The Role of Electrical Anharmonicity in the Association Band in the

Article pubs.acs.org/JPCB

The Role of Electrical Anharmonicity in the Association Band in the Water Spectrum Anne B. McCoy* Department of Chemistry and Biochemistry, The Ohio State University, Columbus, Ohio 43210, United States S Supporting Information *

ABSTRACT: The origin of the intensity of the feature in the spectrum of liquid water near 2100 cm−1 is investigated through calculations of the spectra of water clusters based on low-order expansions of the potential and dipole surfaces in internal and normal mode coordinates. The intensity near 2100 cm−1 is attributed to combination bands involving the HOH bend and intermolecular vibrations that break the hydrogen bonding network. Further, the leading contribution to the intensity reflects large second derivatives of the dipole moment with respect to the internal coordinates that are excited, or electrical anharmonicity. This picture changes if the derivatives of the potential and dipole surfaces are taken with respect to normal modes. In the normal mode representation, the second derivatives of the dipole moment are often vanishingly small, while the mixed third and fourth derivatives of the potential become quite large. On the basis of this result, mechanical anharmonicity appears to be responsible for the intensity in the 2100 cm−1 region. This strong dependence of the interpretation of the origins of the intensity in the 2100 cm−1 region of the water spectrum is investigated and discussed.

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INTRODUCTION

intensity of this band relative to the HOH bend is sensitive to the environment around the water molecules.3,4 In more ordered environments, the band is found to shift to the blue and increase in relative intensity. Studies dating back more than 80 years assign this feature to combination bands involving the HOH bend and librations of water molecules.5 What is less clear is the mechanism for a combination band to carry intensity. In a recent joint experiment and theory study of complexes of H3O+ with three argon atoms, nitrogen, methane, or water molecules,6 we found that the spectra of all of these complexes contained a persistent feature near 2100 cm−1. The position of this feature varied from 1939 to 2247 cm−1, while the relative intensity of this band, compared to the HOH bend fundamental, increased when the interaction strength between H3O+ and the solvating molecules was increased. In that study, we attributed this feature to large quadratic terms in the expansion of the dipole moment of the complex. The large magnitudes of the second derivatives of the dipole moment with respect to the HOH bend and the hindered rotation of the H3O+ molecule about its symmetry axis reflected a modulation of the transition moment of the HOH bend fundamental as hydrogen bonds to the adjacent solvating atoms or molecules were broken. Specifically, in a hydrogen-bonded geometry, the transition moment of the HOH bend is suppressed relative to

Studies of the spectrum of water have a long history. The gasphase spectrum of an isolated water molecule is characterized by a strong asymmetric stretch transition, a weaker symmetric stretch, and the bend fundamental at 3756, 3657, and 1595 cm−1, respectively. Due to a strong Fermi resonance between the symmetric stretch and the bend, an additional peak at 3152 cm−1 is attributed to the first overtone in the bend.1 The spectrum of condensed phase water contains an additional weaker feature near 2100 cm−1, which is plotted in red in Figure 1, and is identified as the association band.2 The

Special Issue: James L. Skinner Festschrift Received: February 15, 2014 Revised: March 31, 2014 Published: April 1, 2014

Figure 1. The spectrum of liquid water as reported by Bertie and Lan.2 © 2014 American Chemical Society

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dx.doi.org/10.1021/jp501647e | J. Phys. Chem. B 2014, 118, 8286−8294

The Journal of Physical Chemistry B

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its value in bare H3O+. While the conclusions were based on an analysis of protonated water complexes, the importance of the hydrogen bonding network in the mechanism led us to conclude that similar behavior should be responsible for the association band at 2100 cm−1 in the spectrum of liquid water shown in Figure 1. The importance of such non-Condon effects has also been invoked by Skinner and co-workers in their analysis of the line shapes of the OH stretch transitions in the spectrum of water.7 In the present study, we revisit this interpretation of the origin for intensity in the 2100 cm−1 region of the water spectrum by investigating the spectra of 13 water complexes composed of six or fewer water molecules. The study aims to address three issues. The first is to determine if the combination bands that were identified in the protonated water complexes in our earlier study6 can be found in the harmonic spectra of neutral water clusters where the frequencies and intensities are obtained from quadratic expansions of the potential and dipole moment surface. As we investigated this question, we found a surprisingly large dependence of the answer on the coordinates that were used to expand the potential and dipole surface. Specifically, when the potential and dipole surfaces were expanded in normal modes that are obtained from linear combinations of displacements of Cartesian coordinates, the calculated spectra either do not contain intensity in the 2100 cm−1 region, or the calculated intensity in this region exceeds that of the HOH bend. Both of these results are inconsistent with experiment. Of additional concern is the observation that very similar cluster structures are found to show qualitatively different spectral features in the 2100 cm−1 region. In contrast, if the potential and dipole surfaces are expanded to second order in normal modes that are derived from linear combinations of displacements of internal coordinates, a band near 2100 cm−1 is found in all 13 water clusters that were studied. The second issue that this study aims to address is why the spectra that are calculated from quadratic expansions of the potential and dipole surfaces are this sensitive to the choice of coordinates. Finally, when second-order perturbation theory is used to calculate the spectrum, including both electrical and mechanical anharmonicities, the coordinate dependence of the results is considerably smaller. The third focus is to understand why this is the case. As we discuss these points, we will use the Cl−·H2O complex, where electrical anaharmonicity was invoked to explain the large intensity of the out-of-plane bend,8,9 and Ar3·H3O+ 6 as well as the water clusters. It has long been recognized that internal coordinates provide a more appropriate set for expanding potential functions than the normal modes. The work of Sibert, Hynes, and Reinhardt further demonstrated that the choice of coordinates is critical when bending motions are involved, as is the case in their analysis of the origins of the Fermi resonance in CO2.10 The importance of the choice of coordinates will become even more critical when we consider molecular clusters where the lowfrequency intermolecular modes undergo large amplitude displacements even at the zero-point level.11,12 To illustrate this, consider the out-of-plane bend in the Cl−·H2O complex. This vibration is illustrated in an internal and Cartesian representation in Figure 2. As is seen, the internal coordinate representation more closely follows the natural motions of the complex, while the Cartesian displacement also leads to a significant increase in the length of the bonded OH bond. In fact, the OH bond length is increased by 0.024 Å relative to its

