Unraveling Anharmonic Effects in the Vibrational ... - ACS Publications

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Unraveling Anharmonic Effects in the Vibrational Predissociation Spectra of H5O2þ and Its Deuterated Analogues Timothy L. Guasco and Mark A. Johnson* Sterling Chemistry Laboratory, Yale University, P.O. Box 208107, New Haven, Connecticut 06520, United States

Anne B. McCoy* Department of Chemistry, The Ohio State University, Columbus, Ohio 43210, United States

bS Supporting Information ABSTRACT: The nature of anharmonic couplings in the H5O2þ “Zundel” ion and its deuterated isotopologues is investigated through comparison of their measured and calculated vibrational spectra. This follows a recent study in which we reported spectra for H5O2þ, D5O2þ, and D4HO2þ from ∼600 to 4000 cm-1, as well as H4DO2þ in the OH and OD stretching regions [J. Phys. Chem. B 2008, 112, 321]. While the assignments of the higher-energy transitions associated with the fundamentals of the exterior OH and OD motions are relatively straightforward, difficulties arise in the assignment of the lower-frequency regions that involve displacement of the bridging proton, especially for the isotopically mixed species. Here we revisit the Ar-tagged isotopomers, and report the low energy action spectrum of H4DO2þ for the first time, as well as present significantly improved spectra for the D4HO2þ and D5O2þ systems. Band assignments are clarified in several cases using IR-IR hole-burning. We then investigate the physical origin of the anharmonic effects encoded in these spectra using a recently developed technique in which the anharmonic frequencies and intensities of transitions (involving up to two quanta of excitation) are evaluated using the ground state probability amplitudes [J. Phys. Chem. A 2009, 113, 7346] obtained from diffusion Monte Carlo simulations. This approach has the advantage that it is applicable to low-symmetry systems [such as (HDO)Hþ(OH2)] that are not readily addressed using highly accurate methods such as the multiconfigurational time-dependent Hartree (MCTDH) approach. Moreover, it naturally accommodates an intuitive evaluation of the types of motion that contribute oscillator strength in the various regions of the spectrum, even when the wave function is intrinsically not separable as a product of low-dimensional approximate solutions. Spectra for H5O2þ, D5O2þ, H4DO2þ, and D4HO2þ that are calculated by this approach are shown to be in excellent agreement with the measured spectra for these species, leading to reassignments of two of the bands in the intramolecular bending region of D4HO2þ.

I. INTRODUCTION The protonated water dimer, H5O2þ, has been the subject of considerable interest since its existence and role in aqueous proton transport were first proposed.1,2 In the liquid, most agree that dynamical fluctuations drive the speciation of the excess proton between two limiting configurations, one where it is trapped primarily by two oxygen atoms in the so-called “Zundel” form (illustrated in Figure 1) and another in which it is accommodated in a trihydrated hydronium ion, or “Eigen” form.3 Because the range of behavior in the liquid varies continuously between these arrangements, there has recently been a flurry of activity to isolate the key structures in the gas phase,4-10 where they can be cooled into minimum energy structures and treated with the theoretical and experimental machinery developed for the study of floppy polyatomic molecules.11,12 The Zundel ion has received particular attention, where the important low-energy region of the spectrum (600-2000 cm-1) has been recorded using mass-selective action spectroscopy,13,14 involving photocleavage of weakly bound rare gas (Ne and Ar) mass “tags” or messengers.9 r 2011 American Chemical Society

The Ne-tagged H5O2þ spectrum is particularly useful since the Ne atom only weakly perturbs the H5O2þ molecular structure and is reproduced here in Figure 2A. The problem is deceptively complex, as routinely available harmonic calculations predict an overall pattern quite similar to that observed, with two strong, widely spaced absorptions below 2000 cm-1. These arise from motion of the shared proton along the OO axis and the intramolecular bending vibrations of the flanking water molecules, respectively, which are indicated by the normal mode displacement vectors at the top of Figure 2. The difficulty becomes evident when one applies the anharmonic correction to these values using the perturbation treatment (evaluated at the same level of theory and basis) available within the same program Special Issue: Victoria Buch Memorial Received: October 19, 2010 Revised: November 15, 2010 Published: January 07, 2011 5847

dx.doi.org/10.1021/jp109999b | J. Phys. Chem. A 2011, 115, 5847–5858

The Journal of Physical Chemistry A package (e.g., Gaussian 03). This procedure yields a negative frequency for the “refined” value of the bridging proton stretch,9 indicating that the perturbation expansion either is nonconvergent or requires many higher-order terms. Of the many anharmonic treatments5,9,15-26 that have been employed to address this situation, Meyer finally obtained a satisfactory agreement with the Ne spectrum using a full 15 dimensional MCTDH (multiconfiguration time-dependent Hartree) calculation.27,28 This analysis not only recovered the locations of the strong absorptions at low frequency, but also reproduced the doublet structure displayed by each of the features in Figure 2A. Interestingly, the 1000 cm-1 absorptions are both derived largely from the bridging proton stretch, which is split by a Fermi-type interaction

Figure 1. Calculated structures of H5O2þ generated from the potential surface in ref 19 at the (A) equilibrium position and (B) second-order D2h saddle point.

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with a nominally dark state involving OO stretch and OOH2 wag excitations.27,29 The larger of the two peaks in the 1800 cm-1 region is mostly due to the bend fundamentals of the water molecules, while the weaker feature is traced to a combination of excitations involving the OO and central proton stretches. Interestingly, more recent classical studies of Jordan, Bowman, and their co-workers also reproduce these splittings.30 The qualitative origins of the anharmonic couplings in the Zundel ion clearly arise from the fact that, as the central proton is displaced along the OO axis, the system is driven toward a geometry that corresponds to H3Oþ solvated by a single water molecule. The breakdown of perturbation theory also reflects the large diagonal quartic terms in the expansion of the potential, particularly in cuts involving displacements of the central shared proton. Here we are primarily concerned with the evolution of these bands upon H/D isotopic substitution.22 This is an extension of our earlier report in which we explored the curious propensity for Ar to preferentially attach to the exterior D atoms in the partially deuterated isoptologues.31 That study was limited to the Ar-tagged complexes because of experimental complexities arising from the near mass-degeneracy between 20Ne and D2O (i.e., within the resolution of our photofragmentation mass spectrometer, m/Δm ∼ 500). Like the Ne complex, the Ar atom attaches to an exterior OH group in a quasi-linear H-bond, and although the overall structure of the H5O2þ 3 Ne spectrum remains intact in the H5O2þ 3 Ar spectrum (displayed in Figure 2B), the stronger interaction with Ar splits the degeneracy in the exterior OH stretches and pulls the bridging proton band toward higher energy by about 40 cm-1.9 The Ar-tagged isotopomers not only display

