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2 Journal of Molecular Spectroscopy 245 (2007) Torsion rotation global analysis and database for the CH 3 18 OH isotopomer of methanol J. Fisher a, G. Paciga a, Li-Hong Xu a, *, S.B. Zhao b, G. Moruzzi c, R.M. Lees a a Centre for Laser, Atomic, and Molecular Sciences (CLAMS), Department of Physical Sciences, University of New Brunswick, Saint John, NB, Canada E2L 4L5 b Department of Physics, University of New Brunswick, Fredericton, NB, Canada E3B 5A3 c Dipartimento di Fisica dell Università di Pisa, Via Filippo Buonarroti 2, I Pisa, Italy Received 21 May 2007; in revised form 11 June 2007 Available online 4 July 2007 Abstract Fourier-transform far-infrared spectra of CH 3 18 OH in the cm 1 region have been analyzed by means of the Ritz assignment program. The far-infrared data have been combined with the literature microwave and millimeter-wave measurements in a full global fitting of the first three torsional states (m t = 0, 1, and 2) of the CH 3 18 OH ground vibrational state. The fitted dataset includes 550 microwave and millimeter-wave lines and more than Fourier-transform transitions covering the quantum number ranges J 6 30, K 6 15, and m t 6 2. With incorporation of 79 adjustable parameters, the global fit achieved convergence with an overall weighted standard deviation of 1.072, essentially to within the assigned measurement uncertainties of ±50 khz for almost all of the microwave and millimeter-wave lines and ±6 MHz ( cm 1 ) to ±15 MHz ( cm 1 ) for the Fourier-transform far-infrared measurements. Based on the global fit results, a database has been compiled containing transition frequencies, quantum numbers, lower state energies and transition strengths. This database will provide support for present and future astronomical studies, such as the on-going Orion surveys in preparation for the launch of the Herschel Space Observatory, in identifying isotopic methanol contributions to interstellar spectra. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Methanol; CH 3 18 OH; Torsion; Millimeter-wave spectra; Fourier-transform far-infrared spectrum; Global fitting; Interstellar molecules 1. Introduction This spectroscopic study and global analysis of the torsion rotation spectrum of the CH 3 18 OH isotopic species of methanol is motivated by recent molecular surveys of the Orion nebula [1,2] carried out in preparation for the forthcoming launch of HIFI (Heterodyne Instrument for the Far-Infrared) on board the Herschel Space Observatory [3], the flying of SOFIA (Stratospheric Observatory For Infrared Astronomy) [4] and the commissioning of ALMA (Atacama Large Millimeter/Submillimeter Array) [5]. Like its parent, the oxygen-18 methanol isotopomer has rich spectra and was first detected in the interstellar medium * Corresponding author. Fax: address: lxu@unbsj.ca (L.-H. Xu). towards Sgr B2 in 1989 [6]. Most recent Orion surveys have revealed thousands of unknown transitions. Thus, identifications of CH 3 18 OH transitions in the survey spectra will help to sort out this interstellar grass of weak lines to aid the search for new interstellar molecules, and will also be of interest in their own right for the understanding of isotope ratios and hence the interstellar chemistry. On the spectroscopic front, the present work is the latest in a series of global fitting studies of the microwave (MW), millimeter-wave (MMW) and far-infrared (FIR) spectra of methanol isotopologues, including normal CH 3 OH [7,8], 13 CH 3 OH [9], CD 3 OH [10], CH 3 OD [11], and CD 3 OD [12,13]. In relation to these previous works, the quantum number range of the present dataset has been extended up to the second excited torsional state (m t_max = 2) and to a maximum J value of 30, as compared to the previous /$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi: /j.jms

