Signatures of Large-Amplitude Vibrations in the Spectra of H5+ and

Nov 26, 2012 - Signatures of Large-Amplitude Vibrations in the Spectra of H 5 + and D 5 + .... Origin of the diffuse vibrational signature of a cyclic...
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Signatures of Large-Amplitude Vibrations in the Spectra of H+5 and D+5 Zhou Lin and Anne B. McCoy* Department of Chemistry and Biochemistry, The Ohio State University, Columbus, Ohio 43210, United States S Supporting Information *

ABSTRACT: H+5 is a weakly bound molecular ion, which is formed from the reaction of H+3 and H2 and that has a very rich vibrational spectrum. In this work, diffusion Monte Carlo (DMC) approaches are used to explore the nature of vibrationally excited states of the proton-transfer mode in H+5 . On the basis of these calculations, alternative assignments of the recently reported infrared multiphoton dissociation spectra of H+5 and D+5 [J. Phys. Chem. Lett. 2012, 3, 3160−3166] are suggested. In the proposed assignments, progressions of transitions in the proton-transfer mode with up to nine quanta of excitation are invoked. Reduced dimensional calculations of the spectra of H5+ and D5+ are used to provide an understanding of why such high overtones should be observable through absorption spectroscopy. Implications of how excitations of this mode can provide insights into the H+3 + H2 reaction are also discussed. SECTION: Spectroscopy, Photochemistry, and Excited States

H

While H+5 has not been identified in the interstellar medium, the results of several studies of the vibrational spectrum of this molecule have been reported. The earliest of these was a lowresolution action spectrum reported by Okumura, Yeh, and Lee.6 In this experiment, H 3+ was detected following fragmentation of vibrationally excited H+5 into H+3 and H2. As such, the experiment was limited to transitions to states that are higher in energy than the dissociation threshold of H+5 . On the basis of this work, three broad features were reported in the 3500−4350 cm−1 region. More recently, Duncan and co-workers revisited this spectrum at higher resolution and over a larger spectral range.7 They also reported the spectrum for D+5 . In addition to observing the three peaks in the spectrum for H+5 that were reported by Lee and co-workers, they found a fourth peak at 2603 cm−1. The spectrum of D+5 consisted of a similar progression of three transitions, shifted to the red by the expected factor of roughly 2−1/2. The fourth peak was not observed for D+5 as its energy is below the dissociation energy of D+5 . Very recently, the lower-energy spectrum (below 2200 cm−1) was recorded using the FELIX free electron laser.8 Again, infrared absorption was detected by monitoring H+3 or D+3 generation from dissociation of H+5 or D+5 , respectively. Because the energies that were probed in this study are well below the dissociation threshold of these ions, the signal reflects multiphoton processes. Consequently, sources of the intensities are more complicated than those for single-photon absorption experiments. Also, the bands may be broadened and their

+ 5

is a molecular ion that has been of continuing interest for experiment, theory, and observation since its first laboratory observation in 1962.1 Much of this interest derived from observations of H+3 in the interstellar medium.2 From studies of the relative abundances of the ortho and para forms of H+3 and H2 in the interstellar medium, it was found that the rotational distributions of these two molecules predict different temperatures.3,4 In addition, the relative abundances of the partially deuterated forms of H+3 do not follow the natural abundance of deuterium.5 On the basis of this, it seems that the low-temperature equilibrium abundances of various forms of H+3 and H2 are determined by more than their energies. The exchange of a proton or a deuteron between H2 and H+3 occurs through a bound H+5 intermediate. The vibrational mode in H+5 that is most closely related to the exchange of a proton between H+3 and H2 is the H2−H+−H2 asymmetric stretch (R1− R2 in Figure 1). Near the equilibrium geometry, shown in Figure 1, this corresponds to a chattering of the shared proton between the two H2 groups, while at higher energy, it is better described as a large-amplitude vibration of a H2·H+3 complex.

