A QUANTUM BAND MODEL OF THE ν₃ FUNDAMENTAL OF METHANOL (CH₃OH) AND ITS APPLICATION TO FLUORESCENCE SPECTRA OF COMETS

For the ground vibrational state (Figure 1), we adopted the energies E''(J, K, T_s) reported by Mekhtiev et al. (1999) and Xu et al. (2008), and used them to estimate the parameters for the upper vibrational state (ν₃). Using the ν₃ parameters reported by Hunt et al. (1991) for some K sub-bands, and additional line identifications determined by Xu, we searched for deviations of the energy structure with respect to the ground state, and created progressions according to τ leading to slower and smoother variation with K. The torsional index (τ = 1, 2, 3) is related to the aforementioned torsional symmetry T_s by the following rule: mod(K + τ)/3 = 0 for E₁, 1 for A, and 2 for E₂. The torsional index (τ), associated with symmetry, should not be confused with the torsional quantum number (ν_t). The torsional energies for the ground state (Figure 2) were estimated using the following relation:

$$\begin{eqnarray} E\left({K,\tau } \right)_{{\rm tors}} &=& E\left({J = K,\tau } \right) - \left[ {\left({\frac{{B + C}}{2}} \right)J\left({J + 1} \right)} \right] \nonumber\\ &&- \left[ {\left({A - \frac{{B + C}}{2}} \right)K^2 } \right], \end{eqnarray} \tag{ 1 }$$

where A, B, and C are the main rotational constants for the ground vibrational state (A = 4.2537233, B = 0.8236523, C = 0.7925575; Xu et al. 2008). This approximation neglects high-order effects and perturbations, but it does provide a useful representation of the general variations of energy for each torsional mode (Figure 2). We obtained experimental fits to the torsional energies, in a fashion similar to that of Wang & Perry (1998), using

$$\begin{equation} E\left({K,\tau } \right)_{{\rm tors}} = c_0 + c_1 K + c_2 K^2 + c_3 \cos \left({c_4 K + (2\pi /3)(\tau - 1)} \right), \end{equation} \tag{ 2 }$$

where c_0–4 are the five coefficients describing the torsional energy pattern. Using this equation and considering all torsional sub-bands having K ⩽ 17, we derived the five constants of Equation (2) (c_0–4; see Table 2). The values are defined with respect to the bottom of the potential well, which (following Mekhtiev et al. 1999) we consider to be 128.10687 cm^‑1 below the first rotational level (J = 0, K = 0, A⁺).

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For the upper vibrational state, we created differential progressions in J for every K sub-band based on experimental data and extrapolated to other sub-bands using the torsional diagram. The differential progressions are defined as

$$\begin{eqnarray} E^{\prime}(J,K,\tau) &=& E^{\prime\prime}(J,K,\tau) + \nu _0\nonumber\\ && + \left[ {\delta \nu (K,\tau) + \delta B(K,\tau)J(J + 1)} \right], \end{eqnarray} \tag{ 3 }$$

where ν₀ is the band center (2844.2 cm⁻¹), δν is the torsional displacement of the K sub-band with respect to the torsional–rotational energy of the same K sub-band in the ground state, and δB is the differential rotational constant for a K sub-band. Using 709 ro-vibrational assignments of the ν₃ band, we retrieved 19 origins and 19 differential rotational constants for sub-bands with K' ⩽ 6 (see Table 1). For all other K sub-bands (marked with an asterisk ''*'' in Table 1), δB was set to ‑0.0048371871, which is the median of the 19 extracted values.

