MP4 Study of the Anharmonic Coupling of the Shared Proton

Article pubs.acs.org/JPCA

MP4 Study of the Anharmonic Coupling of the Shared Proton Stretching Vibration of the Protonated Water Dimer in Equilibrium and Transition States G. Pitsevich,*,† A. Malevich,† E. Kozlovskaya,† E. Mahnach,† I. Doroshenko,‡ V. Pogorelov,‡ Lars G. M. Pettersson,§ V. Sablinskas,∥ and V. Balevicius∥ †

Department of Physical Optics, Belarusian State University, Nezavisimosti ave., 4, 220030 Minsk, Belarus Taras Shevchenko National University of Kyiv, Volodymyrska str., 64\13, 01601, Kyiv, Ukraine § Department of Physics, AlbaNova University Center, Stockholm University, S-106 91 Stockholm, Sweden ∥ Faculty of Physics, Vilnius University, Sauletekio al. 9-3, LT-10222 Vilnius, Lithuania ‡

ABSTRACT: The structure and harmonic and anharmonic IR spectra of the protonated water dimer (PWD) were calculated in C1, C2, and Cs symmetry at the MP4/acc-pVTZ level of theory. We found that structure and IR spectra are practically identical in C2 and C1 symmetry, demonstrating that an equilibrium C1 configuration of the PWD is not realized. Anharmonic coupling of the shared proton stretching vibration with all other modes in the PWD in C2 and Cs symmetry was the focus of this investigation. For this purpose, 28 two-dimensional potential energy surfaces (2D PES) were built at the MP4/acc-pVTZ level of theory and the corresponding vibrational Schrödinger equations were solved using the DVR method. Differences in the coupling of the investigated mode with other modes in the C2 and Cs configurations, along with some factors that determine the red- or blue-shift of the stretching vibration frequency, were analyzed. We obtained a rather reasonable value of the stretching frequency of the bridging proton (1058.4 cm−1) unperturbed by Fermi resonance. The Fermi resonance between the fundamental vibration ν7 and the combined vibration ν2 + ν6 of the same symmetry was analyzed through anharmonic secondorder perturbation theory calculations, as well as by 3D PES constructed using Q2, Q6, and Q7 as normal coordinates. A significant (up to 50%) transfer of intensity from the fundamental vibration to the combined one was found. We have estimated the frequency of the bridging proton stretching vibration in the Cs configuration of the PWD based on calculations of the intrinsic anharmonicity and anharmonic double modes interactions at the MP4/acc-pVTZ level of theory (1261 cm−1). The structure of H5O2+ has been determined from experiments using X-ray diffraction of crystal hydrates, and it was found that the protonated water dimer (PWD) has C2 symmetry.9 In addition, early experimental works have revealed that the excess positive charge in PWD is localized to a symmetrically hydrated proton.8,10 In the IR spectra11,12 of a number of crystal hydrates, broad and intense absorption bands were observed in the spectral range 1000−2000 cm−1. Similar spectral studies of acidic and alkaline solutions, presented in a series of papers,10,13−15 have confirmed the presence of strong absorption bands in this spectral range. Measuring the IR absorption spectrum of the PWD in the gas phase has been challenging. Only in the late 1980s the group of Lee has developed two methods of measuring the IR spectra of the PWD in the gas phase. The first method is based on use of a carrier gas such as hydrogen or some other inert gas.16 The second technique involves infrared multiphoton dissociation

1. INTRODUCTION Water is the key element for life and, therefore, attracts strong attention by the scientific community. Properties of water are investigated and discussed constantly; however, scientists are still far from their full understanding.1−6 There are two outstanding properties of water which determine its unique role in nature and, in particular, in physics and chemistry: strong intermolecular hydrogen bonding, due to which water is in the liquid state under ambient conditions, and the possibility for water complexes to dissociate and form hydronium (H3O+) and hydroxide ions (OH−) which in turn become hydrated. Clusters of the H3O+·(H2O)n type play an important role among such associates. Studies of the hydrated proton in bulk water or in compounds containing water are important for many areas of chemistry and biology.7 The protonated water dimer H5O2+ or Zundel ion8 takes a special place among these clusters. It is one of the simplest protonated water clusters, and its properties are important for a better understanding of mechanisms of proton transfer as well as of properties of the hydrogen bond. © 2017 American Chemical Society

Received: January 17, 2017 Published: February 10, 2017 2151

DOI: 10.1021/acs.jpca.7b00536 J. Phys. Chem. A 2017, 121, 2151−2165

Article

The Journal of Physical Chemistry A (IRMPD).17,18 However, the infrared spectral range used in this work was very narrow (3500−3800 cm−1). Later IR absorption spectra of the PWD in the gas phase were obtained in the spectral region of vibrations of the central proton (600−2000 cm−1).19,20 Theoretical studies of the PWD structure have a very long history.7,21−27 It is notable that calculations in the early 1970s predicted an O−O distance in PWD equal to 2.4 Å, which is in good agreement with the value obtained in more recent calculations. A significant contribution to the understanding of the structural flexibility of the PWD was made by Fritz Schaefer and co-workers who showed28,29 that the structural parameters of the equilibrium configuration of the PWD and, in particular, its symmetry are very sensitive to the choice of basis set and method of including electron correlation. Taking into account symmetry restrictions, equilibrium configurations having Cs, D2h, and C2 symmetry were found using different levels of theory.28,29 Without any restrictions on the symmetry, the optimizations led to the C1 conformer, but depending on the level of theory used, this C1 conformer was close to either Cs or C2. More recent calculations predict the most stable conformer of the PWD to have C2 symmetry.28−30 However, the calculations also predict that the difference in energy of the C2 and Cs configurations is very small (about 60 cm−1), and that the potential energy surface (PES) for the central proton movement along the hydrogen bridge is very flat, which leads to a large amplitude in the stretching vibration.28−30 It should be kept in mind that all of the methods predict the C 2 configuration to be a true minimum, while the other configuration (Cs) is a transition state. It means that even at increased temperature both configurations cannot exist simultaneously. The application of more advanced basis sets and more precise methods of accounting for electron correlation leads to the conclusion that, even without symmetry restrictions in the optimized configuration of the PWD, the proton is closer to the center and the geometry is closer to C2 symmetry, although, as noted in ref 29, this behavior is not trivial. Theoretical analysis of the IR spectra of the PWD is undoubtedly more challenging than its structural analysis. It was shown that IR spectra of the PWD for Cs and C2 configurations calculated in the harmonic approximation differ substantially. 28−30 In particular, in the case of the C s configuration, the calculations predict four absorption bands in the region of the stretching vibrations of free O−H groups, while, in the case of the C2 configuration, only two intense spectral bands are expected. This fact allows establishing that the “carrier gas” method of the PWD IR measurement involves stabilizing the Cs configuration due to the interaction with the carrier molecules, while usage of the second technique retains the C2 symmetry of the PWD geometry.18,31 However, it is clear that the central proton vibrations cannot be properly described in the harmonic approximation due to strong anharmonicity of the PES.29,30 In ref 32, calculations of the IR spectrum of the PWD were carried out in the anharmonic approximation with the cc-VSCF method. It was shown that accounting for anharmonicity effects leads to a blue-shift of the stretching vibration of the central proton (from 871 cm−1 in the harmonic approximation to 1209 cm−1). The authors of ref 32 comparing results of their calculations with experimental data noted that the experimental values are in the range between those obtained in the harmonic and anharmonic approximations. In ref 33, anharmonic calculations of the IR spectrum of