Figure 1. Equilibrium structure of H+5 . The two H2 groups are composed of atoms 1 and 2 and of atoms 4 and 5, while the central proton is atom 3. The two vectors R1 and R2 connect the central proton to the center of mass of one of the H2 groups. The coordinates of interest in this study are R1 and R2. © 2012 American Chemical Society

Received: November 1, 2012 Accepted: November 26, 2012 Published: November 26, 2012 3690

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frequencies shifted from where they would occur if it were a single-photon process. All three of these spectroscopic studies used the current state-of-the-art theory to aid in the assignments of the observed transitions. H+5 has two low-frequency vibrations that are expected to be large-amplitude. These are the shared proton chattering modes depicted in Figure 1 and the torsion of the terminal H2 groups. In addition, the barrier for exchange of the central hydrogen atom and one of the hydrogens in the H2 groups is less than 1600 cm−1.9,10 The existence of these three potentially large-amplitude motions led to doubts that the spectrum can be assigned based on a harmonic analysis at the potential minimum. More sophisticated treatments would be required. At the time of the 1988 study of Lee and co-workers,6 the best available theory that had been performed on H+5 was a detailed investigation of the energetics and harmonic frequencies by Yamaguchi et al.9 On the basis of these studies, the three observed features were assigned as an overtone of a 1746 cm−1 mode, the out-of-phase stretch of the terminal H2 groups, and a combination band involving the H2 stretch and the chattering of the central proton. The two more recent studies used vibrational configuration interaction (VCI) calculations to aid in the assignments of the spectra.7,8 The VCI calculations did a reasonable job of reproducing the most prominent features in the spectrum. The wave functions that corresponded to the calculated bands were decomposed into a product basis that was made up of functions of displacements of individual normal modes. From this analysis, the three bands that were observed in both the H+5 and D+5 spectra above 2000 cm−1 were assigned to the fundamentals in the outer H2 stretches, with significant activity in the mode that corresponds to the chattering of the central proton between the two outer H2 groups. This was specifically noted for the highest-energy peak. The lowest-energy band in the H+5 spectrum (at 2603 cm−1) was assigned as the second overtone (the transition to the ν = 3 level) of the shared proton chattering mode, described above. Likewise, the lower-energy infrared multiphoton dissociation (IRMPD) spectra (below 2000 cm−1) of H+5 and D+5 were assigned using the VCI calculations.8 On the basis of this analysis, a band at 379 cm−1 in the H+5 spectrum was assigned to the fundamental in the shared proton chattering mode. Other higher-energy transitions were assigned to combination bands involving the shared proton chattering mode and other lowfrequency vibrations. While the agreement between the experimental and calculated spectra is very good, especially when one considers the complexity introduced to H+5 by the multiple largeamplitude motions, the assignment of the second overtone to an energy that is more than 7 times the energy of the fundamental is surprising even for a highly anharmonic vibration. The first overtone (which is optically dark) is calculated to have an energy of 1718 cm−1. This is close to 2/3 of the calculated energy for the state with ν = 3 (2751 cm−1) and 4.5 times the calculated energy for ν = 1 (383 cm−1). Similar unusual frequency progressions were also predicted for D+5 . On the other hand, if we look at the spectrum for H+5 below ∼2000 cm−1, we see what appears to be a vibrational progression. This is shown with the vertical blue arrows in Figure 2. As was mentioned above, the peak at 379 cm−1 in the H+5 spectrum has been assigned to the fundamental in the shared proton chattering mode. A progression of much weaker

Figure 2. The IRMPD action spectrum for H+5 (top) and D+5 .8 The vertical arrows are used to show the progression in the proton-transfer mode in the measured spectrum. Dashed arrows are used for the two higher-energy features in the D+5 spectrum as the signal is weak in this region. The numbers in black are the experimental line positions, reported in ref 8, while numbers in blue are calculated DMC energies from the present study. The red ∗ indicate bands that were attributed to the third harmonic of FELIX and correspond to the strong transitions at 940 and 1399 cm−1 in the H+5 spectrum.8