Figure 2. Illustrations of the out-of-plane bend in the Cl−·H2O complex when the displacement is made in the normal mode based on Cartesian and internal coordinates.

equilibrium value when this mode is displaced to the turning point in the harmonic potential at an energy that corresponds to the zero-point level, while for the state with two quanta in this mode, the OH bond is lengthened by 0.12 Å. In this paper, we return to the question of the origin of the features in the water spectrum at roughly 2100 cm−1, focusing on the spectra of water clusters. We find that the feature arises from quadratic terms in the internal coordinate expansion of the dipole moment. As noted above, either no or unphysical large intensity is obtained when a quadratic expansion of the dipole moment in normal mode coordinates is used for the analysis. The origin of this sensitivity of the results to the choice of coordinates is explored by investigating a similar effect in Cl−·H2O.8,9 We then turn our attention to the association band in the water spectrum, and demonstrate that a variety of different structural motifs found in clusters all have intensity in the region of the association band in the liquid water spectrum.

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THEORY In the harmonic oscillator linear dipole approximation, the vibrational Hamiltonian is replaced by a quadratic expansion in the coordinates of interest. Likewise, the dipole moment is expanded to first order in the same set of coordinates. Normal coordinates are then defined as the linear combinations of displacements in the chosen coordinates that render the vibrational Hamiltonian separable at second order. In this representation, the Hamiltonian consists of 3N − 6 uncoupled harmonic oscillators, and the vibrational energies and intensities can be evaluated analytically. By truncating the expansion of the Hamiltonian at second order and the dipole at first order, the results of this harmonic oscillator/linear dipole analysis are independent of the coordinates in which the normal modes are expanded. If perturbation theory is applied to an expansion of the rotation−vibration Hamiltonian in which the terms are sorted by powers of ℏ, e.g., linear terms in ℏ define H(0), while terms that are of the order ℏ(2n+1)/2 are in H(n), the resulting coefficients in the expression of the energy in terms of polynomials in the harmonic oscillator quantum numbers will be independent of the coordinates used to define the normal modes. 13 For other approximate approaches or when resonances are accounted for in the perturbation theory, once higher-order corrections are introduced in the expansion of the Hamiltonian, the choice of coordinates will affect the final results. This will be most important in highly anharmonic systems. 8287



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Figure 3. Slices through the potential and y-component of the dipole surface for Cl−·H2O, plotted as functions of the normal modes that are based on displacements of the internal (a, c, e, g) and Cartesian coordinates (b, d, f, h). In panels a and e, the results for a harmonic potential (a) and linear expansion of the dipole surface (e) in terms of the OH bond length and rotation of the OH bond out of the plane of the complex are plotted. In parts c and g, these are expanded to fourth and third order, respectively. The remaining four panels show expansions of these surfaces to fourth (b and d) or third (f and h) order in the normal modes based on Cartesian displacements. The plots of the x- and z-components of the dipole moment as well as the coefficients for the expansions of the potential and dipole surfaces are provided in the Supporting Information.

plane, as is shown in Figure 2a. The coefficients in the expansions of the potentials that are plotted in Figure 3 are provided in the Supporting Information. This leads us to the second set of normal mode coordinates. Instead of starting from the second derivatives of the potential with respect to Cartesian coordinates, we turn to expansions through second order in displacements of the internal coordinates. Unlike the Cartesian expansion, the choice of internal coordinates is not uniquely defined. On the other hand, following simple sets of rules,14 one can generate a rational set of valence coordinates in which to expand the Hamiltonian. While the plots in Figure 3 show that the expansion of the potential converges more rapidly in these coordinates, it comes at the expense of coordinate-dependent masses and cross terms in the kinetic energy

