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J. Phys. Chem. A 2010, 114, 6818–6828
Vibrational Coupling Pathways in Methanol As Revealed by Coherence-Converted Population Transfer Fourier Transform Microwave Infrared Double-Resonance Spectroscopy Sylvestre Twagirayezu, Trocia N. Clasp, and David S. Perry* Department of Chemistry, The UniVersity of Akron, Akron, Ohio 44325-3601
Justin L. Neill, Matt T. Muckle, and Brooks H. Pate Department of Chemistry, UniVersity of Virginia, McCormick Road, CharlottesVille, Virginia 22904 ReceiVed: March 4, 2010; ReVised Manuscript ReceiVed: May 17, 2010
Coherence-converted population transfer infrared-microwave double-resonance spectroscopy is used to record the infrared spectra of jet-cooled CH3OH and CH3OD. Population transfer induced by a pulsed IR laser is detected by Fourier transform microwave spectroscopy background-free using a two-MW pulse sequence. The observed spectrum of CH3OH in the ν3 symmetric CH stretch region contains 12 interacting vibrational bands, whereas in CH3OD, only one vibrational band is observed in the same interval (2750-2900 cm-1). The bright state, responsible for the transitions observed in this region, is not just ν3 but also contains an admixture of the binary CH bending combinations, particularly 2ν5. The lack of interacting bands in CH3OD confirms that in CH3OH the binary combinations of the OH bend (ν6) and a CH bend (ν4, ν5, ν10) act as doorway states linking the bright state to higher order combination vibrations involving torsional excitation. A time-dependent interpretation of the frequency-resolved spectra reveals a fast (∼200 fs) initial decay of the bright state followed by a slower (1-2 ps) redistribution among the lower frequency modes. I. Introduction As the smallest molecule with both a facile internal rotation coordinate and a methyl group, methanol continues to be a benchmark system for studies of torsional motion in a 3-fold potential.1-3 Methanol is also a benchmark system for studies of intramolecular vibrational redistribution (IVR)4-6 and for the investigation of the couplings between the large-amplitude torsional motion and the other small-amplitude vibrations.7-11 The methanol vibrational frequencies and approximate mode descriptions are summarized in Table 1. Infrared laser-assisted photofragment spectroscopy (IRLAPS) of the OH overtones (v1 ) 3-8) has revealed multiple time scales for IVR in methanol ranging from about 100 fs to 10 ps.4,5 The fastest time scales arise from low-order resonances of the OH stretch with a CH stretch (5ν1 S 4ν1 + ν2) and with the overtone of the OH bend (7ν1 S 6ν1 + 2ν6). The OH stretch fundamental, the lower OH overtones, and the excited torsional bands built on these vibrations (reaching up to 3ν1 + 2ν12 at 7800 cm-1 above the ground state) showed multiple local perturbations but did not show extensive IVR, and the spectra could be fit with the same form of the torsion-rotation Hamiltonian used for the ground vibrational state, but at infrared accuracy. The asymmetric CH stretches, ν2 and ν9, are more complicated, with torsional tunneling splittings between the A and E levels that are inverted relative to the ground state.11-13 The effect results from the variation of the CH stretch force constant with the torsional angle has been the subject of several different theoretical papers.7,8,10,11 Sibert and Castillo-Chara10 have used a combination perturbation-variation method to compute the quantum nuclear dynam* Corresponding author. E-mail:
[email protected].
TABLE 1: Experimental Vibrational Frequencies of Methanol (cm-1) CH3OH
CH3OD
vibration States
descriptiona
exp. gasb
exp. liq.e
exp. gasf
exp. liq.e
ν1 ν2 ν3 ν4 ν5 ν6 ν7 ν8 ν9 ν10 ν11 ν12
ν (OH) ν (CH)a ν (CH)s δ (CH)a δ (CH)s δ (OH) F| (CH3) ν (CO) ν (CH)a δ (CH)a F⊥ (CH3) torsion
3682.5 3003.7 2844.7 1483.5 1453.3 1335.2 1070.2 1033.7c 2961.1 1474.2 1156.5 199.8d
3328 2980 2834 1480 1450 1418 1115 1030 2946 1480
2717.6 3000.1 2841.7g 1479.2h 1455.0 864.0 1224.5 1044.4 2970.0 1463.0 1142.0 177.2i
2467 2978 2838 1469 1449 940 1231 1038 2951 1469
655
475
ν ) stretch; δ ) bend; F ) rock; a ) asymmetric; s ) symmetric. b E-symmetry vibrational band origins are relative to the J′′ ) 0, K′′ ) 0 E level of the ground state. Data from tabulation of ref 10 unless otherwise indicated. c Refs 57, 58. d Ref 59. e Ref 60. f Ref 34 unless otherwise indicated. g This work, E-symmetry vibrational band origin. h Average of the two vibrational frequencies (1480.0, 1477.5 cm-1) given in ref 34. i Data from ref 29. a
ics of methanol on an ab initio potential energy surface. The calculations successfully accounted for the torsional tunneling behavior for all of the vibrational fundamentals and for two combination bands for which the data were available. They used correlation diagrams in which the mass of the internal rotor was varied to elucidate the torsion-vibration coupling. Like most variation treatments of the nuclear motion in polyatomic molecules, they considered only the rotationless molecule (J ) 0). The computational demands of such variational treatments increase dramatically with the dimension of the problem, and even without the inclusion of rotational motion, Sibert and
10.1021/jp1019735 2010 American Chemical Society Published on Web 06/08/2010
Vibrational Coupling Pathways in Methanol
J. Phys. Chem. A, Vol. 114, No. 25, 2010 6819
Figure 1. Schematic diagram of the FTMW-IR apparatus. Two microwave pulses are produced by mixing the outputs from an arbitrary function generator and a microwave synthesizer in a 30 MHz single-sideband modulator. These microwave pulses are amplified and coupled by a circulator and antenna into a Fabry-Perot microwave cavity. The microwave pulses and a tunable infrared pulse interact with jet-cooled methanol molecules in the center of the microwave cavity. The microwave free induction decay (FID) is detected with the same antenna, coupled out through the circulator and amplified with a low noise amplifier. Then the FID is mixed down to 30 MHz and recorded on a 5 Gs/s oscilloscope.
