1 in Ref. 28 is based on the assumption of no significant absorption below 330 cm−1 . Given that our calculation identified the PTM around 300 cm−1 suggests that this assumption should be checked by extending the measurements to below 330 cm−1 . Furthermore, the identification of wu at 409 cm−1 is uncertain, wg was not identified and the NN stretch is not IR-active. With these reservations in mind, one can still check the identification of the series PTM+NN, PTM+2NN and PTM+3NN by Asmis and collaborators. 28 These combination bands were all observed in our AIMD results, see Table 6. If the experimental PTM identification is correct and band A’ is PTM+NN, one can obtain the “experimental” NN stretch frequency by subtraction. Then this series can indeed be estimated by frequency summation as in Table 8, which corroborates the suggestion of Asmis and collaborators, and validates the summation property. While the 6D quantal calculations for this series (Table II in Ref. 28), also ap38 ACS Paragon Plus Environment

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Table 7: Vibrational frequencies (in cm−1 ) for gas-phase N2 H+ 7 at 180 K from the VACF (various SI figures) and DACF spectra generated from a 20 ps AIMD/B3LYP-D3 trajectory. Assignment i-rot wg − wu PTM NN wu wg PTM+NN PTM+wu 2wu wg + wu 2wg sU aU wg + 2wu sU +NN U⊥ (NH+ 4) b(NH3 ) b(NH+ 4) U⊥ (NH+ 4 ) + wu b(NH+ ) 4 + wu b(NH3 ) + wg b(NH+ 4 ) + wg 2sU 2U⊥ (NH+ 4) 2b(NH3 ) ss(NH+ 4) ss(NH3 ) 2b(NH+ 4) as(NH+ 4) as(NH3 ) a b c

VACF suma DACF Figure b 66 fund. inactive S13 209 209 208 S13 287 fund. – S14(B) 311 fund. inactive S14(D) 368c fund. 356 S11 c 577 fund. 581 S11 597 598 594? S15 649 655 637 S15 744 736 753? S15 949 945 936 S15 1137 1154 1142 S15 1220 fund. 1232 S12 1269 fund. 1255 S12 1310 1313 1293? S15 1495? 1531 1495 S15 1528 fund. 1521 S10(C) 1605? fund. 1595 S10(C) 1699 fund. 1687 S11(C) 1885 1896 1894? S15 2071 2067 2054? S15 2178 2182 2180 S15 2274 2276 2271 S15 2394 2440 ? S15 3009 3056 3016 S15 3208 3210 3225 S15 3309 fund. 3310 S9(C) 3319 fund. 3315 S9(B) 3386 3398 3384 S15 3405 fund. S9(C) 3407 fund. 3409 S9(B)

Sum of the fundamental VACF frequencies appearing in this table. fund. = fundamental band. Average of two VACF frequencies, see text.



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Table 8: Suggested series of combinations of the PTM with NN harmonics. 28 band

assignment

A’ C’ F’

PTM+NN PTM+2NN PTM+3NN

Experiment 6D model freq.a Sumb freq.c Sumd 743 743 819 905 1097 1112 1118 1339 1451 1481 1429 1773

a

Experimental values, see Figure 4(A). Arithmetic sum of the corresponding experimental frequencies, assuming PTM and NN frequencies of 374 and 369 cm−1 , respectively. c Restricted dimensionality quantum calculation (6D model). 28 b Arithmetic sum of the corresponding 6D model PTM and NN frequencies of 471 and 434 cm−1 , respectively. 28 b

pear to agree with experiment, they do so by complete violation of the summation property (last column in Table 8). The assignment of band F’ to PTM+3NN is particularly doubtful. Another alternative is to utilize the VPT2 frequencies, which are based on full-dimensional ab initio, rather than on partial-dimensional nuclear dynamics. For this end, we select from the VPT2 results in Table S3 combinations that were found in our AIMD calculations (Table 6). For triple combinations, we simply add the frequency of the third fundamental to that of the binary VPT2 combination. These results are collected in Table 9. The agreement with experiment is better than 60 cm−1 . In comparison with Asmis and coworkers, it agrees with their identification of PTM+NN as band A’ and PTM+2NN as band C’. Between these two bands, the table tentatively identifies four additional low-frequency combinations involving the wags, with smaller experimental peaks marked by us in Figure 4(A). The 6D model could not have identified these, because it did not include the wags. At higher frequencies there is disagreement: The PTM+3NN combination appears at a lower frequency than in the 6D model. Its frequency coalesces with that of band D’, rather than F’. Band D’ was previously assigned to the aU mode. 28 However, both AIMD/B3LYP-D3 and VPT2/MP2 calculations located aU at a 40 ACS Paragon Plus Environment

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Table 9: Combination band frequencies (in cm−1 ) involving the PTM, NN and wu modes of gas-phase N2 H+ 7 from VPT2 (at the MP2(full)/6311++G(d,p) level), as compared with the IRMPD experiment. 28 mode Tbl. S3 20+14 20+15