Figure 2. Vibrational predissociation spectra of (A) H5O2þ 3 Ne and (B) H5O2þ 3 Ar adapted from ref 9. The peaks labeled νs and νa in (A) correspond to the symmetric and asymmetric OH stretches, which, due to the perturbation induced by the Ar tag, are split into four bands (designated νAr Ar-OH for NB the Ar bound OH stretch, νAr and νNB OH for the OH stretch of the hydrogen that belongs to the same water molecule as the tagged H, and νs a for the water molecule not bound to Ar) in (B). The locations of the water bend and parallel shared proton vibrations are indicated as νb and νsp, respectively. The insets show the normal mode displacements of these motions within the D2h symmetry structure. Of note, the direction of motion of the H atoms of the flanking water molecules relative to that of the shared proton is opposite in these two vibrations; i.e., the flanking H atoms move closer to each other as the shared proton moves away in the lower-energy parallel shared proton vibration, but in the higher-energy bending mode the flanking H atoms move closer as the shared proton moves toward them. 5848

dx.doi.org/10.1021/jp109999b |J. Phys. Chem. A 2011, 115, 5847–5858

The Journal of Physical Chemistry A very different spectra when an H or D atom is in the central shared position but also exhibit more subtle variations in these intrinsic patterns according to which of the multiple possible Ar binding sites is occupied in the partially deuterated complexes.31 To anchor the band assignments in two of the key systems (D4HO2þ and H4DO2þ), we carried out isotopomer-selective, IR-IR double-resonance (IR2DR) spectroscopy to sort out the patterns of each isotopomer.32-34 In the course of acquiring these new data, we also obtained considerably higher-quality spectra of the previously reported D5O2þ 3 Ar and D4HO2þ 3 Ar complexes, which are also reported and analyzed here. The vibrational assignments have been made in the case of the symmetrical systems [(H2O)Dþ(OH2) and(D2O)Hþ(OD2)], which have been treated with high-level anharmonic theory.35,36 Due to the smaller frequency difference between the mode assigned as the displacements of the shared hydrogen and the out-of-phase DOD bends in (D2O)Hþ(OD2), the states obtained in the MCTDH calculations are all highly mixed, making it challenging to associate oscillator strengths of the various bands to particular motions. The lower-symmetry species present an even larger challenge for the MCTDH approach, requiring us to turn to more generally applicable approximate methods to understand their band patterns. Specifically, we invoke the recently introduced theoretical technique37 in which anharmonic frequencies and intensities of transitions involving up to two quanta of excitation are evaluated using the ground state probability amplitudes (GSPA) obtained from diffusion Monte Carlo (DMC) simulations and will be referred to as the GSPA approach. This method proved effective in understanding the anharmonic bands in the related OH- 3 H2O complex37 and exploits the fact that the DMC ground state wave function captures the intrinsic anharmonicity (e.g., nonseparability) of the vibrational motion on the “real”, extended potential surface. We then construct linear combinations of bond displacement coordinates, analogous to the usual normal mode vectors, using the explicit character of the DMC ground state probability amplitude. The excited states are then generated by taking products of polynomials in the constructed coordinates and the ground state wave function. This approach differs from the more traditional basis set approaches as the mode-mode couplings are incorporated into the definitions of the coordinates, thereby facilitating assignment of the spectral features as well as the analysis of the underlying origins of any intensity borrowing. We first test the efficacy of this approach on the well-characterized homogeneous systems (H5O2þ and D5O2þ) and then employ it to unravel the bands in the mixed isotopomers in which the heteroatom occupies an exterior position. One curious aspect of the assignments that has not been completely rationalized is that, in H5O2þ, the feature nominally associated with the intramolecular bending mode of the flanking water molecules appears near 1750 cm-1, about 150 cm-1 above the bend in the bare H2O molecule, while the analogous band in D5O2þ falls much closer to the value in bare D2O.26 We will remark on the origin of this effect in the context of the new theoretical results.

II. EXPERIMENTAL SECTION Vibrational predissociation spectroscopy of the H5O2þ ion and its isotopomers were taken by monitoring Ar loss from their size-selected binary complexes with a single Ar atom. This was accomplished with the time-of-flight photofragmentation mass

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spectrometer described previously.38 The cluster ions were generated by counter propagating a 1 keV pulsed electron beam pointed directly into the nozzle of a pulsed supersonic jet arising from a stagnation pressure of 4 atm. The cluster ions were generated by passing a mixture of ∼5% hydrogen gas in Ar gas over a reservoir of water held at 2 C. This procedure forms cationic clusters by proton transfer from H3þ (or D3þ as described below). For the homogeneous isotopomers, H2 was added to the Ar and passed over liquid H2O to create H5O2þ, while a D2/Ar gas mix and liquid D2O were employed to generate D5O2þ. For the mixed isotopomers, we used an H2/Ar carrier gas over liquid D2O (for D4HO2þ), and H4DO2þ was prepared using the D2/ Ar/liquid H2O combination. Laser excitation was carried out with table-top, Nd:YAG pumped parametric converters (LaserVision), which generate relatively high power in the mid-infrared (>5 mJ/pulse from 2400-4000 cm-1) and more modest output (50-500 μJ) in the lower-frequency region from 600-2400 cm-1. The lower-frequency region was generated by mixing the signal and idler beams form the mid-IR OPO/OPA system in a 7 mm AgGaSe2 crystal. The spectra result from the accumulation of many individual scans (typically 10) and are displayed as the action signal normalized to the laser output energy (per pulse), which varies considerably over the course of the scan. The laser fluence was controlled to avoid saturation of the strong peaks that occur in the range above 3000 cm-1, where output from the laser is particularly strong. II.A. Overview of the Isotope-Dependent Predissociation Spectra of the Ar-Tagged Zundel Ion. Figure 3 presents a broad overview of the isotopologue spectra for the homogeneous D5O2þ and H5O2þ species (Figure 3A,D, respectively, obtained in this work) as well as those for the mixed D4HO2þ and H4DO2þ complexes (Figure 3B,C, respectively). The bands form four main groups that are color-coded in Figure 3: (i) the OH stretches around 3600 cm-1 (green), (ii) the OD stretches near 2600 cm-1 (purple), (iii) the intramolecular water bends that range from 1200 to 1700 cm-1 (blue), and (iv) the parallel vibration of the bridging proton from 700-1200 cm-1 (red). As noted in the Introduction, the two bands arising from the exterior OH (or OD) stretches in the Ne cluster are split into multiplets due to the perturbation by the Ar atom in both the OH and OD stretching regions, with the Ar-bound OH(OD) band appearing lowest in energy in each mulitplet (νOH(D)-Ar). Interestingly, the pattern of bands appears remarkably similar to that traced to the H5O2þ moiety in condensed phase HCl hydrates,39 in which all four exterior OH groups were bound to chloride ions. In that case, however, the bands were analyzed in the context of the strong perturbations induced by subtle variations of the counterion locations, which obviously cannot play a role here. Very recently, Baer et al.40 raised the intriguing possibility that some of the additional complexity displayed by the H5O2þ 3 Ar may reflect the population of higher-energy isomers with differing degrees of perturbation according to the binding site. Here we focus our attention on the lowest energy form where Ar attaches to an exterior OH(OD) group. II.B. Band Assignments in the Mixed Isotopologues Using IR-IR Double Resonance. Incorporation of a single deuterium atom in the Zundel ion leads to two intrinsic isotopomers depending on whether the D atom resides between the oxygen atoms or in an exterior position. Because we are focusing on their complexes with an Ar atom, however, there are additional isotopomers in play because, when the D is exterior, the Ar atom can bind to the flanking HDO molecule at either D or H, as well as to 5849


