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IR and FTMW-IR Spectroscopy and Vibrational Relaxation Pathways in the CH Stretch Region of CH3OH and CH3OD Sylvestre Twagirayezu, Xiaoliang Wang,† and David S. Perry* Department of Chemistry, The University of Akron, Akron, Ohio 44325, United States
Justin L. Neill, Matt T. Muckle, and Brooks H. Pate Department of Chemistry, University of Virginia, McCormick Road, Charlottesville, Virginia 22904, United States
Li-Hong Xu Department of Physics, Centre for Laser, Atomic and Molecular Studies (CLAMS), University of New Brunswick, Saint John, New Brunswick E2L 4L5, Canada
bS Supporting Information ABSTRACT: Infrared spectra of jet-cooled CH3OD and CH3OH in the CH stretch region are observed by coherence-converted population transfer Fourier transform microwaveinfrared (CCPT-FTMW-IR) spectroscopy (E torsional species only) and by slit-jet single resonance spectroscopy (both A and E torsional species, CH3OH only). Twagirayezu et al. reported the analysis of ν3 symmetric CH stretch region (27502900 cm1; Twagirayezu et al. J. Phys. Chem. A 2010, 114, 6818), and the present work addresses the more complicated higher frequency region (29003020 cm1) containing the two asymmetric CH stretches (ν2 and ν9). The additional complications include a higher density of coupled states, more extensive mixing, and evidence for Coriolis as well as anharmonic coupling. The overall observed spectra contain 17 interacting vibrational bands for CH3OD and 28 for CH3OH. The sign and magnitude of the torsional tunneling splittings are deduced for three CH stretch fundamentals (ν3, ν2, ν9) of both molecules and are compared to a model calculation and to ab initio theory. The number and distribution of observed vibrational bands indicate that the CH stretch bright states couple first to doorway states that are binary combinations of bending modes. In the parts of the spectrum where doorway states are present, the observed density of coupled states is comparable to the total density of vibrational states in the molecule, but where there are no doorway states, only the CH stretch fundamentals are observed. Above 2900 cm1, the available doorway states are CH bending states, but below, the doorway states also involve OH bending. A time-dependent interpretation of the present FTMW-IR spectra indicates a fast (∼200 fs) initial decay of the bright state followed by a second, slower redistribution (about 13 ps). The qualitative agreement of the present data with the time-dependent experiments of Iwaki and Dlott provides further support for the similarity of the fastest vibrational relaxation processes in the liquid and gas phases.
I. INTRODUCTION The three CH stretch fundamentals of CH3OH (ν2, ν3, ν9) have been previously recorded at high resolution and analyzed.13 In this prototype internal rotor molecule,48 two of these bands (ν2 and ν9) have upper state torsional tunneling splittings that are inverted relative to the ground state.2,3 Several authors have published theoretical work dealing with the inverted torsional structure of certain vibrationally excited states.3,915 The single resonance absorption spectra of methanol, whether static gas1 or jet spectra,2,3 contain a great many lines that could not be assigned to one of the three CH stretch fundamentals. This is the second of two papers16 presenting the spectroscopy of the additional vibrational bands in the CH stretch region. These additional vibrational bands, which borrow intensity from the CH stretch fundamentals, carry information about the couplings between vibrational modes and consequently about the time-dependence of intramolecular vibrational energy redistribution (IVR). r 2011 American Chemical Society
To simplify this very congested and complicated spectral region, we use coherence-converted population transfer Fourier transform microwaveinfrared spectroscopy16 to obtain rotationally selected infrared spectra of jet-cooled CH3OH and CH3OD. The technique is abbreviated CCPT-FTMW-IR or just FTMW-IR. In this paper, we also present additional CH3OH data from the single-resonance slit-jet spectra.2 Reference 16 documented 12 interacting vibrational bands in the ν3 symmetric CH stretch region (27502900 cm1) of CH3OH. In that region, the binary combinations of the OH bend (ν6) and a CH bend (ν4, ν5, ν10) act as doorway states linking the Special Issue: David W. Pratt Festschrift Received: March 2, 2011 Revised: May 20, 2011 Published: June 13, 2011 9748
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The Journal of Physical Chemistry A ν3 bright state to higher-order combination vibrations involving torsional excitation. On the other hand, in CH3OD, only a single band assigned to the ν3 symmetric CH stretch was observed in this region because deuteration reduces the OD bend frequency shifting the doorway states out of resonance.16 In the absence of the doorway states, the coupling of the higher order combination states to the bright state was too weak for them to be observed. A time-dependent interpretation of the frequency-resolved CH3OH spectra below 2900 cm1 revealed a fast (∼200 fs) initial decay of the bright state followed by a slower (12 ps) redistribution among the lower frequency modes.16 This interpretation of the gas phase data agrees qualitatively with the prior liquid-phase time-resolved data reported by Iwaki and Dlott,17 suggesting that the gas and liquid phases share common coupling mechanisms for the fastest vibrational energy transfer processes. In this paper, we test the generality of the doorway state mechanism and further test the commonality of the gas and liquid phase vibrational relaxation mechanisms by extending the analysis to the more complicated higher frequency region of CH3OH and CH3OD that includes the two asymmetric CH stretches (ν2 and ν9). The additional complexity in this spectral region, 29003020 cm1, arises from the denser manifold of coupled states including the six binary combinations of the CH bends, the involvement of rotationally mediated coupling mechanisms, and the more relaxed spectroscopic selection rules. The spectral data and assignments are presented in sections III.A.C. followed by an analysis of the upper state torsional structures (section III.D.), the vibrational coupling mechanisms (section III.E.), and the vibrational relaxation time scales (section III.F.).