Castillo-Chara had to treat some of the vibrational interactions perturbatively. For this reason, it is desirable to develop an experimental method that can measure energies of the rotationless band origins without having to carry out a complete assignment and analysis of the whole rotational structure. In this paper, coherence-converted population transfer Fouriertransform microwave infrared (CCPT-IR-FTMW) spectroscopy is applied to obtain rotationally state-selected infrared spectra of CH3OH and CH3OD in the CH stretch fundamental region. The present analysis will focus on the ν3 symmetric CH stretch region (2750-2900 cm-1). In the previous high resolution work,14 only one band, identified as ν3 and centered at 2844.69 cm-1, was rotationally assigned in this region. Both lowresolution gas phase spectra15 and IRLAPS of jet-cooled methanol (Figure 5e of ref 16) provide an indication of at least one perturbing band on the low-frequency side of the ν3 band. Recent matrix isolation spectra of methanol in N2 and Ne matrixes found three bands on the low frequency side of ν3 that were assigned as combinations of the CH and OH bends: ν4 + ν6, ν5 + ν6, ν10+ ν6.17 In this work, we report a total of 12 interacting bands in the ν3 region of CH3OH. By contrast, the spectra of CH3OD in the same region contain only one strong vibrational band. We will show below that the newly observed CH3OH vibrational structure reflects vibrational couplings through several tiers of vibrational states. II. Experimental Section A detailed description of the spectroscopic technique employed here, coherence-converted population transfer Fourier-
transform microwave infrared spectroscopy (CCPT-FTMW-IR), will be published elsewhere.18 The technique is a variant of the FTMW-IR technique described by Douglass et al.19 in which the population change induced by a laser pulse that precedes the microwave polarization pulse is detected by its effect on the microwave signal measured using a miniature cavity FTMW spectrometer20 based on the Balle-Flygare design.21 This simple approach of monitoring a microwave signal while the laser frequency is scanned is susceptible to large baseline excursions caused by pulse-to-pulse variations in the molecular beam number density from successive sample injections by the solenoid valve (General Valve Series 9 with a 1 mm nozzle diameter). The CCPT-FTMW-IR technique is similar to the doubleresonance method introduced by Nakajima et al. for electronic spectroscopy and makes use of the coherent nature of the FTMW signal.22 A schematic diagram of the spectrometer is shown in Figure 1. The microwave system employs a two-pulse sequence to excite the gas sample (0.1% methanol in an inert gas mixture that is ∼80% neon and 20% helium). The pulse energy of the first, resonant microwave pulse (with 500 ns duration) is chosen to achieve the maximum signal in the FTMW spectrometer. In a Bloch model description, this pulse is chosen to be a “π/2pulse”, although the variation of the transition dipole moment with the angular momentum projection quantum number (MJ) precludes true “π/2” excitation of the rotational transition. The second pulse, also with 500 ns duration, is applied to the sample after a 50 ns pulse delay. The phase of the second pulse is shifted 180° relative to the first pulse. In a Bloch model analysis of
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this excitation sequence, the second pulse is a “-π/2-pulse” and simply counter-rotates the Bloch vector, returning the coherence back into a population. Therefore, there is no detectable coherence in the spectrometer following this sequence. In practice, fine-tuning of the phase and amplitude of the second pulse can “null” the FTMW signal to less than 1% of the normal, single pulse level. To measure the laser-induced absorption spectrum, the pulsed IR laser (Continuum Mirage OPO/OPA with ∼10 ns pulse duration) interacts with the molecular gas in the 50 ns interval between the microwave pulses. When the laser frequency is resonant with an infrared transition from one of the two rotational levels involved in the rotational transition, it induces a population difference. In the Bloch model description, the laser-induced population difference will point along either the +z-axis or -z-axis, depending on whether laser excitation takes place out of the upper or lower energy rotational level in the monitored microwave transition. When a laser-induced population difference is created, the second microwave pulse (“-π/2pulse”) converts the population to a detectable coherence. Therefore, a microwave signal is observed only when a laserinduced transition has occurred. This microwave signal is proportional to the population transferred by the laser pulse and is, therefore, the absorption spectrum of the molecule. Furthermore, the phase of the laser-induced free induction decay observed after the second microwave pulse indicates whether the laser excitation occurred from the upper or lower energy rotational level in the monitored rotational transition. Like the previous IR-FTMW method,19 we observe the laser-induced absorption spectrum through changes in the population difference of the two rotational levels; however, the measurement is now background-free. The undesired OPO/OPA output frequency (near 1.5 µm) is filtered out by three mirrors at Brewster’s angle. The laser beam is then introduced into the vacuum chamber