Anharmonicity of Vibrational Modes in Hydrogen ChlorideâWater

Subscriber access provided by Washington University | Libraries

Thermodynamics

Anharmonicity of vibrational modes in hydrogen chloride-water mixtures Eva Perlt, Sarah A. Berger, Anne-Marie Kelterer, and Barbara Kirchner J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.8b01070 • Publication Date (Web): 27 Feb 2019 Downloaded from http://pubs.acs.org on March 3, 2019

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 45 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

Journal of Chemical Theory and Computation

Anharmonicity of vibrational modes in hydrogen chloride-water mixtures Eva Perlt,∗,† Sarah A. Berger,‡ Anne-Marie Kelterer,‡ and Barbara Kirchner¶ †Department of Chemistry, University of California, Irvine, 1102 Natural Sciences II, Irvine, California 92697-2025, United States ‡Institute of Physical and Theoretical Chemistry, Graz University of Technology, NAWI Graz, Stremayrgasse 9, 8010 Graz, Austria ¶Mulliken Center for Theoretical Chemistry, University of Bonn, Beringstrasse 4, D-53115 Bonn, Germany E-mail: [email protected] Abstract A thorough analysis of molecular vibrations in the binary system hydrogen chloride/water is presented considering a set of small mixed and pure clusters. In addition to the conventional normal mode analysis based on the diagonalization of the Hessian, anharmonic frequencies were obtained from the perturbative VPT2 and PT2-VSCF method using hybrid density functional theory. For all normal modes, potential energy curves were modeled by displacing the atoms from the minimum geometry along the normal mode vectors. Three model potentials—a harmonic potential, a Morse potential and a fourth order polynomial—were applied to fit these curves. From these data it was not only possible to characterize distinct vibrations as mainly harmonic, anharmonic or involving higher order terms, but also to extract force constants k and anharmonicity constants xe . By investigating all different types of intramolecular vibrations including

1

ACS Paragon Plus Environment

Journal of Chemical Theory and Computation 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

covalent stretching or bending vibrations and intermolecular vibrations such as librations, we could demonstrate that while vibrational frequencies can be obtained applying scaling factors to harmonic results, useful anharmonicity constants cannot be predicted in such a way and the usage of more elaborate vibrational methods is necessary. For each particular type of molecular vibration, we could however determine a relationship between the wavenumber or wavenumber shift and the anharmonicity constant which allows to estimate mode dependent anharmonicity constants for larger clusters in the future.

1

Introduction

The evaluation of vibrational spectra of liquid phases from theoretical methods is still a demanding task. 1,2 If a normal mode analysis using quantum chemical approaches is performed on a minimum geometry, system sizes are limited due to computational demands of the applied method. Furthermore, the presence of different configurations in a liquid phase is not accounted for. If classical molecular dynamics approaches are applied, in contrast, the phase space sampling enables a proper weighting of configurations given that the phase space can be thoroughly explored and the large system sizes account for solvent effects and intermolecular vibrations. However, the dependence of the dynamics on force field parameters limits the predictive power of such spectra. In particular, the harmonic potential which is the basis for intramolecular interactions in most force fields is not capable of reproducing the correct potential energy surface for intramolecular vibrations. For small systems, the effort of parameterizing the high-dimensional potential energy surface using highly accurate methods has been made. 3 Ab initio molecular dynamics (AIMD) provide an excellent alternative regarding the prediction of vibrational spectra, as they allow both configurational sampling along the trajectory and an accurate potential energy surface. Furthermore, the availability of the 2


Page 2 of 45

Page 3 of 45 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


electron density along the trajectory simplifies the calculation of spectra from molecular dipole moments and polarizabilities 1,4 . However, the disadvantage is that AIMD is restricted to affordable electronic structure methods such as GGA-DFT due to computational costs and data handling (file sizes) can easily become impractical. Furthermore, standard AIMD neglects nuclear quantum effects since it is still based on the classical equations of motion. Fermi resonances cause a problem which is due to different error sources for the frequencies. Often frequencies of overtones are shifted in AIMD which leads to an overall bad description. As an example in Ref. 5 it is shown for methanol that while experimental IR and Raman spectra show two intensive bands for the C-H vibration at 3000 cm−1 , one obtains only one intense band in AIMD. Especially with regard to thermodynamic data, a promising alternative to the aforementioned methods is the quantum cluster equilibrium (QCE) method. This method is based on the cluster gas approximation describing the thermodynamics of a fluid and vapor phase by individual clusters with the help of the total partition function that is modified by mean field parameters accounting for intermolecular interactions of the clusters. 6–12 In 2011 it has been extended to treat binary mixtures. 13 A considerable source of errors in this method is the use of the harmonic approximation, which is so far applied to both the vibrational partition function and the vibrational frequencies used to evaluate it. While in a recent publication, the implementation of an anharmonic partition function in the QCE method was presented and the impact on thermodynamic properties was discussed, 14 anharmonic frequencies and anharmonicity constants for the individual molecular vibrations were not available, so far. The inadequacy of the harmonic approximation for thermodynamic quantities has already been mentioned elsewhere. 15 As shown earlier in water clusters 16,17 the vibrational entropy Svib as a very sensitive measure for anharmonicity constitutes a critical point in the description of the thermodynamics. Previous studies on water, water-methanol and HF 16,18–21 included the standard harmonic model for the calculation of Svib 16 with the limitation of underestimating the mixing enthalpies.

3



The harmonic approximation which is made in nearly all theoretical approaches has a considerable effect on calculated frequencies which enter the vibrational partition function. In quantum chemical frequency calculations usually the so-called double harmonic approximation is employed 2 , in which the potential energy of nuclear motion is described in the harmonic approximation and intensities are evaluated from first order derivatives. A number of theoretical methods have evolved to perform vibrational analysis beyond the harmonic approximation. Therein, the anharmonicities are treated on different theoretical levels comparable to correlation effects in electronic structure calculations such as the vibrational SCF or vibrational PT2 methods. One approach presented by Panek and Jacob 22,23 is based on localized vibrational modes 24,25 rather than normal modes making the method affordable for larger biomolecules such as proteins. Within the vibrational second-order perturbation theory (VPT2) the harmonic Hamiltonian characterizes the unperturbed system and anharmonicity is introduced as a perturbation. In order to include cubic and semi-diagonal quartic force constants, numerical derivatives of analytical Hessians are obtained. 26,27 VPT2 has been shown to describe the anharmonic frequencies of water adequately, and has been recommended for low-lying vibrational levels. 27 Moreover, it has been successfully applied to a series of complexes including water as a donor or as an acceptor, but mainly for the large wavenumbers. A second approach for anharmonicity is the vibrational self-consistent field (VSCF) method that solves the vibrational Schrödinger equation. For weak couplings, it is followed by second-order perturbation (PT2-VSCF) including pairwise and truncated higher order mode-mode couplings. This method was shown to reproduce the anharmonicity of water clusters well. In particular infrared intensities of low-frequency soft modes (e.g. intermolecular torsions) are affected by the different shapes of the harmonic and the VSCF wavefunctions. 28 For strong couplings, the vibrational configuration interaction (VCI) can be used to correct for correlation of the VSCF solution. Since the clusters of water and HCl exhibit stronger and weaker intermolecular interactions, and both strong and weak couplings

4


Page 4 of 45

Page 5 of 45 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


of the modes may appear, we will test the following three approaches VPT2, PT2-VSCF and VCI-VSCF for the dimers in this work. However, all of these methods are computationally more demanding and thus, these methods are limited to small system sizes, as well. The present paper deals with a detailed analysis of vibrational modes of small clusters of hydrochloric acid, which is a well-known and very abundant chemical product. The aqueous solution of hydrogen chloride has a number of applications in industry such as steel pickling 29 , in synthesis of organic products like vinyl chloride or derived plastic materials and inorganic products. Hydrochloric acid is the main component of gastric acid and maintains the pH value in the stomach. Experimentally, small hydrogen chloride-water clusters have been investigated by infrared cavity ringdown laser absorption spectroscopy by Huneycutt in 2003, 30 nicely providing experimental vibrational O−H and H−Cl stretching frequencies in gas phase. In particular for the clusters, theoretical data for the intermolecular interactions were also available in literature by different authors. 30,31 In the present study, we examine in total 148 molecular vibrations of twelve binary waterHCl clusters with regard to their anharmonicity. The paper is organized as follows: After a brief introduction of our Methods in Sec. 2, we first assess the accuracy of VPT2, PT2-VSCF, and VCI-SCF approaches for the monomers and dimers to choose a proper reference method for the second part, in which we model potential energy curves for all molecular vibrations by performing single point energy calculations on grid points along the vibrational normal modes. These are then fitted to a harmonic potential, a Morse potential, and a fourth order polynomial to extract force constants and anharmonicity constants, which are then compared to reference data. Finally, an attempt is made to group the molecular vibrations according to type or symmetry and to establish certain rules, which can be transferred to estimate anharmonicity data of vibrational modes in larger clusters.