II. EXPERIMENTAL SECTION Coherence-converted population transfer FTMW-IR spectroscopy is described in ref 16. All experimental conditions are unchanged, except that the infrared laser source (Continuum Mirage OPO/OPA) is scanned through the higher wavenumber region, 29003020 cm1. The sample gas (0.1% methanol in an inert gas mixture of ∼80% neon and 20% helium) is introduced as a jet into an FTMW cavity18,19 through a General Valve Series 9 0.9 mm pinhole nozzle at a pressure of ∼50 kPa. Two microwave (MW) pulses, tuned to the same methanol rotational transition, are applied. Each pulse has a duration of 500 ns, and they are separated in time by 50 ns. The first MW pulse produces a rotational coherence that would result in a free induction decay (FID) signal, and its amplitude is adjusted to create an optimal “π/2” excitation. The second MW pulse, 180 out of phase with the first, is carefully tuned in amplitude and phase to be a “π/2” pulse, resulting in no pure rotational FID. The infrared pulse (∼10 ns duration) is applied in the time interval between the two MW pulses. When the IR laser wavenumber is resonant with an infrared transition from one of the two rotational levels involved in the rotational transition, it induces a population difference. The second MW pulse converts that population difference to a detectable rotational coherence, and a laser-induced FID is detected background-free. To record rotationally selected infrared spectra, the magnitude and phase of the FID are recorded as the infrared wavenumber is scanned. The phases of the FIDs induced when the laser is resonant with the upper and lower rotational states are opposite. A rotationally state-selected spectrum is acquired by plotting the imaginary part of the Fourier transform of the FID at the frequency of the methanol MW transition as a function of IR laser wavenumber. For example, in
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Figure 1a, the upward pointing infrared transitions originate from the ground vibrational state rotational level JK = 30, and the downward pointing lines from 2þ1. The measurement of relative IR intensities requires consideration of the variable line shape of the OPO/OPA system. The observed IR lineshapes in our spectra appear to be a single mode, ≈0.04 cm1 fwhm, accompanied by a second mode of lower intensity displaced by about 0.06 cm1 to higher or lower wavenumber. The relative intensities of the two modes varied across the spectrum, sometimes being almost equal, and occasionally more than two modes are observed. The reported intensities are integrated intensities over an interval (0.05 cm1 relative to the line center, where the line center is defined to be the wavenumber at which the integrated area is a local maximum. The microwave transitions were chosen to be low-J, low-K transitions in the range (826 GHz) accessible to our microwave equipment. We use the same E-species transitions as in ref 16: 20 r 31 at 12.179 GHz and 30 r 2þ1 at 19.967 GHz for CH3OH and 11 r 10 at 18.957, 21 r 20 at 18.991, and 31 r 30 at 19.005 GHz for CH3OD. The lowest-frequency rotational transition of the A-species populated in a supersonic jet is the R(0) line (∼48 GHz), which could not be reached in this work. We employ the symmetric rotor notation JK with a signed value of K rather than the asymmetric rotor notation JKa,Kc because the torsional and K-rotational angular momenta are strongly coupled and the E-species rotational wave functions are polarized according to the relative directions of these two angular momenta. The signal-to-noise ratio relative to the largest peak in Figure 1 is about 200:1. In addition, single-resonance slit-jet infrared absorption spectra were recorded as previously described.2 A total of 1218 lines were observed in the region 29452985 cm1. The continuouswave infrared laser source was a Burleigh FCL-10 laser. The pulsed slit expansion (6% CH3OH in 65 kPa Ar) was surrounded by a multipass cell to increase the infrared absorption path length. The residual Doppler width in the slit-jet was 0.0025 cm1. Relative wavenumber calibration was achieved with a temperature-controlled, sealed 150 MHz marker etalon with a precision of ∼0.0002 cm1 as judged by the standard deviation of combination loops involving a set of more intense infrared lines. Absolute wavenumber calibration was established by a fit of the marker etalon fringe positions to ethylene gas cell absorption lines20 (standard deviation 0.0005 cm1).2 Lines that were observed in both the high-resolution slit-jet spectra and also in the FTMW-IR spectra are indicated in Figure 1. Because of the higher resolution and the higher density of observed lines, there are on average 34 lines in the slit-jet spectrum within the (0.05 cm1 measurement uncertainty of each line in the FTMW-IR spectra. Therefore, extensive work on the slit-jet assignments was necessary to identify the corresponding features in the two kinds of spectra. All lines expected to appear in the FTMW-IR spectra based on the slit-jet assignments were in fact observed and the transition wavenumbers agreed within a standard error of 0.05 cm1.
III. RESULTS AND DISCUSSION A. Overview of the Spectra. Together the two experimental methods have resulted in rotationally assignable spectra of both CH3OH and CH3OD from 27503020 cm1. The lower part of this range below 2900 cm1 was reported in ref 16 and the detailed assignments of ν2 CH stretch band of CH3OH at 9749
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Figure 1. Overview of representative coherence-converted population transfer FTMW-IR spectra. (a) Infrared spectrum of CH3OH in the CH-stretch region resulting from the microwave transition 30 r 2þ1 at 19.967 GHz. Markers indicate lines assigned from single-resonance spectra: slit-jet spectra for ν9, 2ν4 (present work), and ν2 (ref 2), and gas phase FTIR for ν3 (ref 49). (b) Infrared spectrum of CH3OD resulting from 11 r 10 at 18.957 GHz.
2999 cm1 have also been published.2 In the whole spectral region, there are a large number of interacting vibrational bands. As we show below, 28 bands are identified for CH3OH and 17 for CH3OD. With some residual uncertainties, the upper state rotational quantum numbers J0 , K0 , and the A/E torsional symmetries are assignable. The vibrational identity of each upper
state is harder to assign because each upper state is a mixed state that results from the interaction of the three CH stretch fundamentals (ν2, ν3, and ν9) and the various bath states made up of combinations of the lower frequency normal modes. The frequencies of the infrared fundamentals for both CH3OH and CH3OD are listed in Table 1 of ref 16. To facilitate 9750
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Figure 2. An example of the detailed assignments, given here for a section of the CH3OD spectra in the ν9 asymmetric CH-stretch region. The spectra result from the microwave transitions: 11 r 10 at 18.957 GHz, 21 r 20 at 18.991 GHz, and 31 r 30 at 19.005 GHz. The ground state rotational levels for the upward and downward pointing lines are indicated at the right side of the figure in the form, J00 K0 0 . The upper state rotational assignments, indicated above and below the spectra, apply to both upward and downward pointing lines as appropriate. Solid lines mark the positions of ΔK = 0 transitions and dashed lines indicate the positions of ΔK = (1 transitions.
the discussion, we use the CH3OH normal mode numbering for the equivalent modes in CH3OD as well. Because transitions from the ground state to each of the three CH stretch modes is allowed by the usual harmonic oscillator selection rules, transitions to these modes will be responsible for most of the oscillator strength in the observed spectra. There are three strong bands with about the expected frequencies that we identify nominally as the CH stretch fundamentals, but, in fact, each contains some admixture of the various bath states. For these three “fundamentals”, we are able to identify the different K0 upper state rotational levels that belong to each. However, for most of the other upper state vibrations, which borrow intensity from the CH stretches, we are not able to make specific vibrational assignments. In fact, for most of them, we are not even able to track a particular upper state vibration from one K0 to the next. This is not just a technical difficulty with the assignments, but a more fundamental issue that arises because the spacing between the interacting vibrational states is comparable to the change in the torsional energies from one K0 to the next. It means that the admixture of zeroth-order vibrations that comprise a particular observed vibrational upper state changes substantially from one K0 to the next. Accordingly, in most cases, we handle the upper state vibrations for each K0 separately. Generally, the J0 - dependent structure (that is P, Q, R spacings) is more regular and can be handled in the usual way.
B. FTMW-IR Spectra. Representative FTMW-IR spectra for both CH3OH and CH3OD are shown in Figure 1. Whereas ΔK = 0 (parallel) transitions dominate the ν3 region,16 ΔK = (1 (perpendicular) transitions also appear with comparable intensity above 2900 cm1. The selection rules ΔJ = 0, (1 apply, except ΔJ 6¼ 0 for K0 = K00 = 0. Therefore, even with the lower state quantum rotational numbers (J00 , K00 ) and the symmetry completely specified, some effort is required to assign the upper state quantum numbers (J0 , K0 ) of each infrared transition in Figure 1. As was the case in the ν3 region,16 the high density of interacting bands in the region 2900 2980 cm1 makes the pattern of the upper state K0 levels for each vibrational band a priori unpredictable. Even the P, Q, R spacings are not completely regular. The wavenumber measurements with the pulsed infrared laser are relatively imprecise ((0.05 cm1), so the ground state combination differences do not, by themselves, provide definitive assignments. Because of the available microwave pump transitions and because of the more regular spectra, the CH3OD assignments were easier to accomplish and are likely more secure than those for CH3OH. Accordingly, we present the CH3OD assignments first before moving on to CH3OH. The available CH3OD MW transitions occur in the convenient sequence, 11 r 10, 21 r 20, and 31 r 30 (Figure 2). These lower states allow the observation of the pattern of the infrared P, Q, R spacings over the range J00 = 13. Because 9751
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Table 1. Summary of FTMW-IR Assignments: Numbers of Assigned Lines and Upper Vibrational States for CH3OH and CH3OD in the Range (29003020 cm1) CH3OH
CH3OD
assigned
vibrational
assigned
vibrational
K0
lines a
states b
lines a
states b
2
14(14)
25
5(0,5,0)
1
75(5)
14(7,5,2)
137(5)
15(12,1,2)
0
98(7)
16(8,4,4)
118(5)
16(11,4,1)
þ1 þ2
54(5) 6(6)
11(5,2,4)
56(2)
11(0,10,1)
a
The number in parentheses represents how many of the assigned lines are not included in the fits to eq 1 because of possible J0 -localized perturbations or uncertain assignments. b The three numbers in parentheses identify the numbers of upper vibrational states for which the assignments are characterized as “solid”, “good”, and “less certain”, respectively.