through a calcium fluoride window, after which it enters a plane-parallel multipass cell, which consists of two 7.5 cm square gold-plated mirrors on either side of the FTMW expansion. The unfocused laser beam makes ∼15 passes across the expansion region. The wavenumber of the infrared light was determined by measurement of the OPO/OPA signal wave with a Coherent Wavemaster (model 33-2650) wavemeter. The wavenumber is recorded every 10 frequency steps and then linearized. The scan rate of the laser, at 20 signal averages per frequency step, is 10.6 cm-1/h. The spectra, shown in Figure 2, are obtained by plotting the imaginary-part of the Fourier transform of the laser-induced freeinduction decay at the center frequency of the rotational transition as the infrared laser wavenumber is scanned (where the phase reference is determined from the pure rotational signal in a single microwave pulse measurement). Using this analysis approach, instead of the commonly used magnitude Fourier transform, preserves the phase information in the microwave signal and generates an infrared spectrum of the two connected rotational levels with the transitions from one rotational level pointing up and from the other pointing down. We confirmed the linearity of the infrared transitions by measuring the relative intensities of weak and strong lines at different infrared pulse energies (1, 2, 4, and 8 mJ) in the range 2838-2845 cm-1. Although there was significant scatter, we did not find any systematic evidence of infrared saturation. For the previously assigned lines in Figure 2a and b, the rootmean-square deviation of the present wavenumbers from the previous values is 0.05 cm-1, and the mean deviation is -0.006 cm-1. Since the previous slit-jet experiments employed a
Twagirayezu et al. precisely calibrated ((0.0005 cm-1) continuous wave laser, the observed deviations reflect the uncertainties in the present wavenumber measurements. All lines expected to appear in the present spectra-based on previous high-resolution slit-jet12 and FTIR14,23 spectra-were observed and are labeled in Figure 2a and b. The measurement of relative intensities required consideration of the variable line shape of the OPO/OPA system. The lineshapes observed in our spectra appeared to be a single mode, ≈0.04 cm-1 fwhm, accompanied by a second mode of lower intensity displaced by about 0.06 cm-1 to higher or lower frequency. This is consistent with an etalon monitor in the system, which shows intermittent double-mode behavior. The relative intensities of the two modes varied across the spectrum, sometimes being almost equal. The reported intensities are integrated intensities over an interval (0.05 cm-1 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 (8-26 GHz) accessible to our microwave equipment. Several convenient E-species transitions were available, of which we used 20 r 3-1 at 12.179 GHz and 30 r 2+1 at 19.967 GHz for CH3OH and 1-1 r10 at 18.957, 2-1 r 20 at 18.991, and 3-1 r 30 at 19.005 GHz for CH3OD. Here, 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 in Figure 2a is ∼200:1 relative to the largest peak. III. Results and Discussion a. Overview of the 3 µm Region. The CH3OH spectrum from 2750 to 3015 cm-1 shown in Figure 2a and b indicates that assignments for only a small fraction of the lines are available from previous single-resonance spectra.12,14,23 Since the three CH stretches (ν2, ν3, and ν9) are the only infrared fundamentals in this spectral region, the majority of the lines in Figure 2a and b must reach upper vibrational states that are overtones and combinations of the lower frequency vibrations. For convenience, we will refer to these as “extra” lines. The midinfrared fundamentals that could contribute to binary combinations in this spectral range are all much weaker than the CH fundamentals, and the binary combinations are likely to be even weaker. Higher-order combination vibrations will be weaker still. Therefore, to facilitate the initial assignments, we will use the operational assumption that the combination vibrations appearing in the CH3OH spectrum gain their intensity by mixing with one or more of the CH stretch fundamentals. This assumption, which is necessary for the interpretation of the data in terms of IVR couplings, will be revisited in section IIId, below. The CH3OH spectrum falls naturally into two distinct regions, shown in Figure 2a and b, respectively. Below 2900 cm-1, there are, in addition to the lines assigned to the strong ν3 fundamental, many “extra” transitions of weak or moderate intensity spread sparsely over a wide range. Throughout this range, the observed transitions are predominantly parallel (∆K ) 0) in nature. This is the sort of pattern expected for combination bands that borrow their intensity from ν3, a predominantly parallel band. Above 2900 cm-1, the perpendicular and parallel transitions appear to be comparable in number and strength. The region between 2905 and 2980 cm-1 is densely packed “extra” features attributable
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Figure 2. Coherence-converted population transfer FTMW-IR spectra of CH3OH and CH3OD: (a, b) Observed infrared spectra of CH3OH in the CH-stretch region that result from the microwave transition 20 r 3-1 at 12.179 GHz. (c) Observed spectra of CH3OD that result from 1-1 r 10 at 18.957 GHz, and 2-1 r 20, at 18.991 GHz. Previous assignments are from refs 12, 14, and 23.