Before discussing how the choice of coordinates affects the calculated spectra, it is useful to define the two sets of coordinates that will be used in the present study. The first is the traditional normal mode coordinates. These coordinates are defined as the linear combinations of mass-weighted displacements of the Cartesian coordinates that are the eigenfunctions of the Hessian, e.g., -i , j =

∂ 2V ∂yi ∂yj

(1)

mi (xi − xi,e)

(2)

where yi =

While these coordinates provide an effective description of stretching modes, expansions of the potential and dipole surfaces can be slow to converge when they are used to describe bending motions. This is illustrated by the expansions of cuts through the potential surface for the Cl−·H2O complex that are plotted in Figure 3a−d. In panels a and c, quadratic and quartic expansions of cuts through the Cl−·H2O potential are plotted as functions of coordinates that are proportional to displacements in the length of the bonded OH bond and out-of-plane bend angle illustrated in Figure 2b. In panels b and d, the potentials in panels a and c are expanded through fourth order in the normal modes defined by the eigenvectors of the Hessian in eq 1. As is seen, a potential that is uncoupled in internal coordinates (panel a) shows substantial coupling when normal modes are used to expand the potential (panel b). The distorted shapes of the potentials plotted in Figure 3b and d reflect large cubic and quartic terms in the expansion of the potential that result from the increase in the OH bond length when the hydrogen is displaced perpendicular to the molecular

T=−

ℏ2 2

3N − 6

∑ i,j=1

∂ ∂ Gi , j(r) ∂ri ∂rj

(3)

The transformation from the Cartesian to the internal coordinate kinetic energy operator is easily derived using 3N

Gi , j =

∑ n=1

∂ri ⎛ 1 ⎞ ∂rj ⎜ ⎟ ∂xn ⎝ mn ⎠ ∂xn

(4) 11,14,15

where ri represents one of the internal coordinates. The necessary G-matrix elements have been reported and may be obtained either from these tabulations or by numerical evaluation of the partial derivatives in eq 4.16 Unless otherwise noted, all calculations were performed at the MP2/aug-cc-pVDZ level of theory and basis. The initial structures for the water clusters were obtained from the earlier work of Shields and co-workers.17 While higher levels of 8288



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overtone in the out-of-plane bend of Cl−·H2O to changes in the dipole moment of the complex along the O−Cl axis.8 The large dependence of the y-component of the dipole moment on the out-of-plane rotation of the OH bond was ascribed to this motion, leading to a loss of the ionic hydrogen bond between the water molecule and the chloride ion, which leads to an increase in the localization of the excess negative charge on the chlorine atom.8 At about the same time as we reported these findings, Rheinecker and Bowman22 performed a full-dimensional variational calculation of the spectrum of this complex using analytical fits of the six-dimensional potential and dipole moment surfaces. These calculations were carried out in the normal coordinates that are derived from the displacements of the Cartesian coordinates of the individual atoms as described by eqs 1 and 2. These calculations reproduced the intensity for the out-of-plane bend overtone, and Rheinecker and Bowman attributed the intensity to large contributions from the HOH bend fundamental and a combination band involving the bonded OH stretch and two quanta in the out-of-plane bend to the description of the molecular eigenstate that corresponded to two quanta of excitation in the out-of-plane bend. While both treatments of the intensity of the out-of-plane bend in the spectrum of Cl−·H2O were able to account for the intensity of this transition, the internal coordinate description attributes the intensity to a quadratic term in the expansion of the y-component of the dipole moment, while the normal mode description ascribes the intensity to the mixed nature of the vibrational wave function. This mixed nature arises from cubic or higher-order terms in the potential that couple the v = 2 state in the out-of-plane bend to harmonic levels that carry substantial oscillator strength. While the results of fulldimensional variational treatments must be independent of the coordinates used in the calculation, the difference in interpretation of the mechanism by which this transition gains intensity is unsettling. It is in light of this result that we embark on a deeper understanding of why the choice of coordinates has such a large impact on the interpretation of the origin of intensity in the overtone in the out-of-plane bend in Cl−·H2O. To investigate this question, we turn to the results of perturbation theory based on the full-dimensional system performed in the normal coordinates based on Cartesian displacements, as can be obtained from Gaussian 09.19 On the basis of the results of these calculations, the fundamental in the HOH bend, out-of-plane bend, and overtone in the out-ofplane bend have intensities of 72.9, 54.5, and 72.3 km mol−1, respectively. This is consistent with the near equal intensities of these bands observed experimentally.8 On the other hand, if we use the quadratic term in the expansion of the dipole moment in these coordinates, the predicted intensity of the out-of-plane bend drops to 0.15 km mol−1, while linear terms in the dipole moment anticipate the intensities of the two fundamentals to be 101.6 and 63.0 km mol−1 for the HOH bend and out-ofplane bends, respectively. The discrepancy in these results points to higher-order terms in the potential leading to intensity in the overtone in the out-of-plane bend in Cartesianbased normal mode representation. To better understand the origins of the difference in the interpretations provided by analysis of the results of calculations based on internal and Cartesian coordinates, we turn to a reduced dimensional model of the out-of-plane bend in the chloride water complex in which we consider the twodimensional motions of the bonded OH bond shown in Figure 2. As such, the two normal modes correspond to displacements