parallel and perpendicular infrared transitions are both observable, the available MW transitions provide a large number of combination loops. Proceeding in this way, we are able to identify sub-bands reaching the upper states K0 = 2, 1, 0, and þ1. An example of these infrared assignments is shown in Figure 2. For K0 = 0, a total of 16 vibrational bands were found, and a summary of the assignments is given in Table 1. The complete CH3OD assignments are listed in Table S1 of the Supporting Information. For CH3OD, 11% of the observed lines between 2900 and 3020 cm1 remain unassigned, representing 3% of the total intensity. For CH3OH, only two MW pump transitions were used, 20 r 31 and 30 r 2þ1, for a total of four different (J00 , K00 ) lower states from which infrared transitions could originate. In addition, there was more irregularity in the J0 -dependence of the upper state term values. As in ref 16, the upper state term va0 0 lues νK J i were fit to the expression 0 0
0
0
νK J i ¼ νK0 i þ BKeffi J 0 ðJ 0 þ 1Þ
ð1Þ
Multiple upper vibrational states for each K0 are observed, and the index i is introduced to label them. Only in a few cases (i = ν2, ν3, ν9, and 2ν4) are we able to track the identity of individual vibrations from one K0 to the next; in other cases, the vibrations of each K0 are indexed separately. Because most of the data in this paper is for the E species, all term values in this paper are given relative to the ground state J00 = 0 E level, which is 9.122 cm1 above the J00 = 0 A level and about 137.229 cm1 above the bottom of the torsional potential.6 Four criteria were used in evaluating the tentative assignment of a set of infrared lines to a given upper state (K0 ,i): (i) observation of at least 0one P-R, 0P-Q, or P-R spacing with the correct multiple of 2BKeffi with BKeffi within about 5% of the ground state 00 ; (ii) observation of infrared transitions rotational constant, Beff from at least three of the available (J00 , K00 ) lower states; (iii) observation of at least one combination loop, that is, two or more infrared transitions reaching the same (J0 ,K0 ,i) upper state; and 0 (iv) a good fit to eq 1 such that the standard error of νK0 i is 1 e0.05 cm . The assignments are considered to be “solid” if all 4 criteria are met; “good” if 3 are met and “less certain” if 2 are met. The distribution of assignments over these three quality levels is summarized in Table 1. The detailed assignments are
given in Table S2 and Figure S1 of the Supporting Information. In some cases, additional lines were identified as possibly belonging to a (K0 , i) upper state vibration, but which were outliers not included in the fits to eq 1. The presence of such outliers could result from J-localized perturbations. For CH3OH, 15% of the lines remained unassigned, representing 6% of the total intensity. Many of the unassigned lines likely represent the K0 = 2 and þ2 upper states, for which the assignments are incomplete and uncertain. For completeness, the quality levels of the CH3OD assignments are also included in Table 1. The situation for CH3OD is much better than for CH3OH with fewer unassigned lines and fewer outliers. Although there are some “less certain” assignments for CH3OD according to the four criteria, the fact that the overall assignments for the CH3OD spectra are rather more complete provides an additional level of confidence in these assignments. For spectra with a large number of interacting vibrational states and a limited data set, residual uncertainties in the assignments are a fact that must be acknowledged, and there are inevitably a certain number of errors in the detailed assignments. However, the assignments presented here represent our best effort at characterizing the data and should provide a reliable basis for developing a global interpretation of the spectra. C. Slit-Jet IR Absorption Spectra. Examples of our methanol slit-jet spectra have been published.2 The observed transitions in the 29452985 cm1 range are depicted in Figure S2 of the Supporting Information. Because there is no state-selection in this spectrum, assignments, as previously,2 rely on the method of ground state combination differences, which is based on the precise knowledge of the ground state term values.6,21 When a set of transitions originating from different J00 and different K00 all match the known differences in the ground state energies within the 0.0002 cm1 precision of the measurement, one is confident of the J00 , K00 assignments. The upper state J0 , K0 are then assigned based on the usual ΔJ = 0, (1, ΔK = 0, (1 selection rules. Additional information about K0 comes from the requirement K0 e J0 . However, some ambiguity about K0 is possible because K0 is not a rigorous quantum number and states coupled by b- or c-type Coriolis coupling have mixed K0 character. A total of 709 lines in slit-jet spectra were assigned to two strong vibrational bands centered near 2957 cm1. The assigned lines are indicated in Figure S2 and tabulated in Table S3 of the Supporting Information. The ν9 asymmetric CH stretch fundamental (A00 symmetry in CS or A2 in G6) is expected in this region. The other band is likely one of the CH bend binary combinations, probably 2ν4 (A0 or A1) because ν4 has the highest fundamental frequency of the three CH bends (ν4, ν5, and ν10). One of the assigned vibrational bands, which we tentatively label as ν9, displays the rotational selection rules that we expect for an A00 band. That is, for the A species lines, the K00 = 0 levels connect to the lower member of the K0 = 1 asymmetry doublet, and the expected pattern also follows through the other ν9 subbands. Similarly, the other of the two assigned bands, which we call 2ν4, follows the pattern expected for an A0 band, which is connection of the K00 = 0 levels to the upper member of the K0 = 1 asymmetry doublet. For K0 = 2, 3, and higher, the asymmetry splittings are rather small, and in the ν4 fundamental band, the observed reversed ordering of the asymmetry doublets, has been attributed to accidental perturbations.22 It is less likely that the larger K0 = 1 asymmetry splittings would be reversed by an accidental perturbation, but this finding22 does mean that caution is appropriate 9752
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when making assignments based on the ordering of asymmetry doublets. For the E-species, both A0 and A00 bands have the same selection rules. This means that we cannot distinguish the A0 and A00 vibrational symmetries in the present FTMW-IR spectra, which probe only the E species. As expected for an A0 band, both parallel (ΔK = 0) and perpendicular (ΔK = (1) components are observed for 2ν4. However, an A00 band should be purely perpendicular, but for ν9, a parallel component (12% of the total intensity) is observed along with the expected perpendicular transitions. This anomalous parallel component will be discussed further below. D. Torsion-Rotation Level Structure. In addition to producing the A/E tunneling splittings at J = 0, the torsional motion interacts strongly with the rotational angular momenta to produce patterns of rotational energy levels different from those of a rigid molecule. Because in methanol the internal rotation axis is nearly aligned with the a rotational axis, the torsional motion interacts much more strongly with rotation about that axis than with rotation about the b and c axes. Accordingly, the J-dependent rotational structure, that is, the P, Q, R spacings, is qualitatively similar to that of rigid molecules, but the K-dependent structure, that is, the pattern of the sub-band origins, is qualitatively different. In this section, the spectroscopic parameters derivable from the present data are characterized, and the patterns of the sub-band origins for the upper vibrational states, ν2, ν3, ν9, and 2ν4, are compared to the well-established pattern5 for the ground state of methanol. For each of these bands, the torsional tunneling is characterized as normal or inverted and compared to theoretical calculations. 0 The sub-band origins, νK0 , obtained from the fits of the to eq 1, are converted to reduced observed (J0 , K0 ) upper states 0 subband term values νKR by subtraction of the remaining rigidsymmetric-rotor energy, 0
0
νKR i ¼ νK0 i ½A 00 ðB00 þ C00 Þ=2K 02
ð2Þ
0 For the FTMW-IR data, the derived reduced term values, νKR , for
K00 = 1, 0, and þ1 and the corresponding relative intensities, IK i, are plotted in Figure 3a for CH3OH and 0 in Figure 3b for rotational constants, BKeffi, associated with CH3OD. The effective K0 i each of the νR are plotted in Figure 4a. The relative parallel 0 versus perpendicular intensity in each of the observed νKR i subbands is characterized by the quantity 0
∑
0
)
βK i ¼ ð I K i K0 i
∑I^K i Þ=IK i 0
0
ð3Þ
0 0 = ∑ IK|| i þ ∑ IK^ i and the sums are over the observed
where the I parallel or perpendicular0 lines reaching the (K0 , i) upper state. As shown in Figure 4b, βK i varies from þ1 for a purely parallel subband to0 10 for a purely perpendicular subband. The full set of the 0 0 νKR i,IK i, BKeffi, and β K i for CH3OD and CH3OH, as obtained from the FTMW-IR spectra, are listed in Tables S4 and S5, respectively. Some data for K0 = 2 of CH3OD are also included. The corresponding fitted parameters obtained from the CH3OH slitjet spectra are listed in Table S6. Only for the upper vibrational states, ν2, ν3, ν9 (both isotopomers), and 2ν4 (CH3OH only), were we able to track the nominal vibrational assignment through all of the observed values of K0 . The patterns of the torsion-rotation structure are evident in the plots of the reduced energies versus K0 and are compared to the characteristic ground state patterns in Figures 5 and 6. At K = 0 for each vibrational state, there are two points (A and E symmetries) split by the torsional tunneling splitting. For each
Figure 3. E-symmetry sub-band term values for K0 = þ1, 0, and 1 relative to the J00 = 0, K00 = 0 E level of the vibrational ground state.