5



2

Page 6 of 45

Methods

In order to perform a normal mode analysis from quantum chemical calculations, the Born– Oppenheimer potential energy is expanded in a Taylor series around a minimum energy geometry R0 where the vector x denotes the displacement vector x = R − R0 :

V (x) = V (0) +

X ∂V i

∂xi

1X xi + 2 i,j 0

∂ 2V ∂xi ∂xj

xi xj + . . . ,

(1)

0

which is usually truncated after the second order term. 2 Since the gradient

∂V ∂xi

vanishes 0

in the minimum and the absolute value V (0) can be chosen as reference energy, this equation can be rearranged and solved to yield mass-weighted normal mode vectors and eigenvalues from which vibrational frequencies can be extracted. Due to the truncation to second order this is called the harmonic approximation, which is widely applied due its simplicity which also extends to the evaluation of the partition function and derivatives thereof.

2.1

Potential energy curves and potential models

For each normal mode distorted geometries have been obtained by displacing the atoms along the normal mode vector. The maximal displacement has been chosen to be |∆Qi,max | = ±0.1 with intervals of 0.02, where Qi denote mass-weighted normal modes and the index i denotes the vibrational degree of freedom and runs from 1 to 3N − 6 (3N − 5 for linear molecules) for N -atomic molecules. Cartesian displacement vectors xi have been obtained by dividing √ the normalized mass-weighted normal modes by the square root of the reduced mass µi : ∆Qi ∆xi = √ , µi

with ∆Qi = −0.1, −0.08 . . . , 0.0, . . . , 0.08, 0.1 ,

6


(2)

Page 7 of 45 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


where in the Gaussian program the reduced mass for N -atomic molecules is evaluated according to: µi =

3N X

!−1 2 Icart,k,i

(3)

k

with Icart is the Cartesian moment of inertia tensor. Single-point energy calculations have been performed for each of the distorted geometries, whereas the energy reference has been chosen to equal the energy at the equilibrium geometry for convenience’s sake. Consequently, the minimum of all potential curves equals zero at zero displacement. First, a harmonic potential has been applied as the most obvious function to fit to the potential energy curve, yielding the harmonic force constant k harm as well as the harmonic frequencies, conveniently expressed in terms of wavenumbers ν˜harm . 32 1 harm 2 k x 2 s

V harm (x) =

1 2πc

ν˜harm =

k harm , µ

(4) (5)

where c denotes the speed of light. The second potential has been chosen to be the Morse potential which constitutes the most popular anharmonic approximation: 32

V Morse (x) = De (1 − exp (−ax))2 .

(6)

Apart from the dissociation energy De , which can be directly extracted as a fit parameter, again the fundamental wavenumber ν˜Morse and the force constant k Morse can be determined using the exponential factor a according to

k Morse = 2De a2 s a 2De ν˜Morse = . 2πc µ

7


(7) (8)


Page 8 of 45

Furthermore, the anharmonicity constant xe can be derived, which is required to determine the vibrational energy levels of each mode: hc˜ ν Morse 4De 2 1 1 Morse = v+ hc˜ ν − v+ hc˜ ν Morse xe . 2 2

xe = Ev

(9) (10)

Therein, h denotes Planck’s constant. Finally, a fourth order polynomial has been used as a potential function. This choice provides a more flexible function as it contains more parameters and it provides higher order terms which are independent of each other and in contrast to the Morse potential. The choice to include terms up to fourth order is in agreement with anharmonic force fields, which are usually quartic force fields, 15 whose parameterization can in the simplest case be accomplished by such a potential energy fit. Assuming that the energy in the minimum was set to zero and the gradient was expected to vanish in the minimum geometry zeroth and first order terms are not included in the fitting procedure. This is consistent with the fact that slight deviations from the exact minimum position are not corrected in the harmonic normal mode analysis, either. The final polynomial for the fitting procedure is given in Eq. 11: V poly = c2 x2 + c3 x3 + c4 x4 .

(11)

The coefficients which are obtained as fitting parameters can be directly transferred to force constants by multiplication with the prefactors arising from the series expansion:

k poly = 2c2 .

(12)

Knowing the force constants, the fundamental wavenumber ν˜poly is once more evaluated according to ν˜poly

1 = 2πc 8

s

k poly . µ


(13)

Page 9 of 45 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


To fit the single-point energy data to the model potentials, a non-linear least squares fit using the scipy routine curve_fit has been applied. The fitting was performed on the grid outlined above and all data points were equally weighted. All parameters have been included in the optimization procedure. When using the VPT2, PT2-VSCF, or VCI-VSCF methods, anharmonic frequencies, overtones and force constants can be obtained from the calculations. To get an estimate of the level of anharmonicity, the anharmonicity constant is computed from harmonic ν˜harm and anharmonic wavenumbers νãnh as well as the wavenumber of the corresponding overtone mode νõvertone :

xanh e

2˜ ν anh − νõvertone = 2˜ ν harm

.

(14)

The experimental anharmonicity constant xexp was calculated using equation (14) using the e experimental fundamental frequency for both harmonic and anharmonic wavenumbers.

2.2

Computational Details

Harmonic and anharmonic VPT2 calculations have been performed with the program Gaussian09. 33 Within the latter ones, the perturbative treatment is based on quartic, cubic and semi-diagonal quartic force constants. Optimizations and frequency calculations applied standard parameters and tight convergence criteria. Specific definitions were used for the DFT grid via the keyword Int(Grid= 150590) and for normal mode calculations with the keyword Freq=(Anharm,hpmodes) to print five digits of the normal mode displacements. For (HCl)2 additionally to the standard VPT2 approach using two displacements, a calculation with four displacements for numerical differentiation was done with the keyword Freq=(FourPoint, Numeric) to try to overcome the negative anharmonic frequency. The program GAMESS was used for the PT2-VSCF and VCI-VSCF (including the ground state and the first and second vibrationally excited state in the CI expansion) calculations applying

9



standard parameters and 16 grid points in 2 dimensions. 34,35 The hybrid density functional B3LYP with empirical dispersion correction 36 using the Becke-Johnson damping function 37 (B3LYP-D3(BJ)) has been applied throughout this study as implemented in the programs. It is noted that the default B3LYP functional of GAMESS uses the VWN5 correlation compared to the default in Gaussian09 using VWN3. However, since only closed shell molecules are considered in this study, this does not impact the results. Basis sets of triple zeta quality (def2-TZVP) have been applied for the comparison of the anharmonic models VPT2, PT2-VSCF and VCI-VSCF. 38 For the interpretation of the deflection curves, VPT2 single point energy calculations along the distorted normal mode vectors around the optimized geometries have been performed with the triple-zeta def2TZVP and quadruple zeta def2-QZVP basis set. 38 The post-processing of the deflection curves including fittings of the different model potentials has been performed using an inhouse python code.

3

Results

In this study pure and mixed clusters of water and hydrogen chloride up to a total size of three monomer units have been studied. In total 148 vibrations of twelve clusters or molecules shown in Fig. 1 have been investigated by the procedure mentioned above. Within this discussion we first assess the quality of different vibrational wavefunction approaches before presenting a very detailed analysis of the intramolecular vibrations and turning to the intermolecular modes at lower wavenumbers. A visualization of each normal mode together with the potential energy plot and the fitting potentials is shown in the Supporting Information. Furthermore, all fit parameters are listed in Tables in the SI.

10


Page 10 of 45

Page 11 of 45 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 1: Ball-and-stick representation of cluster structures investigated in this study as obtained by B3LYP-D3(BJ)/def2-TZVP. Oxygen atoms: red; Chlorine atoms: green, and hydrogen atoms: white.

