E The Astrophysical Journal Supplement Series, 142:145–151, 2002 September # 2002. The American Astronomical Society. All rights reserved. Printed in U.S.A. ACETONE: LABORATORY ASSIGNMENTS AND PREDICTIONS THROUGH 620 GHz FOR THE VIBRATIONAL-TORSIONAL GROUND STATE Peter Groner Department of Chemistry, University of Missouri–Kansas City, Kansas City, MO 64110-2499 Sieghard Albert,1 Eric Herbst,2 and Frank C. De Lucia Department of Physics, Ohio State University, Columbus, OH 43210-1106 Frank J. Lovas Optical Technology Division, National Institute of Standards and Technology, Gaithersburg, MD 20899-8441 and Brian J. Drouin and John C. Pearson Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109-8099 Received 2002 January 23; accepted 2002 April 16 ABSTRACT The acetone molecule has a complex rotational spectrum arising from the internal rotation of two methyl tops and their interaction with each other and the rigid-body rotation of the molecule. Here we report the measurement and assignment of a large number of new rotational-torsional transitions of acetone in the vibrational-torsional ground state at frequencies of up to 620 GHz. The newly measured lines have been combined with previously measured transition frequencies and fitted by a procedure used successfully for dimethyl ether. From the fit to the global data set, we have extracted parameters that have enabled us to predict accurately the frequencies and intensities of over 10,000 lines of possible astronomical importance at frequencies of up to 620 GHz. The detection of interstellar acetone was first reported by Combes et al. based on two transitions with partially resolved internal rotation structure. The present work will aid further interstellar studies to verify acetone as an interstellar constituent. Subject headings: ISM: molecules — molecular data — molecular processes — radio lines: ISM On-line material: machine-readable tables dant interstellar ions. Making what appeared to be reasonable assumptions about the rates of the reactions and the amount of methyl ion, Combes et al. (1987) were able to show that the calculated abundance ratio of acetone to acetaldehyde is approximately equal to what they surmised to exist in Sgr B2. The rate of reaction (1), however, had not yet been studied in any detail. In 1990, Herbst, Giles, & Smith (1990) investigated the radiative association between methyl ions and acetaldehyde theoretically, and they successfully tested their theory with experimental results on the analogous ternary reaction in which the collision complex is stabilized by an inelastic collision with a third body. They found that reaction (1) is much slower than previously estimated and that the amount of interstellar acetone produced by the sequence of reactions (1) and (2) is far too little to explain the results of Combes et al. (1987). More recently, interferometric studies of Sgr B2 have shown that saturated (hydrogen rich) oxygen- and nitrogencontaining organic molecules are typically, but not universally, confined to a particular region associated with highmass star formation known as Sgr B2(N-LMH) that was originally called the ‘‘ compact region in Sgr B2(N) ’’ (Miao et al. 1995; Mehringer et al. 1997; Pei, Liu, & Snyder 2000). Sgr B2(N-LMH) is one of many small regions rich in saturated organic and inorganic molecules embedded in giant interstellar clouds. Known as hot cores, these condensates possess gas densities and temperatures in excess of standard values—densities are typically 106 cm3 rather than 103–104 cm3, and temperatures are closer to 300 K than to 10 K 1. INTRODUCTION Acetone (CH3COCH3), a molecule with two internal rotors, was first reported to be in the interstellar medium by Combes et al. (1987), who tentatively detected weak and blended rotational lines in the source Sgr B2. They estimated a column density of 5
1013 cm2, corresponding to an average fractional abundance with respect to H2 of 5