electronic structure theory would provide more accurate treatments of the harmonic frequencies, the purpose of the study was to evaluate origins of intensity in the water spectrum between 1900 and 2250 cm−1. For the smaller clusters, additional electronic structure calculations were performed using an aug-cc-pVTZ basis to confirm that a larger basis would not affect the overall findings. For each cluster, the geometry was optimized, and a normal mode calculation was performed within the Gaussian 03 or Gaussian 09 program package.18,19 The Hessians in Cartesian and internal coordinates were used to develop the normal modes in the Cartesian and internal coordinate representations, and the n normal modes that correspond to the HOH bend were identified. Two additional calculations were performed for each bend normal mode to determine the dipole derivatives at displaced geometries Δq = ±0.20 or ±2.00 in dimensionless units. The results of these calculations were used to obtain the mixed second derivatives of the dipole moment with respect to the bend and each of the other normal modes. We also investigated the spectra for the water dimer and for the chloride water complex using second-order vibrational perturbation theory, as implemented in Gaussian 09.19 This allowed us to explore the role of both higher-order terms in the expansion of the potential and dipole moment on the calculated spectrum. To further investigate the origins of intensity in this complex, we developed a two-dimensional model consisting of only the bonded OH stretch and out-of-plane bend in Cl−·H2O.

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RESULTS Role of Coordinates in the Mechanism for Intensity. In earlier studies, we interpreted unexpected intensities in bend overtones and combination bands in hydrogen-bonded complexes based on higher-order terms in the expansion of the dipole surface.6,8,9 In all cases, the intermolecular vibrations were treated in an internal coordinate representation. Before pursuing this approach for our study of signatures of the association band in the spectra of water clusters, we investigate the extent to which the interpretation of the intensity of these features as reflecting higher-order terms in the expansion of the dipole surface depends on the coordinates used for the expansion. For this discussion, we will focus on the overtone in the outof-plane bend in the chloride water complex. The spectrum of the chloride water complex shows comparable intensity in the first overtone in the out-of-plane bend, as is seen in the fundamentals in the water bend and in the out-of-plane bend.8 This large overtone intensity is surprising for two reasons. First, based on harmonic oscillator models for molecular vibrations, the general expectation is that the intensity will be largest for fundamental transitions and will decrease with vibrational excitation.20 While other exceptions to this trend have been noted,21 the case of the out-of-plane bend in the chloride water complex is particularly striking when one considers that this vibration has A″ symmetry. Therefore, the intensity of the fundamental in this mode arises from changes in the dipole moment of the complex in the z-direction in Figure 2. The wave function for the v = 2 level transforms as A′ symmetry, and the intensity arises from changes in the dipole moment of the molecule in the molecular plane as the bonded hydrogen atom is displaced along a vector that is perpendicular to this plane. On the basis of a one-dimensional internal coordinate model, we were able to attribute this unexpectedly large intensity of the 8289



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Table 1. Calculated Harmonic and Anharmonic Frequencies and Intensities for Cl−·H2O frequency (cm−1) nOH, nθ

intensity (km mol−1) b

harm.

anharm.

second/first

728 1456 3371 6742

718 1406 3123 5976

41.76 0.00 1106 0.00

c

second/second

third/first

third/second

41.76 0.15 1106 39.40

41.76 43.12 1106 42.34

41.76 41.97 1106 0.27

41.76 26.53 1106 39.40

41.76 1.77 1106 42.34

41.76 41.97 1106 0.27

VPT2d

Cartesiana 0, 0, 1, 2,

1 2 0 0

64.53 73.31 1012 0.30

internale 0, 0, 1, 2,

1 2 0 0

728 1456 3371 6742

718 1406 3123 5976

41.76 0.00 1106 0.00

a

Normal modes expanded in Cartesian coordinates. bVPT2 based on the two-dimensional model Hamiltonian described in the text. cmth/nth provides the second-order perturbation theory results based on an mth-order expansion of the potential and an nth-order expansion of the dipole surfaces. dVPT2 results for all six vibrations as implemented in Gaussian 09.19 eNormal modes expanded in internal coordinates.

of the OH bond vector in either Cartesian (y, z) or internal (r, θ) coordinates, and the Hamiltonian becomes H=−

ℏ2 ⎛ ∂ 2 ∂2 ⎞ ⎜ 2 + 2 ⎟ + V (y , z ) 2μOH ⎝ ∂y ∂z ⎠

(5)