K > 0, there are three points: A, E1, and E2. The E1 and E2 points are both E symmetry, but the subscripts represent the relative signs of the torsional and rotational angular momenta. The reduced energies for each species (A, E1, and E2), when plotted against K, fall on oscillatory curves. As shown in the bottom panel of Figure 5, the oscillations vary more slowly with K and are visualized more easily, when curves are plotted for each value of 9753
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Figure 4. (a) Effective rotational constants and (b) the amount of parallel versus perpendicular intensity for the E-species sub-bands 0 observed in the FTMW-IR spectra. Here, β K I = þ1 or 1 for purely parallel and perpendicular subbands, respectively.
the parameter τ = 13. The parameter τ is defined4 by the equation τ = K þ 3N þ Γ, where Γ = 0, 1, 2 for E1, A, and E2 respectively, and N is an integer chosen to ensure that τ = 1, 2, or 3. For convenient reference, the patterns of the τ curves for the vibrational ground states are represented in Figures 5 and 6. The τ curves presented in this paper are obtained by fits of reduced energies to cosine curves of the form ντ,R i ¼ Ci1 þ Ci2 K Ci3 cosðCi4 K þ ðτ 1Þ 2π=3Þ
ð4Þ
where the Cij are the fitted constants for the band i. The τ curves are included in Figures 5 and 6 only as a guide to the eye to highlight the patterns of the reduced energies. Figure 5 shows that for CH3OH the reduced term values for three CH stretch fundamentals and also for 2ν4 follow patterns that are qualitatively similar to the ground state. However, there are significant differences. The patterns for ν2 and ν9 are inverted
Figure 5. Reduced term values (νKR ) of CH3OH for the CH stretch vibrational upper states and for the ground state. The A-symmetry points are indicated with circles, and the E-symmetry points with squares for K g 0 (E1) and triangles for K < 0 (E2). The plotted term values are relative to the J00 = 0 E level of the ground state. The ground state data are from refs 6 and 21, the ν2 data are from ref 2, and the ν3 data are the combined set from ref 49 and from the present FTMW-IR data. The τ curves are fits of the experimental points to the eq 4. The point in brackets [0] was not included in the τ-curve fit.
relative to those for 2ν4, ν3, and the ground state. The “normal” ordering at K = 0 is A below E, but for the two bands, ν2 and ν9, we find E below A. The patterns for K0 > 0 are altered accordingly. In terms of eq 4, Ci3 is positive for 2ν4 (þ4.85 cm1) and ν3 (þ6.23 cm1) but negative for ν2 (2.52 cm1) and ν9 (3.49 cm1). This is the inverted torsional tunneling behavior that has been the subject of significant theoretical effort.3,915 An internal-coordinate Hamiltonian was able to account for the three vibrational frequencies and the three torsional tunneling splittings in CH3OH (Figure 7).3 The model contained three adjustable parameters, the local CH stretch frequency ω = 2934.0 cm1, the 9754
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Figure 7. Experimental reduced term values for K0 = 0 for CH3OH and CH3OD are compared to calculated values. The ab initio calculated values are from Table 7 of ref 14. The internal coordinate model calculation from ref 3 for CH3OH is extended to CH3OD in the present work. The plotted term values are relative to the J00 = 0 E level of the ground state.
Figure 6. Reduced term values (νKR ) of CH3OD for the CH stretch vibrational excited states and for the ground state. The A-symmetry points are indicated with circles, and the E-symmetry points with squares for K g 0 (E1) and triangles for K < 0 (E2). The plotted term values are relative to the J00 = 0, E level of the ground state (ref 33). For the upper state, hollow markers indicate the values for the perturbed K0 = 1, 1, and 2 sub-bands. Filled markers are used for the K0 = 0 sub-bands and the deperturbed K0 = 1, 1, and 2 sub-bands. To facilitate the comparison of the patterns in the ground and vibrationally excited states, dashed lines are shown to connect corresponding E symmetry points. For ground state, the τ curves are solid lines fitted to the experimental points using eq 4.