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Figure 3. Predissociation spectra of (A) D5O2þ 3 Ar, (B) D4HO2þ 3 Ar, (C) H4DO2þ 3 Ar, and (D) H5O2þ 3 Ar. Bands predominantly assigned to νsp activity are indicated in red, while those in blue, purple, and green correspond to the HOH(DOD) bends, OD stretches, and OH stretches, respectively. The most red-shifted bands in the OH and OD stretch region are due to the Ar bound H(D) and have been labeled νOH(D)-Ar.

the H2O constituent. To address the vibrational band assignments in these mixed isotopomers, we carried out a series of holeburning experiments on the H4DO2þ and D4HO2þ systems, which yields isomer-selective spectra as described in detail in refs 32 and 33. Briefly, this is a vibrational double-resonance scheme that is carried out with two independently tunable pulsed infrared lasers in a configuration where the populations of isotopomers responsible for particular features in the predissociation spectrum are depleted by photodissociation with a powerful pump laser. Isomer-selective spectra are recovered as dips in the action signal generated by a fixed-frequency probe laser, which interrogates the population of ions responsible for selected transitions, while the pump laser is scanned through the band. Implementation of this scheme requires three stages of mass selection and two laser interaction regions. This particular mode of the DR method is only available where the pump laser power is sufficiently high to deplete a significant fraction (typically about 30%) of the isomer population responsible for particular features in the spectrum. As a result, we first sorted out the assignments in the higher-energy region (2400-4000 cm-1) associated with the

exterior OH and OD stretching fundamentals and then linked the lower-energy features to specific transitions in the OH(OD) stretching region, thus identifying the isotopomer responsible for each band interrogated by the pump. The performance of the double-resonance approach in its application to isotopomers of the Zundel ion is illustrated in Figure 4, which presents the results on the H4DO2þ 3 Ar isotopologue in the OH stretching region. The middle trace (Figure 4C) arises from contributions of all isotopomers. As discussed in an earlier paper,31 this species strongly favors the light isotope in the bridging position, so the most abundant isotopomers arise from the differing locations of the Ar atom among the exterior OH or OD groups. Of these, the one in which Ar is attached to the dangling OD group is uniquely associated with the most strongly red-shifted OD band in Figure 3C, also indicated in the Figure 4 inset (left side) by the blue asterisk. To reveal the specific transitions in the OH stretching region that are also associated with this isotopomer, we fixed the probe laser on the asterisked transition and monitored the modulation in this action signal as the pump laser was scanned through the OH region of the spectrum, 5850



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Figure 4. OH stretching region of the vibrational predissociation spectrum of H4DO2þ reproduced from Figure 3C in (C), flanked by the IR2DR spectra (B) and (D) obtained by probing the transitions indicated by the red dagger (†) and blue asterisk (*), respectively. Calculated harmonic [MP2/ 6-311þG(d,p)] spectra are shown for the (A) Ar bound to H isotopomers and the (E) Ar bound to D isotopomer. The calculated spectra have been scaled by 0.9460.31 The yellow trace in (A) corresponds to the isotopomer in which the deuterium is on the free end, while the green trace is due to the one in which the deuterium resides on the water molecule to which the tag is attached, as shown in the insets. The superscript in the transition labels indicates whether a given band is due to an H2O or HOD molecule. The subscript denotes if the band is due to a symmetric or asymmetric stretch (s and a, respectively), a single OH stretch, or an Ar bound OH(D) stretch.

with the result presented in trace 4D. The bands arising from the OD-bound Ar isotopomer appear as a series of dips. As such, this dip spectrum does not exhibit any activity near the broad, redshifted feature near 3520 cm-1, which has previously been assigned to the Ar-bound OH stretch.31 The scaled, harmonic spectrum (MP2/6-311þG(d,p), factor = 0.946031) of the ODbound isotopomer is included in Figure 4E for comparison, which readily reveals the assignments of three of the bands to the decoupled OH stretch in the HOD molecule (νHDO OH ) and the symmetric and asymmetric stretches of the flanking H2O component 2O 2O (νH andνH , respectively). Interestingly, two more bands s a toward higher energy (denoted by b) are also associated with this species and are likely due to frustrated internal rotations as discussed earlier in the metal ion complexes by Duncan and McCoy.41 We also obtained double-resonance data by fixing the probe laser at the peak of the broad, OH-Ar peak at 3520 cm-1 (indicated by † in insert on left of Figure 4), with the resulting dip spectrum displayed in Figure 4B. Note that the peak at ∼3640 cm-1, which appears as a broad asymmetrical feature in the nonselective trace, is clearly resolved into two peaks, each due to separate isomers, using the DR method. The specific assignments are more difficult in this case, however, as there should actually be three isotopomers arising from Ar attachment to an exterior OH,