18+16 7+18 20+17 2 × 18 7+20

VPT2a IRMPD freq. freq. band NN + aU 1476 1451 F’ NN + sU 1472 1415 E’ PTM+3NN 1329b 1325c D’ PTM+2NN 1058d 1097 C’ wg + wu 1025 998 B2 e PTM + wu 939 938 B’ NN + wg 889 904 A4 2wu 809 824 A2 PTM+NN 787 743 A’ assignment

a

See Table 6. By addition of the NN frequency to PTM+2NN. c Possibly a FR between PTM+3NN and aU . d By addition of the NN frequency to PTM+NN. e VPT2 violates the summation rule here, as the sum of the PTM and wu frequencies is 893, lower than 939. b

frequency that is lower by 60 – 80 cm−1 than band D’ (Table 4). From the AIMD calculation, PTM+3NN occurs near 1200 cm−1 , as opposed to 1329 cm−1 in VPT2. If the true answer was inbetween these two values, it would be close to the calculated aU frequency leading, perhaps, to yet another FR. Indeed, PTM+3NN and aU share not only the same A1 symmetry, but also the same asymmetry toward inversion along the NN axis. Thus, in this resonance, the umbrella and PTM get mutually enhanced, leading to the particularly strong and blue shifted band D’ in the spectrum. The F’ band, in turn, is now assigned to NN+aU , whereas the previously unassigned band E’ is NN+sU . Although the difference between the NN+aU and NN+sU frequencies in the VACF calculation is only a few wavenumbers, in our VACF/AIMD results this difference is about 70 cm−1 (the average of the two calculations is 35 cm−1 , in agreement with the experimental difference between



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the F’ and E’ bands). Which assignment is correct then? Clearly, until the measurements cover the whole spectral range and theory generates the correct fundamental frequencies (PTM and wag), we might not be in a position to settle this question. Here, the study of isotopologues may be of help.

Isotope Effects In an effort to understand why this small cluster has such a complex spectrum, we have investigated an isotopologue with deuterium (D+ ) replacing the central proton (H∗+ ). With the increase in mass, D+ will sit deeper in its well on the ammonium side, and the distinction between the two parts of the cluster will increase. Under this situation one might expect less coupling between the two moieties (ammonia and ammonium), and as a result a simplified IR spectrum. Indeed, all the intense modes are now fundamentals, none is a combination band. The VPT2 and AIMD (DACF and VACF) fundamental frequencies are summarized in Table 10. Figure 12 shows the AIMD stretching modes in the high frequency region. As expected, these are very similar to the undeuterated case in Figure 3, with the four peaks red-shifted by about 15 cm−1 . In comparison, the VPT2 frequencies are only barely shifted. The overtone of the ammonia bend is slightly blue-shifted here, so it appears inbetween the two as bands. Figure 13 shows the remainder of the AIMD/DACF spectrum, with three VACF spectra to help in identifying the peaks. As compared to the undeuterated cluster, for which the b(NH+ 4 ) band was not seen clearly in the DACF spectrum, but deduced from the bending VACF to occur around 1700 cm−1 (Figure 6), the b(NH3 D+ ) band is now seen as a weak DACF peak at 1705 cm−1 , corroborating our assignment. Indeed, most of the frequencies do not change much, agreeing with Figure 6 in Ref. 54, which shows that D in the center causes only little red-shift 42 ACS Paragon Plus Environment

Page 42 of 61

+

N D

N H

D A C F 3

3 3 1 3

3

2 b (N H 3

D

+

) 3 4 0 4

×1 0 3

3 3 3 0

0 .5

5 0 K

3 4 1 7

H

3 4 2 7

(A ) 1 .0

3 3 0 0

3 3 2 0

0 .0 H

3 3 2 9

(B ) 1 .0 ss(N H 3

3

N

)

1

+

N D

N H 3

H s tre tc h

V A C F

5 0 K

(N H

m o ie ty ) 3

3 4 2 0

0 .5 a s(N H

) 3

ss(N H 3

D

+

)

3 3 2 9

(C ) 1 .0

ss(N H 3

3 4 0 5

0 .0 3 3 1 3


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


3 4 1 1

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) a s(N H

×1 0

0 .5

3

D

+

H N 6

3

N D

+

N H 3

V A C F

H s tre tc h (N H 3

5 0 K D

+

m o ie ty )

)

3

×3 5

0 .0 3 2 5 0

3 3 0 0

3 3 2 0

3 3 0 0

3 3 2 0

3 3 0 0

3 3 4 0

3 3 5 0

3 4 0 0


-1

3 4 5 0

3 5 0 0

)

Figure 12: The simulated AIMD/B3LYP-D3 DACF IR spectrum of the H3 ND+ NH3 cluster in the range above 3200 cm−1 (panel A) is decomposed into the contributions from the NH stretch of the NH3 (panel B) and NH+ 4 (panel C) moieties, respectively, using partial VACF analysis. (sometimes even a blue shift). The exception are the U⊥ (NH3 D+ ) and sU modes, which red shift by over 100 cm−1 (Figure 14). In comparison, it is difficult to decide how credible is the VPT2 spectrum in this case (Table S4 in the SI). It produced infinite intensities for 14 out of 21 fundamental modes, an imaginary i-rot frequency and a deuterium transfer mode (DTM) that moved up to 882 cm−1 . Thus a major effect of deuterating the excess proton on the AIMD spectrum is on diminishing the intensity of all the peaks near the U⊥ (NH3 D+ ) band. Here the most conspicuous is the absence of the 1642 cm−1 band from the spectrum. This band, which is the most intense in the N2 H+ 7 DACF spectrum [Figure 4(B)], was attributed by us to a FR between aU + wu (1637 cm−1 ) and b(NH3 ) (1613 cm−1 ). Here, the aU + wu combination is the strongest peak in the NDN VACF spectrum 43 ACS Paragon Plus Environment

6 9 9

×1 0 7

×1 0 4

3 0 0

0 .0 1 0 0

×3 0

7 0 0

2 0 0

2 0 0

8 0 0

1 6 2 3

1 5 2 9

×6 0

×1 0 6

1 5 2 7

1 6 1 3

1 7 1 5

b (N H 3)