the PWD were carried out using second-order perturbation theory (PT2). However, this was applied only to the stretching vibrations of the O−H bonds of peripheral water molecules. It is commonly understood34−36 that in the case of very anharmonic PES (like for the PWD) the PT2 anharmonic approximation could be insufficient to correctly estimate vibrational frequencies. In such cases, calculations of PES with restricted set of the vibrational coordinates are needed.37−39 The 1D and 2D PES for the stretching vibrations of the central proton and the stretching O···O vibrations of the hydrogen bridge were for the first time calculated in ref 40. The subsequent numerical solution of the Schrödinger equation revealed that for the stretching vibrations of the central proton of the PWD both strong intrinsic anharmonicity and strong anharmonic interaction with the stretching O···O vibration of the hydrogen bridge are typical. In refs 41 and 42, in addition to the two coordinates used in ref 40, one and two bending coordinates of the central proton were taken into consideration, respectively. Thus, the dimensionality of the vibrational problem to be solved increased to 3D and 4D. Using the MP2/T(O)DPZ level of theory, the authors of refs 41 and 42 showed that an increase of dimensionality of the vibrational problem leads to a further reduction in the frequency of the stretching vibration of the shared proton (1438, 1226, and 1158 cm−1 for 1D, 3D, and 4D, respectively). The fact that the calculated frequencies of the bending vibrations of the central proton (968 and 1026 cm−1 for 4D) are lower than the frequency of its stretching vibration was quite surprising. The authors of ref 20, comparing their results with the experimental data obtained in ref 42, note that the absorption bands at 990 and 1163 cm−1 can be assigned to the bending and stretching vibrations involving the central proton, respectively. However, the absorption band of medium intensity at 1337 cm−1 has no counterpart in the spectra presented in ref 42, neither among the fundamental vibrations and their overtones nor the combined frequencies. In a later work,43 a full-dimensional PES of the PWD was used (it was obtained earlier in refs 40 and 44 and abbreviated as OSS3(p) after the names of the authors of refs 40 and 44) to find the vibrational frequencies using the MULTIMODE program.45 Frequency values of the central proton vibrations, calculated in the frame of this approach, were very close to the harmonic ones (914, 1369, and 1380 cm−1 for the stretching and two bending vibrations, respectively). Later in ref 46, the OOS3(p) PES was used again. Frequencies of the PWD vibrations were determined using the correlation function quantum Monte Carlo (CFQMC) method. A significant contribution from the central proton stretching vibrations was observed for the normal vibrations at 303, 737, and 870 cm−1. For all three modes, the contribution of the wagging vibrations of water molecules is substantial. According to the calculations, the most intense infrared absorption band should be at 737 cm−1. The authors of ref 46 assigned this vibration to the very low intensity experimental band at 788 cm−1.19 However, the most intense experimental band at 921 cm−1 19 was assigned to the calculated normal mode at 870 cm−1. The experimental bands of similar intensity at 1043 and 1317 cm−1 were not assigned in the frame of the CFQMC approach, though the authors note that one of the bending vibrations of the central proton should be observed at 1353 cm−1. The OSS3(p) PES was used once again in ref 47. The IR spectrum of the PWD was calculated (1) in the framework of the 4D model using the same vibrational coordinates as in ref 42 but without adiabatic separation of 2152


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approach. Theoretical calculations of the IR spectra of the PWD and PWD+Ar at the MP2/acc-pVDZ level of theory, using normal coordinates as vibrational coordinates, were performed in ref 64. It was shown that an Ar atom significantly perturbs the equilibrium configuration of the PWD and causes a blue-shift of the shared proton stretching vibration by almost 200 cm−1. Raman spectra of the PWD in the region of free O− H bond-stretching vibrations were analyzed in ref 65. The above review of previous works has clearly revealed the unabated or even rapidly increasing interest into H5O2+ as a key unit for the theory and understanding of hydrogen-bonded systems with a large amplitude motion of the bridging proton culminating with complete proton transfer. Any novel data on multidimensional PES, structure and vibrational dynamics of the dimer, as well as measurement of new IR absorption spectra in the gas phase are very welcome. The available literature data indicate extremely strong anharmonic mode coupling of the shared proton vibrations. Ojamäe, Shavitt, and Singer40 were the first to propose strong anharmonic interaction of the shared proton stretching vibration and stretching O···O vibration. Works of the Sauer group41,42 indicate the presence of anharmonic interaction between the shared proton stretching vibration and its bending vibrations. In works of the Vendrell group,54−58 a significant anharmonic interaction between the analyzed mode and the antisymmetric bending vibration of the O−H bonds in water molecules was noted for the first time, as well as the fact that the interaction with the stretching vibrations of O−H bonds in water molecules is insignificant. The degree of anharmonic interaction between the shared proton stretching vibration and other modes (first of all, lowfrequency modes) has so far not been investigated. Earlier it was noted17,50 that in carrier spectra experiments some carriers can stabilize the Cs configuration of the PWD. However, IR spectrum calculations of the PWD in the Cs configuration were performed only in the harmonic approximation. Anharmonic calculation of the PWD+Ar IR spectra was performed in ref 50, but the basis set that was used there (acc-pVDZ) seems to be substantially incomplete. Recently, the PWD+Ar IR spectrum was analyzed in the frame of driven molecular dynamics at the MP2/acc-pVDZ level of theory.64 Thus, there are no reliable data on the frequency of the shared proton stretching vibration in the Cs configuration of the PWD so far, and there are no data on the degree of anharmonicity in the interaction of this vibration with other vibrational modes. In the current work, we present calculations of the PWD geometry with and without restrictions on the symmetry. The cases of C2 and Cs symmetry were analyzed at the MP4/acc-pVTZ level of theory. Anharmonic coupling of the bridging proton stretching vibration with all other modes for C2 and Cs configurations of the PWD was analyzed. For the C2 PWD configuration, the Fermi resonance was simulated in order to elucidate its role in the appearance of the doublet band structure near 1000 cm−1. The column of the “effective” anharmonic matrix for the bridging proton stretching vibration was obtained for both PWD configurations, which gives opportunities to estimate the frequencies of the bridging proton stretching vibrations while taking into account all anharmonic double-mode couplings.

variables and (2) in the framework of the 15D VCI (vibrational configuration interaction) method. In the former case, the obtained frequency values for the bending and stretching vibrations of the central proton are 1344, 1328, and 1185 cm−1, and for the latter case, they are 1388, 1354, and 902 cm−1, respectively. The authors made a detailed analysis of the results obtained in ref 42 in the context of the (2 + 2)D approach and noted that, for such a complex system as the PWD, calculations of vibrations with limited dimensionality, excluding interaction with all remaining vibrational degrees of freedom, even when using the more precise PES, may not be reliable enough. Calculations in the CCSD(T)/acc-pVTZ approximation of the full-size PES (by analogy to OSS3(p), let it be called HBB PES), presented in ref 48, can be considered as a new step in the study of the PWD. This PES was used in ref 49 to calculate the frequencies of the stretching O−H vibrations of water molecules in the PWD. The HBB PES was also used in ref 50 to calculate vibrational frequencies of the central proton using the diffusion Monte Carlo (DMC) and MM/VCI methods. The frequency values of the stretching vibrations, calculated using these two methods, are near 995 and 1070 cm−1, respectively. Calculated frequency values for the bending vibrations unfortunately were not given. Moreover, the authors of ref 50 carried out a number of experiments on measuring IR spectra of the PWD using the “carrier gas” method. This time, in addition to the experiments with Ar gas, similar studies were performed using Ne as carrier gas. The authors convincingly show that, due to the smaller perturbation in the case of Ne, the IR spectrum is closer to the C2 configuration than in the case of Ar, where the PWD presumably has the Cs configuration. Two of the most intense absorption bands in the measured IR spectrum of the PWD+Ne (928 and 1047 cm−1), however, have no satisfactory description in the frame of the DMC and MM/VCI approaches, since the band at 928 cm−1 (as the authors note) can be associated only with the combination band at 1056 cm−1, the calculated intensity of which is extremely low. Subsequently, researchers have increasingly focused on the study of the IR spectra of the PWD using molecular dynamics methods at different levels.51−53 The HBB PES was used again in a number of works.54−58 Using modern approaches of quantum dynamics and the HBB PES, the authors managed to reach good agreement between the calculated and experimental50 spectra. In particular, the observed doublet of IR absorption bands near 1000 cm−1 is explained by the authors58 as a Fermi resonance doublet. New experimental IR spectra investigations of the PWD with helium atom as the carrier59 are found close enough, in terms of basic characteristics, to the IR spectra of the PWD+Ne. In particular, the doublet of bands around 1000 cm−1 is red-shifted by only 7 cm−1 (to 921 and 1040 cm−1). Taking into account the decrease of the perturbation effects caused by a helium atom compared to neon, the authors of ref 59 suggested that the PWD+He IR spectrum is practically equivalent to the free PWD IR spectrum. Recently, the IR spectrum of the PWD was simulated by effective mode analysis (EMA),60,61 but the region of the central proton stretching vibrations in the calculated spectrum is in contradiction with the data from ref 50. An interesting approach was proposed in ref 62, where the “clusters-in-liquid” (CLA) method was used for the analysis of IR spectra of the PWD but the results of the investigations did not agree well with the experimental data for the free PWD. In ref 63, structural and electronic properties of the PWD were analyzed in the frame of the quantum Monte Carlo (QMC)

2. CALCULATIONS Equilibrium configurations of the PWD and its IR absorption spectra were calculated in the harmonic approximation for the cases of C2 and Cs symmetry as well as without symmetry limitation (C1) using the acc-pVTZ basis set66 at the 2153


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Figure 1. Equilibrium configuration of the PWD (C2 symmetry) from two different points of view (a and b) and the configuration of the transition state of the PWD (Cs symmetry) from two different points of view (c and d).