locallocal coupling λ = 42.2 cm1, and the lowest-order torsion-vibration coupling μ = 19.35 cm1,23 plus, the ground state torsional potential. The torsional inversion phenomenon is a general one2426 that can be understood with use of the
concepts of symmetry12 and geometric phase.13 The vibrational eigenstates computed by Sibert and Castillo-Chara14 on a full dimensional ab initio potential energy surface are also in good agreement with the experimental data except that the tunneling splitting for ν3 is too small (Figure 7). The vibrational levels calculated with MULTIMODE on a similar full-dimensional ab initio potential15 are also in qualitative agreement with the experiment, but here the ν3 splitting is more accurate and the ν2 and ν9 splittings are too small. The body of theoretical work3,915 on CH3OH produces consistently the correct signs for the torsional tunneling splittings, with ν3 normal and ν2 and ν9 inverted. Whereas the ground state term values of CH3OH (Figure 5) are well fit by simple cosine formulas (eq 4), there is noticeable scatter relative to the curves for the vibrational excited states that is likely caused by perturbations. In addition, the pattern of the τ curves is sloped upward for ν2 and ν9 but downward for 2ν4. In terms of eq 4, Cν22 = þ0.33 cm1 and Cν29 = þ2.09 cm1, but C22ν4 = 1.72 cm1. One possible cause is our use of the ground state rotational constants in eq 2, whereas the upper state rotational constants will be slightly different, but such a discrepancy would cause a quadratic K0 -dependence of the τ curves. The weighted-average term origin for ν9 is C1i = 2954.30 cm1 at 9755
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The Journal of Physical Chemistry A K0 = 0, which is close to that for 2ν4 (2954.07 cm1). Thus, the bands are almost degenerate at K0 = 0 and then diverge linearly with increasing K0 . This is the signature one would expect for an a-type Coriolis interaction between the two bands. Indeed, an a-type Coriolis interaction is symmetryallowed between an A1A2 pair of vibrational states. In the absence of a global fit to an interacting band Hamiltonian, we can only say that the patterns of the ν9 and 2ν4 level structures have the characteristics one might expect for an a-type Coriolis interaction. The slight upward slope of the pattern for ν2 could be the result of an off-resonant a-type Coriolis interaction with ν9. The torsion-rotation structures of the CH stretch fundamentals of CH3OD are compared to the ground state in Figure 6. Because of the available rotational transitions accessible to the FTMW-IR experiment, we have upper state term values for only the E-species. Several of the upper states are split by near-resonant perturbations. To obtain a “best estimate” of the torsion-rotation structure, these near-resonant multiplets were deperturbed using the standard two-level formulas or, where appropriate, the Lehmann implementation27 of Lawrance and Knight deconvolution.28 For ν9, the K0 > 0 levels are split into pairs of levels observed with nearly equal intensity. The splitting of the pairs increases with |K0 |, which is a possible indicator of an a-type Coriolis interaction similar to that seen in CH3OH. The splitting of the K0 = 2 levels is about half of that seen in CH3OH. A significant difference from CH3OH is the lack of a nearresonant splitting of K0 = 0 in CH3OD. For K0 = þ1 of ν2, there is a quartet of levels, one of which was much more intense than the others (Figure 3); therefore, the deconvolved term value is rather close to the largest feature. Because the A-species has not been observed in CH3OD, the K0 = 0 torsional tunneling splittings are not directly observable. However, the relationship between the K0 = (1 (E1, E2) term values provides this information indirectly. For the ν3 and ν9 vibrations, the energy ordering is E1 < E2, which is the same as the ground state, but for ν2, it is inverted, E1 > E2 (Figure 6). In both CH3OH and CH3OD, the pattern for ν3 is normal, and for ν2, it is inverted. However, the two isotopomers differ in the case of ν9, where the pattern is inverted for CH3OH, but normal for CH3OD. By comparing the E1, E2 term values to those for the ground state, as was done in ref 16, estimates of the A-species J0 = 0 term values for CH3OD can be obtained, and they are shown in parentheses (A) in Figure 7. Figure 7 compares the experimental term values and torsional tunneling splittings for both CH3OD and CH3OH with ab initio14 and model calculations. The same internal coordinate model3 used for CH3OH was also applied to CH3OD, and the model parameters (ω = 2933.13 cm1, λ = 43.18 cm1, μ = 20.38 cm1) obtained by fitting the E-species data are similar. The signs of the torsional tunneling splittings of the ab initio and model calculations agree in all cases. They also agree with the experiment in all cases except for ν9 of CH3OD where the tunneling splitting is predicted to be inverted by both kinds of theory but is found to be normal by the experiment. We could find no reasonable combination of model parameters that would give a normal tunneling splitting for ν9. There are three possible reasons for the discrepancy for ν9 of CH3OD between the model and the experimental data. (i) The model Hamiltonian, as implemented, includes only the lowest-
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order interaction term between the torsion and the CH stretches. Appropriately symmetrized, that interaction term is 1 1 2π cos γ þ v2 þ cos γ μ v1 þ 2 2 3 1 2π cos γ þ ð5Þ þ v3 þ 2 3 where γ is the torsional angle, vi are the local mode CH stretch vibrational quantum numbers, and subscripts label the three CH bonds. Higher-order terms have been derived3 but were not implemented because the experimental data were insufficient to constrain their values. (ii) The model includes only the torsion and the CH stretches, neglecting the other eight vibrational degrees of freedom. As is evident from the spectra of both isotopomers, ν9 interacts strongly with nearly resonant binary combinations of the CH bends. Because the tunneling splittings are about three times smaller in CH3OD than in CH3OH, perturbation of one or both of the A and E species could more easily affect the ordering of the levels. The use of the pattern of the K0 = (1 E term values to predict the K0 = 0 A term value may be inaccurate in the presence of such perturbations. (iii) Finally, the deperturbation approach that we have used to estimate the K0 = (1 E-species term values could be erroneous because of interfering oscillator strength from other zeroth order states or because of missed levels that ought to have been included in the deperturbation. The criticisms (i) and (ii) do not apply in the same way to the ab initio theory,14 which employs a fulldimensional potential energy surface. Although the ab initio theory includes anharmonic interactions up to fourth order with all of the near-resonant combination vibrations, small errors in the relative frequencies can have a large impact on the spectral shifts that result from those interactions and could possibly affect the A/E ordering. Given that both the ab initio and model calculations predict inverted tunneling for ν9, and given the difficulties of inferring the K0 = 0 A term value from the K0 = (1 E term values, direct measurements of the A-species would be desirable when the experimental capabilities allow it. For both isotopomers, the upper state effective rotational constants (Figure 4a) cluster around the ground state values. The deviations of the effective rotational constants from the ground state values found for the various upper state vibrations are too large to reflect changes in the vibrationally averaged geometry of the excited state vibrations. Instead these deviations reflect J0 -dependent perturbations of each vibrational upper state. Such J0 -dependence can arise from b- and c-type Coriolis interactions but also from the J0 tuning of anharmonic and a-type Coriolis interactions closer to further from resonance as J0 increases. The detailed analysis29 of perturbations in the CH3OH ν1 band for J0 up to 28 provides examples of how analysis of only the low J0 levels available in a jet can give rise to a range of effective rotational constants for individual subbands. The relative amount of parallel versus perpendicular intensity observed for each of the upper state vibrations (Figure 4b) provides an indication of which of the CH stretch fundamentals is contributing to its oscillator strength. The band nominally CH stretch ν3 is domassigned in CH3OH as the symmetric 0 inantly parallel in character (β K ν3 ≈ þ0.8) because the a-principal axis is nearly aligned with the symmetry axis of the methyl group. On the other hand, for the asymmetric CH stretches, 0 the perpendicular intensity is greater (β K i < 0). The observed bands in CH3OH below 2900 cm1 are all predominantly 9756