0 7 ×1

1 4 1 2

b

1 7 0 4 + (N H 3 D ) ×1 0 4

×1 0

5

1 6 0 9

b (N H 3)

1 5 2 8

a U + N N

U

1 4 1 2 + ⊥( N H 3 D )

sU + P T M

1 7 0 5

1 6 3 8

15 50

15 00 14 00

1 2 4 2

×1 0 4

1 6 0 3

×1 0

×1 2 0

1 2 4 7

13 00

1 1 4 3 ×4 5 0

1 0 7 0 1 1 2 9

1 1 4 0

15 50

2 w u + N N

1 0 6 7

×1 0 6

1 1 0 0

6 0 0

1 5 9 8

2 w g

×1 0 4 1 2 0 0

1 3 0 0

9 0 0

4 0 0

4 0 0

1 7 0 0

5 0 K

b e n d

2 w u

0 .5

1 7 0 0

a U + w u

V A C F 3

1 2 0 0

1 4 3 5

N H 6

1 1 0 0

1 7 0 0

1 6 0 0

15 00

+

1 0 0 0

1 3 8 1

N D D *N

9 0 2

3 1

9 0 0

15 00

5 6 8

N

8 0 0

3 6 8

2 2 0

7 0 0

H

w g - w u

(D )

4 0 0

a U

×2 0

9 2 0

3 0 0

w u + 2 N N

2 0 0

×1 0 4

w u + w g

×1 0 3

w g ×1 0

0 .0

sU

1 5 0 0

1 3 5 1

w u 0 .5

9 5 7

(5 0 K )

1 4 0 0

1 2 4 1

V A C F 3

b e n d

1 2 9 1

N H

×1 0 4

1 3 0 0

×1 5 0

1 7 0 0

1 5 0 0 1 5 5 0

×1 0 3

1

1 6 0 0

1 1 4 0

N 6

1 2 0 0

0

+

N D

8 5 1

3 4 6

2 8 6

3

1 1 0 0

1 5 0 0

a U

1 0 0 0

1 1 2 6

H H N

9 0 0

1 0 6 7

(C )

8 0 0

1 2 0 0

1 1 2 9

7 0 0

1 1 0 0

10 50 11 0

4 0 0

9 2 0

3 0 0

sU

×3 0 0

w g

w g + N N

×1 0 0

(5 0 K )

×1 0 6

×1 0 6 3 0 0

V A C F 3

b e n d

2 w u + N N

N H

6

9 5 9

3 5 8

1

1 2 0 0

1 0 0 0

w u + w g

N

w u

0 .0

1 .0

+

N D 3

H N

9 0 0

9 5 9

2 8 5

H

2 0 0

8 5 0 9 0 0

8 0 0

2 w g - w u

(B )

0 .5

7 0 0

7 0 0

7 8 0

0 .0

5 0 0

5 7 2

4 0 0

1 .0

6 0 0

6 0 0

1 1 0 0

2 0 0

1 .0

5

×1 0 4

2 8 6

×4 5 0

9 2 3

×1 0 5

2 1 6 1 7 2

×1 0 5

0 .5

1 4 1 7

(5 0 K )

Page 44 of 61

15 50

5 6 6

3 5 0

D A C F 3

1 3 5 3

N H

1 1 3 3

+

N D 3

9 0 5

H

7 8 2 8 5 4

(A ) 1 .0


8 0 0

1 0 0 0


1 4 0 0

1 2 0 0 -1

1 5 0 0

1 4 0 0

1 6 0 0

1 8 0 0

)

Figure 13: The simulated AIMD/B3LYP-D3 DACF IR spectrum of the H3 ND+ NH3 cluster (panel A) is decomposed into the contributions from the HNN bends (panels B and C) and the ND∗ N bend (panel D), using partial VACF analysis. 1 8 0 0 1 6 0 0

+

N 2

H

H 3

N D

7 +

N H 3

)

1 4 0 0

F re q u e n c y (c m

-1

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

) b( N

M o d e

b( N

H

N

H

4

U

3

aU

w

w

N

sU

u

g

+

)

⊥

2 0 0

PT 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

6 7 8


Figure 14: AIMD/DACF frequencies of the H3 ND+ NH3 cluster (red circles) as compared with those of the N2 H+ 7 cluster (blue squares).


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Table 10: Fundamental vibrational frequencies (in cm−1 ) for gas-phase H3 ND+ NH3 at 50 K from VPT2 (at the MP2(full)/6-311++G(d,p) level), DACF and/or partial-VACF spectra from AIMD with the B3LYP-D3 functional. assn. VPT2a DACF VACF as(NH3 ) 3444 3417 3420 as(NH3 D+ ) 3430 3404 3405 ss(NH3 ) 3335 3330 3329 + ss(NH3 D ) 3320 3313 3313 b(NH3 D+ ) 1605 1705 1704 b(NH3 ) 1616 1603 1613 U⊥ (NH3 D+ ) 1274 1417 1412 aU 1264 1247 1242 sU 1247 1070 1067 wg 607 572 572 wu 376 350 346 NN 263 – 286 DTM 882 286 – a

VPT2 results here are suspect, because most of the anharmonic intensities (14 of the 21 fundamentals) are infinite, see Table S4 in the SI.