11



3.1

Choosing the Reference Method

Before discussing the results of potential energy fits, the anharmonic frequencies and the anharmonicity constants from three models implemented in standard programs were considered: the frequently applied VPT2 method, the VSCF approach with perturbation correction (PT2-VSCF), and the VCI-VSCF variant are compared in Tables S1 - S5 in the Supporting Information for the monomers of water and HCl as well as for the pure and mixed dimers, respectively. Reference data from literature are also given. 17,30,39–46 Based on computed anharmonic wavenumbers in comparison to experimental data, we can select the most reliable method in terms of anharmonicity for the rest of this work. From Tables S1 - S5 in the SI it is evident, that the relative deviation between calculated and experimental frequencies is largest for small wavenumbers. For water clusters, it was already shown by Temelso and Shields 47 that the low frequency modes contribute most to the zero-point vibrational energy and to the vibrational entropy underlining the importance of these modes. For the water dimer (Table S5 in the SI), VPT2 describes the frequencies within 75 cm−1 from the experiment underestimating the high frequencies and overestimating the low frequencies. PT2-VSCF and VCI-VSCF show only slightly larger deviations for the high frequencies above 1000 cm−1 , but the low frequencies are strongly overestimated. The best agreement of calculated anharmonic modes within the low frequency regime is found for the intermolecular O-O stretching vibration (˜ ν exp = 143 cm−1 ) with all three methods. In Ref. 48 the authors show, that the PT2-VSCF method gives very good agreement of the water dimer low frequencies and of the entropy of water dimerization with experimental values when the grid points are calculated at internal coordinate displacements. On the other hand, it was shown by Ruth L. Jacobsen et al. 49 on the basis of 176 different molecules that unscaled anharmonic VPT2 frequencies agree better with experimental data. They stress, however, that scaled harmonic frequencies have comparable RMSD values to anharmonic ones. For HCl, (HCl)2 , and HCl−H2 O (Tables S1, S3 and S4 in the SI), the large wavenumbers 12


Page 12 of 45

Page 13 of 45 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


are well described by all three approaches having somewhat larger deviations for PT2-VSCF. But the few available low frequencies are again much overestimated by both PT2-VSCF and VCI-VSCF methods. Overall, the best agreement in terms of the wavenumbers is observed for the VPT2 method, in particular in the lower wavenumber regime. It should be emphasized that there are further error sources in addition to the vibrational analysis such as the quality of the underlying potential energy surface as described by B3LYP or the basis set so that a substantial amount of error cancellation might be present. However, this should apply equally to the trimer clusters which are not included in Tables S1 - S5 in the SI so that we rely on these data when selecting the reference method. For the few cases where reference values for the anharmonicity constants are available, again a very good agreement with the VPT2 method is observed which gives us confidence to apply it as a suitable reference in the second part of the study. One shortcoming of VPT2 data is the negative anharmonic frequency for the (HCl)2 dimer, which also does not disappear if a four-point numerical differentiation is used for the anharmonic calculations. Therefore, and because we want to characterize the various normal mode vibrations for the dimers and trimers of HCl and water, a fitting of the normal mode deflections around the minimum geometries will be performed in the next section. Based on these observations, VPT2 data were chosen as the reference for these fits.

3.2

Potential Energy Fits along the Normal Modes

Before we shall investigate the different types of vibrations, a few general remarks are worth mentioning. First, it should be mentioned that the investigation of molecular vibrations based on single-point energy data along the normal modes is a very poor approximation and neglects important contributions such as inter-mode couplings. However, the aim is to characterize the anharmonicity in larger clusters which can then be used in statistical thermodynamic models such as the QCE method to describe liquid phases. Thus, such a naive 13



approach is studied in the following despite its severe approximation. For all vibrational modes without any exception, the fourth order polynomial function provided an excellent fit in terms of the root-mean-square deviation (RMSD, see Tables S6 - S17 in the SI) to the energy data obtained from single-point calculations. It is noticed that especially for vibrations which induce only small energy changes and flat energy profiles, the exact minimum of the fitted energy curve may deviate from the position of zero deflection. This is due to a finite convergence criterion in the geometry optimization and as a consequence, an artificial asymmetry of the energy function is observed which results in incorrect anharmonicity parameters. Although that problem could be fixed easily by adding one parameter to shift the model potential, this shall not be done in this study since such corrections are not performed in the standard normal mode analysis procedure, either. Consequently, these curves are not considered in the discussion to follow. To judge the reliability of the data obtained by the fitting approach—particularly anharmonicity constants—VPT2 numbers are given for comparison. The wavenumber which is obtained from the fourth order polynomial agrees very well with the wavenumber obtained by the harmonic normal mode analysis. Further, it is emphasized that while considerable deviations with respect to higher order force constants or anharmonicity constants are observed, most of the harmonic force constants agree reasonably well comparing the three model potentials for each of the investigated normal modes. Thus, errors due to the lack of higher order terms for the modeling of the potential energy profile are expected to be small and the major error source in harmonic vibrational analysis is the neglect of inter-mode couplings. In the following, all investigated vibrations are grouped according to their normal mode. The discussion will be focused on the anharmonicity of the particular vibrations according to the potential curve fits. In particular, the symmetry of the vibrational modes will be considered with regard to its influence on the anharmonicity. With regard to the wavenumber of the vibration, a comparison to the full anharmonic data as obtained with the VPT2 method

14


Page 14 of 45

Page 15 of 45 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


will be done so that anharmonicity constants for the different groups of vibrations can be suggested to be used for larger species in future work.

3.3

Intramolecular Vibrations

3.3.1

HCl stretching vibration

For the monomer, a wavenumber of ν˜ = 2941.92 cm−1 is observed from the usual normal harmonic mode analysis using B3LYP-D3(BJ). Experimental data in literature give numbers from 2886 cm−1 in He droplets 50 to 2991 cm−1 in gas phase. 51 In Table 1 all parameters of the H−Cl stretching vibration in larger HCl and mixed clusters are collected. Table 1: Harmonic (˜ ν harm ) and VPT2 (˜ ν VPT2 ) wavenumbers using B3LYP-D3(BJ) and experimental reference (˜ ν exp ) as well as anharmonicity constants from Morse potential fits ) and from VPT2 calculations according to Eq. 14 for H−Cl stretching vibrations. (xMorse e All wavenumbers in cm−1 , xe is unitless.