The definition of δν beyond the measured sub-bands is extremely difficult without a full Hamiltonian that includes perturbations and interactions with other vibrational and torsional states. In addition, the empirical data are limited to values that span K' ⩽ 6, and some of the measurements indicate strong perturbations (e.g., K' = 3). We extrapolated to other K ladders, by fitting the five coefficients (c_0–4; see Equation (2)) of the torsional diagram to the measured torsional band centers (δν, see Table 2). In the fit, we excluded the perturbed values for K' = 3. We thus define the torsional–rotational displacement for those K sub-bands having no experimental data to be δν = E'_tors–E''_tors, where E'_tors was calculated using Equation (2) with (fitted) c_0–4. E''_tors was calculated using Equation (1) with values from Mekhtiev et al. (1999) and Xu et al. (2008). See synthetic spectra and energy spectral distribution at 100 K in Figure 3.

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Even though great advances have been made in characterizing the energy structure of methanol, current spectroscopic databases differ greatly in their description of the partition functions, and this could ultimately lead to significant errors in the computed band intensities. For example, the HITRAN database tabulates Z_tot = 82,469 at 296 K in the total partition table (''parsum.dat''), and 35,314 at 296 K in the molecular table (''molparam.txt''). At 300 K, the JPL Molecular Spectroscopy database for methanol (Pearson & Xu 2010) reports Z_tot = 37,892 (after correcting for a spin multiplicity of 4), while the analytical method of Dang-Nhu et al. (1990) estimates Z_tot = 35,065 at 296 K. Furthermore, none of these sources lists partition functions at 1 K intervals in the 10–400 K temperature range, required for computing fluorescence efficiencies for comets and absorption coefficients for planetary atmospheres. We found the method of Dang-Nhu et al. (1990) to be appealing because of its simplicity and possible high-dynamic range. Nevertheless, when we compared results obtained using this method with values derived using more rigorous calculations, we were discouraged by the large differences, in particular in the low temperature range.

In order to compute partition functions we combined the energy characterization obtained by Mekhtiev et al. (1999) for the ground state (which spans ν_t ⩽ 8 and J ⩽ 22) with a simplified Hamiltonian for high energies (J > 22). We assumed that the torsional–rotational structure of the ground state also applies to all other vibrational modes (see vibrational energies in Dang-Nhu et al. 1990) and that the rotational levels with J > 22 follow the progression listed in Equation (1). Using this method we created 2,193,894 levels (ν_t ⩽ 8 and J ⩽ 100), in addition to the 9522 levels listed in Mekhtiev et al. (1999), and we computed the partition function by explicitly summing over all energy levels generated by this calculation (Table 3). Our partition functions are in excellent agreement with the latest numbers given by Pearson & Xu 2010 (see revision after 2011 November, now listed on the JPL Molecular Spectroscopy Web site)—the differences are typically smaller than 0.2% (Figure 4).

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The most recent report of the integrated intensity for the ν₃ band of methanol is more than 30 years old, when Rogers (1980) performed a comprehensive study of infrared intensities of alcohols and ethers. Band intensities from this work were used by Reuter (1992) and Bockelée-Morvan et al. (1995; referred to as BM95 hereafter) to derive cometary fluorescence efficiencies, although the authors differ in their integrated fluorescence efficiencies (g_band) by ∼20%. Considering the formalism by Crovisier & Encrenaz (1983), we derive g_band = 1.47 × 10⁻⁴ s⁻¹, consistent with the value presented in BM95. However, the derivation of the intensity of the ν₃ fundamental presented in BM95 is only valid for the low temperatures (T < 100 K) typically measured in comets, i.e., for temperatures at which torsional and vibrational excitations (and thus hot bands) are negligible (see below).