Table 1. Total Energies (Hartree) and Structural Parameters of the Bridging Hydrogen Atom of the PWD Calculated at Different Levels of Theory and for Different Configurations structural parameters of the hydrogen bridge level of theory B3LYP/acc-pVQZ CCSD(T)/acc-pVTZ MP4/acc-pVTZ

lH4O1 (Å) 1.2022 1.1976 1.1977

MP2/acc-pVTZ wB97X/acc-pVTZ tHCTHhyb/acc-pVTZ mPW3PBE/acc-pVTZ mPW1PW91/acc-pVTZ B3LYP/acc-pVTZ PBEh1PBE/acc-pVTZ B3LYP/acc-pVQZ CCSD(T)/acc-pVTZ MP4/acc-pVTZ B3LYP/acc-pVTZ MP2/acc-pVTZ PBEh1PBE/acc-pVTZ B3LYP/acc-pVQZ CCSD(T)/acc-pVTZ MP4/acc-PVTZ

lH4O5 (Å)

1.1972 1.1986 1.2018 1.1990 1.1965 1.2030 1.1972 1.2021 1.1966 1.1975 1.2422 1.2633 1.2305 1.2404 1.2762 1.2754

lO1O5 (Å)

∠O1H4O5 (deg)

Optimization without Symmetry Limitations (C1) 1.2012 2.4014 174.32 1.1956 2.3896 173.71 1.1975 2.3915 173.62 Optimization for C2 Symmetry 2.3906 173.57 2.3942 174.24 2.4003 174.06 2.3948 174.10 2.3899 174.20 2.4029 174.24 2.3913 174.14 2.4012 174.22 2.3898 173.74 2.3913 173.60 Optimization for Cs Symmetry 1.1600 2.4005 175.65 1.1318 2.3933 175.65 1.1590 2.3878 175.65 1.1601 2.3988 175.65 1.1219 2.3964 175.65 1.1237 2.3971 175.41

B3LYP,67−69 MP2,70,71 MP4,72 and CCSD(T)73,74 levels of theory, as well as in the B3LYP/acc-pVQZ approximation66 using Gaussian 09.75 The calculations of the equilibrium geometry and IR spectra in the harmonic approximation for the C2 PWD configuration were also carried out using the accpVTZ basis set and mPW1PW91, 7 6 mPW3PBE, 7 6 PBEh1PBE,77 tHCTHhyb,78 and wB97X79 hybrid DFT methods. The anharmonic calculations of the IR spectrum of the PWD for the configurations with C2, Cs, and C1 symmetry were carried out using B3LYP/acc-pVQZ, B3LYP/acc-pVTZ, and MP2/acc-pVTZ. In the latter case, the standard secondorder perturbation theory (PT2) model for inclusion of anharmonicity effects is used.80,81 It was implemented in Gaussian75 owing to the work of Barone.82 For the configurations with C2 and Cs symmetry, calculations of the IR spectra in the anharmonic approximation were also carried out using the acc-pVTZ basis set and mPW1PW91, mPW3PBE, PBEh1PBE, tHCTHhyb, and wB97X hybrid DFT functionals. The calculations of 1D−3D PESs for some vibrational coordinates of the PWD with C2 and Cs symmetry were carried out at the MP4/acc-pVTZ level.

lH4O1 − lH4O5 (Å)

energy

0.0010 0.0020 0.0002

−153.27311903 −153.01229145 −153.01411966

0 0 0 0 0 0 0 0 0 0

−152.98426485 −153.21412812 −153.23024400 −153.16346798 −153.21867370 −153.26078609 −153.10857905 −153.27312085 −153.01229129 −153.01411966

0.0822 0.1315 0.0715 0.0803 0.1544 0.1517

−153.26030850 −152.98352720 −153.10742846 −153.27268817 −153.01157391 −153.01336476

3. ANALYSIS OF STRUCTURAL AND ENERGETIC PARAMETERS OF THE PWD The equilibrium configuration of the PWD with C2 symmetry and the transition state with Cs symmetry, calculated at the MP4/acc-pVTZ level, are presented in Figure 1. The total energies and the main structural parameters involving the bridging hydrogen obtained at different levels of theory and for different configurations of the PWD are presented in Table 1. Analyzing structures of the PWD of different symmetry, one can conclude that the most stable structure is that predicted by Valeev and Schaefer29 and that, with increasing basis set size and using more precise methods of accounting for electron correlation, the results of calculations without symmetry limitations should tend to the parameters of the configuration with C2 symmetry. In particular, it was noted29 that going from CISD to CCSD(T) and using a TZ2P basis set the difference between the lengths of “short” and “long” O−H bonds decreases from 0.1894 to 0.0517 Å. As is seen from Table 1, the prediction of Valeev and Schaefer was correct. The results of calculations of the equilibrium configuration of the PWD without symmetry constraints for all levels of theory according to the criteria determined in ref 29 are more realistic than those presented in ref 28. The results obtained using the MP4/accpVTZ approximation are especially impressive. As is seen from 2154


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Table 2. Calculated Frequencies (cm−1) and Intensities (km/mol) of Infrared Absorption Bands of the PWD Configurations with Different Symmetries level of theory N

assignment

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

twist H2O s(d) wagging H2O + stretching O···H···O as(d) wagging H2O s(d) rock + twist H2O + bending O···H···O s(d) rock + twist H2O + bending O···H···O as(d) stretching O···O s(d) stretching O···H···O + wagging H2O as(d) bending O···H···O as(d) bending O···H···O s(d) bending H−O−H s(d) bending H−O−H as(d) stretching O−H s(m) as(d) stretching O−H s(m) s(d) stretching O−H as(m) as(d) stretching O−H as(m) s(d)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15


MP4/acc-pVTZ sym.

harm. freq.

CCSD(T)/acc-pVTZ

intens.

sym.

harm. freq.

Optimization without Symmetry Restrictions (C1) A 167.2 n/a A 165.7 A 344.7 n/a A 336.8 A 462.6 n/a A 462.3 A 534.6 n/a A 535.1 A 541.3 n/a A 542.6 A 623.5 n/a A 625.5 A 859.1 n/a A 850.3 A 1476.8 n/a A 1481.2 A 1558.4 n/a A 1563.5 A 1714.1 n/a A 1719.0 A 1763.7 n/a A 1768.5 A 3726.3 n/a A 3739.5 A 3733.2 n/a A 3747.0 A 3823.0 n/a A 3831.6 A 3823.3 n/a A 3832.6 C2 Symmetry A 165.3 n/a A 166.0 B 344.7 n/a B 335.3 A 460.5 n/a A 461.5 B 533.4 n/a B 533.1 A 541.8 n/a A 542.1 A 623.5 n/a A 625.3 B 859.8 n/a B 850.1 B 1477.5 n/a B 1481.7 A 1557.7 n/a A 1563.4 A 1713.9 n/a A 1718.7 B 1763.9 n/a B 1768.5 B 3726.7 n/a B 3739.8 A 3733.6 n/a A 3747.2 B 3823.3 n/a B 3831.9 A 3823.7 n/a A 3832.4 Cs Symmetry A″ −321.2 n/a A″ −317.4 A″ 187.8 n/a A″ 190.2 A′ 428.6 n/a A′ 428.9 A′ 486.6 n/a A′ 484.6 A″ 495.5 n/a A″ 496.4 A′ 583.2 n/a A′ 583.3 A′ 1211.2 n/a A′ 1218.0 A′ 1398.2 n/a A′ 1401.4 A″ 1605.9 n/a A″ 1612.2 A′ 1684.2 n/a A′ 1689.2 A′ 1818.8 n/a A′ 1824.2 A′ 3714.4 n/a A′ 3726.2 A′ 3773.5 n/a A′ 3787.9 A″ 3804.5 n/a A″ 3812.0 A′ 3876.3 n/a A′ 3885.8