to combination bands, and many of these features are comparable in intensity to the lines assigned to the ν2 and ν9 asymmetric CH stretch fundamentals. A preliminary analysis of Figure 2a reveals many combination loops, which are indicative of perpendicular (∆K ) (1) transitions. This is the pattern expected for combination bands that borrow their intensity from the ν9 or ν2 or both. Prominent among the interacting combination bands expected in this region are the six binary combinations of the CH bends (ν4, ν5 and ν10). Conveniently, there is a “gap” region 2880-2905 cm-1 with only a few rather weak lines that separate the bands that appear to interact with ν3 from the bands that appear to interact with ν9 and ν2. The separation of the spectrum into two regions is an approximation that facilitates the analysis, but significant interactions between the regions are to be expected and in the end will have to be taken into account. In this paper, we focus on the analysis and interpretation of the spectral region below 2900 cm-1. To aid in the interpretation, spectra of CH3OD were also recorded in the range 2750-2900 cm-1 (Figure 2c). This spectral region of CH3OD contains only a small fraction of the number of lines found in CH3OH. In fact, we will show below
that almost all of the CH3OD lines are assignable to a single vibrational band, whereas in CH3OH, there is evidence for 12 interacting vibrations in this region. b. Rotational Assignments for CH3OD. The detailed rotational assignments for CH3OD are shown in Figure 3 and are listed in Table S1 in the Supporting Information. Each infrared transition originates in one of the rotational levels selected by the microwave pulses and reaches rotational levels in the upper vibrational state according to the selection rules ∆J ) 0 and (1, and ∆K ) 0 and (1. As for CH3OH, the parallel component (∆K ) 0) is strongest. Whereas very few perpendicular transitions (∆K ) ( 1) are observed in CH3OH, many weak perpendicular transitions are seen in the spectrum of CH3OD. Therefore state selection on K′′ ) 0 and -1 rotational levels provides access to K′ ) -2, -1, 0, and +1 in the upper vibrational state. Since the K′ ) -1 and 0 may each be reached from both K′′ ) -1 and 0, there are combination loops in the spectra of Figure 3 that confirm the K′ assignments. Thus, it was possible to make confident K′ assignments without making any assumptions about the pattern of the K′-dependence of the upper state energies. This capability is very useful in methanol because the K′ dependence is dominated by the very strong
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Figure 3. Detailed assignment of CH3OD spectra in the ν3 CH-stretch region that result from the microwave transitions 1-1 r 10 at 18.957 GHz, 2-1 r 20 at 18.991 GHz, and 3-1 r 30 at 19.005 GHz. Solid lines mark the positions of ∆K ) 0 transitions, and dashed lines indicate the positions of ∆K) ( 1 transitions.
torsion-rotation coupling, and the K′ pattern is inverted for some vibrationally excited states and may be irregular when there is mixing with other vibrations.12,24-26 For K′ ) -1 (Figure 3), each rotational (J′) transition is split into a triplet consisting of two comparably intense lines split by about 0.6 cm-1 and a weaker one about 0.2 cm-1 on the high frequency side. Neither of these splittings depends appreciably on J′, as would be expected for a perpendicular (bor c-type) Coriolis interaction. Therefore, these interactions might be either anharmonic or parallel (a-type) Coriolis in origin. Aside from the splittings of the K′ ) -1 levels, there are no perturbations evident in the CH3OD spectra. Almost all of the observed lines in Figure 3 are thus assigned, leaving only a few very weak features unassigned. Upper state term values νK′J′ were determined from the known ground state wavenumbers27 and fit to the expression K′ νK′J′ ) νK′ 0 + BeffJ′(J′ + 1)
(1)
The resulting subband origins ν0K′ were converted to reduced subband term values νRK′ by subtraction of the remaining rigidsymmetric-rotor energy, 2 νRK′ ) νK′ 0 - [A′′ - (B′′ + C′′)/2]K′
)
-1 νK′ × K′2 0 - 2.9206 cm
(2)
The derived reduced term values, νRK′, for K′ ) -1, 0, and +1 and the corresponding relative intensities, IK′ are plotted in K′ are in Figure Figure 4b. The effective rotational constants Beff
5. These data together with those for K′ ) -2 are listed in Table S2 of the Supporting Information. The reduced term values follow a pattern similar to the ground state (see for example Figure 6 of ref 12) with the K′ ) 1 subband origin at slightly higher wavenumber than K′ ) 0 (∆ν ) 0.75 cm-1) and K′ ) -1 shifted further away on the low side (∆ν ) 1.8 cm-1). Since the K′ ) -1 subband appears as a triplet, a Lawrance and Knight devoncolution28 was applied to obtain a deperturbed subband origin for that subband. Even though we do not observe the A-species J′ ) 0 level in this experiment, we are able to use the splitting between the K′ ) +1 and -1 subband origins to estimate the J′ ) 0 torsional tunneling splitting as +2.3 cm-1. This is slightly smaller than the CH3OD ground state torsional tunneling splitting of +2.6 cm-1.27,29 The explicit “+” signs are used to indicate normal tunneling splitting with A below E. c. Rotational Assignments for CH3OH. There are two factors that make the detailed rotational assignments for CH3OH more difficult than for CH3OD. First, there are many interacting bands in the region 2750-2880 cm-1. Second, the perpendicular (∆K ) (1) transitions are relatively weaker, and only a few are observed, which means that only a few combination loops were available to confirm the assignments. Accordingly, rotational assignments for the dominant parallel (∆K ) 0) transitions were made by identifying pairs or triples of lines with the P, Q, R spacings appropriate to the selected rotational states. For CH3OH, two microwave pump transitions were used, providing access to infrared transitions originating in the following ground state rotational levels: 3-1, 2+1, 20, and 30. The parallel transitions with K ) 0, which have no Q-branch, could be confirmed by the pattern of P and R transitions
Vibrational Coupling Pathways in Methanol
Figure 4. E-symmetry subband term values of (a) CH3OH and (b) CH3OD at K′ ) +1, K′ ) 0, and K′ ) -1. The labels, “a”, “b”, “c”, and “d”, are used to identify different groups of subbands.