The laboratory measurements of Rogers (1980) consisted of integrating over the complete C–H stretching region (2650–3175 cm⁻¹), which was further divided into the different vibrational modes (ν₂, ν₃, ν₉) by considering the following relative contributions: fν = 28%, 22%, 50%, respectively. We compared the C–H value in Rogers (1980) (127 ± 10 (km mol⁻¹) or 211 × 10⁻¹⁹ (cm molecule⁻¹)) with recently calibrated spectra (Sharpe & Sams 2000) of the Pacific Northwest National Laboratory (PNNL) and obtained excellent agreement (Table 3). However, these laboratory measurements were performed at room temperature, where torsional and hot bands are particularly active and therefore contribute significantly to the observed intensities. The first torsional state lies just 294.45 cm⁻¹ above the ground state. From our partition function calculations, we estimate the contribution of hot bands to be f_hot = 33% at 296 K, similar to the findings of 31% by Xu et al. (2004) for the C–O stretching region. Taking the intrinsic intensity of the hot bands to be the same as the fundamental, the contribution of the hot bands is estimated to be f_hot = 1–(Z_vibZ_tors)⁻¹. Accordingly, we estimate the intensity of the ν₃ fundamental to be S_v3(296 K) = (f_v3S_CH)(1 − f_hot) = 31 × 10⁻¹⁹ (cm molecule⁻¹); see Table 3.

We computed line intensities (S_v) for all allowed transitions connecting upper and lower states (ΔJ = J'–J'' = +1, 0, ‑1; ΔK = 0; A ± ↔ A± (ΔJ ≠ 0), A± ↔ A ∓ (ΔJ = 0), E ↔ E) as follows:

$$\begin{eqnarray} S_v (T = 296{\kern 1pt} \,{\rm K}) &=& (v/v_0)L_{{\rm HL}} S_{v3} (T)\,[ {1 - \exp (- hcv/kT)} ]\nonumber\\ &&\times [ {w^{\prime\prime}\exp (- E^{\prime\prime}hc/kT)}]/Z_{{\rm rot}} (T), \end{eqnarray} \tag{ 4 }$$

where ν is the line frequency (E'–E'' (cm⁻¹)), L_HL is the Hönl–London factor, w'' is the statistical weight of lower state (=4(2J''+1), including the spin multiplicity of 4), and Z_rot(T) is the rotational partition function (24,558 at 296 K). The Hönl–London factors for a parallel band are defined as (Herzberg 1945, p. 422):

$$\begin{eqnarray} \Delta J &=& 1\qquad L_{{\rm HL}} = \frac{{(J^{\prime\prime} + 1)^2 - K^{\prime\prime 2} }}{{(J^{\prime\prime} + 1)(2J^{\prime\prime} + 1)}} \nonumber\\ \Delta J &=& 0\qquad L_{{\rm HL}} = \frac{{K^{\prime\prime 2} }}{{J^{\prime\prime}(J^{\prime\prime} + 1)}} \\ \Delta J &=& - 1\qquad L_{{\rm HL}} = \frac{{J^{\prime\prime 2} - K^{\prime\prime 2} }}{{J^{\prime\prime}(2J^{\prime\prime} + 1)}}. \nonumber \end{eqnarray} \tag{ 5 }$$

This approximation for the computation of line intensities neglects high-order interactions coupling torsional and vibrational modes, which have been reported to be active in the 3 μm spectral region (Brooke et al. 2003; Xu et al. 1997). Einstein-A coefficients were computed following Šimečková et al. (2006):

$$\begin{equation} A_{21} = \frac{{8\pi cv^2 Z_{{\rm tot}} (T)S_v (T)}}{{[ {1 - \exp (- hcv/kT)} ][ {\exp (- E^{\prime\prime}hc/kT)} ]I_a w^{\prime}}}, \end{equation} \tag{ 6 }$$

where Z_tot(T) is the total internal partition sum (36,761 at 296 K), I_a is the isotopic abundance (I_a = 0.98593 for normal CH₃OH), and w' is the statistical weight of the upper state.

Following the above guidelines, we computed 2948 spectral lines with J_max = 23 in the 2802 cm⁻¹ to 2892 cm⁻¹ frequency range. The database has been organized according to the HITRAN 2008 format with line identifications following the G₆ point group model, which is accessible by our GFM and planetary radiative transfer models (LBLRTM and MRTM; Clough et al. 2005; Villanueva et al. 2011). The vibrational indicators are V3, GROUND; while the local quantum numbers are J, K, and T_S. Line shape coefficients were defined following Xu et al. (2004): air-broadened half-width γ_air = 0.100 cm⁻¹ atm⁻¹, self-broadened half-width γ_self = 0.400 cm⁻¹ atm⁻¹, temperature-dependence exponent η_air = 0.75, and pressure-induced line shift δ_air = 0.00 cm⁻¹ atm⁻¹.