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The Journal of Physical Chemistry A parallel bands (Figure 4b), consistent with intensity borrowing mostly (or entirely) from ν3. Above 2900 cm1, the band types for both isotopomers are distributed widely over the whole 0 range, 1 e βK i e þ1, indicating a substantial contribution from the asymmetric CH stretches ν2 and ν9 in this region. The fact0 that some of the upper states above 2900 cm1 have β K i ≈ þ1 indicates the likely contribution of ν3 to the intensity in this region as well. In fact, ab initio theory30,31 predicts ν3 to be strongly anharmonically coupled to 2ν5, which is expected at about 2916 cm1. Therefore, the vibrational bands observed in the region 29002960 cm1 may each derive their intensity in some proportion from each of the three CH stretch fundamentals. In addition, we cannot rule out the possibility that some of the bands, other than the CH fundamentals, might have some intrinsic oscillator strength. As noted above in section III.C., the ν9 band is not a strictly perpendicular band as would be expected from the CS point group selection rules, but it has 12% of its intensity in parallel (a-type) transitions in the slit-jet spectra (Table S3). Unexpected a-type transitions have been found previously for another A00 band in methanol, ν11.24 The explanation offered in that case, borrowing of a parallel transition moment by coupling with other modes, can also explain the present situation. To take the largeamplitude torsional tunneling into account, we use the G6 molecular symmetry group that is based on permutation-inversion symmetry.32 In this group, the c-component of the transition dipole responsible for ν9 transforms as A2, which gives the vibrational selection rules A2 r A1 and E r E and the rotational selection rules ΔK = (1, ΔJ = 0, (1. Thus, the primary oscillator strength of the ν9 band follows perpendicular selection rules for both the A and E species. At J0 = 0, only anharmonic coupling is allowed and might possibly lend an a-type transition moment to the ν9 band. However, an A2 vibration like ν9 can mix anharmonically only with other A2 vibrations, and all A2 vibrations are reached from the ground state by the c-component of the transition dipole and cannot lend a-type oscillator strength. On the other hand, the E-species levels can mix anharmonically with all other J0 = 0 E-species levels,3,10,23 some of which correlate to A0 levels in the CS point group and have a-type transition moments. This argument agrees with our experimental observation of an a-type transition reaching the J0 = 0 E level but no a-type transition reaching the J0 = 0 A2 level. For J0 > 0, Coriolis coupling can mix the A2 vibrational upper state with A1 vibrations and thereby impart an a-type transition moment. For K0 = 0 and J0 > 0, only b-type Coriolis couplings can mix A2 and A1 vibrational states. Because no a-type transitions were observed reaching the ν9 K0 = 0 A levels, we conclude that b-type Coriolis coupling is weak or absent from this sub-band. However, a-type transitions reaching some ν9 K0 > 0 A levels are observed, suggesting that a-type Coriolis coupling is active. Indeed, we have already noted above evidence for a strong a-type Coriolis interaction between ν9 and 2ν4. E. Vibrational Coupling Pathways. In ref 16, our interpretation of the intramolecular vibrational coupling pathways in the lower energy region (27502900 cm1) relied upon a comparison of the observed band origins (Figure 3) to the estimated zeroth-order energies of the possible combination vibrations that lie in that energy region. That comparison is shown in Figure 8 where the energy range is now extended up to 3020 cm1 to include the new data as well as the previous data. The development in this section will proceed as follows: First, we consider the zeroth-order states expected in the extended energy range and
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compare to the number of observed vibrational bands to the number of zeroth-order states in each part of the range. Then, based on the assumption that the intensity of the observed bands is primarily borrowed intensity from the three CH stretch fundamentals, we make deductions about the vibrational coupling pathways in each part of the extended energy range. Figure 8 displays the estimated E symmetry term values for the zeroth-order combination vibrations of both CH3OH and CH3OD that fall into the range 27503020 cm1. These combination-vibration states are arranged into tiers. The CH stretch fundamentals are labeled as “bright states”, and each successive tier has one more quantum of vibration. That is, the states in tier 1 are binary combinations; those in tier 2 are ternary combinations, and so on. Each estimated term value in Figure 8 is the sum of the relevant fundamental frequencies (as given in Table 1 of ref 16), except that the contribution of the very anharmonic torsional mode ν12 is computed differently. As shown in Table 2 for CH3OD, the pattern of the torsional levels deviates substantially from the evenly spaced levels of a harmonic oscillator. Furthermore, for some vibrations of both isotopomers, the torsional tunneling splittings are inverted relative to the normal pattern. In addition, the A/E splittings in all of the torsionally excited states built on such vibrations are also expected to be inverted. Because of the strength of the torsionvibration interaction, we can only make rough predictions of the excited torsional states of general combination bands. The spread in Table 2 between the normal and inverted torsional energies at each torsional overtone provides a measure of the uncertainty for the estimated term values of CH3OD. The uncertainty for CH3OH is larger16 because of the smaller torsional moment of inertia and consequently larger torsional spacings. The uncertainty in the torsional energies becomes large in the range ν12 to 3ν12 where the torsional excitation is comparable to the torsional barrier height (373 cm1 for CH3OH6 or 366 cm1 for CH3OD33). In tier 1, the term values are likely accurate within a few cm1, but the tier 2 states include one quantum of torsion and may be in error by tens of cm1. Some of the term values in tier 3 and higher could be in error by 100 cm1 or more. Therefore, Figure 8 provides only a reasonable estimate of the number and type of higher tier vibrational states existing in the region of the present spectra. For the purpose of characterizing the distribution of the observed band origins and the inferred coupling pathways, we divide the overall range 27503020 cm1 into four spectral regions, labeled I, II, III, and IV in Figure 8. In this discussion, we focus on the bands assigned to K0 = 0, but the same qualitative aspects also apply to the other observed K0 as well. In region I (27502850 cm1), 12 CH3OH bands were observed, but only one band was found in CH3OD. In ref 16, this difference was attributed to the isotopic shift in CH3OD of the tier 1 doorway states out of resonance. The band count for CH3OH can be compared to the 13 zeroth-order vibrational states expected in region I. In the other regions, both isotopomers exhibit the same qualitative behavior: no assigned band origins in region II (28502900 cm1), a high density of assigned bands in region III (29002960 cm1), and only a single band, the ν2 fundamental,2 in region IV (29603020 cm1). Region III (29002960 cm1) contains the new assignments presented in this paper. In total, region III has 15 assigned bands for CH3OH, and 15 for CH3OD. The number of observed bands can be compared to the number of zeroth-order vibrations in this region in Figure 8: 8 for CH3OH and 16 for CH3OD. If the 9757
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Figure 8. Comparison of estimated zeroth-order term values for combination-vibration states (left) to the observed K0 = 0 upper state values (right). In both (a) and (b), the band origins are calculated using the harmonic approximation for all modes except the torsion, ν12. The zeroth-order vibrational states are arranged into tiers according to the coupling order relative to bright state. The spectral regions I, II, III, and IV labeled at the far right are used to facilitate the discussion in the text. 9758
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Table 2. CH3OD E Species J = 0 Torsional Term Values (cm1) torsional states a
m
normalb
ma
0
1
2.6
0.5
0
ν12
2
179.9
2.5
198.7
2ν12 3ν12
4 5
372.9 522.5
3.5 5.5
316.5 610.9
4ν12
7
934.9
6.5
817.3
5ν12
8
1195
8.5
1059.7
6ν12
10
1490.5
9.5
1337.5
nν12
a
invertedb