a

VPT2 ν˜TZ

VPT2 ν˜QZ

ν˜exp

xMorse e

xVPT2 e,TZ

xVPT2 e,QZ

2941.92

2839.20

2843.66

2991 51 2886 50

0.020

0.018

0.018

donor

2857.04

2774.57

2768.32

0.023

0.022

0.022

2930.04

2824.76

2827.36

2857 41 2852 52 2880 41 2887 52

acceptor

0.020

0.017

0.018

(HCl)3 -r

sym asyma

2743.07 2791.51

2660.57 2701.63

2667.55 2706.99

— 2810 53

0.008 0.002

0.013 0.014

0.013 0.014

(HCl)3 -c

center donor acceptor

2798.94 2846.57 2924.77

2706.59 2750.92 2819.49

2712.25 2764.09 2841.24

0.021 0.018 0.020

0.023 0.021 0.018

0.023 0.020 0.018

HCl−H2 O

H−Cl

2650.02

2590.79

2594.90

0.030

0.037

0.038

(H2 O)2 −HCl-r

H−Cl

2311.79

2198.70

2228.71

0.046

0.085

0.083

(H2 O)2 −HCl-u

H−Cl

2305.98

2194.69

2224.02

2464 50

0.046

0.086

0.083

H2 O−(HCl)2

Cl−H · · · O 2421.93

2327.39

2342.13

0.040

0.066

0.065

Cl−H · · · Cl 2726.95

2627.82

2643.45

2508 54 2580 50 2774 54

0.026

0.030

0.030

Cluster

assignment

HCl

H−Cl

(HCl)2

ν˜harm

twofold degenerate, average values are given

15


2714 52 2723 30 2724 50


For all HCl stretching vibrations listed in Table 1 the wavenumbers obtained in the harmonic approximation show an acceptable agreement with experimental references and with VPT2 data. This good performance of B3LYP with relatively small basis sets has already been noted in 55 where scaling factors for harmonic frequencies have been determined to be close to one for B3LYP/def2-TZVP and def2-QZVP. For the two stretching modes in the HCl dimer, red-shifts of 30 cm−1 and 6 cm−1 with respect to the monomer wavenumber have been observed for the hydrogen bonded and the free molecule, respectively, as determined by Karpfen and coworkers. 51 The computed shifts compared to the isolated monomer amount to 84.88 cm−1 and 11.88 cm−1 for bonded and free molecules, respectively, in the harmonic approximation and 64.63 cm−1 (75.34 cm−1 ) and 14.44 cm−1 (16.30 cm−1 ), respectively, for VPT2/def2-TZVP (def2-QZVP) which is in reasonable agreement with the values reported above. While the anharmonicity constants obtained from VPT2 calculations seem converged at the triple-zeta basis set and do not change if the quadruple-zeta basis is applied, significant deviations are observed if anharmonicity constants are determined from potential energy fits (see Tables in the Supporting Information). The VPT2 anharmonicity constant for the monomer (xe = 0.018, xe ν˜ = 51.11cm−1 ) is in very good agreement with experimental data (xe = 0.018, 26 xe ν˜ = 52.01cm−1 ). 56 For all clusters the formation of hydrogen bonds leads to a weakening of the covalent H−Cl bond which also leads to a red shift of the corresponding wavenumbers. A clear relation between strength of the hydrogen bond, red shift of the stretching vibration and anharmonicity from the VPT2 approach can be observed. Hydrogen bonds to water molecules are stronger than those to other hydrogen chloride units. The largest effect is observed if the hydrogen chloride molecule is donating a hydrogen bond to a water molecule and accepting a hydrogen bond from another water molecule as in both (H2 O)2 −HCl clusters. In order to be able to predict anharmonicity constants from standard harmonic vibrational analyses in future studies, we investigate the dependence of xe on the wavenumber shift with respect to the isolated monomer in Fig. 2 (a). Relating the

16


Page 16 of 45

Page 17 of 45 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


anharmonicity of stretching vibrations to their wavenumber shift turns out to be a reasonable choice. In particular for hydrogen bonded systems the red shift of the covalent X-H bond has been correlated with the strength of the hydrogen bond very early, 57 and has more recently been applied for the detection of hydrogen bonds and as an estimator for the hydrogen bond energy. 58,59 A linear relationship between the anharmonicity constant and the shift of the vibrational wavenumber is observed:

xe = xe,0 + f × ∆˜ ν ,

(15)

where ∆˜ ν = ν˜1 − ν˜j indicates the red shift of the wavenumber with respect to the isolated monomer value. The parameters in our case are determined to be xe,0 = 0.0123 and f = 1.07 × 10−4 (r=0.985). As denoted by the circles in Fig. 2 (a) the stretching vibrations of the cyclic HCl trimer cannot be described by the linear regression. The reason for that lies in the cluster symmetry which causes the molecules to vibrate together in asymmetric or symmetric movements of hydrogen atoms within the ring. This of course affects the anharmonicity so that these vibrations deviate from the linear regression. In order to investigate vibrations in such clusters further, larger cyclic structures will need to be considered in future studies. The quality of anharmonicity constants determined from potential energy fits is strongly dependent on the cluster which is considered. In pure, linear hydrogen chloride clusters, where the stretching vibrations can be attributed to certain monomer units, the anharmonicity constants determined from the Morse potential match the VPT2 data quite well. For the cyclic trimer, in contrast, where all three vibrations are superpositions or in mixed clusters with water, these numbers may deviate by a factor of two or more. This deviation is most likely due to combination bands, i.e., coupling to other modes, which is not considered in the potential energy function of a single mode. While this failure is comprehensible but

17



Figure 2: VPT2 (B3LYP-D3(BJ)/def2-QZVP) anharmonicity constants xe as a function of the red shift of the wavenumber with respect to the monomer for a) the H−Cl stretching vibration, b) the symmetric stretching vibration, and c) the asymmetric stretching vibration in water. Cyclic trimer values are indicated by circles, chain-like fragments by ’plus’ symbols. Linear regression is only applied to the latter.

18


Page 18 of 45

Page 19 of 45 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


hardly predictable, the determination of anharmonicity constants from the potential energy of a deflected normal mode is unambiguous.

3.3.2

Water Bending Vibration

The bending vibration is determined to occur at 1648.47 cm−1 from rotational-vibrational spectra 60 or 1595 cm−1 using data from HITRAN. 61 In literature, many computational results are available for the water monomer. Coleman and Howard employed MP2 with basis set extrapolation and CCSD(T) using a quintuplezeta basis to obtain 1631 cm−1 and 1650 cm−1 , respectively. 62 Another article is in excellent agreement of the former experimental value calculating a number of 1648 cm−1 . 60 A density functional theory study using the B2PLYP functional 63 suggests 1598 cm−1 including anharmonicity while simulations yield 1578 cm−1 from Born–Oppenheimer simulations and 1560 cm−1 from Car–Parrinello simulations, respectively. Within the double harmonic approach, we obtain a wavenumber of ν˜ = 1617.01cm−1 and with VPT2 we get ν˜ = 1567.31cm−1 or 1577.57 cm−1 with triple and quadruple zeta basis sets, respectively, for the water monomer. Wavenumbers and anharmonicity constants for the bending vibration of water in the clusters investigated in this study are listed in Tab. 2. For all clusters – pure and mixed with hydrogen chloride – the water bending vibration is found in a relatively shallow wavenumber regime with only small shifts as compared to the monomer. Using the conventional normal mode analysis with B3LYP-D3(BJ) values from 1615 cm−1 to 1673 cm−1 are obtained, while VPT2 predicts wavenumbers in the range from 1567 cm−1 to 1608 cm−1 (def2-TZVP) or 1577 cm−1 to 1613 cm−1 (def2-QZVP). Overall, the anharmonic VPT2 method is able to reproduce the experimental reference data better, but the wavenumbers obtained within the double harmonic approximation using B3LYP-D3(BJ) are also reasonable. Anharmonicity constants for this vibration as predicted from potential energy fits are in the order of ≈10−3 so that nearly harmonic behavior is predicted. In contrast, anharmonicity

19



Page 20 of 45

Table 2: Harmonic (˜ ν harm ) and VPT2 (˜ ν VPT2 ) wavenumbers using B3LYP-D3(BJ) and experimental reference (˜ ν exp ) as well as anharmonicity constants from Morse potential fits Morse ) and from VPT2 calculations according to Eq. 14 for H2 O bending vibrations. All (xe wavenumbers in cm−1 , xe is unitless. Cluster

assignment

ν˜harm

VPT2 ν˜TZ

VPT2 ν˜QZ

ν˜exp

xMorse e

xVPT2 e,TZ

xVPT2 e,QZ

H2 O

H2 O

1617.01

1567.31

1577.57

1648 60 1595 61

0.002

0.011

0.012

(H2 O)2

H2 O don H2 O acc

1619.11 1639.94

1577.29 1597.85

1585.50 1598.43

1599 45 1616 45

0.001 0.001

0.010 0.011

0.011 0.012

(H2 O)3 -r

asyma sym

1640.14 1672.95

1587.36 1607.04

1587.98 1612.40

0.000 0.000

0.007 0.007

0.007 0.007

(H2 O)3 -u

u(+)u(−) 1636.24 d 1641.30 u(+)u(+)d(+)1662.75

1593.78 1600.11 1605.96

1594.86 1602.13 1604.52

0.000 0.000 0.001

0.008 0.010 0.008

0.008 0.009 0.007

HCl−H2 O

H2 O

1615.21

1577.02

1585.13

0.002

0.011

0.011

(H2 O)2 −HCl-r

HO−H · · · Cl 1627.53 HO−H · · · O 1642.56

1576.06 1585.99

1583.32 1587.87

0.001 0.001

0.012 0.015

0.012 0.015

(H2 O)2 −HCl-u

asym sym

1625.98 1637.49

1576.52 1581.01

1589.04 1593.95

0.000 0.001

0.007 0.009

0.008 0.010

H2 O−(HCl)2

H2 O

1621.38

1578.60

1583.47

0.002

0.013

0.013

a

1608 64


constants as obtained from VPT2 calculations are one order of magnitude larger and range from 0.007 to 0.015. A major contribution of inter-mode coupling for this type of vibration seems to underestimate the anharmonicity in the analysis of the potential energy along that single mode. In agreement with experimental data, the water bending vibration is only slightly affected by cluster formation. 62,65 Due to the fact that the bending vibration is hardly shifted upon cluster formation, no attempt is made to relate anharmonicity to wavenumber shifts. However, also the anharmonicity constants are all predicted to be around 0.01 so that this is considered to be a reasonable choice to characterize the anharmonicity of the water bending vibration.