We applied our quantum band model to high-resolution (λ/Δλ ∼ 25,000) infrared spectra of comet C/2001 A2 (LINEAR) (A2 hereafter), acquired in 2001 July with NIRSPEC (Near-InfraRed Echelle SPECtrograph; McLean et al. 1998) at the Keck II Telescope in Hawaii. In 2001 July, comet A2 displayed intense lines of CH₃OH (Figure 5), owing to its particularly rich organic chemistry (Magee-Sauer et al. 2008). The abundances of ethane, acetylene, hydrogen cyanide, and methanol were enriched by factors of 2–3 with respect to most comets in our database (e.g., Mumma & Charnley 2011; DiSanti & Mumma 2008).

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From A2 spectra acquired on 2001 July UT 9.5, we retrieved a global production rate [Q(CH₃OH)] of (11.3 ± 0.04) × 10²⁶ s⁻¹ and a rotational temperature of 78 ± 5 K (using a chi-square minimization technique). The revised production rate is smaller by about 20% than the published value of (14.5 ± 0.4) × 10²⁶ s⁻¹ (Magee-Sauer et al. 2008), owing mainly to the revised g-factor for the integrated Q-branch. A re-analysis of the H₂O, C₂H₆, and HCN spectra using our new analytical methods, reveals similar temperatures (T_rot near 80 K) for these three species and consistent with the methanol value. Further details on the application of our new water model to this data set will be discussed elsewhere.

Our new production rate (Table 4) is very consistent with the value retrieved the previous night (2001 July UT 8.3) through radio observations (Biver et al. 2006), further validating our revised analytical methodology for computing non-LTE fluorescence efficiencies for this IR band system. We also explored the possibility of retrieving the nuclear spin temperature (T_spin) from these data, but uncertainties in line-by-line intensities introduced by the symmetric rotor approximation and by current unknowns of the CH₃OH energy structure for K > 6 could introduce systematic uncertainties associated with such retrievals. Formally, our results indicate T_spin > 18 K for CH₃OH, consistent with the findings of T_spin = 23⁺⁴₋₃ K for H₂O in comet A2 (Dello Russo et al. 2005), and with T_spin derived for methanol in comet Hale–Bopp (T_spin > 15–18 K; Pardanaud et al. 2007).

Comet C/2004 Q2 (Machholz) (Q2 hereafter) was particularly active in 2005 January, with a very high water production rate (∼3 × 10²⁹ mol s⁻¹; Bonev et al. 2009; Kobayashi & Kawakita 2009). Like A2, Q2 presented a rich chemistry. On 2005 January 19, we obtained 8 minutes of integration with the KL1 setting of NIRSPEC, which samples the ν₃ band of methanol in order 22. As shown in Figure 6, the methanol lines are noticeably strong and thus pose an excellent test for our newly developed model.