and indeed the widths of the dips in this case are broader than in Figure 4D as would be anticipated for blended transitions. While the disentangling of all bands in this spectral region is not the central theme of this paper, it is likely that the dip spectrum in Figure 4B results from two overlapping patterns indicated by the scaled harmonic spectra in trace A. These two isomers are associated with Ar attachment to the OH in the HDO molecule (green) or to an OH on the H2O partner (yellow), with the structures indicated in the upper trace of Figure 4. While the high-energy OH(OD) stretching bands did not present any surprises, the situation regarding the low-energy bands is less straightforward. The most complex case is presented by the D4HO2þ ion (Figure 3B), which is expanded in Figure 5. At the lowest-energy side, the broad transitions assigned to the bridging deuteron are clear around 800 cm-1,26 as is the characteristic band associated with the shared proton close to 1000 cm-1. The fact that these features occur with similar intensities indicates both species are present in abundance. As such, we expect a more complex scenario regarding the (nominally) intramolecular bending modes next higher in energy. To facilitate the discussion, Figure 5 includes the locations of the bending fundamentals (dashed lines) for the isolated D2O, HDO, and H2O molecules. Three strong peaks are observed in 5851



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Figure 5. Vibrational predissociation spectrum of D4HO2þ 3 Ar reproduced from Figure 3B in black with the position of the HOH, HOD, and DOD bends in the isolated molecule indicated. The red, green, and blue traces display IR2DR laser-on/laser-off difference photofragment mass spectra of D4HO2þ 3 Ar with the pump laser fixed at 3681 cm-1, the location of the nonbonded OH stretch (labeled νNB OH), and the probe laser fixed at the frequency of the colored symbols. The downward peak corresponds to depletion of the probe photofragment while the upward peak to the production of the pump photofragment. The dips in the blue and green traces indicate that the peaks at 1268 and 1488 cm-1 are due to an isotopomer with the unique hydrogen located in an exterior position. The lack of dip in the red trace indicates that the transition at 1548 cm-1 is due to the isotopomer with the hydrogen in the shared position. The inset displays the two possible positions of the unique hydrogen atom in bare D4HO2þ.

this region, one just above the D2O band and an asymmetric doublet just above the HOD limiting value. In our first report,26 we suggested that the band close to D2O was mainly due to the (D2O)Hþ(OD2) configuration, while the two bands just above the HDO asymptote were tentatively assigned to the (D2O)Dþ(OHD) arrangement with no attempt to define which motions were most important. This scheme was called into question, however, in light of the MCTDH results of Meyer and coworkers35,36 on the symmetrical (D2O)Hþ(OD2) system, in which they report an intense band in their calculated spectrum at 1564 cm-1. On the basis of the underlying bend frequencies, we previously ascribed activity in this region to the HDO bending mode in the D-bridged isotopomer. In their fully coupled analysis, the transitions arise from complex admixtures of zero-order motions involving the intramolecular bends as well as combination levels built on O-O stretch and wag motions in conjunction with the shared proton asymmetric stretch. If this is correct, then we must conclude that the observed band near the D2O bend at 1300 cm-1 arises from the isotopomer with the shared D arrangement. Unfortunately, calculations of this asymmetric (D2O)Dþ(OHD) are more complicated than the already difficult calculations of the spectrum for the (D2O)Hþ(OD2) isomer, so the bands arising from that structure were not calculated, thus precluding a complete evaluation of the internal consistency of this assignment scheme. To further explore the assignment of the low-energy bands in D4HO2þ, we turned to the double-resonance approach discussed above to experimentally determine which bands arise from isotopomers in which the H resides in an exterior position. Application of the method to these lower-energy bands is

somewhat more challenging, however, because of the much lower laser energy available to probe them as a result of the extra stage of nonlinear mixing required to access that spectral range. To overcome this problem, we carried out the hole-burning measurements in a fixed point mode where we sequentially fixed the probe laser on a low-energy band and obtained many averages of the probe photofragment mass peak response to the pump laser (tuned to a known transition in the OH stretch region). Typically 5000 pump laser on/off cycles were averaged to build up sufficient signal-to-noise to conclusively establish whether the probed features respond to depletion of population by the pump. (See Supporting Information for a more detailed description.) Since the main objective was to identify activity in the lowenergy region arising from the D4HO2þ with the H exterior, we fixed the pump laser on the strong free OH band (labeled νNB OH on the right of Figure 5). This arrangement effectively probes most isotopomers in which the hydrogen is in an exterior position because of the well-known prevalence for the Ar to bind to an exterior deuterium when available.31 The response of three key bands indicated by (*, †, and )) were monitored by the probe laser, with the results summarized in the box at the top of Figure 5. Note that these are the averaged responses of photofragment mass peaks, where a dip indicates reduction of the population probed by the pump laser. The results demonstrate that the * and † bands are mostly due to the species with a bridging D atom, with both bands falling above the asymptotic values for D2O and HDO by 90 and 86 cm-1, respectively. The band highest in energy ()) of the three probed features does not arise from the same isotopomer, however, and we thus infer 5852



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that it is due to the shared H form, contrary to our initial assignment.26 This is consistent with the MCTDH results mentioned above, where the dominant band above the shared H fundamental band appeared far above the region expected for the D2O bend. The blue and green traces result from probing the two lower-energy transitions in this region, as indicated in the spectrum at the bottom of Figure 5. We note in passing that we also explored the response of the sharp peak nominally assigned to the bridging H at 939 cm-1 when the pump laser depleted the isomers with an exterior H as above. As expected, the bridging proton peak was unresponsive, confirming this assignment (see Supporting Information).