ℏ2 ⎛ ∂ 2 1 ∂2 1 ⎞ ⎜ 2 + 2 2 + 2 ⎟ + V (r , θ ) 2μOH ⎝ ∂r r ∂θ 4r ⎠

(6)

oscillator/linear dipole treatment. The results identified as third/first and second/second provide results when higherorder terms are introduced to the potential or dipole surface, respectively. On the basis of the above analysis, the origins of the intensity in the out-of-plane bend in the chloride water system are primarily ascribed to electrical anharmonicity in an internal coordinate representation, while it is attributed to mechanical anharmonicity in a Cartesian coordinate representation. This leads to two questions. First, why does the interpretation of this band depend on the choice of coordinates used to express the normal modes, and second, does one descriptor provide a “better” picture of the system? Starting with the first question, we consider the expansions of the dipole moment in dimensionless normal coordinates based on the Cartesian and internal coordinates. These are provided in the Supporting Information. In the present definition of the coordinates, a displacement in the internal coordinate r corresponds to a displacement along the y-axis, while a displacement in θ corresponds to lowest order to a displacement in the z-direction. As is seen in the results reported in Table S2 (Supporting Information), the y-component of the μθθ term is comparable in magnitude to the z-component of the μθ term, while the y-component of μyy is nearly zero. The μyy and μθθ terms in the expansion of the dipole moment are the ones that provide the leading contribution to intensity to the out-of-plane bend overtone. To understand the differences in the size of the quadratic terms in the expansions of the y-component of the dipole moment in the internal and Cartesian coordinates, we look at how we evaluated μyy from the derivatives of the dipole moment with respect to internal coordinates. We find that μ μ μyy Δy 2 ≈ r Δr + θθ2 Δθ 2 re 2re (7)

or H=−

with y = r cos θ and z = r sin θ. For these calculations, electronic energies and dipole moments were obtained at the MP2/aug-cc-pVDZ level of theory using a finite difference scheme based on the internal coordinates. These functions were then reexpanded to the same order in Cartesian coordinates. For the internal coordinates, the effective mass for the bend, Gθ,θ = 1/μOHr2, is expanded through second order in the displacement of r from its equilibrium value and only the constant contribution to V′ = −ℏ2/(8μOHr2) is included. Following standard partitioning16,23 of the perturbation expansion, the terms that are quadratic in the coordinates or momenta are included in the zero-order harmonic Hamiltonian, cubic terms are placed in H(1) while quartic terms go into H(2). The only exception is V′ where the constant contribution is put into H(2). As expected, when second-order perturbation theory is applied to the expanded Hamiltonians in eqs 5 and 6, identical results are obtained.13 We then use the expansion of the wave function to transform the dipole moment to the new representation, through second order. In this study, we follow the work of McCoy and Sibert24 and put the constant term in the expansion of the dipole moment in μ(0), the linear terms in the expansion are in the first-order correction, and quadratic terms come in at second order. This differs from the approach used by Bioino and Barone25 and which is implemented in Gaussian 09.19 On the other hand, it is consistent with the partitioning of the potential energy surface. Additional details of this treatment are provided in the Supporting Information. In Table 1, we report the calculated frequencies and intensities for the fundamental and first overtones in the OH stretch and out-of-plane bend obtained by the harmonic oscillator/linear dipole treatment and from perturbation theory. The intensities reported in Table 1 are organized by the order of the expansion of the potential and dipole surface, so the column labeled second/first gives the results for the harmonic

where re represents the equilibrium OH bond length, μr/re = 0.0962, and μθθ/(2re2) = −0.0949. The opposite signs of μr and μθθ arises from the fact that, as the hydrogen atom is displaced perpendicular to the plane, the OH bond stretches. The first term in eq 7 reflects the fact that displacement of the hydrogen atom perpendicular to the plane of the complex will lead to an increase in the OH bond length. This in turn will increase the magnitude of the dipole moment along the y-axis, as the hydrogen atom will be moved toward the OH− + HCl geometry. Motion of the hydrogen atom perpendicular to the 8290



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Before moving to a discussion of the water clusters, a few more observations are of note. The first is the very small intensity of the bonded OH stretch overtone. Specifically, while the bonded OH stretch fundamental is more than 50 times more intense than the free OH stretch (1013 vs 20.54 km mol−1), the overtone intensities for both modes are nearly equal (0.29 and 0.82 km mol−1). The decrease in intensity in the overtone of the bonded OH stretch compared to the fundamental is 3 orders of magnitude, while the intensity of the free OH stretch decreases by a factor of roughly 25. Similar behavior has been noted in other hydrogen-bonded systems.26−29 The origin of the loss of intensity of the overtone in the hydrogen-bonded OH stretch results from a near exact cancellation of the contributions to the intensity of this mode from electrical and mechanical anharmonicity, as is shown in the perturbation theory results reported in Table 2. While not