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For 2005 January 19, we retrieve a methanol production rate of (4.2 ± 0.2) × 10²⁷ mol s⁻¹ and a rotational temperature of 75 ± 5 K. The rotational temperature is similar to that derived for other molecules (78–83 K) by Kobayashi & Kawakita (2009). Considering the co-measured Q(H₂O) derived by Bonev et al. (2009) of 2.7 × 10²⁹ mol s⁻¹, we obtain a methanol mixing ratio of 1.54% ± 0.07%. Kobayashi & Kawakita (2009) did not sample the ν₃ band of methanol in their survey, but they retrieved methanol production rates from certain ν₉ methanol features in the 2890–2934 cm⁻¹ using empirical g-factors. The ν₉ empirical g-factors were derived by scaling the observed fluxes measured in comet Lee (Dello Russo et al. 2006) and by considering a ν₃ Q-branch g-factor of 2.17 × 10⁻⁵ s⁻¹ derived from BM95. On the other hand, our new model predicts the g-factor of the ν₃ Q-branch to be 1.55 × 10⁻⁵ s⁻¹ at 70–80 K; see Table 3. The difference mainly originates because some K-ladders (e.g., K' = 3) are heavily perturbed and their Q-lines (ΔJ = 0) do not fall within the region normally defined as the Q-branch for this band system (2843.5–2845.0 cm⁻¹). The fact that the new model considers a realistic torsional–rotational structure for the ν₃ vibrational state solves this problem (see comparison in Figure 3 and line identifications in Figures 6 and 7). Correcting the methanol value reported by Kobayashi & Kawakita (2009) with a scaling factor of 1.4 (i.e., 2.17/1.55) brings their published mixing ratio (1.65% ± 0.09%) into agreement with our value (see Table 4).

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The data available for comet 8P/Tuttle (8P hereafter) are extremely rich, since the comet was observed using two powerful high-resolution spectrometers (NIRSPEC/Keck II and CRIRES/VLT), by two independent teams and at different times during its apparition in 2007–2008. We re-processed our methanol data acquired with NIRSPEC (on 2007 December 22; Bonev et al. 2008) and CRIRES (on 2008 January 29; Böhnhardt et al. 2008), and compared the revised results with those acquired on 2008 February 4, with CRIRES (Kobayashi et al. 2010). The new model provides excellent agreement with 8P data recorded at high spectral resolutions (Figure 7), even when compared to data acquired using the narrowest slit in CRIRES (0.2''), which delivers an extremely high resolving power (λ/Δλ ∼ 98,000). Of particular significance is the high level of agreement between model and data near the Q-branch, which is the region used by astronomers to derive g-factors and whose differing models led to the disagreement among methanol values obtained for the same comet.

Our originally reported methanol production rates for 8P were reduced by ∼20% when the new model was applied, leading to a revised methanol abundance of 1.9%–2.5% with respect to water (see Table 4). The measurements of methanol by Kobayashi et al. (2010) targeted certain ν₂ lines, but this time the empirical ν₂ g-factors were scaled relative to the ν₃ Q-branch after adopting a g-factor of 1.17 × 10⁻⁵ s⁻¹ (DiSanti et al. 2009). Herewith they too should be revised downward by 20% (see Table 4). Their revised methanol abundance (2.52% ± 0.34%) then agrees very well with our value (2.51% ± 0.19%) measured a week earlier. The improved confidence levels (by removing modeling error) also suggest that the mixing ratio of methanol did change from December 22 to January 29.

The use of different g-factors to extract methanol abundances in comets became particularly problematic in 103P/Hartley 2, as two teams presented very different methanol abundances in this comet (Mumma et al. 2011; Dello Russo et al. 2011), even though most of the other abundances were consistent, and both teams used the same spectroscopic technique (NIRSPEC at Keck II) to sample methanol in this comet (ν₃ band of methanol). As nicely summarized in Meech et al. (2011), the methanol abundances differed by a factor of ∼2. If we correct the abundances by applying the new g-factor for the methanol Q-branch of 1.55 × 10⁻⁵ s⁻¹, both measurements now indicate a mixing ratio of ∼1.8% (see Table 4). More importantly, the new developed model not only brings consistency to the values of methanol measured in comets using different techniques and methods, but allows us to better understand this important spectral region (where many hydrocarbons are radiatively active) and it provides us with a robust tool to retrieve accurate rotational temperatures for this molecule from infrared spectra. The comparison of rotational temperatures among various species permits inferences regarding their common physical distributions and is needed when determining nuclear spin temperatures from the respective populations of individual nuclear spin species.