20


Page 21 of 45 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


3.3.3

Water Symmetric Stretching Vibration

The water monomer exhibits a considerable anharmonic effect for the symmetric stretching vibration which conserves the molecular symmetry, whereas both methods yield similar anharmonicity constants from the potential curve fits. Experimental values for the wavenumber are 3832 cm−1 from rotation-vibration spectroscopy 60 and 3657 cm−1 from HITRAN database. 61 While conventional normal mode analysis is in good agreement with the former value, VPT2 reproduces the latter one better. Wavenumbers and anharmonicity constants for the symmetric stretching vibration of water in the clusters investigated in this study are listed in Tab. 3. Table 3: Harmonic (˜ ν harm ) and VPT2 (˜ ν VPT2 ) wavenumbers using B3LYP-D3(BJ) and experimental reference (˜ ν exp ) as well as anharmonicity constants from Morse potential fits (xMorse ) e and from VPT2 calculations according to Eq. 14 for symmetric H2 O stretching vibrations. All wavenumbers in cm−1 , xe is unitless.

a

VPT2 ν˜TZ

VPT2 ν˜QZ

ν˜exp

xMorse e

xVPT2 e,TZ

xVPT2 e,QZ

3785.59

3617.71

3639.17

3832 60 3657 61

0.013

0.011

0.011

H2 O don

3669.81

3525.08

3544.56

0.024

0.023

0.024

3781.96

3612.81

3632.27

3602 44 3730 42 3651 44

H2 O acc

0.012

0.011

0.011

(H2 O)3 -r

sym asyma

3516.43 3584.41

3385.41 3434.03

3417.56 3462.23

0.009 0.002

0.015 0.016

0.014 0.016

(H2 O)3 -u

sym (ring-H) 3492.68 asym 3565.76 asym 3575.29

3350.82 3407.00 3419.16

3382.93 3434.91 3446.26

0.010 0.001 0.007

0.016 0.017 0.017

0.015 0.018 0.017

HCl−H2 O

H2 O

3778.64

3624.52

3641.68

0.012

0.011

0.011

(H2 O)2 −HCl-r

OH2 · · · O OH2 · · · Cl

3543.48 3707.47

3396.60 3546.97

3423.51 3571.87

0.028 0.020

0.033 0.021

0.033 0.020

(H2 O)2 −HCl-u

OH2 · · · O OH2 · · · Cl

3519.87 3704.98

3359.53 3536.81

3388.86 3560.74

0.029 0.020

0.034 0.021

0.034 0.020

(H2 O)−(HCl)2

H2 O

3710.95

3548.12

3567.71

0.019

0.019

0.019

Cluster

assignment

H2 O

H2 O

(H2 O)2

ν˜harm

3472 64 3518 64 3530 64


As it has already been observed for the stretching vibration of hydrogen chloride, the formation of hydrogen bonds apparently alters the covalent bond and thus impacts the covalent 21



stretching vibration. Once more a red shift of the respective wavenumbers is accompanied by an enhanced anharmonic character if hydrogen bonds are formed. This effect is more pronounced for hydrogen bond donor molecules. To get a more detailed picture of that relation, the anharmonicity constant is plotted as a function of the wavenumber shift in Figure 2 (b). Again, the values in the figure follow a linear trend if pure cyclic clusters are not taken into account. The parameters for the regression according to Eq. 15 are xe,0 = 0.0115 and f = 9.12 × 10−5 (r=0.989). As found for the H-Cl stretching vibration, the different behavior of the cyclic trimer cluster’s OH stretching vibrations is explained by the fact that in this case, all vibrations are superpositions. Anharmonicity constants for these rings are smaller compared to the remaining clusters. For the trimers, values are scattered around 0.015. The anharmonicity constants which are evaluated from potential energy fits provide a reasonable estimate for all vibrations except those from the cyclic pure water trimers. Especially for the asymmetric superpositions of vibrations in those clusters, the potential energy fitting approach completely fails due to the neglect of important couplings between modes.

3.3.4

Water Asymmetric Stretching Vibration

The asymmetric stretching vibration is slightly higher in energy than the symmetric one for all observed clusters. Again, the harmonic approximation is in better agreement with the wavenumber obtained from rotation-vibration spectra 60 (3943 cm−1 ) and the VPT2 wavenumber reproduces the value from HITRAN database 61 (3756 cm−1 ). From the potential energy plots, harmonic behavior of this normal mode is predicted for the water monomer. However, considerable anharmonic effects are observed from VPT2 calculations indicating couplings to other modes. This is also the case for the hydrogen bond accepting monomer unit in the pure water dimer and in the mixed dimer. Wavenumbers and anharmonicity constants for the asymmetric stretching vibration of water in the clusters investigated in this study are listed in Tab. 4.

22


Page 22 of 45

Page 23 of 45 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: Harmonic (˜ ν harm ) and VPT2 (˜ ν VPT2 ) wavenumbers using B3LYP-D3(BJ) and experimental reference (˜ ν exp ) as well as anharmonicity constants from Morse potential fits Morse ) and from VPT2 calculations according to Eq. 14 for asymmetric H2 O stretching (xe vibrations. All wavenumbers in cm−1 , xe is unitless.

a

VPT2 ν˜TZ

VPT2 ν˜QZ

ν˜harm 3890.78

3708.91

H2 O don

3860.66

H2 O acc (H2 O)3 -r

ν˜exp

xMorse e

xVPT2 e,TZ

xVPT2 e,QZ

3727.21

3943 60 3756 61

0.000

0.012

0.012

3679.52

3698.26

0.019

0.019

0.019

3882.86

3701.45

3719.02

3730 44 3753 42 3745 44 3760 42

0.000

0.012

0.012

asyma sym

3862.21 3863.99

3685.16 3685.87

3704.70 3706.10

3718 42

0.002 0.007

0.012 0.004

0.015 -0.003

(H2 O)3 -u

d u(+)u(-) u(+)u(+)

3853.06 3857.01 3858.80

3669.45 3680.94 3679.65

3689.64 3693.37 3697.07

3724 64 3726 64

0.021 0.013 0.016

0.020 0.018 0.011

0.020 0.015 0.005

HCl−H2 O

H2 O

3881.75

3699.47

3714.93

3763 42

0.000

0.012

0.012

(H2 O)2 −HCl uu

OH2 · · · O OH2 · · · Cl

3845.22 3862.57

3667.87 3689.33

3684.35 3706.34

3708 42 3731 42

0.023 0.016

0.021 0.018

0.021 0.018

(H2 O)2 −HCl ud

OH2 · · · O OH2 · · · Cl

3841.39 3855.84

3661.93 3675.55

3679.78 3692.77

0.023 0.016

0.021 0.018

0.021 0.018

H2 O−(HCl)2

H2 O

3847.96

3671.87

3687.67

0.013

0.017

0.017

Cluster

assignment

H2 O

H2 O

(H2 O)2

3723 42


It can be seen from Table 4 that the Morse anharmonicity constant obtained from the potential energy fitting approach is unreliable for many vibrations including the simple water monomer. While relatively accurate for some cluster vibrations, this method is inadequate for others and should not be applied to calculate anharmonicity constants. Also for the asymmetric stretching vibration, anharmonicity constants have been plotted against the shift of the wavenumber in Figure 2 (c). Again, the data points of all clusters but the pure cyclic water trimers follow a close to linear trend (r = 0.906) with xe,0 = 0.0114 and f = 1.88 × 10−4 . As for the other vibrations, the anharmonicity constants of cyclic water trimer are scattered over a large data range, presumably due to superposition of monomer vibrations to

23



cluster vibrations in these rings. Interestingly, the symmetric superposition in the case of the (H2 O)3 −r trimer is predicted to be nearly harmonic as indicated by very small anharmonicity constants. However, a certain disagreement between VPT2 calculations with triple and quadruple zeta basis sets is notable, here, and indicates some geometry convergence issue so that these data have to be taken with care.

To conclude the section on intermolecular vibrations, it is noteworthy that the anharmonicity of stretching vibrations in clusters depends on the modification of the particular bond due to cluster formation and can be related to the resulting wavenumber shift. Anharmonicity constants predicted from potential energy fitting completely lack inter-mode couplings and thus are not able to predict anharmonic constants correctly. Observed wavenumbers are affected by anharmonic treatment in terms of the VPT2 approach. However, the shifts of the wavenumbers due to cluster formation and the relative position of vibrational bands to each other is predicted very well in the harmonic approximation so that wavenumbers can be obtained in the harmonic approximation and scaled with the corresponding scaling factor.