B3LYP/acc-pVQZ

intens.

sym.

harm./anharm. freq.

intens.

n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a

A A A A A A A A A A A A A A A

175.0/139.1 346.0/210.4 422.7/188.2 499.1/396.0 513.4/432.6 608.4/457.9 923.2/1375.5 1449.6/1274.0 1512.5/1342.6 1680.7/1594.0 1743.8/1820.8 3724.5/3561.6 3733.1/3570.3 3812.9/3634.7 3813.2/3634.3

27.2 257.1 218.6 6.2 25.6 0.2 2812.2 236.7 101.3 0.4 1017.1 227.9 6.4 240.8 312.4


A B A A B A B B A A B B A B A

175.4/150.7 345.4/200.3 422.4/183.0 498.4/396.9 512.5/436.3 608.5/451.5 922.9/1363.6 1450.6/1273.9 1511.7/1340.5 1680.7/1593.7 1744.0/1818.4 3724.6/3562.4 3733.3/3570.5 3813.0/3636.8 3813.3/3636.5

27.3 257.7 219.9 5.9 25.1 0.2 2810.1 236.8 101.1 0.4 1018.4 227.9 6.5 242.0 311.5


A″ A″ A′ A′ A″ A′ A′ A′ A″ A′ A′ A′ A′ A″ A′

−281.6/−535.4 186.6/181.6 413.7/305.1 486.8/386.3 489.4/477.9 569.9/591.6 1021.3/547.1 1410.9/1404.9 1528.2/1286.6 1662.4/1609.9 1762.8/1586.9 3719.9/3575.0 3760.2/3572.9 3802.6/3647.2 3852.9/3649.2

300.3 36.4 210.7 24.8 19.9 215.6 2230.2 186.4 64.7 40.9 1371.9 150.0 97.0 290.9 286.7

note that in the MP4/acc-pVTZ approximation calculations of the energies without symmetry limitations and with C2 symmetry give exactly the same numerical values and that the energy of the transition state with Cs symmetry is only 165.68 cm−1 higher. These results allow stating that there is no potential minimum for the free PWD with C1 symmetry. Comparing the energies of the C2 and Cs configurations calculated at different levels of approximation shows that the Cs configuration is higher in energy by 52.53 cm−1 in the case of

Table 1, the difference between the lengths of the two hydrogen bonds is less than 0.0002 Å and the geometry parameters of the PWD obtained without symmetry constraints are in a good agreement with the results obtained for the case of C2 symmetry. For the optimization in the case of C2 symmetry, the structural parameters obtained using the PBEh1PBE/acc-pVTZ approximation have similar values as those obtained in the MP4/acc-pVTZ approximation. Comparing energies of the optimized configurations, it is important to 2155


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The Journal of Physical Chemistry A the PBEh1PBE/acc-pVTZ approximation; by 104.82 cm−1 in the case of the B3LYP/acc-pVTZ approximation; by 94.96 cm−1 in the case of the B3LYP/acc-pVQZ approximation; by 161.89 cm1 in the case of the MP2/acc-pVTZ approximation; by 165.68 cm −1 in the case of the MP4/acc-pVTZ approximation; and by 157.45 cm−1 in the case of the CCSD(T)/acc-pVTZ approximation. Thus, all levels of theory used for the calculations predict that the configuration with C2 symmetry is energetically more preferable than the configuration with Cs symmetry and that geometry optimization at a high enough level of theory results in C2 symmetry also without symmetry constraint.

3614.7, 3624.8, 3697.3, and 3716.8 cm−1 for the frequencies of the stretching vibrations, again indicating the possibility to observe four absorption bands in the IR spectrum of the PWD in the case of stabilization of the Cs configuration, as could be the case when using a carrier gas (see below). The determination of configurations of the PWD in experiments by using a carrier gas thus becomes debatable. However, it should be noted that anharmonic calculations for the more accurate approximations (MP4/acc-pVTZ and CCSD(T)/accpVTZ) are absent. Let us have a closer look at the stretching vibration of the central proton in the case of the C2 configuration. Analysis of the potential energy distribution (PED)83 of the ν7 vibration in the harmonic approximation (see Table 2) reveals that the central proton stretching vibration gives the major contribution to the PED of this vibration (up to 80%). The rest of the potential energy comes from the antiphase wagging vibrations of water monomers. There is a tendency of decreasing frequency of this vibration when the level of theory is increasing. Therefore, in the series of approximations B3LYP/ acc-pVTZ, B3LYP/acc-pVQZ, MP2/acc-pVTZ, MP4/accpVTZ, and CCSD(T)/acc-pVTZ, the frequency of this vibration is 925.5, 922.9, 911.2, 859.8, and 850.1 cm−1, respectively. It means that the higher the level of calculation, the smaller the curvature of the calculated PES near the minimum in the direction along the bridging proton. All anharmonic calculations predict a significant blue-shift of the frequency of this vibration. The calculated frequencies of ν7 are in the range from 1363 to 1464 cm−1. The opposite situation is for the Cs configuration. When calculated in the harmonic approximation, the frequency increases when the level of theory is improved. Thus, in the series B3LYP/acc-pVTZ, B3LYP/accpVQZ, MP2/acc-pVTZ, MP4/acc-pVTZ, and CCSD(T)/accpVTZ, the frequency of this vibration is 1026.5, 1037.9, 1168.2, 1211.2, and 1218.0 cm−1, respectively. Accounting for the anharmonicity in the frame of PT2 leads to a significant redshift of the frequency of this vibration (calculated values vary from 219.2 to 734.8 cm−1). The vibration ν2 is the second component of the pair of vibrations appearing as the result of mixing of the stretching vibration of the central proton (with a contribution in PED up to 20%) and the antiphase wagging vibrations of water monomers (contribution in PED up to 70%). The two bending vibrations of the central proton for the C2 configuration of the PWD (ν8 - normal to the C2 axis and ν9 - along the C2 axis) in the harmonic approximation have significantly larger frequency. In the series B3LYP/acc-pVTZ, B3LYP/acc-pVQZ, MP2/acc-pVTZ, MP4/acc-pVTZ, and CCSD(T)/acc-pVTZ, the frequencies of these vibrations are equal to 1451.3 and 1511.1, 1450.5 and 1511.7, 1472.5 and 1550.6, 1477.5 and 1557.7, and 1481.7 and 1563.4 cm−1, respectively. This is opposite to the trend of the stretching vibration, since with increasing level of theory the frequencies of the bending vibrations are seen to increase rather than decrease as was the case for the stretching vibration. If we compare the corresponding anharmonic calculations (B3LYP/acc-pVQZ), we notice that accounting for anharmonicity effects leads to the frequencies of the bending vibrations (1273.9 and 1340.4 cm−1) becoming lower than the frequency of the central proton stretching vibration (1363.3 cm−1). For the Cs configuration, the situation with the central proton bending vibrations is similar to that for the C2 configuration both in the harmonic and anharmonic approximation. The peculiarity of the PES of the PWD with the C2 configuration