features is also seen in the D+5 spectrum. The apparent progression that is seen in the H+5 spectrum and possibly in the D+5 spectrum led us to wonder if there might be an alternative way to think about the assignment of the spectra. In addition, over the past several years, we have been using fixed-node diffusion Monte Carlo (DMC) approaches with good results to predict the energies of vibrational states of very anharmonic systems including H5O+2 ,11,12 H3O−2 ,13,14 and CH+5 .15,16 We wanted to see if this approach could reproduce the unusual vibrational progression of the shared proton chattering mode seen in the VCI calculations. In this Letter, we report the results of these DMC calculations. On the basis of these results, we propose alternative assignments for several of the transitions in the IRMPD spectra for H+5 and D+5 . Specifically, we assign the progression of states that carry oscillator strength to overtones with odd numbers of quanta in the large-amplitude vibration of the H2·H3+ complex. Additionally, we discuss why high overtones in this mode are observed in the spectrum and how these states aid our understanding of the proton-transfer dynamics between H+3 and H2. In Table 1, we compare the energies of the ν = 0, 1, 2, and 3 levels of H+5 obtained from these DMC calculations to those reported previously. Table 2 provides analogous comparisons for D+5 . All of the calculations reported in Tables 1 and 2 were performed using the potential surface developed by Xie et al.10 There are two sets of DMC values for the zero-point and ν = 1 energies. One set was reported by Cheng et al.,7 while the other provides the results of the present study. As is seen, the two DMC calculations are in near exact agreement for the zero3691

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moving them farther away from the values reported in ref 7. In addition, on the basis of our difficulties in reproducing the previously reported ν = 1 DMC energies, we evaluated the energies for the ν = 1 state of H+5 using two DMC codes that were developed independently. Both provided the same energy, within the statistical uncertainties reported in Table 1. On the basis of the agreement among our calculated energies for the ν = 1 state of H+5 , we have good confidence in the value that we report for the DMC energy. A second interesting feature of the results reported in Table 1 is seen in the ν = 2 and 3 energies. The DMC and MCTDH18 calculations give energies for the ν = 2 state of H+5 that differ by less than 1 cm−1, and both are roughly 1/3 of the energy reported for the ν = 2 state based on the VCI calculations.7 Similar disagreement is seen between the VCI and DMC energies for the ν = 3 level of H+5 and both the ν = 2 and 3 energies of D+5 . While there are large deviations between the energies of the assigned states with ν = 2 and 3 from the VCI calculations and those from the other two calculations, there are VCI energies in the list of calculated states8 that have significant shared proton chattering character, the correct symmetry, and that are close in energy to the values obtained from the other calculations. These are reported in the fifth column of Tables 1 and 2. The question naturally arises as to how two different assignments can be used to describe the same set of states. The answer can be found in the wave functions obtained from a two-dimensional treatment of H+5 and D+5 . In these calculations, the terminal H2 groups are treated as unified atoms with the mass of H2 or D2, and the H2−H+−H2 system is constrained to be collinear. A similar approach was recently employed by Sanz-Sanz et al. in a study of the spectra of H+5 and D+5 above the dissociation threshold.19 The resulting wave functions for the four lowest energy states (identified as νpt = 0−3) and two more highly excited states of H+5 (identified as νa = 2 and 3) are plotted as functions of s = 2−1/2(R1 + R2) and a = 2−1/2(R1 − R2) in Figure 3 along with the potential surface used for the calculation. The plots for D+5 are very similar to the ones for H+5 , and are provided in the Supporting Information. To aid in the discussion that follows, we will use two types of quantum number labels for the states. The first is the usual normal-mode labels for the symmetric (s) and antisymmetric (a) combination of R1 and R2, defined above. In this picture, the nodal surfaces are expected to lie roughly parallel to the axes in the plots in Figure 3. A second notation is used to recognize that for many of the states, the nodes lie perpendicular to the minimum-energy contour along the potential surface, which is roughly V-shaped in Figure 3. When we count the number of nodes that lie perpendicular to this contour we will identify these states using νpt, as largeamplitude motion along this contour correspond to transferring a proton from one H2 molecule to the other one. Such motion corresponds to the proton chattering discussed above. As we consider the wave functions, we will divide the potential into three regions. The region where s ≤ 2 Å corresponds to the H+5 minimum. The region in which s > 2 Å and a > 0 represents the product channel in which H5+ dissociates into H2 + H+3 , and hydrogen atoms 1 and 2 in Figure 1 make up the H2 molecule. Finally, the region with s > 2 Å and a < 0 represents the second product channel in which H+5 dissociates into H+3 + H2, and hydrogen atoms 4 and 5 in Figure 1 comprise the H2. While the ground state and first excited state remain localized near the potential minimum,

Table 1. Calculated Energies for the Ground State and First Three Excited States in the Shared Proton Chattering Mode of H+5 , Obtained in the Present Study, Compared to Values Reported Previouslya state g.s. ν=1 ν=2 ν=3