The correspondences between the ν12 vibrational quantum number n and the internal rotation angular momentum quantum number m are documented in ref 23. b The calculated values are relative to the lowest torsional state. For the normal pattern, the lowest torsional state is A1, and for the inverted case, it is E. In both cases, the ground state torsional potential is used to solve for the torsional energies and wave functions. In the inverted case, 4π boundary conditions are applied (ref 23) and the torsional wave functions ϕi that are antisymmetric in the torsional angle γ are used; that is, ϕi f ϕi when γ f γ þ 2π. a
reference range is extended to 28802980 cm1, then the numbers observed bands remains the same, but the numbers of zeroth-order states increase to 10 and 22, respectively. We conclude that the number of observed bands is comparable to the total number of zeroth-order vibrational states in the 2880 2980 cm1 region. It is puzzling that for CH3OH, the number of observed bands is actually higher than the counted number of zeroth-order vibrational states. There are a number of possible reasons for this: (i) Errors in the calculation of the energies of the zeroth-order states could mean that there are more zeroth-order states in region III than are represented in Figure 8a. Figure 8a has a higher density of states in tiers 26 just outside of region III, and some of those states may belong within it. (ii) Residual uncertainties in the assignments could have a slight effect on the number of observed bands. (iii) Coriolis b- and c-type couplings have selection rules ΔK0 = (1 and can cause the upper states to have mixed K0 character. With successive stages of Coriolis coupling, the 2J0 þ 1 different K0 states can all be mixed and, therefore, could show up in the spectrum increasing the total number of observed states. Such effects can be detected when a given K0 upper state can be reached by a larger-than-expected number of K00 lower states. For example, a K0 = þ2 upper state with some K0 = þ1 character would be reachable with perpendicular selection rules from K00 = 0, þ1, þ2, and þ3. With the present FTMW-IR data, such Coriolis mixing is hard to detect because of the limited number of K00 accessible. In the slit-jet spectrum, one sub-band is tentatively assigned to K0 = 2 r K00 = 0 because of the lack of R(0) and P(2) lines and the distribution of intensity between the branches. That assignment could not be confirmed by K0 = 2 r K00 = 3 transitions because of the limited thermal population of K00 = 3 levels in the jet. At this point, we cannot quantify the extent to which Coriolis b- and c-type couplings contribute to the density of observed bands. In ref 16, evidence was presented that all or almost all of the observed intensity in region I was attributable to the oscillator strength of the CH stretch “bright states.” In CH3OH, the first tier states (ν4 þ ν6, ν5 þ ν6, and ν10 þ ν6) might have had enough intrinsic oscillator strength to appear in the observed spectra without borrowing intensity from ν3 or another of the
CH stretch fundamentals. However, in CH3OD, the first tier states, which contain a quantum of COH bend were shifted well out of resonance with ν3 and were not observed in the reported spectra. It was concluded that the intrinsic intensity of these tier 1 states was 1% or less than that of the CH stretch fundamentals, and it is likely that the higher tier states are even weaker. In region III, many of the bands other than ν9 have substantial intensity. The relevant tier 1 states here are the binary combinations of the CH bends (ν4, ν5, and ν10) with little amplitude on the OH group, and accordingly, deuterium substitution in CH3OD does not shift then out of resonance with ν9. Therefore, we do not have any direct measure of the intrinsic intensity of these tier 1 states. Nonetheless, it is likely, given the evidence from region I, that a substantial part (if not most) of the observed intensity in region III is also attributable to the oscillator strength of one or more of the CH fundamentals. With this assumption, the present spectra provide evidence for the mixing of the CH stretch fundamentals with the tier 1 and higher “bath” states. Figure 8 shows that in regions where there are near-resonant tier 1 states, a high density of vibrational bands is observed. In CH3OH, tier 1 states are present in regions I and III, and a total of 27 vibrational bands have been identified in these two regions. In regions II and IV, there are no near-resonant tier 1 states and only the ν2 CH stretch fundamental is observed. In CH3OD, there are tier 1 states only in region III where 15 bands are observed. In regions I, II, and IV of CH3OD, there are no tier 1 states and only the CH stretch fundamentals, ν2 and ν3 are observed. For both isotopomers, in the places where a high density of vibrational states is found, the number of observed states is substantially higher than the number of tier 1 states and is comparable to the total density of vibrational states in each respective region. Therefore, the tier 1 states can be regarded as doorway states. They not only couple to the bright states, but also couple to the higher tier states to produce the observed number of coupled bands. One can think of two limiting coupling schemes compatible with the observed data. The first is a strict tier model in which only adjacent tiers are coupled. The pathway for energy flow from the bright states would be first to one or more of the tier 1 doorway states, then to tier 2 states, and, subsequently, to tier 3 states, and so on. This coupling scheme is restricted to sequential steps of low-order coupling.34,35 The second kind of coupling scheme is similar to the first except that the doorway states would be allowed to couple directly to many or all of the higher tier states. This second coupling scheme allows direct high-order coupling between the doorway states and all of the higher-tier “bath” states. More realistic than either of these two limiting schemes is an intermediate scheme where the low-order couplings are supplemented by high-order couplings to a greater or lesser extent.36,37 Because the tier 2 and higher states all involve excitation of the very anharmonic torsional mode ν12, a larger contribution of direct high-order couplings is expected relative to a more rigid molecule.23,37 While the present data are not sufficient to distinguish between these schemes, it is evident that once the doorway states are accessed, all or nearly all of the higher tier states are coupled strongly enough to appear in the spectrum. F. Time-Dependent Intramolecular Vibrational Redistribution (IVR). Information about the IVR time scales can be derived from the present data by considering hypothetical coherent excitations of certain sections of the spectrum. The procedure for calculating the time-dependent probability of the initially prepared state following such a hypothetical coherent 9759
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Figure 9. Survival probabilities computed for coherent excitations of the indicated spectral ranges of the stick spectra in Figure 3. The heavy black curves are the averages of the curves for K0 = 1, 0, and þ1. For the averaged curves, the initial fast decay times, τIVR, and the dilution factors, ϕd, are indicated. A second time scale is evident in each of the averaged curves as the decay of the recurrences over interval 13 ps.
excitation is well established.3840 In ref 16, the computed timedependent dynamics following a coherent excitation of region I were compared to the liquid-phase time-dependent experiments of Iwaki and Dlott17 with excitation at 2870 cm1. Iwaki and Dlott also reported time-dependent measurements with excitation at 2970 cm1, and now, with the present data, it is also possible to simulate coherent excitations of gas-phase jet-cooled methanol in that part of the spectrum. The resulting timedependent survival probabilities are presented in Figure 9. In the upper panels, the excitation bandwidth corresponds to regions III and IV together; in the lower panels, only region III is excited. Comparison of the upper and lower panels gives an appreciation for the dependence of the dynamics on the excitation bandwidth. The survival probabilities for CH3OH and CH3OD, compared in the left and right panels of Figure 9, are qualitatively similar. Within each panel, curves for K0 = 1, 0, þ1 are plotted and the average of the three curves is given. The discussion that follows focuses on the common features of all four averaged survival probability curves. One of the challenges in this work is to determine the vibrational state that is prepared at t = 0 immediately following the hypothetical coherent excitation by a short pulse of light. If only a single zeroth-order state had any oscillator strength from the ground state and if all of the eigenstates that shared that intensity were excited with a uniform intensity, then the initially prepared vibration would be that zeroth-order bright state.