4. ANALYSIS OF IR ABSORPTION SPECTRA OF THE PWD IN THE HARMONIC AND ANHARMONIC APPROXIMATIONS For all approximations and configurations of the PWD, after equilibrium geometry and the transition state determination, calculations of IR spectra in the harmonic and anharmonic approximations were carried out. An exception was made for the CCSD(T) and MP4 levels of theory, since this option is not provided within Gaussian 09.75 The calculated IR frequencies and intensities of the PWD in the harmonic and anharmonic approximations are presented in Table 2 (Note that vibrations can be treated as symmetric or antisymmetric in relation to the whole dimer molecule or to the single monomer molecule. In order to take into account this peculiarity, we introduce the following notation: s(m), as(m), s(d), and as(d), where the first letter designates whether the vibration is symmetric or antisymmetric and the letter in brackets designates the relation to which part this symmetry was definedmonomer or dimer.) It is notable that differences between harmonic frequency values are less than 1 cm−1 between calculations, carried out without symmetry limitations and in C2 symmetry in the case of the MP4/acc-PVTZ, CCSD(T)/acc-pVTZ, and B3LYP/accpVQZ approximations. It means that these approximations and optimization algorithms incorporated in Gaussian 09 successfully cope with calculations of the harmonic PES even without any symmetry limitations. In the anharmonic approximation, the calculated IR spectra of the C2 and C1 configurations (B3LYP/acc-pVQZ) have significantly larger frequency differences (up to 12 cm−1), which are due to the PES analysis by necessity being performed far from the equilibrium position in comparison with the harmonic approximation. The calculated IR spectra for the C2 and Cs configurations of the PWD, as was noted earlier in refs 28 and 29, are very different. The presence of one imaginary frequency in the IR spectra of the Cs configuration at all levels of theory indicates that we deal with a first-order transition state. Analysis of the region of the O−H stretch vibration reveals that the harmonic calculations at all levels of theory agree with literature data: for the Cs configuration, four nonoverlapping intense absorption bands are predicted, while, for the C2 configuration, only two absorption bands are expected due to energetic overlap and the low intensity of some of them. The anharmonic calculations (B3LYP/acc-pVQZ) for the C2 configuration give the same results, while in the case of the Cs configuration the situation is not so simple (see Table 2), since due to accounting for anharmonicity effects the frequencies are grouped in pairs (3647.2 and 3649.2 cm−1) and (3572.9 and 3575.0 cm−1). In this case, only two absorption bands will be observed in the IR spectrum too. The results of anharmonic calculations of the IR spectrum of the PWD at the MP2/acc-pVTZ level predict 2156


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Table 3. Calculated Frequencies of the Shared Proton Stretching Vibrations in the C2 PWD Configuration, Obtained Taking into Account Interactions with Other Normal Modes, Calculated Values of the Column Elements of the “Effective” Anharmonic Constants Matrix, Obtained at the MP4/acc-pVTZ Level of Theory, and Column Elements of the Anharmonic Constants Matrix, Obtained at the MP2/acc-pVTZ and PBEh1PBE/acc-pVTZ Levels of Theory normal mode number Qi Q1 Q2 Q3 Q4 Q5 Q6 Q7 (ω7 = 859.8; ν̃1D 7 = 1348.1) Q8 Q9 Q10 Q11 Q12 Q13 Q14 Q15

calculated 2D(Q ,Q ) ν̃7 7 i (cm−1)

column of the effective anharmonic constants matrix (MP4/acc-pVTZ) (cm−1)

column of the anharmonic constants matrix (MP2/acc-pVTZ) (cm−1)

column of the anharmonic constants matrix (PBEh1PBE/acc-pVTZ) (cm−1)

1361.7 1460.9 1346.5 1348.1 1339.7 1120.6

27.2 225.5 −3.18 −0.02 −16.78 −455.11 244.15

−9.416 520.826 −44.899 −8.012 −55.621 −347.342 251.014

−68.66 153.02 −39.00 −48.07 −18.90 −178.07 204.39

1306.8 1325.4 1328.9 1275.0 1342.0 1346.8 1340.7 1340.8

−82.44 −45.34 −38.36 −66.12 −12.12 −2.64 −14.81 −14.64

−65.037 −54.609 −142.949 157.507 0.313 0.809 −0.182 −0.181

−64.29 −61.12 −45.04 100.87 −1.41 −1.50 −2.52 −2.42

is40 that accounting for anharmonicity effects in the frame of the standard model leads to a blue-shift of the frequency of the stretching vibration and to a red-shift of the frequencies of the bending vibrations. In many works, however, it is noted that accounting for anharmonicity effects in the standard PT2 model leads to overestimating both stretching and bending frequencies.32,36 At the same time, usually the true value of the frequency of the fundamental vibration is between the calculated harmonic and anharmonic values. Taking into account this remark and the values of the stretching and bending vibrations obtained in the harmonic and anharmonic approximations (for the B3LYP/acc-pVQZ level of theory), one can expect that the stretching vibrations of the central proton should be lower than the bending ones. The mentioned frequency shifts for stretching and bending vibrations depending on the level of theory confirm this assumption. It is interesting that, according to ref 32, the harmonic values of ν7, ν8, and ν9 are equal to 871, 1511, and 1552 cm−1, respectively, while the anharmonic frequencies of these vibrations obtained using the cc-VSCF method32 are 1209, 1442, and 1494 cm−1, in complete agreement with this assumption. One can also state that the ν7 vibration is highly anharmonic, while the bending vibrations ν9 and ν9 have medium anharmonicity. It is confirmed by the values of diagonal and nondiagonal anharmonicity constants obtained at different levels of theory. The differences in the numerical values of anharmonicity constants for vibrations with different degrees of anharmonicity can be estimated comparing the data from refs 36, 84, and 85.

Calculations of the 2D PES were carried out using normal coordinates Q7 and Qi, i ≠ 7; the normal coordinates were varied in the range −0.6 ≤ Q7, Qi ≤ 0.6, with a step of 0.1. Since both for the coordinate Q7 and for the coordinate Q2 the central proton shifts, there were some points of the 2D grid where the energy could not be calculated because of a too small distance between the proton and one of the oxygen atoms. We, however, did not reduce the range of variation of the coordinates, but instead the energy at these points was set large enough. The vibrational Schrödinger equation in this case can be written as −

ℏ2 ∂ 2Ψ ℏ2 ∂ 2Ψ − + U (q7 , qi)Ψ = E Ψ 2μQ l0 2 ∂q7 2 2μQ l0 2 ∂qi 2 i

7

where q7 =

Q7 l0

; qi =

Qi l0

(1)

; l0 = 1 Å.

Let us rewrite eq 1 as ∂ 2Ψ ∂ 2Ψ − f + U (q7 , qi)Ψ = E Ψ 2 qi 7 ∂q ∂qi 2 7

−fq

(2)

The numerical solution of eq 1 was carried out using the DVR method.40,86−91 The Hamiltonian matrix elements were calculated as qq

qq

H(i , j),(i ′ , j ′) = Dii ′7 7δjj ′ + δii ′Djji′ i + δii ′δjj ′U (q7i , qi j)

(3)

where D = (D ) F D ; F = fqE ; q ∈ {q7, qi}. Eq is a unit matrix, the dimensionality of which corresponds to the dimensionality of Dq qq

5. ANALYZING THE BRIDGING PROTON STRETCHING VIBRATION ANHARMONIC COUPLING WITH OTHER MODES IN C2 AND CS CONFIGURATIONS OF THE PWD BY CONSTRUCTING 2D PESS For PES calculations, the MP4/acc-pVTZ approximation was used, which, as noted in sections 3 and 4, has allowed obtaining excellent agreement in the harmonic approximation between structural parameters and vibrational spectra calculated without restrictions on the symmetry and for the C2 symmetry.

q T q

q

q

q

( −1)i ′− i ωq ; q ∈ {q7 , qi}; ⎡ π(i ′ − i) ⎤ 2 sin⎢ 2N + 1 ⎥ ⎣ q ⎦ i′, i ∈ {1, 2, ..., 2Nq + 1}; Diiq = 0;

Diq′ i =

ωq = 2157

2π ; Δq = 0.1; Nq = 6 (2Nq + 1)Δq DOI: 10.1021/acs.jpca.7b00536 J. Phys. Chem. A 2017, 121, 2151−2165