1011 . The suggestion that acetone had been detected in Sgr B2 was given some further credence by Snyder et al. (1997). Very recently, Snyder et al. (2002) reported the detection of 13 new acetone emission features assigned to 20 acetone transitions based on the measured and calculated frequencies of the present study. Combes et al. (1987) suggested that their estimated abundance of acetone in Sgr B2 can be explained by gas-phase formation and destruction pathways. Formation occurs by the radiative association reaction between methyl ion and interstellar acetaldehyde, þ CHþ 3 þ CH3 CHO ! CH3 COCH4 þ h ; ð1Þ followed by dissociative recombination with electrons,  CH3 COCHþ 4 þ e ! CH3 COCH3 þ H ; ð2Þ while destruction occurs by reaction with a variety of abun1 Present address: Laboratory for Physical Chemistry, ETH Zurich, HCI Hönggerberg, CH-8093 Zurich, Switzerland. 2 Also at the Department of Astronomy, Ohio State University, Columbus, OH 43210-1106. 145 146 GRONER ET AL. (Blake et al. 1987; Sutton et al. 1995; Ikeda et al. 2001; Liu, Mehringer, & Snyder 2001). Although more distant than the better studied hot-core sources in Orion (the Hot Core and Compact Ridge), Sgr B2(N-LMH) is a good object for the detection of complex molecules because it has so much material. Whether or not acetone will prove to be a standard hot-core molecule or to extend over a wider region like glycolaldehyde (Hollis et al. 2001) is not yet clear; the results of Snyder et al. (1997), as well as the new results (Snyder et al. 2002), indicate that acetone is more abundant in the direction toward Sgr B2(N-LMH). The picture that has emerged concerning molecular formation in hot-core regions is one in which grain surface chemistry plays a major role in a previous era of low temperatures (10 K) before the onset of star formation. It is necessary to invoke surface processes because gas-phase ionmolecule chemistry is generally unable to produce saturated species in sufficient abundance (Brown, Charnley, & Millar 1988). At low temperatures, hydrogen atoms, which migrate rapidly upon sticking to surfaces, can probably convert condensed phase CO into methanol (Charnley, Tielens, & Rodgers 1997); other surface processes leading to even more complex saturated species have been discussed (Brown 1990; Hollis & Churchwell 2001). The grain mantles are eventually lost via thermal evaporation from rising temperatures due to star formation or by sputtering caused by intermittent shock waves. If methanol is the most complex species formed on the grains, it can then be at least partially converted into more complex species in the gas (Blake et al. 1987; Millar, Herbst, & Charnley 1991; Charnley, Tielens, & Millar 1992; Caselli, Hasegawa, & Herbst 1993; Charnley et al. 1995). This picture of the chemistry of hot-core regions has not yet been put on a quantitative basis because of uncertainties in the surface and gas-phase chemistry of production of molecules more complex than methanol and because of an incomplete understanding of the role of shock waves. It is therefore important to look carefully at a saturated species such as acetone and determine how its abundance varies spatially compared with other oxygencontaining organic molecules found in Sgr B2, including HCOOH (formic acid), CH2 OHCHO (glycolaldehyde), HCOOCH3 (methyl formate), and CH3 COOH (acetic acid). To observe acetone unambiguously in space requires knowledge of its spectrum on the ground. The rotationaltorsional spectrum of acetone has been studied previously in the laboratory at microwave frequencies by a variety of groups (Weatherly & Williams 1952; Swalen & Costain 1959; Nelson & Pierce 1965; Peter & Dreizler 1965; White 1975; Oldag & Sutter 1992). In 1986, Vacherand et al. (1986) extended the frequency range of the measured spectrum to 300 GHz (a wavelength of 1 mm). We have been engaged in a program to measure the spectrum of acetone into the submillimeter-wave region of the spectrum. Here we report our completed work on the rotational spectrum of acetone in its vibrational-torsional ground (VTG) state. We have measured a large number of lines and blends through 600 GHz with three different methods and fitted these and most previously measured lines and blends to a high degree of precision. The rotational-torsional parameters determined in the fit have been used to predict the frequencies and intensities of a large number (11,000) of unmeasured lines through 620 GHz in frequency and J ¼ 60 in rotational quantum number. Vol. 142 2. THEORY To fit the data and to predict the spectrum of acetone, an effective rotational Hamiltonian for molecules with two periodic large-amplitude motions (Groner 1997) was used. Its application to the rotational spectrum of dimethyl ether has been described (Groner et al. 1998). From a theoretical standpoint (effective rotational Hamiltonian and group theory), acetone and