DMCb 7205 369 673 983

± ± ± ±

5 5 5 5

DMCc

VCId

VCIe

MCTDHf

expt.g

7210 334

7244 382 1718 2751

7244 382 695 952/989

7210.3 358.7 673.6

379 940

a

The ground-state energy is calculated relative to the potential minimum, while the other energies are reported relative to the ground state. All energies are reported in cm−1. bThis work. cPreviously reported DMC energies.7,8 dVCI state assignments to these levels.7,8 e VCI states that might correlate to the states obtained from DMC. f MCTDH energies.18 gBands in the IRMPD spectrum.8

Table 2. Calculated Energies for the Ground State and First Three Excited States in the Shared Proton Chattering Mode in D+5 , Obtained in the Present Study, Compared to Values Reported Previouslya state g.s. ν=1 ν=2 ν=3

DMCb 5149 241 473 713

± ± ± ±

5 5 5 5

DMCc

VCId

VCIe

expt.f

5152 222

5174 257 1241 1821/1834

5174 257 513 827

679

a

The ground-state energy is calculated relative to the potential minimum, while the other energies are reported relative to the ground state. All energies are reported in cm−1. bThis work. cPreviously reported DMC energies.7,8 dVCI state assignments to these levels.7,8 e VCI states that might correlate to the states obtained from DMC. f Bands in the IRMPD spectrum.8

point energy. The multiconfigurational time-dependent Hartree (MCTDH) zero-point energy also agrees with these values for H+5 , while the VCI zero-point energies for H+5 and D+5 are somewhat larger than the values obtained by the other methods. DMC is a method that should provide an exact value for the ground-state energy, with some statistical uncertainty. We conservatively report a value of 5 cm−1 for uncertainties of all of the DMC energies due to statistics. For the ν = 1 state, the present DMC results are intermediate between the VCI and MCTDH results for H+5 and close to the VCI energy for D+5 . The three values differ by between 10 and 20 cm−1 and are all close to the energy of the 379 cm−1 band in the IRMPD spectrum for H+5 . Such differences between DMC, MCTDH, and VCI energies are comparable to the agreement that we obtained between VCI, MCTDH, and DMC studies of H5O+2 11,12,17 and H3O−2 .13 On the other hand the energies that are obtained from the two DMC calculations for the ν = 1 states of H+5 and D+5 differ by 35 and 20 cm−1, respectively, with the previously reported DMC energies being lower than the results of the present calculations.7 These differences are well outside of the statistical uncertainties of the DMC energies. We are not sure why the two DMC calculations are not in better agreement. One possible source of the difference is the definition of the coordinate along which the node is placed (see the Methods Section). The present DMC calculations use a slightly different definition of the proton chattering coordinate than was described in ref 7. We repeated our calculations for H+5 using the definition for the proton chattering coordinate described in ref 7. We find that our calculated DMC energies increase to 378 ± 5 cm−1 for H+5 and 246 ± 5 cm−1 for D+5 , 3692

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Figure 3. The two-dimensional potential and six wave functions described in the text are plotted as functions of s = 2−1/2(R1 + R2) and a = 2−1/2(R1 − R2). The potential is plotted in increments of 2000 cm−1 up to 15 000 cm−1. The states are identified by the state number as well as either the number of quanta in the proton-transfer mode (νpt) or the number of quanta in the asymmetric stretch (νa). The dashed lines in the νpt = 1, 2, and 3 states indicate the form of the nodal surfaces used in the DMC calculations of these states.

Figure 4. One-dimensional (a) projections and (b) cuts through the probability amplitudes obtained from the two-dimensional wave functions plotted in Figure 3. In panel (a), the probability amplitudes are projected onto a, while in panel (b), cuts through the probability amplitude are evaluated for s = 1.75 Å. In all cases, the curves are scaled so that the integrated areas are unity, and the plots are shifted to facilitate comparison.

nodes in the νpt = 2 wave function do not lie parallel to either the a or s axis, which correspond to two normal modes in this

there is significant amplitude in the H2 + H+3 product channels, even for the state that is labeled as νpt = 2. Additionally, the 3693