However, if two or more zeroth-order states carry oscillator strength, then a linear combination of these states would be prepared at t = 0 by the hypothetical coherent excitation. In this context, we refer to the initially prepared state as an “effective bright state”. In ref 16, it was shown that the dominant oscillator strength in region I comes from the ν3 symmetric CH stretch. The ΔK = 0 rotational selection rules of the bands in region I reflect a vibrational transition moment nearly parallel to the a-axis (βK,i ≈ þ1 in Figure 4b), as expected for bands borrowing intensity from ν3, whereas the ν2 and ν9 CH fundamentals include strong perpendicular (ΔK = (1) transitions (βK,i < 0). In the observed spectra, the bands in region III show the full range of transition moments (1 e βK,i e þ1 in Figure 4b), indicating that the bands acquire their oscillator strength in different proportions from the three CH fundamentals. Certainly in the upper panels of Figure 9, both ν2 and ν9 contribute to the oscillator strength. Even in the lower panels of Figure 9, which correspond to excitation of region III only, the observed “bath” states might have derived some of their oscillator strength from ν2. Although ν3 is somewhat further away in region I, there is evidence from theory30,31 that ν3 mixes with the tier 1 state 2ν5 and likely with some of the other CH bending combinations in tier 1. That is, the effective bright states prepared at t = 0 in the different panels of Figure 9 will involve all three of the CH stretches in some proportion. 9760
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A second source of uncertainty about the nature of the effective bright state comes from the overlap of the excitation bandwidth with the set of molecular eigenstates. If the excitation bandwidth does not include all of the vibrational eigenstates that borrow intensity from one of the CH stretches, then the effective bright state would have some admixture of the tier 1 or higher states. If some of the tier 1 states have intrinsic oscillator strength of their own, then the definition of the effective bright state is more complicated. The comparison with the experiment is further complicated by the experimental pulse shape, which might be a Gaussian centered at the nominal excitation wavenumber. In the liquid, the molecular sample has a dephasing width that contributes to the effective excitation bandwidth. Thus, the spectral features closer to the center excitation wavenumber will be subjected to a greater intensity than those in the wings. It is probably fair to say that the effective bright states in both the present calculation (Figure 9) and in the experiment17 have a strong component of CH stretch character, but a more precise comparison is difficult. Two distinct relaxation time scales are evident from each of the curves in Figure 9. The first time scale is the immediate fast falloff which has a decay time40 τIVR ≈ 200 ((30%) fs. Following the initial decay, recurrences are evident in Figure 9 with periods typically in the range 8001000 fs. The second time scale is the decay of these recurrences in 13 ps. There is also a third decay time scale beyond the range of the time axes in Figure 9. Slower recurrences occur with periods of 1012 ps. The slow recurrences decay on a time scale of tens of ps. The slow recurrence time corresponds approximately to the median spacing of the vibrational states in region III, which is ∼3 cm1. At long times as t f¥, the survival probabilities fluctuate about a time averaged value, which is the dilution factor ϕd given by ϕd ¼ ð
∑i Ii2 Þ=ð∑i Ii Þ2
ð7Þ
The sums in eq 7 are over the intensities of the bands within the coherent excitation bandwidth. Because the relative intensities of the coupled states are greater in region III than in region I, the dilution factor ϕd ≈ 0.120.15 is less than in region I, where ϕd ≈ 0.30.5. This means that the curves in Figure 9 decay to lower values than the curves in Figure 8 of ref 16, reflecting a more extensive redistribution of the bright state energy among the bath states. Iwaki and Dlott’s time-domain measurements of vibrational relaxation in liquid methanol show two different time scales. Whereas Figure 9 presents the survival probability of the bright state, the time-domain experiments detected the arrival of amplitude in the various receiving modes. The time-domain results for excitations at 2870 and 2970 cm1 gave similar results. They found that the CH bending modes (ν4, ν5, ν10) and the OH bend (ν6) appeared first in less than 1 ps and that the methyl rocks (ν7, ν11) and the CO stretch (ν8) appeared in 25 ps. These time scales correspond well with the time scales found in the time-domain interpretation of the frequency-resolved data on gas phase CH3OH (Figure 9 of this paper and Figure 8 of ref 16). The survival probability curves computed from the frequencyresolved spectrum do not directly give information about the identity of the vibrational modes to which the energy is transferred, but some insight can be obtained from the tier structure in Figure 8a. If the IVR is dominated by low-order couplings from
one tier to the next, then one expects the states in the lower tiers to be populated first, and then the successively higher tiers. For excitation of region I in CH3OH, the first tier states involve the bending modes (ν4, ν5, ν10, ν6), but the methyl rocks (ν7, ν11) and the CO stretch (ν8) appear only in the second and higher tiers. In regions III and IV, the first tier states involve only ν4, ν5, and ν10; the OH bend (ν6) first appears in tier 2; and ν7, ν11, and ν8 do not appear until the third and higher tiers. Thus, for both excitation regions, the ordering of the modes in the tier structure corresponds to the order in which the modes are observed in the time-domain experiments. Thus, both the observed time scales and the ordering of the modes suggest a significant commonality between short-time IVR in the isolated gas-phase molecule and the observed time-dependence in the liquid phase. Time-domain studies comparing the gas- and solution-phase vibrational dynamics of a variety of systems4148 have found that the initial relaxation in the solution phase is determined by the isolated molecule dynamics. The comparison of the present data with the liquid-phase results of Iwaki and Dlott provides additional support for that finding. However, the two kinds of experiments and their respective interpretations do have some differences. The rise time of the OH bend (ν6) in the liquid phase is the same whether the excitation is at 2870 or 2970 cm1, but based on the tier model, one would expect it to be slower for excitation at 2970 cm1 because it does not appear until tier 2 in regions III and IV. This could be a limitation of the tier model or an intrinsic difference between the gas and liquid phases. For excitation in the liquid at 2970 cm1, the higher frequency asymmetric CH bends ν4 and ν10 (together notated as δa) appear promptly with the excitation pulse. Iwaki and Dlott suggested that the initially prepared state at this excitation frequency contained some δa character. Their suggestion is consistent with the discussion of the effective bright state above, but it would have been hard to predict the prompt appearance of δa from the frequency-resolved data. The most significant difference between the two phases is what happens at long times. Iwaki and Dlott used CCl4 as a reporter to measure the rate at which energy leaves the initially excited methanol molecule. They found that almost no energy remains in methanol after ≈20 ps because of energy transfer to the solvent. The longest time scale (1050 ps) found in the present gas-phase work is instead purely intramolecular.
IV. SUMMARY AND CONCLUSIONS With the addition of the spectra reported here from 2900 to 3020 cm1, rotationally state-selected infrared spectra have been recorded for both CH3OH and CH3OD throughout the whole CH stretch region 27503020 cm1. The rotational stateselection relies on E-species microwave transitions making available 46 low-J ground state rotational levels. These spectra, together with slit-jet single-resonance spectra above 2945 cm1, have made possible the assignment of 28 vibrational bands for CH3OH and 17 bands for CH3OD distributed throughout the CH stretch region. The pattern of the torsional tunneling splittings for the three CH stretch fundamentals has been compared to a model calculation and to ab initio theory. With the assumption that all of the observed bands borrow their intensity from the CH stretch fundamental bands, the spectra provide information about intramolecular vibrational energy transfer (IVR) pathways. A tier model, in which the three CH stretch fundamentals are the bright states, explains the both 9761
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The Journal of Physical Chemistry A the present (2900 3020 cm1) and previous16 (27502900 cm1) observed spectra. The first tier states are binary combinations of bending modes that are coupled to the bright states by 1:2 Fermi resonance interactions. The first tier states are not distributed evenly over the whole the CH stretch region, but are concentrated into certain subranges of the spectrum. Where the first tier states occur, a high density of vibrational states is observed in the spectra, comparable to the molecules’ total density of vibrational states; elsewhere, only the bright states are observed with modest perturbations. Therefore, the first tier states are doorway states that mediate the coupling of the bright states to essentially all of the higher tier states. The higher tier states are combinations of the lower frequency modes and substantial amounts of torsional excitation. There are also significant differences in the coupling mechanisms in the two spectral regions. In the higher wavenumber region, the asymmetric CH stretches couple to the CH bend doorway states by both anharmonic and Coriolis mechanisms to yield extensive mixing for both CH3OH and CH3OD with a higher density of bath states. The weaker mixing in the symmetric CH stretch region was observed for CH3OH only and is predominantly anharmonic mediated by doorway states that involve the OH bend as well as the CH bends. The time-dependent interpretation of the frequency-resolved spectra shows three IVR time scales. The fastest two time scales (∼200 fs and 13 ps) are in qualitative agreement with the liquid phase time-resolved data, suggesting a common mechanism for the fastest relaxation processes in the liquid and gas phases. At longer times (>5 ps), the processes in the two phases necessarily differ because of energy transfer to solvent molecules in the liquid phase. Both the time-dependent interpretation of the present frequency-resolved data and the time-dependent liquid phase measurements constitute well-defined processes; however, the presence of three bright states means that the precise nature of the initially prepared states is not well understood.
’ ASSOCIATED CONTENT
bS
Supporting Information. Lists of transition frequencies, intensities, and fitted parameters for both CH3OD and CH3OH, and figures of the slit-jet and FTMW spectra of CH3OH. This material is available free of charge via the Internet at http://pubs. acs.org.
’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected]. Present Addresses †
3884 Long Ridge Blvd, Carmel, IN 46074.
’ ACKNOWLEDGMENT The work was supported by the Division of Chemical Sciences, Offices of Basic Energy Sciences, Office of Energy Research, U.S. Department of Energy under Grant No. DEFG02-90ER14151, and by the National Science Foundation (CRIF: ID CHE-0618755). L.-H.X. acknowledges financial support for this research from the Natural Sciences and Engineering Research Council of Canada.
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