dimethyl ether are exactly alike. Both have the b principal inertial axis coinciding with the C2 symmetry axis of the molecule, and both have two equivalent methyl internal rotors whose axes are in the a-b principal inertial plane. Therefore, the calculations for acetone were performed in exactly the same manner with the same program. The principal difference between these molecules occurs in the size of the barrier to internal rotation of a single methyl group. For dimethyl ether and acetone, the barriers are 903.4(70) cm1 (Lovas, Lutz, & Dreizler 1979) and 251.4(26) cm1 (Groner 2000), respectively (type A uncertainties with coverage factor k ¼ 1 are given in parentheses). As a consequence, the effects of the interaction between internal and overall rotation are much larger in acetone. In the theoretical treatment of these interactions, the symmetry numbers 1 and 2 have been used to distinguish sets of internal rotation basis functions. Because of the threefold periodicity of the internal rotations of the methyl groups in acetone, the symmetry numbers k can assume any of the values 0, 1, or 2. They are related to the conventional designations of the torsional substates derived from the original investigation of the symmetry properties (Myers & Wilson 1960). Of all possible combinations of 1 and 2 , (1 ; 2 Þ ¼ ð0; 0Þ is nondegenerate and corresponds to the conventional AA notation for one of the torsional substates. The combinations ð0; 1Þ, ð1; 0Þ, ð0; 2Þ, and ð2; 0Þ are fourfold degenerate and correspond to the EE substate. The doubly degenerate pairs ð1; 1Þ, ð2; 2Þ and ð1; 2Þ, ð2; 1Þ correspond to the AE and EA substates, respectively. Because of the interactions, the substates have slightly different rotational energy levels. This leads to rotational transitions that are split into four different components, one for each substate. These quartets are not always completely resolved; the components ð1; 1Þ and ð1; 2Þ, in particular, blend very often into a single band resulting in characteristic triplets for many transitions. For dimethyl ether, the resulting quartets are either collapsed into a single blended line or spread out over up to 10 MHz. For acetone, the width of the quartet splitting extends from a few megahertz to more than 1 GHz. As a consequence of the symmetry properties, the different rotational quantum numbers and torsional substates have different spin statistical weights, which are factors that take into account the degeneracies of the torsional substates and the allowed nuclear spin functions. In the ð0; 0Þ substate the spin weights are 6 (Ka Kc : ee $ oo) and 10 (Ka Kc : oe $ eo), in the ð0; 1Þ substate the spin weight is 16, in the ð1; 1Þ substate the spin weights are 2 (Ka Kc : ee $ oo) and 6 (Ka Kc : oe $ eo), and in the ð1; 2Þ substate the spin weight is 4 (Myers & Wilson 1960). 3. EXPERIMENT AND SPECTRAL ANALYSIS The experiments at Ohio State University were accomplished with the Fast Scan Submillimeter Spectroscopic Technique (FASSST) system discussed by Petkie et al. No. 1, 2002 ACETONE (1997). This technique uses a voltage tunable backward wave oscillator (BWO) as the primary source of radiation. The BWO is scanned rapidly (104 MHz s1) through a large frequency range, and frequencies are calibrated optically by use of a Fabry-Perot cavity. There is no need for active frequency stabilization because of the fast sweep and data acquisition. The detectors are liquid helium–cooled InSb bolometers. With this system it is possible to measure literally thousands of spectral lines per second, with a frequency accuracy of 0.1 MHz. Spectral transitions of acetone were measured in the regions 260–345, 486–515, and 592–611 GHz. About 750 transition frequencies and blends (with rotational quantum number J of up to 60) belonging to the VTG state were assigned. The experiments at the Jet Propulsion Laboratory (JPL) were accomplished with a klystron-based spectrometer driving a harmonic generator. Tone burst modulation and a liquid helium–cooled InSb detector were used to produce and detect second-derivative line shapes. Friedl et al. (1995) described this spectrometer system in detail. Over 100 transition frequencies were measured in regions between 370 and 430 GHz, as well as in some gaps of the FASSST measurements between 290 and 316 GHz. At the National Institute of Standards and Technology (NIST), spectral measurements were carried out with