Figure 5. Effective rotational constants of CH3OH and CH3OD for the observed subbands.
originating from 20 and 30. Together, these transitions provided access to K′ ) 0, J′ ) 1, 2, 3, and 4 and allowed a fit to obtain
J. Phys. Chem. A, Vol. 114, No. 25, 2010 6823 K′ the effective rotational constants Beff for each vibrational band (Figure 5). The parallel transitions with K ) (1 could not be confirmed in this way but, tentative assignments for K′ ) -1, +1; J′ ) 1, 2, and 3 could still be made, and the BK′ eff, determined (Figure 5). For each of the interacting bands, subband origins ν0K′ and reduced subband term values νRK′ were determined as for CH3OD, except that the rotational constants appropriate to CH3OH ([A′′ - (B′′ + C′′)/2] ) 3.4657 cm-1 1-3) were used. For each K′, the relative intensities, IK′, of the various observed vibrational states were taken as the sum the observed intensities of all of the relevant assigned transitions. A total of 12 reduced subband term values were determined for K′ ) -1 and 0, and 7 for K′) +1 (Figure 4a). Detailed line assignments for CH3OH are listed in Table S3 of the Supporting Information, and K′ complete numerical values for Beff , ν0K′, νRK′, and IK′ are given in Table S4. It is not immediately obvious from Figure 4a which of the K′ ) -1 and +1 reduced term values correspond to each of the 12 new vibrational states found for K′ ) 0. There is a fundamental difficulty in establishing such a correspondence because it is likely that each of the 12 vibrational states is a mixture of a number of normal mode combination states. Because the torsion-rotation coupling in CH3OH causes the νRK′ to vary by as much as 12 cm-1 and because the sign and magnitude of this coupling depends on the nature of the vibrational motion, the relative zeroth-order energies of the various coupled combination states will vary by 12 cm-1 or more as K′ goes from -1 to 0 to +1. Since many of the observed spacings (Figure 4a) are less than 12 cm-1, the mixture of zerothorder combination states that make up each of the observed vibrations will depend, either subtly or dramatically, on K′. However, because the overall range of the term values in Figure 4a is ∼100 cm-1, some correspondences relating the term values for K′ ) -1, 0, +1 can be identified. The large features labeled “a” in Figure 4a correspond to the most intense lines in the spectrum and are the ones previously assigned to the ν3 vibration.14 The smaller features labeled “b” in Figure 4a may correspond to the same perturbing vibrational state. The other features to lower wavenumber in Figure 4a are only labeled as the subgroups “c” and “d”. Figure 4a also contains information about the torsional tunneling splittings of the vibrations “a” and “b”. For the “a” vibration, Hunt et al.14 found a normal J′ ) 0 tunneling splitting with the E level 9.1 cm-1 above the A1 level. In this work, we have only observed the E species, but the tunneling splitting can be estimated from the pattern of the E-species K′ ) 0, +1, and -1 reduced term values as we did above for CH3OD. With this approach, the estimated tunneling splitting is +9.3 cm-1, close to the actual K′ ) 0 value of +9.1 cm-1. If the features “b” do correspond to the same perturbing vibrational state, its torsional tunneling is inverted with K′ ) 0 and +1 below K′ ) -1. The J′ ) 0 tunneling splitting for the “b” state is estimated roughly to be -10.7 cm-1. d. Interpretation of the ν3 Region in Terms of IVR Coupling Pathways. Estimated term values for the zeroth-order combination vibrations of CH3OH (J′ ) 0, E symmetry) that fall into the range 2700-2900 cm-1 are shown in Figure 6. Each estimated term value is the sum of the relevant fundamental frequencies (Table 1), except that the contribution of the very anharmonic torsional mode ν12 is computed differently. As shown in Table 2, the pattern of the torsional levels deviates substantially from the evenly spaced levels of a harmonic oscillator. Furthermore, for some vibrations, the torsional structure is inverted relative to the normal pattern.13,24,30 Because
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Figure 6. A comparison of estimated zeroth-order term values for combination-vibration states (a) to the observed K′ ) 0 upper state energies (b). The term values in part a are computed using the data of Tables 1 and 2. The zeroth-order vibrational states are arranged into tiers according to the coupling order relative to bright state. Tier 1 represents third-order coupling; tier 2 is fourth-order coupling; etc. In part a, the band origins are calculated using the harmonic approximation for all modes except the torsion, ν12. All of the zeroth-order combination vibrations are assumed to have the normal torsional structure, except where there is evidence of inverted torsional structure (refs 7, 10, 25, 26). For the states represented in part a, this affects only ν4 + 4ν12 and ν10 + 4ν12.