Article


Table 4. Calculated Frequencies of the Shared Proton Stretching Vibrations in the Cs PWD Configuration, Obtained Taking into Account Interactions with Other Normal Modes, Calculated Values of the Column Elements of the “Effective” Anharmonic Constants Matrix, Obtained at the MP4/acc-pVTZ Level of Theory, and the Column Elements of the Anharmonic Constants Matrix, Obtained at the MP2/acc-pVTZ and PBEh1PBE/acc-pVTZ Levels of Theory normal mode number Qi Q1 Q2 Q3 Q4 Q5 Q6 Q7 (ω7 = 1211.2; ν̃1D 7 = 1203.3) Q8 Q9 Q10 Q11 Q12 Q13 Q14 Q15

calculated 2D(Q ,Q ) ν̃7 7 i (cm−1)

column of the effective anharmonic constants matrix (MP4/acc-pVTZ) (cm−1)

column of the anharmonic constants matrix (MP2/acc-pVTZ) (cm−1)

column of the anharmonic constants matrix (PBEh1PBE/acc-pVTZ) (cm−1)

1229.6 1212.5 1248.1 1287.2 1183.4 1169.5

52.5 18.4 89.6 167.8 −39.8 −67.8 −3.95

−172.094 2.880 −35.314 −12.302 −13.292 159.980 −121.445

−114.1 0.13 −118.8 3.6 −32.7 74.3 −30.5

1199.8 1150.1 1201.1 1235.3 1202.9 1198.1 1194.4 1192.9

−7.11 −106.4 −4.41 64.00 −0.81 −10.4 −17.8 −21.8

129.28 −282.778 0.183 −244.591 42.604 −23.716 49.710 −18.787

4.6 −232.4 −6.9 −84.6 35.5 −17.4 43.5 −14.7

theory and takes into account PESs of more limited sizes.80−82 Thus, it is no wonder that values of the “effective” anharmonicity constants differ significantly from the corresponding anharmonicity constants, calculated at the PBEh1PBE/acc-pVTZ and MP2/acc-pVTZ levels of theory.

The Hamiltonian matrix (eq 3) was constructed and diagonalized using the program Mathematica.92 In order to obtain the most complete information, besides the 2D vibrational problem, two 1D vibrational problems were solved. The results of the calculations are presented in Tables 3 and 4. Analyzing the shared proton stretching vibration (ν̃7) in the frame of two-mode interactions, we can use a well-known formalism for accounting for the anharmonic interactions:80,81 νj̃ f = ωj + 2χjj +

1 2

6. DISCUSSION As seen from Table 3, the intrinsic anharmonicity of the seventh normal mode is responsible for the strong blue-shift of its frequency, by analogy with the description of this vibration as pure proton transfer.40−43 Anharmonic interactions of ν7 with other modes can be divided into three groups: (1) modes, interactions with which lead to a blue-shift of ν7, (2) modes, interactions with which lead to a red-shift of ν7, (3) modes, interactions with which insignificantly affect the ν7 frequency. The first group includes interaction of ν7 with ν1 and ν2. The blue-shift of the ν7 frequency due to the interaction with the antisymmetric twisting vibration of the water monomers leads to a blue-shift of ν7 by only 13 cm−1, while a similar shift due to interaction with the second mode was not noted previously, although it increases the ν7 frequency by more than 100 cm−1. It should be remembered that the ν2 and ν7 normal modes are mixtures of the shared proton stretching vibration and antisymmetric wagging vibration of the water molecules. With this, the second type of vibrations is prevailing in the first mode, and the first type, in the second mode. Interaction of ν7 with stretching O−H vibrations of the water monomers and with the third, fourth, and fifth normal modes should be included in the third group of the interactions. Interaction with the remaining five modes (ν6, ν8−ν11) decreases the ν7 frequency. The maximal red-shift (more than 200 cm−1) is due to the intermolecular interaction with water monomers. This fact was previously noted in a series of works,40−42 despite the fact that the description was limited to proton movement along the hydrogen bridge. The next, in terms of the value, is a red-shift due to the interaction with the antisymmetric bending vibration of the water monomers (ν11). This fact was previously noted in refs 54−58, but in our case, when normal coordinates are used, the

∑ χij i≠j

(4)

For calculations of the column elements of the “effective” anharmonicity constants matrix, eqs 5 and 6 were used 1D(Q 7)

ν7̃

= ω7 + 2χ77

2D(Q 7 , Q i)

ν7̃

1D(Q 7)

= ν7̃

(5)

+

1 χ 2 i7

(6)

where ω7 is the harmonic frequency of the vibration being 7) analyzed (859.8 cm−1), ν̃1D(Q is the frequency value of the 7 vibration, obtained by solving the 1D vibrational problem for 7,Qi) normal coordinate Q7 (1348.1 cm−1), and ν̃2D(Q is the 7 frequency value of the vibration, obtained by solving the 2D vibrational problem using the 2D PES construction for normal coordinates Q7 and Qi (see the second column in Tables 3 and 4). Values of the “effective” anharmonic constants for the seventh normal mode are presented in the third columns in Tables 3 and 4. The fourth and fifth columns in Tables 3 and 4 present anharmonicity constant values for the seventh mode, obtained respectively at the MP2/acc-pVTZ and PBEh1PBE/ acc-pVTZ levels of theory using ref 75. It should be noted that values of the “effective” anharmonicity constants were obtained on the basis of full accounting of the anharmonic properties of 2D PESs, that have significant magnitudes and were calculated at the MP4/acc-pVTZ level of theory. However, MP4 anharmonic constants are not yet available in Gaussian 09.75 Besides, the standard model of the anharmonicity constants matrix calculations is obtained using second order perturbation 2158


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The Journal of Physical Chemistry A value of the red-shift is significantly smaller (less than 35 cm−1) than the results of ref 56, where the red-shift was found to be more than 120 cm −1 . It is interesting to note that anharmonicity constants, calculated at the PBEh1PBE/accpVTZ and especially at the MP2/acc-pVTZ levels of theory, at least correlate with the values of “effective” anharmonicity constants, but in the case of ν11, there is a notable contradiction between their values (see Table 3). Such a contradiction is probably associated with the intricate topology of the 2D PES, constructed by varying Q7 and Q11, that leads to an incorrect interaction estimation in the frame of the standard model of accounting for anharmonicity effects.80,81 Indeed, the 2D PES for these coordinates is quite flat around the minimum (Figure 2).

account additional corrections to the triple mode anharmonic coupling: νj̃ f = ωj + 2χjj +

1 2

∑ χij

+

i≠j

1 3

∑

χikj

i , k(i ≻ k); i,k≠j

(8)

Similar to the solution of 3N − 7 2D vibrational problems with the seventh normal coordinate for finding all χ7i, in order to find all χ7jk, one needs to solve C142 = 91 vibrational problems with Q7. With this, eqs 5 and 6 should be complemented by one of the following equations: 1 1 1 χ + χk 7 + χjk 7 2 j7 2 3 3D(Q , Q , Q ) 1 1 1 1D(Q ) ν7 7 j k = ν7 7 + χj7 + χk 7 + χjk 7 2 2 3 3D(Q 7 , Q j , Q k) 2D(Q 7 , Q j) 1 1 ν7 = ν7 + χk 7 + χjk 7 2 3 3D(Q 7 , Q j , Q k) 1 1 2D(Q , Q ) ν7 = ν7 7 k + χj7 + χjk 7 2 3 3D(Q 7 , Q j , Q k)

ν7

= ω7 + 2χ77 +

(9)

Extension of this approach to the case of taking into account quadruple and higher modes of anharmonic coupling is obvious. However, it is expected that the higher the levels of interaction that are taken into account, the more insignificantly it will affect the frequency position of the investigated vibration. Returning to the results of the anharmonic calculations of vibrational spectra of the PWD in different approximations, we note that in all cases there is Fermi resonance of the ν7 mode with three combined modes (ν6 + ν2, ν3 + ν2, and ν3 + ν5). However, there is intensity redistribution from fundamental to combined modes only for the combined ν6 + ν2 mode, albeit limited to about 3%. The smallness of this value should not be mistaken for a small interaction, since, without the perturbation of the resonance, the frequency of the fundamental ν7 vibration obtained at the B3LYP/acc-pVQZ level equals 1330 cm−1 and that of the combined vibration ν6 + ν2 is 675 cm−1. As a result of the resonance, the frequencies shift to 1362 and 643 cm−1, respectively. Such a strong interaction of the fundamental vibration ν7 with the combined vibration ν6 + ν2, taking into account the very large difference between the frequencies of these vibrations, is due to a rather big value (−364.7 cm−1 BPEh1BPE/acc-pVTZ) of the cubic force constant k7,6,2, which is responsible for the appearance of the Fermi resonance. In order to analyze the possibility of appearance of the Fermi resonance in more detail, additional calculations of 3D PES and 3D DMS were carried out at the MP4/acc-pVTZ level using normal coordinates Q2, Q6, and Q7. The normal coordinates were again varied in the range −0.6 ≤ Q2, Q6, Q7 ≤ 0.6 with a step of 0.1. Since both for coordinate Q7 and for coordinate Q2 the central proton shifts, there were some points of the 3D grid where the energy could not be calculated because of a too small distance between the proton and one of the oxygen atoms. We, however, did not reduce the range of variation of the coordinates, but instead, the energy at these points was set large enough. The vibrational Schrödinger equation in this case can be written as

Figure 2. 2D PES of the PWD (C2 configuration) calculated for the Q7 and Q11 normal coordinates.