a Fabry-Perot cavity, Fourier-transform microwave spectrometer (Lovas & Suenram 1987; Suenram et al. 1989) of the Balle-Flygare type (Balle & Flygare 1981). A pulsed solenoid valve was used to produce a supersonic molecular beam from a mixture of about 1% by volume acetone entrained in argon carrier gas at a total pressure of 100 kPa (1 atm) behind a 0.5 mm nozzle orifice and injected at the 147 midpoint of the Fabry-Perot cavity and perpendicular to the microwave standing wave field. Molecular beam pulses with 200–400 ls duration were employed with repetition rates of up to 35 Hz. The acetone beam was polarized by a short microwave pulse when the microwave frequency was near resonant (D < 400 kHz) with a rotational transition of acetone. The free induction decay signal from the cavity was digitized in 0.5 ls increments for 512 channels. Typically, 200 pulses were signal averaged, after a background microwave pulse was subtracted from each signal pulse, to yield signal-to-noise ratios of 100 or more. The averaged data were Fourier transformed to obtain the power spectrum in the frequency domain with a resolution element of 3.9063 kHz point1. Molecular transitions observed had line widths of 106 of the pump frequency, i.e., 10 kHz at 10 GHz, and the frequency measurement uncertainties were estimated to be 4 kHz in most cases (type A with coverage factor k ¼ 1), which was the resolution element for the digitization time described above. Some transitions exhibited partially resolved spin-spin hyperfine structure, which caused larger uncertainties in those measurements. A total of 32 frequencies between 11 and 25 GHz were measured with this technique. The lines of the VTG state measured during this investigation as well as lines reported by previous investigators (Peter & Dreizler 1965; Oldag & Sutter 1992; Vacherand et al. 1986) and 35 unassigned microwave frequencies from a NASA report (White 1975) have been combined into a global data set of 1175 frequencies and blends. These lines are listed in Table 1 along with the torsional substate, asymmetric top rotational quantum numbers J, Ka , Kc , and source. (Table 1 can be found in its entirety in the electronic TABLE 1 Assigned and Fitted Transition Frequencies of Acetone in the Vibrational-Torsional Ground State ta J0 Ka0 Kc0 J 00 Ka00 Kc00 Frequency (MHz) OC (MHz) Uncertaintyb (MHz) Sourcec 11........ 12........ 01........ 00........ 11........ 00........ 11........ 01........ 12........ 01........ 00........ 12........ 12........ 11........ 01........ 00........ 12........ 11........ 01........ 00........ 3 3 3 3 3 3 2 3 2 2 2 3 1 1 1 1 2 2 2 2 2 2 2 2 3 3 1 3 1 1 1 3 1 1 1 1 2 2 2 2 1 1 1 1 0 0 1 0 1 1 1 0 1 1 1 1 1 1 1 1 3 3 3 3 3 3 2 3 2 2 2 3 0 0 0 0 2 2 2 2 1 1 1 1 2 2 0 2 0 0 0 2 0 0 0 0 1 1 1 1 2 2 2 2 1 1 2 1 2 2 2 1 0 0 0 0 2 2 2 2 10730.960 10749.920 10751.662 10762.709 11073.970 11199.280 11252.502 11257.731 11265.255 11272.459 11286.144 11468.300 15038.511 15064.927 15074.075 15096.330 15615.732 15736.238 15750.855 15827.738 0.400 0.023 0.009 0.007 0.019 0.040 0.004 0.016 0.014 0.005 0.013 0.054 0.003 0.000 0.001 0.000 0.002 0.003 0.006 0.006 0.0 0.03 0.01 0.01 0.03 0.03 0.004 0.01 0.004 0.004 0.004 0.03 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 O P O O P P N O N N N P N N N N N N N N Other Sourcesc P P P P P P O, P P P P O, P O, P P P O, P O, P Note.—Table 1 is published in its entirety in the electronic edition of the Astrophysical Journal Supplement. A portion is shown here for guidance regarding its form and content. a Torsional substates: 00 ¼ AA, 01 ¼ EE, 11 ¼ AE, 12 ¼ EA. b Estimated experimental uncertainty (coverage factor k ¼ 1; see Table 2 for type A or B). A value of 0.0 in the uncertainties column signifies that the transition has been assigned but not included in the fit. A total of 1002 assigned frequencies were used in the fit. c O: Oldag & Sutter 1992; P: Peter & Dreizler 1965; N: Fourier transform (FT) microwave data at NIST (this work). 148 GRONER ET AL. Vol. 142 TABLE 2 Sources and Uncertainties of Frequency Data Source Type of Data Code Estimated Uncertainty (coverage factor k ¼ 1) (MHz) Peter & Dreizler (1965)........ Vacherand et al. (1986)........ Vacherand et al. (1986)........ Oldag & Sutter (1992) ......... White (1975) ....................... NIST (this work)................. JPL (this work) ................... FASSST (this work)............ FASSST (this work)............ Microwave Double resonance microwave Millimeter wave FT Microwave Microwave FT microwave Submillimeter wave (klystron) Submillimeter wave <400 GHz Submillimeter wave >400 GHz P V V O W N J F F 0.03 (type B) 0.05 (type B) 0.1 (type B) 0.01 (type B) As quoted (0.2–0.02; type B) 0.004 or 0.008 (type A) 0.1 (type B) 0.1 (type B) 0.2 (type B) edition of the Astrophysical Journal Supplement.) The torsional substates are identified by the symmetry numbers (1, 2). Their relation to the conventional notation is discussed in x 2. The origin of the frequency entered into the table is indicated by the code in the column headed ‘‘ source.’’ Other investigations with different or less precise frequencies are coded in the column headed ‘‘ other sources.’’ The codes are defined in Table 2. For lines without severe torsional-rotational interactions, the transitions obey b-type asymmetric top selection rules (DJ ¼ 0, 1; DKa ¼ 1; 3; . . .; DKc ¼ 1; 3; . . .; see Gordy & Cook 1984), and the torsional substate does not change. For large rotational quantum numbers J, the energy levels of asymmetric rotors are practically degenerate for equal and high Ka (a-type degeneracy) or for equal and high Kc (ctype degeneracy, low Ka ). The c-type degeneracy persists for all torsional components of acetone. The a-type degeneracy holds only for the ð0; 0Þ and ð1; 1Þ torsional states, but not for the ð0; 1Þ and ð1; 2Þ states. As a consequence, many of the lines measured in the millimeter- and submillimeterwave region are superpositions of degenerate transitions. In most instances of c-type degeneracies, only one of the degenerate transitions has been used in the fit and listed in Table 1. Weights of individual measurements in the least-squares fit based on the approach in x 2 were determined from the estimated experimental uncertainties, which are also listed in Table 1 and summarized for each source in Table 2. An experimental uncertainty of 0.0 MHz marks assigned transitions whose frequencies were not used in the least-squares fit. As a result of these criteria, 1002 out of 1175 assigned frequencies were used to determine 33 spectroscopic parameters by the least-squares method. The calculated residuals (observed minus calculated frequencies) are shown in Table 1. The spectroscopic parameters (‘‘ constants ’’) determined by the least-squares procedure are listed in Table 3. The dimensionless standard deviation achieved is 1.58. In addition to 14 nontunneling (q ¼ q0 ¼ 0) parameters corresponding to the rotational constants and the quartic and sextic centrifugal distortion constants, the internal rotation parameters  (unitless) and (degrees) were determined, as well as four energy tunneling parameters , eight tunneling parameters associated with the rotational constants, and TABLE 3 Spectroscopic Parameters for Acetone in the Vibrational-Torsional Ground State Parameter Valuea Parameter Valuea ............................................... (deg)...................................... A (MHz) .................................. B (MHz)................................... C (MHz) .................................. DJ (kHz) .................................. DJK (kHz)................................. DK (kHz).................................. J (kHz) ................................... K (kHz)................................... J (Hz) .................................... JK (Hz) .................................. KJ (Hz) .................................. K (Hz).................................... J (Hz)..................................... JK (Hz)................................... K (Hz) .................................... 0.0621760(60) 25.8322(93) 10165.21654(80) 8515.16477(65) 4910.19903(44) 4.9055(25) 3.620(17) 10.245(17) 2.0645(12) 0.7393(56) 0.0506(34) 0.337(20) 0 0.423(20) 0.0254(17) 0.0273(41) 0.2215(83) 10 (MHz)............................. 11 (MHz)........................... 11 (MHz)............................. 20 (MHz)............................. ½A  ðB þ CÞ=210 (kHz) ...... ½A  ðB þ CÞ=211 (kHz) .... ½A  ðB þ CÞ=220 (kHz) ...... ½ðB þ CÞ=210 (kHz) ............. ½ðB þ CÞ=211 (kHz) ........... ½ðB þ CÞ=220 (kHz) ............. ½ðB  CÞ=410 (kHz) ............. ½ðB  CÞ=411 (kHz) ........... ½DJ 10 (kHz).......................... ½DJK 10 (kHz)........................ ½DK 10 (kHz) ......................... ½J 10 (kHz) .......................... ½K 10 (kHz).......................... 