III. THEORETICAL METHODS With secure isotopomer assignments in hand, we turn our attention to the model that will be used to evaluate the types of motion that give rise to them. This model uses ground state probability amplitudes (GSPA), obtained from DMC simulations, to generate excited state wave functions and infrared intensities associated with transitions to these states. The GSPA approach was introduced in an earlier report, in which it was used to unravel the anharmonicities at play in the OH- 3 H2O complex.37 Since the details of how we obtain the Monte Carlo sampling ground state wave function are not necessary for the discussion that follows, they will not be repeated here. The only deviation from the approaches taken in our previously reported9,20,23 DMC studies of H5O2þ is that after each displacement of the atoms in the Monte Carlo simulation, the coordinates are rotated into an Eckart frame,42-44 based on one of the stationary points on the H5O2þ potential. This was done to provide a well-defined Cartesian axis system for the definition of the components of the dipole moment vector. The procedure is entirely analogous to that used in our recent study of H3Oþ.44 This process is important for molecules, like H5O2þ, that contain only two heavy atoms and are, on average, symmetric tops. This results in the complication that the orientation of the axes perpendicular to the symmetry (OO) axis is dependent on the choice of the reference geometry. The DMC simulations provide a set of Monte Carlo samplings of the ground state wave function. As such, integrals involving the wave function are treated as a summation over the n Monte Carlo sampling points: Z n X f ðri Þ ð1Þ ΨðrÞ f ðrÞ dr i¼1

In the case of the normalization integral, n X Ψðri Þ ÆΨjΨæ ¼

ð2Þ

i¼1

The above relationship illustrates the need to evaluate the value of Ψ(r) at each of the Monte Carlo sampling points. This is achieved by using descendent weighting.45 Finally, taken together, the above relationships lead to the following expression for the expectation value of any multiplicative operator n P Ψðri Þ Oðri Þ ÆΨjOjΨæ ¼ i ¼ 1P ð3Þ n ÆΨjΨæ Ψðri Þ i¼1

37

As described previously, excited states are approximated by products of Hermite polynomials and the ground state wave

function. The motivation for this is two-fold. First, we know in many one-dimensional systems, excited state wave functions are products of orthogonal polynomials in the chosen expansion coordinate and the ground state wave function. Second, by using these expressions, matrix elements of multiplicative operators can be evaluated using eq 3. While this will allow us to obtain matrix elements involving the potential and dipole surfaces, the evaluation of matrix elements of the kinetic energy operator, which involves derivatives of the wave function with respect to the internal coordinates, presents a challenge. Since the best zero-order picture to describe vibrations is the harmonic oscillator, we develop the relationship for the kinetic energy associated with putting n quanta of excitation into mode j ÆTænj - ÆTæ0 ¼

p 2 nj Rj 2mj

ð4Þ

where mj is the mass associated with the jth coordinate, qj. To simplify the evaluation, we use mass-weighted coordinates for which mj = 1. Finally, Rj is defined by Æqj 2 æ0 ð5Þ Rj ¼ 4 Æqj æ0 - Æqj 2 æ0 2 In ref 46, we showed that such an approximation is effective for describing the kinetic energy for a Morse oscillator and the nine vibrational modes in H3O2- and D3O2-. The choice of coordinates is critical. For this work, we use mass weighted linear combinations of displacements of internal coordinates, rj - rj,e. The expansion coefficients are generated by determining the eigenvectors of the matrix of mass-weighted second moments, defined as M0 ¼ G - 1=2 MG - 1=2

ð6Þ

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where G is the usual Wilson G-matrix and M is the matrix of second moments, defined in terms of 3N - 6 internal coordinates. The elements of both of these matrices are averaged over the ground state probability amplitude. The specific elements of the matrices are defined as Mj, k ¼ ÆΨ0 jðrj - Ærj æÞðrk - Ærk æÞjΨ0 æ

ð7Þ

+ X 3N Drj 1 Drk Ψ0 Ψ l ¼ 1 Dxl ml Dxl 0

ð8Þ

* Gj, k ¼

where rj represents one of the 3N - 6 internal coordinates, while xl is one of the 3N Cartesian coordinates with associated mass ml. What remains is the definition of the 3N - 6 internal coordinates in H5O2þ. For H3O2-, this choice was relatively straightforward, and we used the six coordinates that define the geometries of the two outer OH units (two OH bond lengths, two HOO angles, the OO bond length, and the HOOH torsion) and the three coordinates that define the displacement of the central hydrogen from the center of the OO bond. For H5O2þ, the choice is less obvious due to the equivalence of the four outer hydrogen atoms. In this work, we use the atom-atom distances between the two oxygen atoms and the four OH bond lengths of the outer water molecules as well as the Cartesian displacements of the central hydrogen atom from the center of the OO bond. The Cartesian components of this vector are evaluated in a bodyfixed axis system and are determined by the embedding of the molecule. The remaining seven atom-atom distances are the distances between the two hydrogen atoms that make up each of 5853



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Table 1. Comparison of Frequencies (freq in cm-1) and Oscillator Strengths (str) for H5O2þ harmonic - min

harmonic - sp

anharmonic - min

anharmonic - spc

anharmonic - spd

freqa

freq

freq

freq

freq

strb

str

str

str

str

MCTDH freq

169

0.080

206

0.000

223

0.000

22

0.000

103

338

0.272

384

0.003

412

0.000

52

0.000

106

str

471

0.058

203

0.000

474

0.011

496

0.012

326

0.001

106

531

0.061

467

0.032

513

0.000

540

0.000

478

0.000

481

0.027

554

0.046

467

0.032

511

0.000

519

0.000

1022

0.000

481

0.027

OO stretch

630

0.000

634

0.000

545

0.006

607

0.013

515

0.000

550

IHB (parallel)

861

1.000

726

1.000

1037

1.000

1062

1.000

1050

1.000

1033

1.000

1494 1574

0.050 0.020

1466 1466

0.016 0.016

1353 1612

0.060 0.000

1317 1424

0.057 0.058

1102 1179

0.076 0.079

1391 1391

0.010 0.010

IHB (perpendicular) IHB (perpendicular)

fixed nodee expt (Ne)f

995

1047

HOH in phase

1720

0.001

1659

0.000

1727

0.062

1613

0.000

1597

0.000

1606

HOH out-of-phase

1770

0.146

1794

0.258

1728

0.323

1754

0.340

1758

0.346

1741

0.241

1440

1763

OH sym out-of-phase

3744

0.019

3781

0.020

3557

0.025

3586

0.023

3586

0.023

3614

0.035

3511

3603

OH sym in phase

3750

0.000

3781

0.003

3578

0.000

3631

0.000

3632

0.000

3607

OH aym

3832

0.021

3885

0.025

3641

0.029

3693

0.034

3693

0.034

3689

0.035

3652

3683

3832

0.026

3885

0.025

3650

0.033

3719

0.043

3719

0.043

3689

0.035

3652

3683

3553

Frequency (cm-1) in increasing order of energy. b Relative oscillator strength - parallel IHB has an oscillator strength of 1 in all cases. c Using distancebased coordinates. d Using Euler-based coordinates. e Results of fixed node DMC calculations, reported in refs 9, 20, and 26. f Reference 9. a