3.4

Intermolecular Intracluster Vibrations

In this section, we shall discuss those vibrations which cannot be identified in the isolated monomer species and originate from translational or rotational motions of the monomer units in the composed clusters.

3.4.1

Translation of Molecular Subgroups

The vibrations which involve translational motions of the monomer sub-units of a cluster will also be termed “breathing” modes, since the monomers vibrate towards the cluster’s center of mass or away from it. Vibrational motions of this kind can be easily identified from their large reduced mass as they involve comparably large displacements of the heavier atoms, 24


Page 24 of 45

Page 25 of 45 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


i.e., oxygen and chlorine, in this case. Most of these vibrational motions give rise to a very flat energy profile with energies less than 1 kJ mol−1 at the maximal displacement investigated in this study. That results in a very broad and flat minimum which is hard to determine exactly from geometry optimizations with the convergence criteria given in 2.2. Thus, it is observed for many of these vibrations that the minimum of the potential energy does not exactly match the position of no deflection, see Tables and Figures in the Supporting Information. On the one hand, this leads to an artificial asymmetry of the potential energy curves and the Morse potential is no longer able to produce a qualitatively good fit resulting in an incorrect anharmonicity constant. On the other hand, this effect also impacts the VPT2 results and we observe outliers in this small wavenumber regime. Since the majority of vibrations involves such intermolecular motions, results will not be discussed in such detail as for the intramolecular vibrations. First, the translational modes in pure hydrogen chloride clusters will be studied. To get a good overview on anharmonicity constants, they have been plotted as a function of the wavenumber in Figure 3 (a). It is observed that with increasing vibrational energy the anharmonicity decreases linearly. Once more, a linear regression can be applied to determine the anharmonicity constant from the wavenumber

xe = xe,0 + f × ν˜ ,

(16)

where xe,0 = 0.0982 and f = −8.54 × 10−4 provide a very good fit (r = −0.990). For pure water clusters another picture is observed. Again the anharmonicity constants obtained from VPT2 calculations are plotted against the harmonic wavenumbers in Figure 3 (b). For the water breathing modes no such linear behavior can be observed. Furthermore, no significant effect of linear vs. cyclic structures can be seen. Since only a small number of data

25



Figure 3: VPT2 (B3LYP-D3(BJ)/def2-QZVP) anharmonicity constants xe as a function of the wavenumber for the breathing modes in a) HCl clusters, b) water clusters, and c) mixed clusters.

26


Page 26 of 45

Page 27 of 45 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


points is available it cannot be judged whether some of the shown data points are outliers. To this point, no systematic relationship between anharmonicity constant and wavenumber can be observed. As already noted, the very flat potential energy surface poses a challenge to such anharmonic analyses so that in future studies, not only further cluster structures should be taken into account, but also a more accurate method and tighter geometry convergence criteria might shed some light on this water breathing modes. In Figure 3 (c) the same relation is plotted for mixed clusters of HCl and H2 O. In this case, the scattering of data points is even more pronounced. In order to be able to distinguish the different data points, the three stoichiometries present in the clusters are denoted by different symbols. However, still no characteristic relation can be seen so that once more, more detailed investigations are necessary.

3.4.2

Rotation of Molecular Subgroups

The largest subgroup of vibrational modes in cluster compounds are those which represent rotational motions of the molecular subgroups. Again, we divided the vibrations of this type into pure hydrogen chloride, pure water, and mixed clusters and provide graphical representations of the calculated anharmonicity constants depending on the corresponding wavenumber. For pure hydrogen chloride clusters, this plot is given in Figure 4 (a). Since anharmonicity constants are again scattered, we distinguish between different clusters. Interestingly, some of the vibrations show negative anharmonicity constants. If positive anharmonicity constants are observed, these are scattered around a value of xe = 0.05. Negative values are only observed for the trimer clusters and visualization of the normal mode vectors shows that these vibrations are out-of-plane motions, which means that the individual molecules are rotated so that the hydrogen atoms move out of the plane in which the chlorine atoms are located. Figure 4 (b) shows the anharmonicity constants of those modes which involve rotation

27



Figure 4: VPT2 (B3LYP-D3(BJ)/def2-QZVP) anharmonicity constants xe as a function of the wavenumber for the rotational modes in a) HCl clusters, b) water clusters, and c) mixed clusters.

28


Page 28 of 45

Page 29 of 45 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


of water monomer units in pure water clusters. Again no systematic dependence can be observed and no grouping according to cluster size or shape is possible. Two outliers are identified for the water dimer with unusually high anharmonicity constants. As already discussed above, these low lying vibrational modes are challenging for vibrational analyses. Apart from these two outliers, the anharmonicity constants are again scattered around a value of xe = 0.05. Finally, vibrations which show rotational motions of monomer units in mixed clusters of hydrogen chloride and water are considered and the corresponding plot is given in Figure 4 (c). For the mixed clusters, the anharmonicity constants are larger as compared to the pure clusters, and again they are broadly scattered. Although the different clusters are identified by different symbols, no systematic relation is found. Some anharmonicity constant are peculiarly large and above 0.1, however, in this case all data points are at larger values of xe so that no outliers can be identified.

4

Conclusion and Outlook

The present paper deals with a very detailed investigation of all vibrational modes in pure and mixed cluster of the system water/hydrogen chloride with special emphasis on anharmonicity. The main goal of the study was to establish a relation of the anharmonicity constant to some characteristics of the respective mode like type of the vibration or symmetry which would enable the estimate of anharmonicity constants for larger clusters to calculate more accurate thermodynamic data, for example within the QCE model. Therefore, the potential energy curves of all molecular modes were modeled and compared to data obtained from PT2-VSCF, VCI-VSCF and VPT2 methods. The best agreement with experimental data for both high and low frequencies was observed using VPT2 which we chose as our reference. The B3LYP-D3(BJ) method was used throughout this study and generally results in very good harmonic wavenumbers. To obtain more accurate wavenumbers, the use of empirical

29



scaling factors has been established and discussed elsewhere. 66 Particularly if thermodynamic data are sought, however, the use of wavenumber dependent scaling factors is advantageous, as vibrations at lower wavenumbers require stronger scaling and additionally, the anharmonic effect on properties is largest for those modes as we demonstrated for the vibrational entropy. Overall, modeling the potential energy profile of each vibration using single point energy calculations does not reproduce the correct anharmonicity since important effects such as inter-mode coupling are not considered. In agreement with a frequency shift of covalent stretching vibrations due to hydrogen bonding observed earlier, 57–59 we could identify a clear relation between this red-shift and the anharmonicity constant. Also for the remaining vibrations, the following general statements can be made: • for stretching vibrations, the anharmonicity constants depend on the wavenumber shift as compared to the isolated monomer related to the strength of the hydrogen bond, xe can be estimated using a linear equation 15; • for water bending vibrations no significant changes in wavenumber or anharmonicity constant were found upon cluster formation, an average value of xe = 0.01 can be used; • the translation of HCl monomer units in a pure HCl cluster (breathing mode) is linearly dependent on the wavenumber, again a linear equation can be used; • for translations of water and mixed cluster as well as for rotational motions no clear trend can be made, and more clusters need to be investigated. Especially for cyclic clusters different numbers were observed, which were also related to the symmetry. However, due to the small size of the clusters considered in this study, only very few cyclic structures were contained so that in future, larger cyclic clusters will need to be further investigated.

30


Page 30 of 45

Page 31 of 45 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


Acknowledgement E. P. gratefully acknowledges funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) grant number 391320977.

Supporting Information Available Harmonic and VPT2, PT2-VSCF, and VCI-VSCF wavenumbers in comparison to literature data for monomers and dimers (Tables S1 - S5) as well as fitting parameters for all vibrational modes of all clusters for the different model potentials (Tables S6 - S17) together with visualizations of the vibrations and plots of the potential energy curves. This information is available free of charge via the Internet at http://pubs.acs.org

References (1) Thomas, M.; Brehm, M.; Kirchner, B. Voronoi dipole moments for the simulation of bulk phase vibrational spectra. Phys. Chem. Chem. Phys. 2015, 17, 3207–3213. (2) Neugebauer, J.; Reiher, M.; Kind, C.; Hess, B. A. Quantum chemical calculation of vibrational spectra of large molecules—Raman and IR spectra for Buckminsterfullerene. J. Comput. Chem. 2002, 23, 895–910. (3) Leforestier, C.; van Harrevelt, R.; van der Avoird, A. Vibration–Rotation–Tunneling Levels of the Water Dimer from an ab Initio Potential Surface with Flexible Monomers. J. Phys. Chem. A 2009, 113, 12285–12294, PMID: 19476322. (4) Thomas, M.; Kirchner, B. Classical Magnetic Dipole Moments for the Simulation of Vibrational Circular Dichroism by ab Initio Molecular Dynamics. J. Phys. Chem. Lett. 2016, 7, 509–513, PMID: 26771403.