TABLE 2: Methanol E Species J ) 0 Vibrational Term Values (cm-1) torsional states nν12a
ma
normalb
ma
invertedb
0 ν12 2ν12 3ν12 4ν12 5ν12 6ν12
1 2 4 5 7 8 10
9.12 208.9 510.3 751.0 1396.0 1795.6 2745.9
0.5 2.5 3.5 5.5 6.5 8.5
0.0 269.7 484.9 1031.3 1389.7 2273.4
a
The correspondences of the ν12 vibrational quantum number n to the internal rotation angular momentum quantum number m are documented in ref 13. 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, and the torsional solutions, φi, that are antisymmetric in the torsional angle γ are used; that is, φi f -φi when γ f γ + 2π.13
of the strength of the torsion-vibration interaction, we can make only 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 considerable uncertainty. The uncertainty in the torsional energies is worst when the torsional excitation is comparable to the torsional barrier height (373 cm-1); that is, in the range ν12-3ν12. At high excitation, the torsion becomes nearly free internal rotation, and the torsional energies are reasonably well determined by the torsional inertial constant (F ) 27 cm-1). In Figure 6, the symmetric CH stretch fundamental ν3 is identified as the bright state, and the combination vibrations are arranged into tiers according to the coupling order relative to ν3. Generally, the vibrations in the higher tiers contain more quanta of torsion. The principal assumption involved in our interpretation of the spectra is that the ν3 CH stretch is the dominant source of intensity below 2900 cm-1. That is, we assume that all of the
other bands in this region only appear in the spectra because of the borrowed intensity that results from mixing with ν3. This is the usual assumption employed in many spectroscopic studies of IVR.31,32 In the present case of spectra in the fundamental region, this assumption might be questioned because there are nearby binary combinations (the tier 1 states) that could carry some of their own oscillator strength; however, there is some evidence to support our identification of ν3 as the bright state. First, we note that the fundamentals ν4, ν5, ν6, and ν10, which contribute to the binary combinations in the first tier, are weaker than ν3, and the intensities of overtones and binary combinations of mid-infrared fundamentals typically decrease by a factor of 100 or so for each additional quantum.33 An additional test comes from the spectra of CH3OD. In CH3OD, the OD bend is shifted down to 864 cm-1 from 1335 cm-1 for the ν6 OH bend in CH3OH.30,34 This means that the first tier vibrations, ν4 + ν6, ν5 + ν6, and ν10 + ν6, that are in near-resonance with ν3 in CH3OH (Figure 6) are shifted in CH3OD to about 2530 cm-1, which is out of resonance with ν3 and outside the tuning range of the present experiment. The disappearance of these first tier states from our CH3OD spectra is evident in Figure 2c. The gas phase FTIR spectrum of CH3OD15 does not reveal any bands in the 2400-2580 cm-1 range, which implies that the intrinsic absorption strength of these first tier combination vibrations is less than 1% of that for ν3. Therefore, to a good approximation, ν3 is the only zerothorder bright state in CH3OH in the region 2700-2900 cm-1. An additional complication in the interpretation of the IVR coupling in this region is the possible coupling to zeroth-order states above 2900 cm-1. Indeed, there is evidence from theory35,36 that such coupling does occur, particularly with the HCH bend overtone 2ν5 at about 2914 cm-1. The implication of such mixing for the present interpretation is that the effective bright state relevant to the 2750-2900 cm-1 region is not the pure symmetric CH stretch (ν3), but is a mixture involving binary combinations of the CH bends. We wish to determine the coupling pathways that link the bright state to the other zeroth-order states in Figure 6a. The 12 band origins in Figure 6b spanning 100 cm-1 are the rotationless vibrational states determined from our analysis of
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Figure 7. Stepwise deconvolution of the observed vibration spectrum (K′ ) 0). “a”, “b”, “c”, “d” are labels used to identify subgroups eigenstates.