the outer water molecules, the distances from each of the four outer hydrogen atoms to the oxygen atom in the other water molecule and the difference between the distances of the four pairs of hydrogen atoms on different water molecules. Specifically, in defining the last coordinate we assume hydrogen atoms 1a and 1b are bound to oxygen atom 1 and hydrogen atoms 2a and 2b are bound to oxygen atom 2, making this coordinate ð9Þ d ¼ r1a, 2b þ r1b, 2a - r1a, 2a - r1b, 2b Linear combinations of these seven distances correlate to the HOH angle, the θ and φ Euler angles that define the orientation of each of the two water molecules, and a torsion angle. We also investigated a second set of coordinates that includes the four OH bond lengths, the OO distance, the two HOH angles, the five Euler angles that describe the orientations of the two flanking water molecules, and the Cartesian coordinates of the displacement of the central hydrogen atom from the center of the OO bond. While the results below focus on the distance coordinates, comparison of the results obtained using the two coordinate definitions are provided in the Supporting Information. Both coordinate definitions rely on the choice of embedding of the body-fixed axis system. Several choices were explored, and the results presented below use the second-order saddle point, shown in Figure 1B. This saddle point is 524 cm-1 higher in energy than the global minimum on the potential surface.19 In addition to using the embedding to define the coordinates of the central proton, they are used to define the orientation of the dipole moment in a body-fixed axis system.

IV. DISCUSSION OF THEORETICAL RESULTS We start this section by focusing on H5O2þ, for which direct comparison of the spectrum and assignments can be made to the work of Meyer and co-workers.27,29 This system has the additional advantage that we can compare our calculated spectrum to the neon-tagged spectrum, for which perturbations by the tagging atom are much smaller than for argon. Having demonstrated

the effectiveness of the approach, we turn our attention to the deuterated isotopologues. IV.A. H5O2þ and D5O2þ. In Table 1, we report the harmonic frequencies and oscillator strengths for H5O2þ, obtained using the potential surface developed by Huang, Braams, and Bowman,19 and evaluated at two stationary points. The first set of harmonic frequencies and intensities were obtained using the minimum energy geometry (Figure 1A), while the second set of data is for the second-order saddle point, depicted in Figure 1B. Using the same potential, Meyer and co-workers performed high-level, MCTDH calculations. On the basis of that work, they obtained the frequencies and oscillator strengths reported in the columns labeled “MCTDH”. The previously reported results of fixednode DMC calculations are reported in the column labeled “fixed node”.9,20,26 Once again these were obtained using the same potential as was employed in the present work. Finally, experimental frequencies, obtained by loss of a neon atom upon vibrational excitation of H5O2þ 3 Ne are reported in the column identified as “(Ne)”.9 When the strengths of the transitions are reported, the values are the normalized oscillator strengths, which are proportional to the square of the matrix element of the dipole moment operator between the ground state wave function and the wave function for the excited state of interest. They have been normalized so that the strength of the most intense fundamental transition (e.g., the displacement of the central proton parallel to the OO axis) is unity. This choice of normalization was made to facilitate comparison to experimental spectra. Due to variations in laser power across the large frequency regions that are reported, the spectra have been normalized with respect to the laser energy/pulse at a particular excitation energy.48 Figure 6 presents a comparison of the convoluted (HWHM = 5 cm-1) harmonic and experimental spectra for H5O2þ and D5O2þ. The scale for the signal strength of the harmonic spectra is consistent throughout the plot, while the low- and highfrequency regions of the experimental spectra are separately normalized because no overlapping bands are available with 5854



Figure 6. Experimental (black) and harmonic spectra for (A) H5O2þ and (B) D5O2þ, calculated at the minimum energy geometry of the molecule (red) and the second-order saddle point, depicted in Figure 1B (blue). The experimental spectra were obtained by H5O2þ 3 Ne and D5O2þ 3 Ar predissociation.

which to link the relative action in each region. For the modes with frequencies above 500 cm-1, the harmonic frequencies are actually quite similar for the two stationary points, and both yield frequencies of the shared proton stretch that are 150-300 cm-1 lower than both experiment and to the more sophisticated calculations. The fine structure on the main bands is not recovered at the harmonic level, however, and the OH stretch frequencies are too high, falling roughly 200 cm-1 above the experimental values. The harmonic HOH bend frequencies are actually in surprisingly good agreement with experiment9 and with the MCTDH calculations,27 while the fixed-node DMC approach9,26 significantly underestimates the frequency of this mode. In light of the poorer agreement with the experimental spectrum provided by both of the harmonic and the fixed-node DMC frequencies, the question arises whether the simplified anharmonic treatment based on the ground state probability amplitude (GSPA), described above, can effectively reproduce the experimental spectra of these species. The results of this treatment are

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Figure 7. Experimental (black) and anharmonic spectra for (A) H5O2þ and (B) D5O2þ. The calculated spectra were obtained using the distance coordinates and two choices of embedding for the body-fixed axis system. Specifically, the spectrum calculated when the molecule is in its minimum energy geometry are plotted in red and the blue spectra were obtained using the geometry of the second order saddle point, depicted in Figure 1B.

compared with previous results in Table 1. The spectra were calculated using the same two reference geometries used for the harmonic treatments and for the two sets of coordinates, described above. Calculations were also performed using the D2h saddle point structure (Figure 1B), and while not reported here, the results are consistent with the results provided in Table 1. In Figure 7, we plot the calculated spectra using the GSPA method for H5O2þ and D5O2þ, obtained by using an embedding based on the geometries of the minimum and second-order saddle point shown in Figure 1. It is evident that, although the proposed anharmonic approach is similar in spirit to a harmonic treatment, the agreement with experiment is dramatically improved by basing the vibrational modes on the anharmonic ground state, especially taking into account that only transitions involving excitation of two or fewer quanta are included in these calculations. As discussed above, the missing peaks have been 5855



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Figure 8. Experimental (black) and anharmonic spectra for (A) H4DO2þ and (B) D4HO2þ, calculated with the unique atom in the bridging position (red) and an external position (blue), with the structures indicated in the inset. The calculated spectra were obtained using the distance coordinates. Of note, the calculations predict that the highest-energy peak in the HOD bending region at 1548 cm-1 is due to the isotopomer with hydrogen in the shared position.