(peak at 1598 cm−1 ), while the b(NH3 ) remains at about the same frequency. The resulting FR is now a weak peak at 1623 cm−1 . This corroborates our assignment of the “bending FR” and also the conclusion of lesser couplings for the deuterated H∗ isotopologue, leading to the much weaker FR. The “umbrella FR” is not seen here at all, because DTM+3NN occurs at 1152 cm−1 (by frequency sum), quite far from the sU at 1070 cm−1 . As a result, they do not interact, and the sU band in the DACF and VACF occur at nearly the same frequency. We have identified overtones and combination bands up to 1900 cm−1 , see Table 11. Not all these identifications are unique, because the frequencies of the DTM and the NN stretch are identical, and exactly half of the wg frequency. In these cases, we try to consult the VPT2 intensities and, if inconclusive, choose the assignments as for the N2 H+ 7 cluster. With this strategy, we find that most 45 ACS Paragon Plus Environment


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of the N2 H+ 7 overtones and combination bands in Table 6 are seen also for its isotopologue. Of interest is the identification of difference bands. These involve de-excitation of a hot band together with excitation of a fundamental. These are rather rare events because both anharmonic coupling and a hot band are involved. The highest probability for observing a hot band at low temperatures is for the lowest frequency mode, wu . Indeed, wg − wu is identified quite clearly around 220 cm−1 . This difference band appears also for N2 H+ 7 , but there it is superimposed on the red tail of the PTM (e.g., Figure 10(A)). Moreover, we find that wg − wu enters into combinations with other low frequency modes (specifically, DTM/NN and wg ).


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Table 11: Overtone and combination band frequencies (up to 1900 cm−1 ) of the H3 ND+ NH3 cluster at 50 K from AIMD/VACF analysis, as compared with VPT2 results. assignmenta

VACF Sumb DACF VPT2c Id Overtones 2wg 1140 1144 1143 1202 17.8 2wu 699 692 678 745 6.9 Combinations aU + wu 1598 1588 1603 1567 0.63 aU +NN 1528 1528 1529 1448 184 sU +NN 1351 1353 1353 1502 0 wg + wu 920 918 923 970 0.1 NN+wg 851 858 854 864 0.99 Higher order aU +2NN 1819 1814 1822 – – sU +2NN 1635 1639 1638 – – DTM+3NN 1129 1144 1133 – – 2wu +NN 959 978 957 – – wu +2NN 902 918 905 – – Difference bands wg − wu 220 226 216 – – wg − wu +DTM 498 512 500 – – 2wg − wu 780 798 782 – – a

It is sometimes difficult to decide whether the NN or DTM mode is involved in a combination band. b Sum of fundamental frequencies from VACF spectra, Table 10. c Averaged VPT2 frequencies (MP2(full)/6-311++G(d,p) level in Gaussian 09) for E-modes (Table S4 in the SI). d Anharmonic intensity from VPT2 analysis (Table S4 in the SI).

Discussion and Conclusions The assignment of the IR spectrum of the protonated ammonia dimer is a challenging task, because it exhibits fundamental and combination bands that are difficult to disentangle, and are further complicated by FRs. Consequently, several facets were debatable here, such as • Should this cluster be treated in D3d or C3v symmetry?



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

• Can the spectrum be calculated using classical nuclear dynamics, or quantum nuclear dynamics is a prerequisite for its analysis? • Where is the PTM? • How to recognize the combination bands and FRs? • Is a complete assignment of this spectrum at all feasible? In this work we have calculated and analyzed anharmonic spectra based on either a static or dynamic approach. In the static approach, one computes HNMs and anharmonic (VPT2) frequencies and intensities. We have used several quantum chemistry methods for this purpose, finding the MP2 results the most useful. This part is standard in quantum chemistry packages, such as Gaussian 09. In the dynamic Born-Oppenheimer approach, one simulates classical trajectories, with forces calculated every timestep using, usually, DFT. This approach allows also to include temperature effects. Here we have calculated 3 trajectories at 50 K, and a fourth one at 180 K. A fifth trajectory was for D replacing H∗ (at 50 K). We have utilized the CP2K/Quickstep program with the B3LYP-D3 functional. From the output we have computed both the DACF and partial-VACF spectra for the OH and NN distances and for the HNH, HNN, NH∗ N and HNNH angles. While the DACF part is rather customary, partial-VACFs are less frequently utilized. In this approach, one selects several bond distances and bond angles, and a dihedral angle, calculates the autocorrelation of the time-derivative of each coordinate and takes the Fourier transforms [Eq. 2]. The result is a series of vibrational bands to which the particular coordinate contributes. This allows to assign the corresponding DACF fundamental bands, in good agreement with the VPT2 assignments (Table 4).


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Theoretically, we have identified all the 21 fundamental bands. The high frequency ones are localized on the ammonia or ammonium fragments, whereas the low frequency modes are delocalized, so we have symmetric and anti-symmetric umbrella and wagging modes. This is displayed by the color codes in Figure 15. We agree with Asmis and coworkers on assigning bands D’ and G’ to the umbrella and perpendicular proton modes (magenta colored aU and U⊥ bands in Figure 15). However, D’ may actually be a FR of aU with PTM+3NN (rather than the aU fundamental). We also obtain a low-frequency PTM in reasonable agreement with experiment (the red band in Figure 15). Thus quantum nuclear dynamics is not required for obtaining these assignments. o f N 2

+

H

a U

c lu s te r 7

P T M

+ 3 N N

F e rm i re so n a n c e

a

N

+

)

+

+

w u

sU

1 3 2 5 D '

aU

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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 2 6 9 C 3