the observed spectra. The corresponding 100 cm-1 region in Figure 6a contains 13 zeroth-order states. Given the uncertainty in calculating the zeroth-order energies, we estimate that there are a total of 12-15 zeroth-order states in the frequency range where we have observed 12 vibrational bands. This means that the bright state is coupled to almost all of the available vibrational states in the region of the observed spectrum. The vibrational selection rules in the CS point group for anharmonic coupling require that the total vibrational symmetry of the coupled states be the same. That is, A′ states can couple to A′, and A′′ to A′′. By this argument, about half of the zerothorder combination states in Figure 6a would be ineligible to couple with ν3. However, in the G6 molecular symmetry group, which is appropriate in the present circumstance involving largeamplitude torsional motion, anharmonic coupling to vibrations of both point group symmetries is allowed for the E-symmetry J′ ) 0 states that are the subject of this paper. The fact of 12 observed coupled vibrations, not limited to just the ∼6 A′ states, further demonstrates the inapplicability of the CS anharmonic selection rules for anharmonic coupling in methanol. The pattern of the experimental spectrum with a relatively intense band at the ν3 frequency, with the most intense perturbing states on the lower frequency side, is consistent with a primary interaction of ν3 with the tier 1 states. Coupling to the states in the higher tiers could occur either directly via higher order couplings or sequentially by successive third-order couplings from the bright states to the first tier, then to the second tier, etc. In CH3OD, where the first tier states are shifted out of resonance, there is no evident coupling to the higherorder tiers, and only the bright state is observed. The signalto-noise in Figure 2c allows us to place an upper limit on the intensity of perturbing states in CH3OD at 2% of the relative intensity of the strongest perturbing state in CH3OH. This analysis confirms a sequential coupling mechanism via the first tier. The first tier states, which are the binary combinations of one of the three CH bends (ν4, ν5, and ν10) with the OH bend ν6, act as doorway states that connect the bright state to the interior states in tiers 2-5. From the observation of 12 coupled vibrations, we deduce that there is coupling to at least 8 interior states in tiers 2 and higher. We see from Figure 6a that there are three nearly resonant states in tier 2 that are related to those in tier 1 by the exchange of 1 quantum of ν6 for one quantum of the out-ofplane methyl rock ν11 plus one of torsion ν12. The analysis of
high-resolution FTIR spectra30 showed that the ν6 fundamental mixes with the first torsionally excited state built on the inplane methyl rock (ν7 + ν12); therefore, we expect ν6 to couple with both torsion-rock combinations, ν7 + ν12 and ν11 + ν12. The former is forbidden for an anharmonic coupling in the Cs point group, but both are allowed in the G6 molecular symmetry group. This would allow the coupling of about four zeroth-order states in tier 2 that fall within the range of the observed spectrum: ν4 + ν11 + ν12, ν10 + ν11 + ν12, ν5 + ν11 + ν12, and ν4 + ν7 + ν12. To reach a total of 12 coupled vibrations, we need coupling to about four additional coupled states from tiers 3, 4, and 5. These higher-tier states have increasing number of torsional quanta, which makes the estimation of their zerothorder term values increasingly difficult. Consequently, we cannot determine whether or how much each of these higher-tier states is contributing to various of the observed eigenstates (Figure 6b). However, this information may be contained in the already completed quantum nuclear motion calculations on ab initio potentials.7,10 A careful comparison of such calculation to the present data may serve at once to test the accuracy of these calculations and also to uncover the detailed coupling pathways. The coupling from the first tier states into the higher tiers might occur by further stages of stepwise third-order couplings or by direct higher-order couplings. Since all of these couplings involve the addition of torsional quanta, the direct higher order couplings will fall off more slowly with coupling order than in rigid molecules,13,37 and consequently, they can be expected to be an important pathway for energy transfer. Likely both lowand high-order couplings have significant roles to play in the higher tiers. Some information about the stepwise coupling matrix elements connecting the states in Figure 6a can be obtained from a two-step Lawrance and Knight deconvolution28,38 of the observed vibrational spectrum (Figures 6b and 7). For this purpose, we divide the coupling into two stages: first, the coupling of the bright state to the first tier states, which results in first-order states that are mixtures of these states. The firstorder states then couple to the “bath” consisting of the nearly resonant states in the higher tiers. In Figures 4 and 7, there is a tentative identification of subgroups of vibrational eigenstates, labeled as “b”, “c”, and “d”, which might represent the coupling of each first-order state to the bath. This identification is reasonable given the consistency of the pattern through the K′ ) -1, 0, and +1 rotational levels, but is really just a guess.
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Figure 8. The time scale of IVR: Decays curves for the ν3 CH stretch calculated from spectra assuming a single bright state.