assigned to higher-order combination bands and are not included in the present treatment. One very encouraging feature is the agreement between the calculated and experimental frequency and transition strength for the water bends in both H5O2þ and D5O2þ. As noted above, for example, fixed node DMC underestimates that frequency in H5O2þ by more than 200 cm-1. One feature that is less well reproduced is the behavior of the two highestfrequency OH stretches in H5O2þ, which, in the untagged species, should be degenerate by symmetry. The 26 cm-1 calculated splitting of these bands in H5O2þ is a reflection of a breakdown of the symmetry in the calculation. In D5O2þ, this splitting is reduced to 8 cm-1, and is not resolvable at the 5 cm-1 HWHM resolution of the plots. Before moving to a discussion of the mixed isotopologues, it is interesting to explore the extent of coupling among the modes that are used in the above calculations. On the basis of the present GSPA-based analysis, we find that for the modes of H5O2þ or D5O2þ with fundamental frequencies in the measured region

(above 700 cm-1), most have leading contributions from a single mass-weighted coordinate of greater than 0.6. The exceptions for H5O2þ are the two symmetric stretches and the out-of-phase bend. For D5O2þ, the two most highly mixed modes are the outof-phase symmetric stretch and the out-of-phase bend of the outer water molecules, both of which have leading contributions of roughly 0.5. In the case of the feature at ∼1750 cm-1 in H5O2þ nominally associated with the bend, the GSPA analysis indicates that the displacement of the central H along the OO stretch also contributes 0.18. For D5O2þ, there is a 0.28 contribution from the out-of-phase symmetric OD stretch and a 0.17 contribution from the displacement of the shared D along the OO axis. The first of these additional contributions to the out-of-phase bend reflects the fact that we are using distance coordinates. The contribution to the out-of-phase bend in both H5O2þ and D5O2þ from the displacement of the central hydrogen or deuterium atom is leading to the large blue shift of the fundamental frequency of this mode. 5856


The Journal of Physical Chemistry A As mentioned in the Introduction, there are large couplings between the displacement of the central hydrogen atom along the OO axis and the out-of-phase bend involving the two flanking water molecules. This can be understood by realizing that as the central hydrogen moves toward one of the flanking water molecules and away from the other, the system is driven toward a configuration that resembles H3Oþ solvated by a single water molecule. While in the equilibrium configuration, the two HOH angles in H5O2þ are equivalent; in the H3Oþ þ H2O geometry the HOH angle in H2O will be smaller than the HOH angle in H3Oþ. The lower-frequency mode that has a leading contribution from the displacement of the central hydrogen or deuterium atom, which has a frequency of roughly 1000 cm-1 in H5O2þ and roughly 700 cm-1 in D5O2þ, has a contribution from the out-ofphase bend that has the same sign as the central hydrogen or deuterium atom displacement. In contrast the signs are different for the vibration that is called the out-of-phase bend. This leads the system to feel a steeper potential when the collective coordinate is displaced than would be felt if the displacement only involved motion of the two HOH bends. The fact that the central hydrogen and HOH bend motions are out-of-phase results in the observed higher frequency of the out-of-phase bend normal mode compared to what would have been anticipated by considering the HOH bend frequencies in H2O or H3Oþ. The extent of the mixing is larger in H5O2þ than in D5O2þ, and consequently the shift in the frequency is also larger, and well beyond what is expected from the usual H/D isotopic shift of individual local oscillators. In addition, as pointed out in ref 26, the large contribution to the so-called out-of-phase bend normal coordinate from the displacement of the central hydrogen or deuterium atom is also responsible for an enhancement in the intensity of the bend fundamental. IV.B. Mixed Species. We next apply the GSPA method to the spectra of the mixed isotopic species, with the results presented in Figure 8. The calculated spectrum for the (D2O)Hþ(OD2) isotopomer is displayed as the upward peaks in Figure 8B, which appear above the experimental spectrum for comparison. We find that the most intense feature in the 1000-2000 cm-1 region is at 1435 cm-1 and is assigned primarily as the DOD bend, consistent with the center of intensity of the strong transitions (1355 and 1564 cm-1) reported by Meyer and co-workers on the basis of their MCTDH calculations. The GSPA results for the (D2O)Dþ(OHD) isomer are presented by the downward peaks in Figure 8B, which recover the DOD and HOD bend frequencies at 1259 and 1541 cm-1, respectively. As noted in the Experimental Section, these values are much more in line with naïve expectations based on the locations of the bands in the isolated water isotopologues as well as the behavior observed for D5O2þ. Taken together, the emerging picture is that transitions derived at least partially from the outer DOD bends blue shift by nearly 250 cm-1 when the central deuterium is replaced by hydrogen. Comparisons of the calculated and experimental spectra for H4DO2þ are shown in Figure 8A, where again the central plot in this panel reproduces the experimental spectrum for comparison. The upward peaks in the top trace correspond to the GSPA spectrum when the central atom is D, while the downward peaks in the bottom trace were obtained for the isomer with a central H. It is clear at a glance that most of the experimental ensemble accommodates H in the center, as expected from the earlier results of the isotope fractionation study by Devlin et al.22 Focusing on the bend region of the calculated spectrum for (H2O) Hþ(OHD), this spectrum is in good general agreement

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with the measured spectrum. The highest-frequency peak (at 1652 cm-1) corresponds to the HOH bend, while the three smaller lower-frequency peaks correspond to the HOD bend at 1516 cm-1, the OO/shared proton combination band at 1447 cm-1, and a convolution of the two perpendicular displacements of the shared proton at ∼1360 cm-1. Finally, the HOH bend fundamental is not as strong in the calculated spectrum as it is observed to be in the argon-tagged spectrum. Similar behavior is seen in the argon-tagged spectrum for H5O2þ in which the intensity of this band is much larger than in the neon-tagged one, shown in Figure 2. Consequently, this difference in intensity is attributed to changes in environment, brought about by the presence of the argon atom. With the present satisfactory performance of the simpler GSPA method in the interpretation of the asymmetrical Zundel isotopomers, we have established two floppy systems (H5O2þ and OH- 3 H2O) where the approach provides an effective strategy to handle intrinsically anharmonic aspects of cluster spectra. As such, one is motivated to treat more complex systems in an effort to establish how robust the method is when confronted with even higher dimensionality and consequently higher densities of vibrational states in the fingerprint region of the infrared.