9 9 8 B 2

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

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Figure 15: Our theoretical assignments of the IRMPD spectrum of the N2 H+ 7 cluster represented by a color/pattern code. Fundamental modes are depicted by solidly filled bands, corresponding to the N2 H+ 7 colors in the inset. High frequency fundamentals are assigned to the ammonium (magenta) or ammonia (orange) moieties, low frequency wags are delocalized (green and purple), whereas the PTM and NN stretch modes involve predominantly H∗ (red) and the nitrogen atoms (blue), respectively. Combination bands are depicted by two colors (outline and pattern) corresponding to the participating fundamentals. The patterns depict stretching motions (horizontal lines), gerade/symmetric motion (“+” pattern), and ungerade/anti-symmetric motion (“×” pattern). The D’ band, which may be either a fundamental or a FR/combination band, is depicted both ways. Because their quantum dynamics was restricted to 6 dimensions, Yang et al. could not have determined with certainty the bending and wagging NMs, which we have now identified. In particular, we suggest that band H’ is the HNH bending in the ammonia fragment (hence its orange fill in Figure 15), and the ungerade 49 ACS Paragon Plus Environment


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wagging mode is their band a’. In addition, a previously unidentified band that we label C3 was assigned by us to the symmetric umbrella motion sU . Overtones and combination bands are even more intriguing. From our AIMD spectrum, we were able to identify an unprecedented number (> 30) of them, mostly in the NH∗ stretch and NH∗ N bend partial-VACFs. These occur across the whole frequency range (200 – 3500 cm−1 ) and some 5 orders of magnitude in intensity. Usually, the frequencies of these bands are within a few wavenumbers from the corresponding DACF bands. We find that they are also close to the sum of the fundamental frequencies involved in the combinations, in spite of the large anharmonicity inherent in this system. This served as useful criteria for identifying them, as summarized in Table 6. The AIMD combinations bands include all the high-intensity VPT2 combinations and additional ones, including higher order combinations that are not reported in the VPT2 output. The AIMD combination bands are characterized by the following two properties: 1. All of them involve either a wag or the NN stretch as one of the components in the combination. 2. For the high intensity ones the direct product of the participating fundamentals includes the totally symmetric representation. Hence for binary combinations both fundamentals have the same symmetry. We agree with Asmis and collaborators on identifying bands A’ and C’ as depicting the progression PTM+NN and PTM+2NN, respectively. The VPT2 frequencies provide us with a tentative assignments for additional experimental peaks (Table 9). For example, we identify band B’ as PTM+wu and band F’ as the NN+aU combination (rather than PTM+3NN as in Ref. 28).


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Most intriguing are the two Fermi resonances (FR). FRs occur between a combination and a fundamental possessing the same symmetry and similar frequencies, leading to frequency and intensity shifts that may confuse the spectrum quite a bit. Cotton noted that “papers reporting hasty and superficial vibrational analyses of complex molecules display a deplorable tendency to employ “Fermi resonance” as an escape from difficulties or inconsistencies without adequate evidence” (Ref. 55, p. 341). We suggest that excessive deviation of the partial-VACF from the DACF frequency might give evidence for a FR shift, because the single-coordinate VACF may not possess the couplings that lead to the FR. In this vein we find an inexplicable shift between the VACF-identified bending mode, b(NH3 ), at 1613 cm−1 , and the tallest peak in the DACF spectrum at 1642 cm−1 . We note that the frequency sum of the aU + wu is 1633 cm−1 , while its symmetry is the same as that of b(NH3 ). So, by FR, aU + wu blue shifts (to 1642 cm−1 ) and gains in intensity, while b(NH3 ) red-shifts and loses in intensity, producing the H’ band at 1596 cm−1 . This example demonstrates the complexity introduced by FRs. Even the most accurate theoretical methods for calculating anharmonic vibrational bands might have errors that are larger than the energy differences that could bring the system in- or out-of-resonance. As a result, one calculation might show a FR at one energy, and another at a different energy or none at all. Indeed, we find that as the temperature is raised (180 K), or D is substituted for H∗ , the 1642 cm−1 FR loses most of its intensity. At 50 K, either aU + wu or sU + wu might get into resonance with either b(NH3 ) or b(NH+ 4 ). Which of these is manifested experimentally? Summation of the experimental frequencies gives 1734 rather than 1633 cm−1 . So this might come −1 into resonance with the b(NH+ rather than with b(NH3 ). The 4 ) band at 1700 cm

FR could then shift aU +wu to around 1750 cm−1 , which is in a spectral window not



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yet probed by experiment. These possibilities suggest that a more comprehensive experimental investigation is in place, which should systematically cover the whole spectral region rather than just segments thereof. With increasing temperature (from 50 to 180 K), we observed three interesting changes in the DACF and VACF spectra: 1. Intensity changes: The PTM lost its intensity, whereas b(NH+ 4 ) gained. Some of the combination bands and FRs decreased in intensity or became unobservable. 2. Anharmonic frequency shifts: Several fundamental bands red shifted (ca. 20 cm−1 ) as the hotter trajectory sampled higher and less harmonic regions of the potential energy surface. 3. Anharmonic splitting: Low frequency doubly degenerate modes (the wags) split into two bands in the bending VACFs. The splitting increased in magnitude with increasing T . The split bands appear to enter into VACF combinations with their average frequency. It would be interesting to test some of these predictions from experiment. In addition, it would be interesting to probe the H3 ND+ NH3 isotopologue experimentally. Theory suggests a simpler spectrum for it, with all major peaks being fundamentals, and only minor frequency shifts compared to N2 H+ 7 at the same T . Thus assignment of the experimental H3 ND+ NH3 spectrum could greatly assist in achieving complete assignment of the N2 H+ 7 spectrum. Finally, it would be interesting to record and analyze the high resolution vibration-rotation spectra of these clusters. Such data might shed new light on the issues discussed here, for example, whether the cluster is best treated in C3v or D3d symmetry.