plane also breaks the existing hydrogen bond (reflected in the second term in eq 7), and leads to a charge redistribution that localizes the excess charge onto the chloride ion. These two competing factors nearly exactly cancel, leading to a negligible contribution of electrical anharmonicity to the intensity when the calculation is performed in Cartesian coordinates. The factors described above are also illustrated in panels e−h of Figure 3, where expansions of the y-component of the dipole moment are plotted in the internal and Cartesian coordinates. Panel e provides the linear approximation to the dipole moment in internal coordinates. As is seen, this contour is simply a plane. Because the y-component of the dipole moment is an even function in θ, the contours form vertical lines. In panel f, this surface is re-expanded through third order in the Cartesian-based normal modes. Note the curvature in the surface. This reflects the shift of the excess charge toward the OH− as the out-of-plane bend is excited. Panel g shows the cubic expansion of the dipole in the internal coordinate. This plot also shows curvature, although in the opposite direction. Since panels f and g include only contributions from mechanical or electrical anharonicity, when both contributions are included in the analysis, the curvatures ascribed to the separate contributions cancel. As a consequence, the results plotted in panel h look very similar to panel e. On the other hand, if we look at the expansions of the potentials in Figure 3c and d, some of the cubic and quartic terms in the expansion of the potential in Cartesian coordinates are much larger than in the internal coordinate expansion. This leads to the expectation that mechanical anharmonicity will be larger in the Cartesian coordinate representation. The larger cubic (and quartic) terms in the expansion of the potential in Cartesian coordinates lead to increased mixing of zero-order harmonic oscillator basis states in this coordinate representation. This is the mechanism that led to the interpretation of the overtone in the out-of-plane bend being ascribable to intensity borrowing (or mechanical anharmonicity) in the earlier work of Rheinecker and Bowman.22 While the differences in interpretation based on calculations in the two coordinate systems are readily understood, it is unsettling that the description of the underlying physics depends on the choice of coordinates. Similar coordinate dependence on the interpretation of Fermi resonances in CO2 was discussed by Sibert et al.10 They concluded that the internal coordinate description was the better one for the purposes of interpretation due to the artificially large cubic terms in the expansion of the potential in Cartesian coordinates due to the inherently curvilinear nature of bending motions. Similar conclusions were reached by Gordon and co-workers in studies in which they used vibrational self-consistent field approaches to study metal cation complexes with molecular hydrogen.12 On the basis of this, we also argue that, while the normal modes may be the easier coordinates in which to expand the Hamiltonian due to the ubiquity of this choice in electronic structure packages and the generality of the description, the larger couplings introduced by this choice render the Cartesianbased normal coordinates a poorer choice when attempting to extract the low-order description of effects like overtone intensities. As such, for the remainder of the discussion, we will focus our attention on the descriptions of the expanded Hamiltonian based on internal displacement coordinates (eq 6).

Table 2. Intensities (km mol−1) of the Bend Fundamental and the Combination Bands with Lilbration Modes That Break the Hydrogen Bond Using Quadratic Expansions of the Dipole Surface Cartesiana

internala

VPT2b

c

system Ar3H3O+ (H2O)2e

comb. (cm−1)

bendd

comb.d

bendd

comb.d

bendd

comb.d

2006 1982 2263 2000 2281

25.7 86.0 86.0 32.3 32.3

0.5 0.7 0.0 0.3 0.2

25.7 86.0 86.0 32.3 32.3

24.3 0.4 5.6 6.3 0.1

65.8 65.8 46.2 46.2

1.4 0.1 8.7 0.2

a

Based on normal mode descriptions derived from the Cartesian or internal coordinate displacements. bVPT2 results as implemented in Gaussian 09.19 cHarmonic frequency of the combination band. dThe intensity of the bend and bend/libration combination band. eThe four values for the water dimer reflect the two HOH bends and the two high frequency librations.

the primary focus of this study, this effect is of interest and appears to be a general somewhat surprising consequence of hydrogen bonding. Further, unlike the intensity of the bend overtone, the stretch displacements follow the Cartesian vectors, making the result independent of the coordinates used to expand the potential and dipole moment functions. Second, the effects described above for Cl−·H2O are found to apply more broadly to hydrogen-bonded systems. In Table 2, we report the intensities of the transitions to the combination band involving the HOH bend and the intermolecular motions that break hydrogen bonds in Ar3H3O+ and in the water dimer. The reported intensities reflect only contributions from electrical anharmonicity, as they are obtained by evaluating the matrix element of the quadratic terms in the dipole surface that are responsible for this transition of interest. Results are reported on the basis of calculations in both internal- and Cartesian-based normal modes. The results of a full-dimensional second-order perturbation theory calculation are also provided for the water dimer. As with the out-of-plane bend in Cl−·H2O, the origin of intensity in the combination bands involving the HOH bends and modes that break the hydrogen bonding structure is highly coordinate choice dependent, and as in Cl−·H2O, we believe that the internal coordinates provide the cleaner description of the effect. In this representation, the intensity of the combination bands reflects large quadratic terms in the dipole surface, which result from changes in the 8291



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Figure 4. Plots of the spectra for (H2O)6 evaluated on the basis of the normal mode wave functions and quadratic expansions of the dipole moment evaluated in terms of displacements of internal (upper panels) or Cartesian (lower panels) coordinates. The three columns represent calculations using the cluster geometry depicted in the inset: the lowest energy prism (left), book (center), and cyclic (right) structures. The grey curves in all of the panels provide the spectrum of liquid water2 for comparison.