V. SUMMARY AND CONCLUSIONS We presented improved experimental spectra and isotopomerselective, double-resonance based band assignments for several H/D isotopologues of the Zundel ion. The band patterns were considered in the context of a theoretical method for evaluating anharmonic spectra based on statistical sampling of the ground state wave function. Explicit consideration is given to the role of the set of internal coordinates (including the three Euler angles) needed to describe the vibrational dynamics. In the present study, we employ diffusion Monte Carlo to obtain the wave functions. With that in place, the evaluation of the spectra, frequencies and intensities, is straightforward, and all of the spectra reported here were calculated from the corresponding ground state wave functions and required several hours of computer time. While the approach is clearly not as accurate as those based on larger basis sets, it provides a way to sample the spectroscopy and dynamics with minimal overhead. In the case of H5O2þ the results of these calculations shed insights into the nature and the strength of the interaction between the shared proton stretch and the water bends as well as how such coupling is manifested in the spectra. It also allowed us to investigate the assignments of the mixed isotopmers of H5O2þ in the mid-IR region of the spectrum and to understand the surprisingly large differences between the overall band contour of H4DO2þ and those of the other three species. While the above approach appears to be reasonably robust, it relies on knowledge of both potential and dipole surfaces to be effective. As we look to the future, it would be desirable to find ways to obtain the necessary ingredients for determining the ground state wave function (or probability amplitude) without reliance on a potential surface. ’ ASSOCIATED CONTENT

bS

Supporting Information. Description of the single point double-resonance technique used to identify the low-energy bands in the spectrum of D4HO2þ as well as mass spectra of H4DO2þ and anharmonic spectra for H5O2þ and D5O2þ calcu-

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The Journal of Physical Chemistry A lated using the Euler-based coordinates are provided. This material is available free of charge via the Internet at http://pubs. acs.org.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: A.B.M., [email protected]; M.A.J., [email protected].

’ ACKNOWLEDGMENT A.B.M. and M.A.J. thank the Chemistry Division of the National Science Foundation Division for support of this work under grants CHE-0848242 (A.B.M.) and CHE-0911199 (M.A.J.). We also acknowledge the many contributions of Victoria Buch to our understanding of H5O2þ, our many discussions of the nonstatistical nature of the populations of the isomers of the partially deuterated isotopologue, and her introduction of one of us to the power of Diffusion Monte Carlo approaches for studying anharmonic species.

’ REFERENCES (1) Zundel, G.; Metzger, H. Z. Phys. Chem. 1968, 58, 225. (2) Zundel, G. Adv. Chem. Phys. 2000, 111, 1. (3) Eigen, M.; Maeyer, L. D. Proc. R. Soc. London 1958, 247, 505. (4) Okumura, M.; Yeh, L. I.; Myers, J. D.; Lee, Y. T. J. Phys. Chem. 1990, 94, 3416. (5) Asmis, K. R.; Pivonka, N. L.; Santambrogio, G.; Br€ummer, M.; Kaposta, C.; Neumark, D. M.; W€oste, L. Science 2003, 299, 1375. (6) Fridgen, T. D.; McMahon, T. B.; MacAleese, L.; Lemaire, J.; Maitre, P. J. Phys. Chem. A 2004, 108, 9008. (7) Headrick, J. M.; Bopp, J. C.; Johnson, M. A. J. Chem. Phys. 2004, 121, 11523. (8) Headrick, J. M.; Diken, E. G.; Walters, R. S.; Hammer, N. I.; Christie, R. A.; Cui, J.; Myshakin, E. M.; Duncan, M. A.; Johnson, M. A.; Jordan, K. D. Science 2005, 308, 1765. (9) Hammer, N. I.; Diken, E. G.; Roscioli, J. R.; Johnson, M. A.; Myshakin, E. M.; Jordan, K. D.; McCoy, A. B.; Huang, X.; Bowman, J. M.; Carter, S. J. Chem. Phys. 2005, 122, 244301. (10) Douberly, G. E.; Walters, R. S.; Cui, J.; Jordan, K. D.; Duncan, M. A. J. Phys. Chem. A 2010, 114, 4570. (11) Pribble, R. N.; Zwier, T. S. Science 1994, 265, 75. (12) Kim, K.; Jordan, K. D.; Zwier, T. S. J. Am. Chem. Soc. 1994, 116, 11568. (13) Bieske, E. J.; Dopfer, O. Chem. Rev. 2000, 100, 3963. (14) Niedner-Schatteburg, G.; Bondybey, V. E. Chem. Rev. 2000, 100, 4059. (15) Chaban, G. M.; Jung, J. O.; Gerber, R. B. J. Phys. Chem. A 2000, 104, 2772. (16) Vener, M. V.; Kuhn, O.; Sauer, J. J. Chem. Phys. 2001, 114, 240. (17) Dai, J.; Bacic, Z.; Huang, X.; Carter, S.; Bowman, J. M. J. Chem. Phys. 2003, 119, 6571. (18) Huang, X.; Cho, H. M.; Carter, S.; Ojam€ae, L.; Bowman, J. M.; Singer, S. J. J. Phys. Chem. A 2003, 107, 7142. (19) Huang, X.; Braams, B. J.; Bowman, J. M. J. Chem. Phys. 2005, 122, No. 044308. (20) McCoy, A. B.; Huang, X.; Carter, S.; Landeweer, M. Y.; Bowman, J. M. J. Chem. Phys. 2005, 122, No. 061101. (21) Sauer, J.; Dobler, J. ChemPhysChem 2005, 6, 1706. (22) Devlin, J. P.; Severson, M. W.; Mohamed, F.; Sadlej, J.; Buch, V.; Parrinello, M. Chem. Phys. Lett. 2005, 408, 439. (23) McCoy, A. B. Int. Rev. Phys. Chem. 2006, 25, 77. (24) Kaledin, M.; Kaledin, A. L.; Bowman, J. M. J. Phys. Chem. A 2006, 110, 2933. (25) Park, M.; Shin, I.; Singh, N. J.; Kim, K. S. J. Phys. Chem. A 2007, 111, 10692.

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Unraveling Anharmonic Effects in the Vibrational ... - ACS Publications

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