763.198(62) 0.0800(83) 1.050(43) 0.767(13) 55.07(64) 1.62(25) 0.87(21) 21.16(56) 1.43(18) 0.31(13) 3.40(27) 0.475(73) 0.03906(34) 0.0998(17) 0.0737(17) 0.01960(18) 0.03427(98) a Standard uncertainties are type A, k ¼ 1 (1 ) and are given in units of last digit in parentheses. No. 1, 2002 ACETONE five tunneling parameters associated with the quartic distortion constants. For the nontunneling parameters, the symbols of the asymmetric rotor Hamiltonian in Watson’s A reduction (Watson 1977) have been used. The internal energy tunneling parameters are designated by qq0 . For the other tunneling parameters, the notation ½X qq0 has been used, where X is a linear combination of rotational constants or distortion constants and the subscript designates the tunneling component. Two sets of frequencies included in Table 1 (listed with an estimated uncertainty of 0.0 MHz) were not used in the fit. The first set consists of frequencies assigned in previous investigations that do not fit as well as other frequencies from the same source. Typographical errors, misassignments, or unrecognized blends with unidentified lines are the most likely causes for the unsatisfactory fit. The second set consists of frequencies measured in this investigation. Some of them have nearly degenerate but unresolved transitions whose assignments are quite certain. But most of them are frequencies of (high Ka , low Kc )–transitions with J > 21 that do not fit very well. A select few tentative assignments of (high Ka , low Kc )–transitions with J < 20 are also marked with zero uncertainty. The causes for the bad fit of some of these lines can only be guessed at. The assignments of at least the ð0; 0Þ and ð0; 1Þ substate lines seem unquestionable because of their proximity to the predicted positions and the qualitative agreement between predicted and observed relative intensities. At room temperature, they are the strongest nondegenerate transitions (both predicted and observed) at frequencies above 400 GHz. Most likely, some of the (high Ka , low Kc ) levels involved in these transitions are perturbed by interactions with levels from the lowest torsional excited states that are 78 and 125 cm1 above the VTG state (Groner 2000). Already at J ¼ 20 the stack of 149 energy levels belonging to the ground state begins to overlap with the stack of the lowest excited state. 4. DISCUSSION With the spectroscopic constants derived from the fit to the global data set, the frequencies were predicted for an additional 11,000 lines belonging to the VTG state through 620 GHz and through rotational quantum number J of 60. These frequencies are listed in Table 4 (found in its entirety in the electronic edition of the Astrophysical Journal Supplement) along with the torsional substate t ¼ ð1 ; 2 Þ, the rotational quantum numbers, the estimated 1  uncertainties (type A, coverage factor k ¼ 1) of the predicted frequencies, the spin weights, the intensities expressed in S values (Townes & Schawlow 1975), and the energies of the upper states in units of cm1. Only transitions with S > 0:50, Eupper < 500 cm1 , and k ¼ 1 uncertainty < 1:00 MHz are included in Table 4. The upper state energies are measured relative to the absolutely lowest level (J; Ka ; Kc Þ ¼ ð0; 0; 0Þ of the substate ð0; 0Þ. They include a small contribution from the internal rotation [2284 MHz for t ¼ ð0; 1Þ, 4571 MHz for t ¼ ð1; 1Þ, and 4574 MHz for t ¼ ð1; 2Þ]. The assigned and fitted lines listed in Table 1 are included in Table 4 (but with their predicted frequencies) so that the additional information concerning these lines (spin weights, intensities, upper state energies) is readily available for astronomical use. The intensities of the lines are given in terms of the S-values so that these numbers must still be multiplied by the square of the b-direction dipole moment (lb ¼ 2:93  0:03 D; Peter & Dreizler 1965) to obtain the proper intensities. In addition, they must be multiplied by the appropriate spin weight. TABLE 4 Predicted Transition Frequencies of Acetone in the Vibrational-Torsional Ground State ta J0 Ka0 Kc0 J 00 Ka00 Kc00 Frequency (MHz) Uncertainty (MHz)b Spin Weight S Eupper (cm1) 11........ 01........ 12........ 00........ 11........ 00........ 01........ 12........ 11........ 12........ 01........ 00........ 11........ 00........ 11........ 01........ 12........ 01........ 00........ 12........ 