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Supporting Information Available PDF with full reference for Gaussian 09; Tables S1 and S2, optimized structural parameters for N2 H+ 7 in the C 3v and D 3d symmetries, respectively; Tables S3 + and S4, harmonic and anharmonic (VPT2) frequencies of N2 H+ 7 and H3 ND NH3

in C 3v symmetry; Figure S1, geometry of N2 H+ 7 in D 3d symmetry; Figure S2, AIMD/DACF spectrum for various window functions; Figure S3, harmonic, static anharmonic (VPT2) and dynamic anharmonic VACF spectra of NH3 and NH+ 4; Figure S4, time evolution of the NH and N1 H∗ distances from 3 N2 H+ 7 trajectories at 50 K, one 180 K trajectory, and one H3 ND+ NH3 trajectory; Figure S5, har+ monic and anharmonic IR spectra of N2 H+ 7 and H3 ND NH3 at the MP2(full)/6-

311++G(d,p) level; Figure S6, harmonic vibrational modes of N2 H+ 7 ; Figure S7, partial VACF spectrum for HNH∗ bending in N2 H+ 7 from AIMD/B3LYP-D3; Figure S8, conserved energies for five trajectories; Figures S9 to S15, partial VACF spectra for the N2 H+ 7 cluster at 180 K from AIMD/B3LYP-D3; Figure S16, comparison of the DACF spectra from three N2 H+ 7 trajectories; Figures S17 to S20, partial VACF spectra for the H3 ND+ NH3 cluster at 50 K; Excel file with harmonic + and anharmonic (VPT2) frequencies of NH3 , NH+ 4 and N2 H7 (in both C 3v and D 3d

symmetries) and H3 ND+ NH3 (C3v symmetry) using different quantum chemistry methods and basis sets. This material is available free of charge via the Internet at http://pubs.acs.org/. Acknowledgment. We thank Knut R. Asmis for the data displayed in Figure 4(A). This research was supported by a Golda Meir Fellowship to H.W. and 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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References (1) Zundel, G. Hydration Structure and Intermolecular Interaction in Polyelectrolytes. Ang. Chem. Intl. Ed. 1969, 8, 499–509. (2) Agmon, N. The Grotthuss Mechanism. Chem. Phys. Lett. 1995, 244, 456– 462. (3) Asmis, K. R.; Pivonka, N. L.; Santambrogio, G.; Br¨ ummer, M.; Kaposta, C.; Neumark, D. M.; Wöste, L. Gas-Phase Infrared Spectrum of the Protonated Water Dimer. Science 2003, 299, 1375–1377. (4) Hammer, N. I.;

Diken, E. G.;

Roscioli, J. R.;

Johnson, M. A.;

Myshakin, E. M.; Jordan, K. D.; McCoy, A. B.; Huang, X.; Bowman, J. M.; Carter, S. The Vibrational Predissociation Spectra of the H5 O+ 2 ·Rgn (Rg=Ar, Ne) Clusters: Correlation of the Solvent Perturbations in the Free OH and Shared Proton Transitions of the Zundel Ion. J. Chem. Phys. 2005, 122, 244301. (5) McCunn, L. R.; Roscioli, J. R.; Johnson, M. A.; McCoy, A. B. An H/D Isotopic Substitution Study of the H5 O+ 2 · Ar Vibrational Predissociation Spectra: Exploring the Putative Role of Fermi Resonances in the Bridging Proton Fundamentals. J. Phys. Chem. B 2008, 112, 321–327. (6) Douberly, G. E.; Walters, R. S.; Cui, J.; Jordan, K. D.; Duncan, M. A. Infrared Spectroscopy of Small Protonated Water Clusters, H+ (H2 O)n (n =2– 5): Isomers, Argon Tagging, and Deuteration. J. Phys. Chem. A 2010, 114, 4570–4579. (7) Guasco, T. L.; Johnson, M. A.; McCoy, A. B. Unraveling Anharmonic Effects in the Vibrational Predissociation Spectra of H5 O+ 2 and Its Deuterated Analogues. J. Phys. Chem. A 2011, 115, 5847–5858. 54 ACS Paragon Plus Environment