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reduced dimensional picture. Instead, they are roughly perpendicular to the equipotential contours. The next state (νpt = 3) extends further into the product channel. Although the nodes for the νpt = 2 and 3 states are not along the asymmetric stretch normal mode (a), the overall nodal structure remains simple. The wave functions that are in Figure 3 resulted from two-dimensional reduced dimensional calculations, and it would be appropriate to question if such nodal structures would persist in higher-dimensional treatments. In their full-dimensional MCTDH study of H5+, Delgado-Barrio and co-workers18 reported projections of the probability amplitudes for the states reported in Table 1 onto R1 and R2. These projections show the same nodal structure as is seen in the plots in Figure 3. The remaining two states that are plotted in Figure 3 are the 6th and 12th excited states. These are the lowest-energy states for which the nodal structure near the potential minimum resembles two and three quanta in the asymmetric stretch. We believe that it is the nine-dimensional analogues of these states that are assigned as the νa = 2 and 3 levels in the VCI studies that are reported in the fourth column in Tables 1 and 2. This assertion is supported by the results reported in Figure 4, where we provide one-dimensional plots of the probability amplitude for the six states shown in Figure 3. In panel (a), we projected the probability amplitudes onto the a-axis, and the number of oscillations in the probability amplitude corresponds to the number of quanta in νpt. If, on the other hand, we take a cut through the probability amplitude at s = 1.75 Å, the picture looks different. The four lowest energy states all show at most one node in this cut through the probability amplitude, while the states identified as νa = 2 and 3 have two and three nodes, respectively, consistent with their assignments. When we consider vibrational progressions, we typically assume that in the absence of near degeneracies, the wave functions that correspond to the assigned states can be described as products of basis functions that depend on the coordinates of one-dimensional oscillators. As the excitation is increased, there will be an increase in the number of nodes in one of the oscillators. This appears to be the model that was used as the basis of the assignments of the spectrum to the VCI calculated states.7,8 Specifically, the VCI states are described in terms of the quantum numbers associated with the three leading terms in the expansion of the wave functions in a basis set that is composed of products of functions of individual normal-mode coordinates. Often, for molecules that display large-amplitude vibrational motions, other types of vibrational progressions emerge at high energy.20,21 One example of this is the well-studied normal- to local-mode transition in X−H stretches in XHn molecules22,23 or the bends in HCCH.24,25 While at low energies the wave functions are roughly separable in a normal-mode basis, above a threshold energy, some of the states are better described by high levels of vibrational excitation of one of the XH bonds in XHn or one of the HCC bends in the case of acetylene. In these cases, the underlying vibrational motion that is being excited is no longer one of the normal modes but some other type of large-amplitude motion. While this transition usually occurs at well above the zero-point energy, in H+5 it appears that this type of transition is taking place at a relatively low energy. Analysis of the wave functions, plotted in Figure 3, shows that in the case of H+5 , the transition from a progression in the normal-mode vibrations (in which the nodes occur at specified values of a or s in Figure 3) to an intermolecular H+3 ·H2

vibration (in which the wave function extends into these two product channels) occurs near νpt = 2. It is this change in the underlying dynamics that is the origin of the difference in the interpretation of the quantum number assignments for the bands in the IRMPD spectrum. Looking at the wave functions obtained from the twodimensional calculation, we find that in many of the states, the amplitude near the potential minimum continues to resemble the νpt = 1 wave function. This can be seen in the wave function for the νpt = 3 level as well as in the wave functions that we assign as νpt = 5, 7 and 9, which are provided in the Supporting Information. A consequence of this is that the intensity of the overtones does not die off very rapidly. While the fundamental clearly carries the most oscillator strength, on the basis of the two-dimensional calculation, even the transition to the 12th excited state, plotted in the lower right panel of Figure 3, has an intensity that is more than 1% that of the fundamental. This leads us to anticipate that a long progression in the H2·H+3 should be observed. Lists of the energies and intensities for transitions from the ground state based on the two-dimensional calculations are provided for both H+5 and D+5 in the Supporting Information. Returning to the IRMPD spectrum, this long progression in the H2·H+3 stretch is consistent with the reported spectrum. We have placed blue arrows above the bands that we believe correspond to the transitions to the states with up to seven quanta in this mode in Figure 2. This interpretation implies that one should be able to access states with considerable excitation in this mode through single-photon absorption. Because this mode is, in essence, the reaction coordinate for the proton exchange between H+3 and H2, further exploration of the nature of these states and other spectral signatures of this mode should help deepen our understanding of the mechanisms that lead to the unusual ortho to para ratios of H+3 and H2. Before concluding, there is one other observation that should be noted. If we look at the spacings between the three highestenergy peaks in the single-photon dissociation spectra,7 we find that these spacings are close to, albeit slightly lower in energy than, the calculated energies of the νpt = 1 and 2 levels of the proton chattering mode. Specifically, in H5+, the energy differences between the energy of the lower-energy peak in the triplet and the two higher-energy ones are 384 and 712 cm−1, while in D+5 , these differences are 269 and 498 cm−1. These values are within 40 cm−1 of the ν = 1 and 2 energies reported in Tables 1 and 2. On the basis of analysis of the VCI wave functions, the calculated states that carry oscillator strength in this spectral region are reported to have significant contributions from the proton-transfer mode.7 The above observation is consistent with these assignments. We have not been able to calculate the frequencies of these bands by DMC, and we leave this simplified description of the states that are assigned of the bands between 3500 and 4000 cm−1 in H+5 and between 2500 and 3600 cm−1 in D+5 to these combination bands as a speculation. We are continuing to explore this possibility.