1 1 1 1 2 2 2 2 3 3 3 3 3 3 2 3 2 2 2 3 1 1 1 1 2 2 2 2 2 2 2 2 3 3 1 3 1 1 1 3 0 0 0 0 0 0 0 0 1 1 1 1 0 0 1 0 1 1 1 0 1 1 1 1 2 2 2 2 3 3 3 3 3 3 2 3 2 2 2 3 0 0 0 0 1 1 1 1 1 1 1 1 2 2 0 2 0 0 0 2 1 1 1 1 1 1 1 1 2 2 2 2 1 1 2 1 2 2 2 1 5253.063 5269.071 5270.904 5276.057 7343.774 7398.953 7399.964 7453.183 10731.360 10749.943 10751.671 10762.702 11073.951 11199.320 11252.506 11257.715 11265.269 11272.464 11286.131 11468.246 0.007 0.002 0.002 0.001 0.009 0.003 0.003 0.007 0.009 0.002 0.002 0.002 0.006 0.005 0.002 0.006 0.002 0.002 0.002 0.015 6 16 4 10 2 6 16 4 6 4 16 10 6 10 2 16 4 16 6 4 1.495 1.496 1.489 1.500 2.012 2.013 1.989 1.926 2.628 2.624 2.633 2.640 2.050 2.041 1.315 1.893 1.313 1.317 1.320 1.693 0.776 0.700 0.776 0.624 2.036 1.887 1.963 2.040 3.588 3.589 3.514 3.440 3.957 3.814 1.791 3.890 1.792 1.716 1.641 3.972 Note.—Table 4 is published in its entirety in the electronic edition of the Astrophysical Journal Supplement. A portion is shown here for guidance regarding its form and content. a Torsional substates: 00 ¼ AA, 01 ¼ EE, 11 ¼ AE, 12 ¼ EA. b Calculated (1 ) uncertainty, type A with coverage factor k ¼ 1. 150 GRONER ET AL. Vol. 142 TABLE 5 Partition Function for Acetone in the Vibrational-Torsional Ground State State (0, 0) AA (0, 1) EE (1, 1) AE (1, 2) EA Total Asymmetric Top Approximation 10 K .......... 20 K .......... 50 K .......... 100 Ka ....... 200 Ka ....... 300 Ka ....... 2082.56 5867.96 23148.33 65450.95 183787 326294 4123.59 11682.27 46231.83 130843.32 367540 652572 1020.60 2907.19 11541.75 32696.17 91876 163138 1020.59 2907.18 11541.73 32696.15 91877 163139 8247.34 23364.61 92463.64 261686.59 735080 1305143 8274.89 23404.92 92516.07 261674.95 740129 1359703 a Sum of states not converged. Only one transition is listed in cases of c-type or a-type degeneracy if the predicted frequencies are degenerate within 0.001 MHz. These cases are identified in Table 4 by the combined spin weights of the degenerate transitions 16, 32, 8, and 8 for the torsional substates ð0; 0Þ, ð0; 1Þ, ð1; 1Þ, and ð1; 2Þ, respectively. For some nearly degenerate energy levels in the substates ð0; 1Þ (EE) and ð1; 2Þ (EA), mixing and level crossings occur so that the pseudo–quantum numbers Ka and Kc , defined according to the standard ordering in asymmetric tops, do not always represent the actual wave functions very well. As a consequence, ‘‘ forbidden ’’ c-type transitions occur as follows. True ‘‘ forbidden ’’ c-type transitions occur when the wave functions mix; they are always accompanied by b-type transitions between the same pairs of levels. False c-type transitions arise from level crossings; they lack b-type partners because they are the mislabeled b-type transitions. The spin weights must be used as factors for line intensities and to determine the total torsional-rotational-nuclear spin partition function as a function of temperature; numerical values for the overall partition function as well as for the individual torsional substates (obtained by restricting the summation to levels belonging to a specific substate) are listed in Table 5 at selected temperatures. It should be emphasized that the partition function for the torsional substates ð0; 1Þ, ð1; 1Þ, and ð1; 2Þ includes a factor that depends on the energy difference of the (J; Ka ; Kc ¼ 0; 0; 0) levels between the substate and the ð0; 0Þ substate given above. A reasonable approximation to the numerical values for the overall partition function can be obtained from the simple asymmetric top expression (Townes & Schawlow 1975) multiplied by the total spin weight ð2I þ 1Þ6 ¼ 64 ðI ¼ 12 Þ and divided by the overall symmetry number for the C2v point group of 2. The result is q ¼ 261:67495T 3=2 ; results at this level of approximation are also shown in Table 5. This approximation should be used for temperatures above 100 K because the direct summation of the partition function is not converged if J is limited to 60. The fractional population f (the ratio of the state column density to the total column density) for any rotational-torsional state of energy E is then given by the expression f ¼ gs ð2J þ 1Þ expðE=kTÞ ; q ð3Þ where the symbol gs stands for the appropriate spin weight as defined in x 2, J is the rotational quantum number, k is the Boltzmann constant, and T is the temperature in kelvins. For example, for the upper level of the first transition in Table 4 [torsional substate (1, 1), J; Ka ; Kc ¼ 1; 1; 0], the spin weight is 6, and the fractional population at 100 K is f ¼ 6:80