31



(5) Thomas, M. Theoretical Modeling of Vibrational Spectra in the Liquid Phase; Springer Theses; Springer, Cham, 2017; Chapter 4. (6) Weinhold, F. Quantum cluster equilibrium theory of liquids: General theory and computer implementation. J. Chem. Phys. 1998, 109, 367–372. (7) Ludwig, R.; Weinhold, F.; Farrar, T. C. Quantum cluster equilibrium theory of liquids part I: Molecular clusters and thermodynamics of liquid ammonia. Ber. Bunsenges. Phys. Chem. 102, 197–204. (8) Ludwig, R.; Weinhold, F.; Farrar, T. C. Quantum cluster equilibrium theory of liquids part II: Temperature dependent chemical shifts, quadrupole coupling constants and vibrational frequencies in liquid ammonia. Ber. Bunsenges. Phys. Chem. 102, 205–212. (9) Weinhold, F. Kinetics and Mechanism of Water Cluster Equilibria. J. Phys. Chem. B 2014, 118, 7792–7798. (10) Kirchner, B. Cooperative versus dispersion effects: What is more important in an associated liquid such as water? J. Chem. Phys. 2005, 123, 204116. (11) Brüssel, M.; Perlt, E.; von Domaros, M.; Brehm, M.; Kirchner, B. A one-parameter quantum cluster equilibrium approach. J. Chem. Phys. 2012, 137, 164107. (12) Kirchner, B.; Spickermann, C.; Lehmann, S. B.; Perlt, E.; Langner, J.; von Domaros, M.; Reuther, P.; Uhlig, F.; Kohagen, M.; Brüssel, M. What can clusters tell us about the bulk?: Peacemaker: Extended quantum cluster equilibrium calculations. Comput. Phys. Commun. 2011, 182, 1428–1446. (13) Brüssel, M.; Perlt, E.; Lehmann, S. B. C.; von Domaros, M.; Kirchner, B. Binary systems from quantum cluster equilibrium theory. J. Chem. Phys. 2011, 135, 194113. (14) von Domaros, M.; Perlt, E. Anharmonic effects in the quantum cluster equilibrium method. J. Chem. Phys. 2017, 146, 124114. 32


Page 32 of 45

Page 33 of 45 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


(15) Császár, A. G. Anharmonic molecular force fields. WIRes: Comput. Mol. Sci. 2012, 2, 273–289. (16) Spickermann, C.; Lehmann, S. B. C.; Kirchner, B. Introducing phase transitions to quantum chemistry: From Trouton’s rule to first principles vaporization entropies. J. Chem. Phys. 2008, 128, 244506. (17) Temelso, B.; Shields, G. The Role of Anharmonicity in Hydrogen-Bonded Systems: The Case of Water Clusters. J. Chem. Theory Comput. 2011, 7, 2804–2817. (18) Friedrich, J.; Perlt, E.; Roatsch, M.; Spickermann, C.; Kirchner, B. Coupled Cluster in Condensed Phase. Part I: Static Quantum Chemical Calculations of Hydrogen Fluoride Clusters. J. Chem. Theory Comput. 2011, 7, 843–851, PMID: 26606336. (19) Spickermann, C.; Perlt, E.; von Domaros, M.; Roatsch, M.; Friedrich, J.; Kirchner, B. Coupled Cluster in Condensed Phase. Part II: Liquid Hydrogen Fluoride from Quantum Cluster Equilibrium Theory. J. Chem. Theory Comput. 2011, 7, 868–875. (20) Perlt, E.; Friedrich, J.; von Domaros, M.; Kirchner, B. Importance of Structural Motifs in Liquid Hydrogen Fluoride. ChemPhysChem 12, 3474–3482. (21) Matisz, G.; Kelterer, A.-M.; Fabian, W. M. F.; Kunsági-Máté, S. Structural properties of methanol-water binary mixtures within the quantum cluster equilibrium model. Phys. Chem. Chem. Phys. 2015, 17, 8467–8479. (22) Panek, P. T.; Jacob, C. R. Efficient Calculation of Anharmonic Vibrational Spectra of Large Molecules with Localized Modes. ChemPhysChem 2014, 15, 3365–3377. (23) Panek, P. T.; Jacob, C. R. Anharmonic Theoretical Vibrational Spectroscopy of Polypeptides. J. Phys. Chem. Lett. 2016, 7, 3084–3090, PMID: 27472016. (24) Jacob, C. R.; Reiher, M. Localizing normal modes in large molecules. J. Chem. Phys. 2009, 130, 084106. 33



(25) Panek, P. T.; Jacob, C. R. On the benefits of localized modes in anharmonic vibrational calculations for small molecules. J. Chem. Phys. 2016, 144, 164111. (26) Barone, V. Vibrational zero-point energies and thermodynamic functions beyond the harmonic approximation. J. Chem. Phys. 2004, 120, 3059–3065. (27) Barone, V. Anharmonic vibrational properties by a fully automated second-order perturbative approach. J. Chem. Phys. 2005, 122, 014108. (28) Jung, J.; Gerber, R. Vibrational wave functions and spectroscopy of (H2 O)n , n=2,3,4,5: Vibrational self-consistent field with correlation corrections. J. Chem. Phys. 1996, 105, 10332–10348. (29) Markit, I. Chemical Economics Handbook ; Chemical Economics Handbook v. 1; IHS Markit, 2015. (30) Huneycutt, A. J.; Stickland, R. J.; Hellberg, F.; Saykally, R. J. Infrared cavity ringdown spectroscopy of acid–water clusters: HCl–H2O, DCl–D2O, and DCl–(D2O)2. J. Chem. Phys. 2003, 118, 1221–1229. (31) Chaban, G. M.; Gerber, R. B.; Janda, K. C. Transition from Hydrogen Bonding to Ionization in (HCl)n (NH3 )n and (HCl)n (H2 O)2 Clusters: Consequences for Anharmonic Vibrational Spectroscopy. J. Phys. Chem. A 2001, 105, 8323–8332. (32) McQuarrie, D. A. Statistical Mechanics, 1st ed.; University Science Books, 2000. (33) 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.; Nakatsuji, H.; Caricato, M.; Li, X.; Hratchian, H. P.; Izmaylov, A. F.; Bloino, J.; Zheng, G.; Sonnenberg, J. L.; Hada, M.; Ehara, M.; Toyota, K.; Fukuda, R.; Hasegawa, J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.; Nakai, H.; Vreven, T.; J. A. Montgomery, J.; Peralta, J. E.; Ogliaro, F.; Bearpark, M.; Heyd, J. J.; Brothers, E.; Kudin, K. N.; 34


Page 34 of 45

Page 35 of 45 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


Staroverov, V. N.; Keith, T.; Kobayashi, R.; Normand, J.; Raghavachari, K.; Rendell, A.; Burant, J. C.; Iyengar, S. S.; Tomasi, J.; Cossi, M.; Rega, N.; Millam, J. M.; Klene, M.; Knox, J. E.; Cross, J. B.; Bakken, V.; Adamo, C.; Jaramillo, J.; Gomperts, R.; Stratmann, R. E.; Yazyev, O.; Austin, A. J.; Cammi, R.; Pomelli, C.; Ochterski, J. W.; Martin, R. L.; Morokuma, K.; Zakrzewski, V. G.; Voth, G. A.; Salvador, P.; Dannenberg, J. J.; Dapprich, S.; Daniels, A. D.; Farkas, O.; Foresman, J. B.; Ortiz, J. V.; Cioslowski, J.; Fox, D. J. Gaussian~09 Revision D.01. 2013; Gaussian Inc. Wallingford CT. (34) Schmidt, M. W.; Baldridge, K. K.; Boatz, J. A.; Elbert, S. T.; Gordon, M. S.; Jensen, J. H.; Koseki, S.; Matsunaga, N.; Nguyen, K. A.; Su, S.; Windus, T. L.; Dupuis, M.; Montgomery, J. A. General Atomic and Molecular Electronic Structure System. J. Comput. Chem. 1993, 14, 73–78. (35) Gordon, M. S.; Schmidt, M. W. In Theory and Applications of Computational Chemistry: the first forty years; C.E.Dykstra„ G.Frenking„ K.S.Kim„ G.E.Scuseria„ Eds.; Elsevier: Amsterdam, 2005; pp 1167–1189. (36) 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. (37) Grimme, S.; Ehrlich, S.; Goerigk, L. Effect of the damping function in dispersion corrected density functional theory. J. Comput. Chem. 2011, 32, 1456–1465. (38) Weigend, F.; Ahlrichs, R. Balanced basis sets of split valence, triple zeta valence and quadruple zeta valence quality for H to Rn: Design and assessment of accuracy. Phys. Chem. Chem. Phys. 2005, 7, 3297–3305. (39) Perrot, M.; Turrell, G.; van Huong, P. Vibrational Anharmonicity of HCl in Solution. J. Mol. Spectrosc. 1970, 34, 47–52. 35