After applying eq 4 to the seventh normal mode, we can write the following: 1 ν7̃ f = ω7 + 2χ77 + ∑ χi7 2 i≠7 (7) On the basis of accounting for intrinsic anharmonicity and anharmonic double modes coupling ν7 with other normal modes, and using column elements of the “effective” anharmonicity constant matrix (see Table 3), we can evaluate the frequency value of the shared proton stretching vibration. Calculated in this manner, the ν7 value becomes 1098.7 cm−1. According to the experimental data, a doublet of the absorption bands, associated with Fermi resonance, in the case of the least perturbing carrier (helium atom)59 appears at frequencies of 921 and 1040 cm−1. If there is a full match of the frequencies of fundamental and combination vibrations, then the frequency of the shared proton vibration unperturbed by the Fermi resonance occurs at 980.5 cm−1. If we take into account that the intensity of the 1040 cm−1 IR band is nearly 2 times higher than intensity of the 921 cm−1 band,59 we can find the frequency of the unperturbed by Fermi resonance shared proton stretching vibrations more accurately using the following formula:93 (2 × 1040 + 1·921)/3 = 1000.3 cm−1. Comparing calculated and experimental frequency values of the investigated vibration, their satisfactory agreement can be noted. One should take into account here that eq 4 restricts consideration of anharmonicity effects by taking into account only the double modes anharmonic coupling. However, it is obvious that this approach can be extended by taking into 2159


Article

The Journal of Physical Chemistry A −

ℏ2 ∂ 2Ψ ℏ2 ∂ 2Ψ ℏ2 ∂ 2Ψ − − 2 2 2 2 2μQ l0 ∂q2 2μQ l0 ∂q6 2μQ l0 2 ∂q7 2 2

6

7

+ U (q2 , q6 , q7)Ψ = E Ψ

where q2 =

Q2 l0

; q6 =

Q6 l0

; q7 =

Fermi resonance are equal, it can be suggested that the frequency of the shared proton stretching vibration and mixed vibration unperturbed by Fermi resonance are close to 1 (1125 + 1262) = 1193 cm−1. Considering this value as 2 7,Q6,Q7) , we can use one of the formulas from eq 9 and ν3D(Q 7 data from Table 3 to find the χ726 value equal to −121.1 cm−1. It means that anharmonic coupling of three modes (ν2, ν6, and ν7) leads to an additional red-shift of the bridging proton stretching vibration frequency by 40.4 cm−1. Since interaction of the three investigated modes is taken into account to the fullest extent by constructing 3D(Q2, Q6, Q7) PES and the following solution of the vibrational Schrödinger equation, then the interaction of the ν7 vibrational mode with the remaining modes was taken into account by the hybrid method proposed earlier in refs 38, 39, and 94. For finding the corrections to ν7, the following formula was used

(10) Q7 l0

; l0 = 1 Å.

Let us rewrite eq 10 as ∂ 2Ψ ∂ 2Ψ ∂ 2Ψ − f − f + U (q2 , q6 , q7)Ψ = E Ψ 2 q6 q7 2 ∂q ∂q6 2 ∂q7 2 2

−fq

(11)

The Hamiltonian matrix elements were calculated as qq

qq

qq

H(i , j , k),(i ′ , j ′ , k ′) = Dii ′2 2δjj ′δkk ′ + δii ′Djj6′ 6δkk ′ + δii ′δjj ′Dkk7′ 7 + δii ′δjj ′δkk ′U (q2i , q6j , q7k )

(12) 3D(Q 2 , Q 6 , Q 7)

ν7̃ f = ν7̃

where Dqq = (Dq)TFqDq; Fq = fqEq; q ∈ {q2, q6, q7}. Eq is a unit matrix, the dimensionality of which corresponds to the dimensionality of Dq

1 2

∑ i ≠ 2,6,7

χi7

(14)

where χi7 were taken from the third column of Table 3. The calculated frequency of the bridging proton stretching vibration is 1058.4 cm−1. This value is in better agreement with experimental data, and therefore, it can be expected that within this approach the calculated frequency of the similar vibration in the Cs PWD configuration will be acceptable. Moving to the analysis of the anharmonic coupling of the shared proton stretching vibrations with other normal modes for the Cs PWD configuration, we should note that the intrinsic anharmonicity of the analyzed mode is very small (χ77 = −3.95 cm−1; see Table 4). With this, the shape of the corresponding potential curve is far from harmonic (Figure 3).

i ′− i

( −1) ωq ; q ∈ {q2 , q6 , q7}; ⎡ π(i ′ − i) ⎤ 2 sin⎢ 2N + 1 ⎥ ⎣ q ⎦ i′, i ∈ {1, 2, ..., 2Nq + 1};

Diq′ i =

Diiq = 0; ωq =

+

2π ; Δq = 0.1; Nq = 6 (2Nq + 1)Δq

A 3D dipole moment surface (DMS) was calculated as well where one should keep in mind that Gaussian 09 internally transforms to a new Cartesian coordinate system different from that used in the input file. In order to transform the calculated dipole moment components to the initial Cartesian coordinate system, the transformation matrix from the new coordinate system to the initial one was determined using Mathematica.92 Then, using this matrix, the values of the dipole moment components in the initial coordinate system were obtained from their values in the new coordinate system. Then, the matrix elements of the dipole moment operator were calculated in accordance to the relations p0i 2 = (p0xi )2 + (p0yi )2 + (p0zi )2 ; p0αi =

∭ Ψ0pα Ψ*i dV ; α ∈ x , y , z

(13) Figure 3. PES of the PWD Cs configuration as a function of the Q7 normal coordinate (green) and harmonic PES, constructed on the basis of the PES behavior near the minimum (blue).

The results of the calculations are presented in the first row of Table 5. Since the intensities of the two components of the Table 5. Frequencies (cm−1) of Some PWD Vibrations Calculated at the MP4/acc-pVTZ Level of Theory Using 3D PES for the (Q2, Q6, Q7) Set of Vibrational Coordinates (3D) and Corrected Frequencies, Calculated Using the Hybrid Method (3D with the Hybrid Method)a 3D 3D with hybrid method IAU

ν2

ν6

2ν2

ν2 + ν6

ν7

545

585

1075

1125

1262 1018

1165

0.135

0.001

0.0001

0.413

0.387

0.0001

Deviations from the harmonic behavior are obvious, but the character of the deviations differs significantly from the same case for the C2 PWD configuration (Figure 4). It is obvious that in the latter case deviations from harmonic behavior on the left and right sides of the minimum come to the steeper slope of a real potential curve compared to the harmonic one, that leads to the abrupt blue-shift of the analyzed mode frequency compared to the harmonic value. In the case of the Cs configuration, the slope of the real potential curve on the right side from the minimum is larger, and on the left side, it is smaller than the slope of the harmonic curve, that leads to the opposite effects on the frequency of the analyzed mode. The

2ν6

a

Calculated intensity values in arbitrary units are presented in the third row. 2160


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Figure 4. PES of the PWD C2 configuration as a function of the Q7 normal coordinate (green) and harmonic PES, constructed on the basis of the PES behavior near the minimum (blue).