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METHODS SECTION The diffusion Monte Carlo calculations are performed using the algorithm developed by Anderson26,27 and with approaches similar to those used in our earlier studies of CH+5 and H3O−2 ,13,28 with a time step of 10 atomic units, an α-parameter of 0.05 H, and an ensemble of 20 000 walkers. Simulations are run for 50 800 time steps, and the reported energies are 3694

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obtained by averaging the results from five independent simulations. The nodes for the excited states are illustrated with dashed lines in Figure 3. For the ν = 1 state and the central node in the ν = 3 state, the node is defined at R1 = R2. The remaining two nodes for ν = 3 and the two nodes for ν = 2 are defined to occur when R1 or R2 is equal to Rnode. The value of Rnode was determined by an adiabatic variant of DMC.29 All of the DMC calculations were performed using the potential developed by Xie et al.10 As in earlier studies,7 to avoid exchange between the central and outer hydrogen atoms, if we find that atom three is no longer in the central position, the hydrogen atoms are reordered in order to restore atom three to the central position. As noted above, the definition of the nodes used in the present work differs from the definition used in earlier studies.7 Specifically, they defined the node “(in Cartesian coordinates) at the midpoint of the vector defined by the centers of mass of the two H2 units, shown in Figure 1 (of that paper).” In other words, they define a vector that connects the centers of mass of the two H2 units, R⃗ and define a second vector that is defined by the center of R⃗ and the position of the fifth hydrogen atom, r.⃗ The node is defined at the point where the component of r ⃗ that is parallel to R⃗ vanishes. To lowest order, the two definitions of this node are the same, but as the system samples larger regions of the potential, the two definitions will give slightly different results.13 The two-dimensional calculations were performed using potential and dipole surfaces that were evaluated at the MP2/ aug-cc-pVTZ level of theory/basis.30 The cuts were evaluated as functions of the distance of the central hydrogen atom from the centers of mass of the two H2 groups, R1 and R2. The ion was constrained to retain C2v symmetry, with a dihedral angle between the two outer H2 groups of 90°. As a result, when R1 = R2, the molecule has D2d symmetry. Using this surface, the vibrational wave functions were evaluated using a twodimensional DVR in the symmetric and antisymmetric linear combinations of R1 and R2 [s = 2−1/2(R1 + R2), a = 2−1/2(R1 − R2)], with 60 grid points in each dimension equally spaced between 0.4 and 3 Å in s and between −1.75 and 1.75 Å in a.

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REFERENCES

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ASSOCIATED CONTENT

S Supporting Information *

Energies, intensities, and all of the wave functions obtained from the two-dimensional calculations. This material is available free of charge via the Internet at http://pubs.acs.org.

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Letter

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Support through grants from the Chemistry Division of the National Science Foundation (CHE-0848242 and CHE1213347) is gratefully acknowledged. We thank Professor Joel M. Bowman for providing us with the codes used to generate the potential and dipole surfaces for H+5 , and we thank Professor Michael A. Duncan for providing us with the spectra from ref 8, shown in Figure 2. This work was supported in part by an allocation of computing time from the Ohio Supercomputer Center. 3695

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