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(8) Sauer, J.; Döbler, J. Gas-Phase Infrared Spectrum of the Protonated Water Dimer: Molecular Dynamics Simulation and Accuracy of the Potential Energy Surface. ChemPhysChem 2005, 6, 1706–1710. (9) Kaledin, M.; Kaledin, A. L.; Bowman, J. M.; Ding, J.; Jordan, K. D. Calculation of the Vibrational Spectra of H5 O+ 2 and its Deuterium-Substituted Isotopologues by Molecular Dynamics Simulations. J. Phys. Chem. A 2009, 113, 7671–7677. (10) Vendrell, O.; Gatti, F.; Meyer, H.-D. Full Dimensional (15-Dimensional) Quantum-Dynamical Simulation of the Protonated Water Dimer. II. Infrared Spectrum and Vibrational Dynamics. J. Chem. Phys. 2007, 127, 184303. (11) Kulig, W.; Agmon, N. Both Zundel and Eigen Isomers Contribute to the IR Spectrum of the Gas-Phase H9 O+ 4 Cluster. J. Phys. Chem. B 2014, 118, 278–286. (12) Kulig, W.; Agmon, N. Deciphering the Infrared Spectrum of the Protonated Water Pentamer and the Hybrid Eigen-Zundel Cation. Phys. Chem. Chem. Phys. 2014, 16, 4933–4941. (13) Wang, H.; Agmon, N. Protonated Water Dimer on Benzene: Standing Eigen or Crouching Zundel? J. Phys. Chem. B 2015, 119, 2658–2667. (14) Kalish, N. B.-M.; Shandalov, E.; Kharlanov, V.; Pines, D.; Pines, E. Apparent Stoichiometry of Water in Proton Hydration and Proton Dehydration Reactions in CH3 CN/H2 O Solutions. J. Phys. Chem. A 2011, 115, 4063– 4075. (15) Thämer, M.; Marco, L. D.; Ramasesha, K.; Mandal, A.; Tokmakoff, A. Ultrafast 2D IR Spectroscopy of the Excess Proton in Liquid Water. Science 2015, 350, 78–82. 55 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

(16) Kulig, W.; Agmon, N. A ‘Clusters-in-Liquid’ Method for Calculating Infrared Spectra Identifies the Proton Transfer Mode in Acidic Aqueous Solution. Nature Chem. 2013, 5, 29–35. (17) Tanner, C.; Manca, C.; Leutwyler, S. Probing the Threshold to H Atom Transfer Along a Hydrogen-Bonded Ammonia Wire. Science 2003, 302, 1736–1739. (18) Manca, C.; Tanner, C.; Leutwyler, S. Excited State Hydrogen Atom Transfer in Ammonia-Wire and Water-Wire Clusters. Intern. Rev. Phys. Chem. 2005, 24, 457–488. (19) Schwarz, H. A. Gas Phase Infrared Spectra of Ammoniated Ammonium Ions. J. Chem. Phys. 1980, 72, 284–287. (20) Price, J. M.; Crofton, M. W.; Lee, Y. T. Vibrational Spectroscopy of the Ammoniated Ammonium Ions NH+ 4 (NH3 )n , (n = 1–10). J. Phys. Chem. 1991, 95, 2182–2195. (21) Meuwly, M.; Karplus, M. Simulation of Proton Transfer along Ammonia Wires: An “ab initio” and Semiempirical Density Functional Comparison of Potentials and Classical Molecular Dynamics. J. Chem. Phys. 2002, 116, 2572–2585. (22) Wang, B.-C.; Chang, J.-C.; Jiang, J.-C.; Lin, S.-H. Ab initio Study of the Ammoniated Ammonium Ions NH+ 4 (NH3 )0−6 . Chem. Phys. 2002, 276, 93– 106. (23) Fouqueau, A.; Meuwly, M. Structures and Dynamics of Protonated Ammonia Clusters. J. Chem. Phys. 2005, 123, 244308.


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(24) Asada, T.; Haraguchi, H.; Kitaura, K. Simulation Studies of Proton Transfer in N2 H+ 7 Cluster by Classical ab Initio Monte Carlo and Quantum Wave Packet Dynamics. J. Phys. Chem. A 2001, 105, 7423–7428. (25) Tono, K.; Bito, K.; Kondoh, H.; Ohta, T.; Tsukiyama, K. Infrared Photodissociation Spectroscopy of Protonated Ammonia Cluster Ions, NH+ 4 (NH3 )n (n = 5 − 8, by Using Infrared Free Electron Laser. J. Chem. Phys. 2006, 442, 206–211. (26) Tono, K.; Fukazawa, K.; Tada, M.; Fukushima, N.; Tsukiyama, K. Infrared Photodissociation Spectroscopy of NH+ 4 (NH3 )n (n = 3 and 4) by Using Free Electron Laser. Chem. Phys. Lett. 2007, 125, 224305. (27) Asmis, K. R.; Yang, Y.; Santambrogio, G.; Br¨ ummer, M.; Roscioli, J. R.; McCunn, L. R.; Johnson, M. A.; K¨ uhn, O. Gas-Phase Infrared Spectroscopy and Multidimensional Quantum Calculations of the Protonated Ammonia Dimer N2 H+ 7 . Ang. Chem. Intl. Ed. 2007, 46, 8691–8694. (28) Yang, Y.; K¨ uhn, O.; Santambrogio, G.; Goebbert, D. J.; Asmis, K. R. Vibrational Signatures of Hydrogen Bonding in the Protonated Ammonia Clusters NH+ 4 (NH3 )1−4 . J. Chem. Phys. 2008, 129, 224302. (29) Yang, Y.; K¨ uhn, O. H/D Isotope Effects on the Geometry and Infrared Spectrum of the Protonated Ammonia Dimer. Chem. Phys. Lett. 2011, 505, 1–4. (30) Scheiner, S. Theoretical Studies of Proton Transfers. Acc. Chem. Res. 1985, 18, 174–180. (31) VandeVondele, J.; Krack, M.; Mohamed, F.; Parrinello, M.; Chassaing, T.; Hutter, J. QUICKSTEP: Fast and Accurate Density Functional Calculations Using a Mixed Gaussian and Plane Waves Approach. Comput. Phys. Commun. 2005, 167, 103–128. 57 ACS Paragon Plus Environment