105 . In performing analyses for line intensities, astronomers should consider the possibility of blending of one or more of the quartet of lines for each pair of rotational quantum numbers. Even if the quartet of torsionally split lines is totally blended, the total spin weight must still be considered since it has a rotational dependence as well as a torsional dependence. For example, if the first quartet of lines in Table 4 were totally blended, its total spin weight would be 6 þ 16 þ 4 þ 10 ¼ 36, whereas the total spin weight of a blended quartet of lines for Ka Kc : ee $ oo transitions such as 2, 2, 0 $ 1, 0, 1 would be 2 þ 6 þ 16 þ 4 ¼ 28. For a similar reason, the spin weights of blended lines of c-type or a-type degenerate transitions are 16, 32, 8, and 8 for the torsional substates ð0; 0Þ, ð0; 1Þ, ð1; 1Þ, and ð1; 2Þ, respectively. We would like to thank NASA for their support of laboratory astrophysics at Ohio State University. S. A. thanks the Alexander-von-Humboldt Stiftung for a Feodor-Lynen research stipend. A portion of this research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under contract with NASA. REFERENCES Gordy, W., & Cook, R. L. 1984, Microwave Molecular Spectroscopy (New Balle, T. J., & Flygare, W. H. 1981, Rev. Sci. Instrum., 52, 33 York: Wiley) Blake, G. A., Sutton, E. C., Masson, C. R., & Phillips, T. G. 1987, ApJ, Groner, P. 1997, J. Chem. Phys., 107, 4483 315, 621 ———. 2000, J. Mol. Struct., 550–551, 473 Brown, P. D. 1990, MNRAS, 243, 65 Groner, P., Albert, S., Herbst, E., & De Lucia, F. C. 1998, ApJ, 500, 1059 Brown, P. D., Charnley, S. B., & Millar, T. J. 1988, MNRAS, 231, 409 Herbst, E., Giles, K., & Smith, D. 1990, ApJ, 358, 468 Caselli, P., Hasegawa, T. I., & Herbst, E. 1993, ApJ, 408, 548 Hollis, J. M., & Churchwell, E. 2001, ApJ, 551, 803 Charnley, S. B., Kress, M. E., Tielens, A. G. G. M., & Millar, T. J. 1995, Hollis, J. M., Vogel, S. N., Snyder, L. E., Jewell, P. R., & Lovas, F. J. 2001, ApJ, 448, 232 ApJ, 554, L81 Charnley, S. B., Tielens, A. G. G. M., & Millar, T. J. 1992, ApJ, 399, L71 Ikeda, M., Ohishi, M., Nummelin, A., Dickens, J. E., Bergman, P., Charnley, S. B., Tielens, A. G. G. M., & Rodgers, S. D. 1997, ApJ, 482, Hjalmarson, A., & Irvine, W. M. 2001, ApJ, 560, 792 L203 Liu, S.-Y., Mehringer, D. M., & Snyder, L. E. 2001, ApJ, 552, 654 Combes, F., Gerin, M., Wootten, A. S., Wlodarczak, G., Clausset, F., & Lovas, F. J., Lutz, H., & Dreizler, H. 1979, J. Phys. Chem. Ref. Data, 8, Encrenaz, P. J. 1987, A&A, 180, L13 1051 Friedl, R. R., Birk, M., Oh, J. J., & Cohen, E. A. 1995, J. Mol. Spectrosc., Lovas, F. J., & Suenram, R. D. 1987, J. Chem Phys., 87, 2010 170, 383 No. 1, 2002 ACETONE Mehringer, D. M., Snyder, L. E., Miao, Y., & Lovas, F. J. 1997, ApJ, 480, L71 Miao, Y., Mehringer, D. M., Kuan, Y.-J., & Snyder, L. E. 1995, ApJ, 445, L59 Millar, T. J., Herbst, E., & Charnley, S. B. 1991, ApJ, 369, 147 Myers, R. J., & Wilson, E. B. 1960, J. Chem. Phys., 33, 186 Nelson, R., & Pierce, L. 1965, J. Mol. Spectrosc., 18, 344 Oldag, F., & Sutter, D. H. 1992, Z. Naturforsch., 47a, 527 Pei, C. C., Liu, S.-Y., & Snyder, L. E. 2000, ApJ, 530, 800 Peter, R., & Dreizler, H. 1965, Z. Naturforsch., 20a, 301 Petkie, D. T., Goyette, T. M., Bettens, R. A. P., Belov, S. P., Albert, S., Helminger, P., & De Lucia, F. C. 1997, Rev. Sci. Instrum., 68, 1675 Snyder, L. E., Lovas, F. J., Mehringer, D. M., Miao, Y., Kuan, Y.-J., Hollis, J. M., & Jewell, P. R. 2002, ApJ, in press Snyder, L. E., Miao, Y., Mehringer, D. M., Kuan, Y.-J., Hollis, J. M., Jewell, P. R., & Lovas, F. J. 1997, in CO: 25 Years of Millimeter-Wave Spectroscopy, ed. W. B. Latter et al. (Dordrecht: Kluwer), 460 151 Suenram, R. D., Lovas, F. J., Fraser, G. T., Gillies, J. Z., Gillies, C. W., & Onda, M. 1989, J. Mol. Spectrosc., 137, 127 Sutton, E. C., Peng, R., Danchi, W. C., Jaminet, P. A., Sandell, G., & Russell, A. P. G. 1995, ApJS, 97, 455 Swalen, J. D., & Costain, C. C. 1959, J. Chem. Phys., 31, 1562 Townes, C. H., & Schawlow, A. L. 1975, Microwave Spectroscopy (New York: Dover) Vacherand, J. M., van Eijck, B. P., Burie, J., & Demaison, J. 1986, J. Mol. Spectrosc., 118, 355 Watson, J. K. G. 1977, in Vibrational Spectra and Structure, Vol. 6, ed. J. R. Durig (Amsterdam: Elsevier), 1 Weatherly, T. L., & Williams, D. 1952, J. Chem. Phys., 20, 755 White, W. F. 1975, NASA Tech. Note D-7904, 121