The two-step deconvolution procedure is described in more detail in the Supporting Information. This approach provides estimates of the matrix elements for coupling of the bright state to the first tier states: 6 ( 4 cm-1 for “b” and 16.5 ( 1.1 cm-1 for “c” and “d”. It also provides a rough estimate of the average matrix element for coupling of the “c” and “d” first tier states to the bath: 2 ( 1 cm-1. Because of the assumptions involved in the stepwise deconvolution, these numbers should be taken as only qualitative estimates of the matrix elements involved. e. Time-Dependent Dynamics. The present data contain information about the IVR time scales that apply in this spectral region. A hypothetical short-pulse coherent excitation of the 2750-2900 cm-1 region would prepare the nonstationary state vibration that we call the bright state.32,39 The bright state decays in time because of the dephasing of the constituent eigenstates. Survival probabilities for the bright state (Figure 8) are calculated from the reduced term values, νi, and intensities, Ii (Figure 4a), using the standard formula.40 The initial decay occurs in about 200 fs and is followed by a partial recurrence near 1 ps and then additional recurrences every ∼1 ps. The long time average survival probability is given by the dilution factor φd ) (∑i Ii2)/(∑i Ii)2, which is 0.41, 0.33, and 0.50 for K′ ) -1, 0, and +1 respectively. For the initial fast decay, the 1/e decay times relative to these long-time limits40 are 150, 150, and 190 fs respectively. A slower time scale (∼1 ps), evident from the diminished size of the first recurrence peak, results from the coupling of the first tier states to the higher tiers. The existence of unassigned or undetected features that also borrow intensity from ν3 would make dilution factors lower and would also affect the decay times. The lower dilution factor for K′ ) 0 than for K′ ) +1, probably just reflects the fact that we have more data for K′ ) 0 and have done a better job of observing and assigning weak transitions. Thus, we conclude that the IVR lifetimes and dilution factors are roughly the same for K′ ) -1, 0, and +1, which is a signature expected for anharmonic coupling. Strong a-type Coriolis coupling in the ν9 region has been observed,41 but it is not evident in the ν3 region. If a-type Coriolis coupling involving the bright state were active, the IVR lifetime would be shortened, and a-type Coriolis coupling involving any of the tiers would reduce the dilution factor reduced for K > 0.42 If b- or c-type Coriolis coupling were active, one would expect greater fractionation, and possibly faster IVR, at high J′; but this is not observed here for J′ up to 4. The analysis of the vibrational spectrum in the ν3 (CH symmetric stretch) region shows that IVR occurs on two distinct
Twagirayezu et al. time scales. The initial redistribution takes place in ∼200 fs and is caused by the strong coupling of the CH stretch to the set of combination bands formed from the OH bend (ν6) and one of the CH bends (ν4, ν5, ν10). All of these strongly coupled states include one quantum of vibrational excitation in the OH bend. The strongest spectroscopic evidence that anharmonic coupling to these vibrational states causes the initial fast IVR is the lack of this time scale in the CH3OD dynamics. For CH3OD, the OH bend (ν6) frequency is lowered by about 470 cm-1 while the CH3 bend frequencies (ν4, ν5, and ν10) are essentially unchanged (Table 1). The isotopic substitution selectively detunes the resonance between the ν3 (CH symmetric stretch) and OH bend + CH bend combination bands, and the resulting spectrum no longer displays perturbations over a broad spectral width that are the frequency-domain signature of fast IVR. The second stage of intramolecular vibrational redistribution takes place on the 1-2 ps time scale. This time scale is identified through the damping of the coherent population oscillations shown in Figure 8. A sequential Lawrance-Knight analysis28 of the frequency-resolved spectrum also reveals this second 1-2 ps time scale as documented in the Supporting Information. From the vibrational level diagram in Figure 6, it is seen that the remaining zeroth-order vibrational states involve excitation in the torsional mode (ν12). In particular, there is a near-resonant path to the set of vibrational states labeled Tier 2 that involve conversion of the OH bend quantum (ν6) into a combination of a CH3 rock (ν7 or ν11) and one quantum of excitation in the torsion (ν12). Beginning in the next tier, excitation appears in the C-O stretch (ν8). In three of the four near-resonant vibrational states with C-O stretch excitation, there is no remaining excitation in the COH bend (ν6). The tier model structure suggests, therefore, that the second time scale of IVR predominantly involves redistribution of vibrational energy from the OH bend into the CH3 rock and torsional modes with some excitation of the C-O stretch also expected. f. Comparison to Liquid Phase Time-Dependent Experiments. The vibrational dynamics of the ν3 CH symmetric stretch have also been studied in liquid and solution using time-resolved Raman spectroscopy.43 In these measurements, the use of a Raman probe following picosecond IR excitation of the CH symmetric stretch made it possible to follow the subsequent population of the vibrational modes of methanol (although the low-frequency torsional mode could not be monitored). By adding a reporter molecule (CCl4) to the neat liquid, the thermal energy in the solvent was also be monitored with picosecond time resolution. The current analysis of the frequency-domain spectrum of methanol sets the stage for a direct comparison of the gas and liquid dynamics to assess the role that the intrinsic intramolecular dynamics play in the liquid. We point out that at the time of the time-domain measurement, much of the literature on gas-phase IVR using frequency-domain spectroscopy had focused on molecules that were only weakly coupled totheintramolecularbath,mostnotablytheterminalacetylenes.39,44-46 From this work, a rough rule-of-thumb had emerged that IVR would be observed when the vibrational state density reached about 10-100 states/cm-1.31,47,48 As seen in Figure 6, the vibrational state density of methanol is only 0.125 states/cm-1 in the vicinity of ν3. Therefore, the previous analysis of the liquid phase dynamics was based on the assumption that all vibrational relaxation was caused by intermolecular interactions. The present experiments show that methanol is a much more strongly
Vibrational Coupling Pathways in Methanol coupled system and that significant IVR is observed for the isolated molecule. There is excellent correspondence in the vibrational relaxation of gas and liquid methanol. The liquid work identified two fast time scales for the vibrational energy redistribution of liquid methanol.43 On a time scale