(40) Huber, K.; Herzberg, G. Molecular Spectra and Molecular Structure. IV. Constants of Diatomic Molecules; Van Nostrand Reinhold Co., 1979. (41) Schuder, M. D.; Lovejoy, C. M.; Lascola, R.; Nesbitt, D. J. High resolution, jet-cooled infrared spectroscopy of (HCl)2: Analysis of ν1 and ν2 HCl stretching fundamentals, interconversion tunneling, and mode-specific predissociation lifetimes. J. Chem. Phys. 1993, 99, 4346–4362. (42) Skvortsov, D.; Lee, S. J.; Choi, M. Y.; Vilesov, A. F. Hydrated HCl Clusters, HCl(H2O)1–3, in Helium Nanodroplets: Studies of Free OH Vibrational Stretching Modes. J. Phys. Chem. A 2009, 113, 7360–7365. (43) Ault, B. S.; Pimentel, G. C. Infrared spectrum of the water-hydrochloric acid complex in solid nitrogen. J. Phys. Chem. 1973, 77, 57–61. (44) Otto, K. E.; Xue, Z.; Zielke, P.; Suhm, M. A. The Raman spectrum of isolated water clusters. Phys. Chem. Chem. Phys. 2014, 16, 9849–9858. (45) Bouteiller, Y.; Perchard, J. The vibrational spectrum of (H2O)2: comparison between anharmonic ab initio calculations and neon matrix infrared data between 9000 and 90 cm-1. Chem. Phys. 2004, 305, 1 – 12. (46) Nizkorodov, S. A.; Ziemkiewicz, M.; Nesbitt, D. J.; Knight, A. E. W. Overtone spectroscopy of H2O clusters in the vOH=2 manifold: Infrared-ultraviolet vibrationally mediated dissociation studies. J. Chem. Phys 2005, 122, 194316. (47) Temelso, B.; Shields, G. C. The Role of Anharmonicity in Hydrogen-Bonded Systems: The Case of Water Clusters. J. Chem. Theory Comput. 2011, 7, 2804–2817, PMID: 26605472. (48) Njegic, B.; Gordon, M. S. Exploring the effect of anharmonicity of molecular vibrations on thermodynamic properties. J. Chem. Phys. 2006, 125, 224102. 36


Page 36 of 45

Page 37 of 45 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


(49) Jacobsen, R. L.; Johnson, R. D.; Irikura, K. K.; Kacker, R. N. Anharmonic Vibrational Frequency Calculations Are Not Worthwhile for Small Basis Sets. J. Chem. Theory Comput. 2013, 9, 951–954, PMID: 26588738. (50) Weimann, M.; Farnik, M.; Suhm, M. A. A first glimpse at the acidic proton vibrations in HCl-water clusters via supersonic jet FTIR spectroscopy. Phys. Chem. Chem. Phys. 2002, 4, 3933–3937. (51) Karpfen, A.; Bunker, P.; Jensen, P. An ab initio study of the hydrogen chloride dimer: The potential energy surface and the characterization of the stationary points. Chem. Phys. 1991, 149, 299 – 309. (52) Ortlieb, M.; Birer, O.; Letzner, M.; Schwaab, G. W.; Havenith, M. Observation of Rovibrational Transitions of HCl, (HCl)2, and H2O–HCl in Liquid Helium Nanodroplets. J. Phys. Chem. A 2007, 111, 12192–12199, PMID: 17985854. (53) Mancini, J. S.; Bowman, J. M. A New Many-Body Potential Energy Surface for HCl Clusters and Its Application to Anharmonic Spectroscopy and Vibration–Vibration Energy Transfer in the HCl Trimer. J. Phys. Chem. A 2014, 118, 7367–7374, PMID: 24444294. (54) Morrison, A. M.; Flynn, S. D.; Liang, T.; Douberly, G. E. Infrared Spectroscopy of (HCl)m(H2O)n Clusters in Helium Nanodroplets: Definitive Assignments in the HCl Stretch Region. J. Phys. Chem. A 2010, 114, 8090–8098, PMID: 20684581. (55) Kesharwani, M. K.; Brauer, B.; Martin, J. M. L. Frequency and Zero-Point Vibrational Energy Scale Factors for Double-Hybrid Density Functionals (and Other Selected Methods): Can Anharmonic Force Fields Be Avoided? J. Phys. Chem. A 2015, 119, 1701–1714, PMID: 25296165. (56) De Almeida, W. B. An ab initio study of the mechanical and electrical anharmonicities in the vibrational spectrum: CO and HCl. Spectrochim. Acta A 1992, 48, 1445 – 1452. 37



(57) Badger, R. M.; Bauer, S. H. Spectroscopic Studies of the Hydrogen Bond. II. The Shift of the O–H Vibrational Frequency in the Formation of the Hydrogen Bond. J. Chem. Phys. 1937, 5, 839–851. (58) Kirchner, B.; Spickermann, C.; Reckien, W.; Schalley, C. A. Uncovering Individual Hydrogen Bonds in Rotaxanes by Frequency Shifts. J. Am. Chem. Soc. 2010, 132, 484–494, PMID: 20028099. (59) Wendler, K.; Thar, J.; Zahn, S.; Kirchner, B. Estimating the Hydrogen Bond Energy. J. Phys. Chem. A 2010, 114, 9529–9536, PMID: 20707378. (60) Benedict, W. S.; Gailar, N.; Plyler, E. K. Rotation-Vibration Spectra of Deuterated Water Vapor. J. Chem. Phys. 1956, 24, 1139–1165. (61) Kjaergaard, H. G.; Garden, A. L.; Chaban, G. M.; Gerber, R. B.; Matthews, D. A.; Stanton, J. F. Calculation of Vibrational Transition Frequencies and Intensities in Water Dimer: Comparison of Different Vibrational Approaches. J. Phys. Chem. A 2008, 112, 4324–4335, PMID: 18407701. (62) Coleman Howard, J.; Gray, J. L.; Hardwick, A. J.; Nguyen, L. T.; Tschumper, G. S. Getting down to the Fundamentals of Hydrogen Bonding: Anharmonic Vibrational Frequencies of (HF)2 and (H2O)2 from Ab Initio Electronic Structure Computations. J. Chem. Theory Comput. 2014, 10, 5426–5435, PMID: 26583226. (63) 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. Theory Comput. 2010, 6, 2115–2125, PMID: 26615939. (64) Tremblay, B.; Madebène, B.; Alikhani, M.; Perchard, J. The vibrational spectrum of the water trimer: Comparison between anharmonic ab initio calculations and neon matrix infrared data between 11,000 and 90cm-1. Chem. Phys. 2010, 378, 27 – 36. 38


Page 38 of 45

Page 39 of 45 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


(65) Masia, M.; Forbert, H.; Marx, D. Connecting Structure to Infrared Spectra of Molecular and Autodissociated HCl–Water Aggregates. J. Phys. Chem. A 2007, 111, 12181– 12191, PMID: 17915839. (66) Scott, A. P.; Radom, L. Harmonic Vibrational Frequencies: An Evaluation of Hartree– Fock, Møller–Plesset, Quadratic Configuration Interaction, Density Functional Theory, and Semiempirical Scale Factors. J. Phys. Chem. 1996, 100, 16502–16513.

39



For Table of Contents Only

40


Page 40 of 45

Page 41 of 45 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





Page 42 of 45

Page 43 of 45


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 44 of 45

Page 45 of 45 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17



Anharmonicity of Vibrational Modes in Hydrogen ChlorideâWater

Recommend Documents

Anharmonicity of Vibrational Modes in Hydrogen ChlorideâWater