the HOH bends is expected to be small.17 The anharmonicities of the intermolecular modes are anticipated to be larger but are difficult to calibrate in a systematic way. As is demonstrated by the plots in the upper panels of Figure 4, the agreement between the spectra for the water hexamers calculated in internal coordinates and the experimental spectrum is good. The overall spectral envelope is qualitatively similar for the three water clusters. This level of agreement and the relative insensitivity to the structure of the cluster provides additional evidence that electrical anharmonicity provides the leading contribution to intensity in this spectral region. As expected, the agreement is less good when Cartesian coordinates are used to expand the dipole moment surface. In the case of the prism and book structures, shown in the lower panels of Figure 4, they display negligible intensity in the 2100 cm−1 region. This is consistent with the results for the water dimer, Cl−·H2O, and Ar3H3O+, described above. In the case of the cyclic water cluster, the spectrum plotted in the lower right panel of Figure 4 shows enhanced intensity in the 2100 cm−1 region. Such increased intensity is seen in all of our Cartesian coordinate-based calculations of the spectra of the clusters with high symmetry investigated in this study. We believe the enhanced intensity reflects additional difficulties of the Cartesian representation when the normal modes are delocalized over the cluster rather than localized on a single water molecule. If we calculate the spectra for two other cyclic structures of (H2O)6 that have lower symmetry using the normal modes based on Cartesian displacements, the intensity in the 2100 cm−1 region vanishes. These results are plotted in Figure S2 of the Supporting Information. The large sensitivity of the spectra calculated using quadratic expansions of the

oscillator strength for the water bend vibration when the hydrogen bonds are broken. The Association Band in the Water Spectrum. Having discussed theoretical treatments of intensities of overtone and combination bands in a variety of small hydrogen-bonded systems, we turn our attention to the band in the liquid water spectrum in the 2100 cm−1 region, plotted in red in Figure 1. This band has been attributed to combination bands involving the HOH intramolecular bend in the individual water molecules and intermolecular vibrations that break the hydrogen bond network.5 On the basis of the discussion above, we expect the intensity to be interpreted in terms of second derivatives of the dipole moment with respect to the corresponding internal coordinates. The analysis presented in Table 2 for (H2O)2 supports this interpretation. In the discussion that follows, we extend those calculations to larger water molecules. While it is reassuring that the intensity in the 2100 cm−1 region of the water spectrum is apparent in the spectrum of the water dimer, the question remains if this feature is found in the spectra of larger clusters. To this end, we evaluated the bend region of the spectra for 13 water clusters composed of six or fewer water molecules. We focused on the cyclic forms of the clusters with three to five water molecules, as well as the prism, book, and cyclic forms of (H2O)6. Since the general results are qualitatively similar for all of the water clusters, we focus on the results for the three structures of the hexamer, which are presented as stick spectra and with red curves in Figure 4. For comparison, the spectrum of liquid water2 is plotted with gray lines in Figure 4. The frequencies used to plot the calculated spectra are the harmonic frequencies, as the anharmonicity of 8292