34 cm−1, confirming the above-mentioned suggestion. Second in terms of the effect on the ν7 mode is the anharmonic coupling with the antisymmetric bending vibration of the shared proton (the ninth mode), which causes a red-shift of the ν7 mode by more than 53 cm−1. Perhaps, the last significant coupling of the analyzed mode is associated with the antisymmetric bending vibration of the water monomers (ν11). The picture of this coupling is opposite the case of the C2 configuration. Coupling in the case of the Cs configuration leads to a blue-shift of ν7 by 32 cm−1, while, in the case of the C2 configuration, there was a significant red-shift. Moreover, while in the case of the C2 configuration anharmonic calculations predict a blue-shift of ν7, in the case of the Cs configuration, anharmonic calculations predict a red-shift of the analyzed vibration (see Tables 3 and 4). It is important to note that, although the contribution of the shared proton stretching vibration to the antisymmetric bending vibration of water monomers takes place for both PWD configurations, for the Cs configuration, it is almost twice that for the C2 configuration. Coupling of ν7 with other normal modes in the Cs configuration affects the frequency of this vibration insignificantly. Now we can evaluate the frequency of the shared proton stretching vibration in the Cs PWD configuration by taking into account the intrinsic anharmonicity of this mode and the anharmonic couplings with all other normal modes using eq 7 and data from Table 4 on the values of the “effective” anharmonicity constant matrix. As a result, the ν7 frequency occurs at 1261 cm−1. It should be noted that this result was obtained (1) neglecting triple and higher anharmonic couplings of the analyzed vibration with other modes and (2) neglecting possible Fermi resonances. Comparing the PWD IR spectra in a series of carriers He, Ne, and Ar and noting the fact that the intensity of the low-frequency component in this series is decreasing, it can be assumed that the second condition is valid. When assuming that additional perturbations, caused by not accounting for the first factor, have, at least in total, an insignificant influence on the ν7 frequency value, we can consider 1261 cm−1 as an upper limit for the frequency of the shared proton stretching vibration in the Cs configuration of the PWD. Note that the results of all anharmonic calculations predict that the frequency of the shared proton stretching

total effect occurs close to zero. In ref 94, a criterion for the degree of PES anharmonicity calculation was presented. To calculate the degree of PES anharmonicity, deviations of the computed using quantum-chemical methods PES from the harmonic PES were calculated using the following expression η=

1 N

N

∑

fxx =

|U (xi) − U h(xi)| U h(xi)

i=1

d2U (x) dx 2

;

U h(x) =

fxx x 2 2

;

; ... (15)

x=0

where N is the number of points, where calculations were executed, and x is the vibrational coordinate: xi = xmin +

xmax − xmin ·i ; N−1

i = 0 ÷ (N − 1)

The PES anharmonicity, calculated using criteria presented in ref 94, in both cases appears to be considerable (3.92 and 1.78 for the C2 and Cs configurations, respectively). According to the data presented in Table 4, it can be stated that in general the anharmonic effect of the other modes on the analyzed vibration in the Cs PWD configuration is smaller compared to the C2 configuration. The maximal influence on the frequency of the shared proton vibration is from the fourth mode, which is associated with the stretching O···O vibration. However, the contribution of the O···O vibration to this mode is considerably smaller than in the case of the sixth mode in the C2 PWD configuration. Cs symmetry allows contribution of the shared proton stretching vibration into the fourth mode, which is indeed the case. This particular case leads not to the red-shift (as in the case of the C2 configuration) but to a blue-shift of the ν7 frequency by 84 cm−1. The contribution of the stretching O···O vibration is present to a lesser degree in the third and sixth modes, and if in the first case there is contribution of the shared proton stretching vibration, in the second case, this contribution is completely absent. With this, as in the case of coupling with ν4, coupling with the third mode leads to the blue-shift of the ν7 frequency by almost 45 cm−1. At the same time, coupling with the sixth mode leads to a red-shift of ν7 by 2161


Article

The Journal of Physical Chemistry A vibration in the C2 PWD configuration is higher than that in the Cs configuration. However, the results obtained using the PES construction turned out to be opposing and exactly this fact is confirmed by the experimental data. Indeed, in a series of IR spectra of the PWD+He, PWD+Ne, and PWD+Ar complexes, we observe a blue-shift of the average frequency of the doublet of absorption bands near 1000 cm−1. With this, both previously noted assumptions and results of quantum-chemical calculations indicate an increase of the PWD configuration shift in this series toward the Cs symmetry. Note that a blue-shift of the shared proton stretching vibration in the PWD+Ar complex in comparison with free PWD was also obtained in a previously published theoretical work.95

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CONCLUSIONS 1. The calculations carried out in this work have shown that the stable conformer of the PWD never possesses C1 symmetry. For the first time, it was shown that calculations of the structure and IR spectra of the PWD in the C2 configuration give practically identical results both when applying the symmetry constraints and without them (at the MP4/acc-pVTZ level of theory). 2. It was determined that, with increasing level of theory, the harmonic frequency of the stretching vibration of the bridging proton decreases in the C2 configuration and increases in the Cs configuration, tending to 850 and 1220 cm−1, respectively. The anharmonicity effects accounted for within the PT2 model predict a blueshift for the ν7 mode in C2 and a red-shift in the Cs configuration. 3. It is clear that any set of vibrational coordinates of limited dimensionality will have drawbacks when solving the vibrational problem. However, it is also clear that, in the case of analyzing the ν7 mode, which according to a normal coordinate analysis is a mixture of central proton stretching vibration and antisymmetric wagging vibrations of the water molecules, the use of Cartesian or natural coordinates of the shared proton as vibrational ones does not allow taking such mixing into account. Thus, the use of the normal coordinates for the vibrational problem seems to be beneficial. This is confirmed by comparison of different models of PES and central proton frequency vibration calculations with the experimental data. 4. The couplings of the analyzed modes with all other normal modes was taken into account using the hybrid method and elements of the “effective” anharmonicity constants matrix. It allowed reducing the differences between results of calculations of limited dimensionality (3D−1D) getting rather reasonable values (1099 and 1058 cm−1) of the stretching frequency of the bridging proton unperturbed by Fermi resonance. The Fermi resonance between the fundamental ν7 vibration and the combined ν2 + ν6 vibration of the same symmetry was analyzed using the data from anharmonic PT2 calculations, as well as by 3D PES constructed using Q2, Q6, and Q7 as normal coordinates. A significant pumping of intensity (up to 50%) from the fundamental vibration to the combined one was revealed. 5. It is established that couplings of ν7 with modes, that contain a contribution of the shared proton stretching vibration, lead to a blue-shift of the ν7 frequency both in

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the case of the C2 and Cs PWD configurations. The fact that the shared proton stretching vibration in the case of the C2 configuration is antisymmetric and, in the case of the Cs configuration, symmetric leads to increasing the number of modes where this vibration can contribute, from 6 to 9 with the transition from the C2 to Cs configuration, which can be the main reason behind the ν7 frequency blue-shift due to the total anharmonic couplings with other modes in the Cs configuration, while, for the C2 configuration, this effect is opposite. 6. The calculated frequency of the shared proton stretching vibration for the PWD Cs configuration (1261 cm−1) has to be considered as an upper limit. In the series of IR spectra of the PWD+He, PWD+Ne, and PWD+Ar complexes, a blue-shift of the average frequency of the doublet of absorption bands near 1000 cm−1 is observed, which correlates with the results of our calculations that predict an increase of the ν7 frequency in the Cs configuration compared to that in the C2 configuration.

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

A. Malevich: 0000-0001-8716-8655 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the State Foundation for Basic Research of Belarusian Republic (Grant No. F16K-047) and by the grant support of the State Fund for Fundamental Research of Ukraine (Project No. F73/114-2016) and Ministries of Education and Science of Ukraine and Lithuania (grants M/492016 and TAP-LU-15-017). Additional support was provided by the Swedish Research Council (Grant No. 348-2013-6720).

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MP4 Study of the Anharmonic Coupling of the Shared Proton

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