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(32) Bloino, J.; Barone, V. A Second-Order Perturbation Theory Route to Vibrational Averages and Transition Properties of Molecules: General Formulation and Application to Infrared and Vibrational Circular Dichroism Spectroscopies. J. Chem. Phys. 2012, 136, 124108. (33) Gaigeot, M.-P. Theoretical Spectroscopy of Floppy Peptides at Room Temperature. A DFTMD Perspective: Gas and Aqueous Phase. Phys. Chem. Chem. Phys. 2010, 12, 3336–3359. (34) Thomas, M.; Brehm, M.; Fligg, R.; Vöhringer, P.; Kirchner, B. Computing Vibrational Spectra from ab initio Molecular Dynamics. Phys. Chem. Chem. Phys. 2013, 15, 6608–6622. (35) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Scalmani, G.; Barone, V.; Mennucci, B.; Petersson, G. A. et al. Gaussian 09 Revision D.01. Gaussian Inc.: Wallingford, CT, 2013. (36) Becke, A. D. Density-Functional Exchange-Energy Approximation with Correct Asymptotic Behavior. Phys. Rev. A 1988, 38, 3098–3100. (37) Lee, C.; Yang, W.; Parr, R. G. Development of the Cole-Salvetti Correlation Energy Formula into a Function of the Electron Density. Phys. Rev. B 1988, 37, 785–789. (38) Grimme, S. Semiempirical Hybrid Density Functional with Perturbative Second-Order Correlation. J. Chem. Phys. 2006, 124, 034108. (39) Zhao, Y.; Truhlar, D. G. The M06 Suite of Density Functionals for Main Group Thermochemistry, Thermochemical Kinetics, Noncovalent Interactions, Excited States, and Transition Elements: Two New Functionals and Systematic Testing of Four M06-Class Functionals and 12 Other Functionals. Theor. Chem. Acc. 2008, 120, 215–241. 58 ACS Paragon Plus Environment

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(40) Adamo, C.; Barone, V. Toward Reliable Density Functional Methods Without Adjustable Parameters: The PBE0 Model. J. Chem. Phys. 1999, 110, 6158–6170. (41) Grimme, S.; Antony, J.; Ehrlich, S.; Krieg, H. A Consistent and Accurate Ab Initio Parametrization of Density Functional Dispersion Correction (DFT-D) for the 94 Elements H-Pu. J. Chem. Phys. 2010, 132, 154104. (42) Goedecker, S.; Teter, M.; Hutter, J. Separable Dual-Space Gaussian Pseudopotentials. Phys. Rev. B 1996, 54, 1703–1710. (43) Martyna, G. J.; Tuckerman, M. E. A Reciprocal Space Based Method for Treating Long Range Interactions in Ab Initio and Force-Field-Based Calculations in Clusters. J. Chem. Phys. 1999, 110, 2810–2821. (44) VandeVondele, J.; Hutter, J. An Efficient Orbital Transformation Method for Electronic Structure Calculations. J. Chem. Phys. 2003, 118, 4365–4369. (45) Heine, N.; Fagiani, M. R.; Rossi, M.; Wende, T.; Berden, G.; Blum, V.; Asmis, K. R. Isomer-Selective Detection of Hydrogen-Bond Vibrations in the Protonated Water Hexamer. J. Am. Chem. Soc. 2013, 135, 8266–8273. (46) Horn´ıˇcek, J.; Kaprálová, P.; Bouˇra, P. Simulations of Vibrational Spectra from Classical Trajectories: Calibration with ab initio Force Fields. J. Chem. Phys. 2007, 127, 084502. (47) Vuilleumier, R.; Borgis, D. Transport and Spectroscopy of the Hydrated Proton: A Molecular Dynamics Study. J. Chem. Phys. 1999, 111, 4251–4266. (48) Kim, J.; Schmitt, U. W.; Gruetzmacher, J. A.; Voth, G. A.; Scherer, N. E. The Vibrational Spectrum of the Hydrated Proton: Comparison of Experiment, Simulation, and Normal Mode Analysis. J. Chem. Phys. 2002, 116, 737–746. 59 ACS Paragon Plus Environment


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(49) Ram´ırez, R.; López-Ciudad, T.; Kumar, P.; Marx, D. Quantum Corrections to Classical Time-Correlation Functions: Hydrogen Bonding and Anharmonic Floppy Modes. J. Chem. Phys. 2004, 121, 3972–3983. (50) Berens, P. H.; White, S. R.; Wilson, K. R. Molecular Dynamics and Spectra. II. Diatomic Raman. J. Chem. Phys. 1981, 75, 515–529. (51) NIST Standard Reference Database 101, Computational Chemistry Comparison and Benchmark DataBase, http://cccbdb.nist.gov/. (52) Gillespie, R. J. Fifty Years of the VSEPR Model. Coord. Chem. Rev. 2008, 252, 1315–1327. (53) Biczysko, M.; Panek, P.; Scalmani, G.; Bloino, J.; Barone, V. Harmonic and Anharmonic Vibrational Frequency Calculations with the Double-Hybrid B2PLYP Method: Analytic Second Derivatives and Benchmark Studies. J. Chem. Theor. Comput. 2010, 6, 2115–2125. (54) Sobczyk, L.; Obrzud, M.; Filarowski, A. H/D Isotope Effects in Hydrogen Bonded Systems. Molecules 2013, 18, 4467–4476. (55) Cotton, F. A. Chemical Applications of Group Theory, 3rd ed.; Wiley and Sons: New York, 1990.


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Graphical TOC Entry o f N 2

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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 2 6 9 C 3

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Complete Assignment of the Infrared Spectrum of ... - ACS Publications

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