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the Best Current Values of the Optical Constants of H2O(l) at 25°C between 15,000 and 1 cm−1. Appl. Spectrosc. 1996, 50, 1047−1057. (3) Pinkley, L. W.; Sethna, P. P.; Williams, D. Optical Constants of Water in the Infrared: Influence of Temperature. J. Opt. Soc. Am. 1977, 67, 494−499. (4) Al-Abadleh, H. A.; Grassian, V. H. FT-IR Study of Water Adsorption on Aluminum Oxide Surfaces. Langmuir 2003, 19, 341− 347. (5) Fox, J. J.; Martin, A. E. Investigations of Infra-Red Spectra (2·5− 7·5μ). Absorption of Water. Proc. R. Soc. London, Ser. A 1940, 174, 234−262. (6) McCoy, A. B.; Leavitt, C. M.; Guasco, T. L.; Olsen, S. G.; Johnson, M. A. Vibrational Manifestations of Strong non-Condon Effects in the H3O+·X1−3 (X=Ar, N2, CH4, H2O) Complexes: Microscopic Analogues of the Association Band in the Vibrational Spectrum of Water. Phys. Chem. Chem. Phys. 2012, 14, 7205−7214. (7) Skinner, J. L.; Auer, B. M.; Lin, Y.-S. Vibrational Line Shapes, Spectral Diffusion, and Hydrogen Bonding in Liquid Water. Adv. Chem. Phys. 2009, 142, 59. (8) Roscioli, J. R.; Diken, E. G.; Johnson, M. A.; Horvath, S.; McCoy, A. B. Prying Apart a Water Molecule with Anionic H-Bonding: A Comparative Spectroscopic Study of the X−·H2O (X = OH, O, F, Cl, and Br) Binary Complexes in the 600 - 3800 cm−1 Region. J. Phys. Chem. A 2006, 110, 4943−4952. (9) Horvath, S.; McCoy, A. B.; Elliot, B. M.; Weddle, G. H.; Roscioli, J. R.; Johnson, M. A. Investigations of the Infrared Intensity Patterns in X−·H2O complexes [X = Cl, Br and I] and its Deuterated Analogues. J. Phys. Chem. A 2010, 115, 1556−1568. (10) Sibert, E. L.; Hynes, J. T.; Reinhardt, W. P. Fermi Resonance from a Curvilinear Perspective. J. Phys. Chem. 1983, 87, 2032−2037. (11) Pesonen, J. Vibrational Coordinates and Their Gradients: A Geometric Algebra Approach. J. Chem. Phys. 2000, 112, 3121−3132. (12) De Silva, N.; Njegic, B.; Gordon, M. S. Anharmonicity of Weakly Bound M+H2 Complexes. J. Phys. Chem. A 2011, 115, 3272− 3278. (13) McCoy, A. B.; Sibert, E. L. Perturbative Calculations of Vibrational (J = 0) Energy Levels of Linear Molecules in Normal Coordinate Representations. J. Chem. Phys. 1991, 95, 3476−3487. (14) Frederick, J. H.; Woywod, C. General Formulation of the Vibrational Kinetic Energy Operator in Internal Bond-Angle Coordinates. J. Chem. Phys. 1999, 111, 7255−7271. (15) Wilson, E. B.; Decius, J. C.; Cross, P. C. Molecular Vibrations; Dover: New York, 1955. (16) Sibert, E. L. Theoretical Studies of Vibrationally Excited Polyatomic Molecules Using Canonical Van Vleck Perturbation Theory. J. Chem. Phys. 1988, 88, 4378−4390. (17) Temelso, B.; Archer, K. A.; Shields, G. C. Benchmark Structures and Binding Energies of Small Water Clusters with Anharmonicity Corrections. J. Phys. Chem. A 2011, 115, 12034−12046. (18) Frisch, M. J.; et al. Gaussian 03, revision C.02; Gaussian, Inc.: Wallingford, CT, 2004. (19) Frisch, M. J.; et al. Gaussian 09, revision D.01; Gaussian, Inc.: Wallingford, CT, 2009. (20) Lehmann, K. K.; Smith, A. M. Where Does Overtone Intensity Come From? J. Chem. Phys. 1990, 93, 6140−6147. (21) Miller, B. J.; Du, L.; Steel, T. J.; Paul, A. J.; Södergren, A. H.; Lane, J. R.; Henry, B. R.; Kjaergaard, H. G. Absolute Intensities of NHStretching Transitions in Dimethylamine and Pyrrole. J. Phys. Chem. A 2012, 116, 290−296. (22) Rheinecker, J.; Bowman, J. M. The Calculated Infrared Spectrum of ClH2O Using a New Full Dimensional ab Initio Potential Surface and Dipole Moment Surface. J. Chem. Phys. 2006, 125, 133206-1−133206-13. (23) Nielsen, H. H. The Vibration-Rotation Energies of Molecules. Rev. Mod. Phys. 1951, 23, 90−136. (24) McCoy, A. B.; Sibert, E. L. Calculation of Infrared Intensities of Highly Excited Vibrational States of HCN Using Van Vleck Perturbation Theory. J. Chem. Phys. 1991, 95, 3488−3493.

potential and dipole surfaces in normal modes based on Cartesian coordinates to small changes in the cluster structure provides additional concern for using these coordinates to interpret spectral features based on low-order expansions. In contrast, the normal modes based on displacements of internal coordinates provide a robust description of the water cluster spectrum in the 2100 cm−1 region.

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CONCLUSIONS In this paper, we have explored the origins of the intensity in the 2100 cm−1 region of the water spectrum. We found that it can be attributed to electrical anharmonicity arising from large mixed second derivatives of the dipole moment with respect to the HOH bend and one of the intermolecular vibrations that breaks the hydrogen bonding network. The coordinates used for this analysis need to be based on displacements of internal coordinates. Although more straightforward to calculate, the Cartesian-based normal mode representation introduces the large couplings between the critical low-frequency intermolecular bend vibrations and intramolecular vibrations in the individual water molecules due to the rectilinear description of vibrations imposed by these coordinates. This leads to large cubic and quartic terms in the expansion of the potential as well as near complete cancellation of the contributions to the second derivatives of the dipole moment with respect to the critical coordinates. Thus, care needs to be taken when these normal modes are used to interpret the underlying origins of anharmonicity in vibrational spectra. These observations support earlier findings of Sibert et al. based on their analysis of the origins of Fermi resonances in carbon dioxide.10

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ASSOCIATED CONTENT

S Supporting Information *

A description of the perturbative expansion of the dipole moment, parameters for the quartic expansion of the potential and dipole surface, plots of the x- and z-components of the dipole moment, and spectra for other structures of the cyclic water hexamer, obtained from Cartesian coordinate expansions of the potential and dipole moment surfaces. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The author gratefully acknowledges support from the Chemistry Division of the National Science Foundation (CHE-1213347) and an allocation of computing time from the Ohio Supercomputing Center. She also acknowledges Professor Mark Johnson and his students and post docs for discussions that inspired this study,

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REFERENCES

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The Role of Electrical Anharmonicity in the Association Band in the

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