3 8 J. Fisher et al. / Journal of Molecular Spectroscopy 245 (2007) 7 20 limits of m t_max = 1 and J max = 20. Coupled with this increased range of data, the set of molecular parameters has also been expanded, with a number of higher order terms introduced into the fitting for the first time. The present and previous global fits have all been successful in modeling the data to within experimental accuracy, using a computer program [14] based on the formalism of Herbst et al. [15]. Here, we will give brief introductory remarks relating specifically to CH 3 18 OH, and refer the reader to the previous literature [7 19] for more general details of the computer program, the Hamiltonian model and the notation for the torsion rotation quantum numbers and transition labeling. There have been several studies of the MW and MMW spectra of CH 3 18 OH [20 24], devoted originally towards determination of structural constants and later with an emphasis on application to molecular astronomy. In the infrared (IR) and far-infrared (FIR) regions, our group has carried out a number of Fourier-transform studies of CH 18 3 OH [25 28] leading to identification of numerous optically pumped far infrared laser transitions [29 31]. Following the initial analysis of FIR subbands with origins up to 150 cm 1 [25,27], the whole spectrum up to nearly 500 cm 1 has now been examined in a systematic study using the Ritz approach [16] that will be discussed in Section 2.2. The first part of the present paper reports our Ritz analysis of over transitions for the m t = 0, 1, 2, and 3 torsional states in which the energies of 4291 torsion rotation levels have been determined via the Rayleigh Ritz combination principle. The second part of the paper deals with the global modeling of assigned MW, MMW, and FTFIR ground-state transitions of CH 18 3 OH involving m t =0, 1, and 2 levels for all observed K rotational states up to a max- v t CH 3 18 OH A Species 4 gr,4; gr,4; gr,4; gr,3; gr,3; gr,3; gr,3; gr,3; gr,3; gr,3; gr,3; gr,3; * gr,3; * gr,3; gr,2; gr,2; gr,2; gr,2; gr,2; gr,2; gr,2; gr,2; gr,2; gr,2; gr,2; gr,1; gr,1; gr,1; gr,1; gr,1; gr,1; gr,1; gr,1; gr,1; gr,1; gr,1; gr,1; gr,1; gr,1; gr,0; gr,0; gr,0; gr,0; gr,0; gr,0; gr,0; gr,0; gr,0; gr,0; gr,0; gr,0; gr,0; gr,0; gr,0; K Fig. 1. Ritz transition sequence diagram for A symmetry. Listed in each box are first the substate labels (gr, m t ; J max ) where gr denotes the ground vibrational state, m t is the torsional quantum number, and J max is the maximum value of J assigned in the Ritz analysis; and second the value of the zero order coefficient a o in the J(J + 1) power series expansion, representing the J-reduced substate energy. Lines between boxes indicate connecting transition subbands that have been assigned between those states. Ideally, all substates should be connected forming one single family with a single reference energy chosen as zero. However, in the present case, the (m t, K, T s ) = (3,8A) and (3,9A) substates are not connected to the main family so have a separate reference energy, marked with *. The same is true for the (2, 8E), (2, 7E), (3, 6E), and (3, 5E) group of substates. v t CH 3 18 OH E Species 3 gr,3; gr,3; gr,3; gr,3; gr,3; gr,3; gr,3; gr,3; gr,3; gr,2; gr,2; gr,2; gr,2; gr,2; gr,2; gr,2; gr,2; gr,1; gr,1; gr,1; gr,1; gr,1; gr,1; gr,1; gr,1; gr,1; gr,1; gr,1; gr,1; gr,0; gr,0; gr,0; gr,0; gr,0; gr,0; gr,0; gr,0; gr,0; gr,0; gr,0; gr,0; Fig. 2. Ritz transition sequence diagram for +K Esymmetry. gr,0; gr,0; gr,0; gr,0; K

4 J. Fisher et al. / Journal of Molecular Spectroscopy 245 (2007) v t CH 3 18 OH E Species 4 gr,4; gr,3; gr,3; gr,3; gr,3; * gr,3; * gr,3; gr,3; gr,3; gr,3; gr,3; gr,2; gr,2; gr,2; * gr,2; * gr,2; gr,2; gr,2; gr,2; gr,1; gr,1; gr,1; gr,1; gr,1; gr,1; gr,1; gr,1; gr,1; gr,1; gr,1; gr,1; gr,1; gr,0; gr,0; gr,0; gr,0; gr,0; gr,0; gr,0; gr,0; gr,0; gr,0; gr,0; gr,0; gr,0; Fig. 3. Ritz transition sequence diagram for K Esymmetry. (For more details see Fig. 1.) gr,0; gr,0; gr,0; K ν t = 3 not in the fit yet not in the fit yet νt t = νt t = 1 ν t = 0 V 3 = cm -1 3 = cm -1 A +K,E -K,E K Values Fig. 4. J-reduced torsion rotation energies for CH 3 18 OH plotted as a function of K. Solid lines are for A torsional symmetry; dashed and dotted lines are for +K and K Etorsional symmetry, respectively. imum rotational quantum number J max of 30. The onedimensional torsion rotation model used for the global analysis was implemented in a computer program [14] well tested on previous methanol isotopologues [7 13] but augmented here, given the expansion in quantum number range, with a number of new high order torsion rotation terms.

5 10 J. Fisher et al. / Journal of Molecular Spectroscopy 245 (2007) 7 20 Table 1 Taylor coefficients of J(J + 1) power-series expansions of Ritz term values for CH 18 3 OH substates of A torsional symmetry a m t K a 0 a 1 a 2 a 3 a 4 a 5 a (13) (36) 4.61(26) (76) (10) (67) (66) (59) 2.844(12) (13) (12) 1.462(26) (28) (69) 1.104(51) (16) (24) (17) (52) (13) 6.0(10) (34) (54) (41) (53) (99) 1.238(56) (14) (16) (89) (13) (95) (18) (39) (65) 1.464(35) (85) (10) (58) (45) (67) 1.285(33) (74) (82) (44) (30) (27) 1.572(72) (73) (99) (14) 1.482(71) (17) (21) (12) (26) (21) 1.655(52) (50) (79) (27) (81) (27) (19) (10) 1.981(16) (38) (40) (95) (16) (16) (38) (62) (46) (85) (12) (11) (30) (28) (15) (37) 1.590(28) (89) (14) (10) (13) (10) 1.733(20) (68) (12) 4.604(64) (15) (17) (96) (18) (13) (22) (39) (36) (94) (92) (56) (92) 1.468(54) (15) (20) (14) (23) (76) (11) (16) 1.162(85) (21) (28) (18) (18) (15) 1.846(38) (40) (83) (40) (72) (35) (49) (24) (26) (27) (69) (62) (21) 2.02(20) (87) (19) (22) (12) (61) (21) (11) (77) (14) (43) (57) 1.127(22) (31) (95) (15) 1.106(77) (16) (155) (87) (33) (10) (16) 1.013(91) (24) (31) (20) (43) (80) 1.111(58) (22) (46) (55) (34) (13) (48) (21) (13) (90) (51) (33) (15) (30) (16) (89) (55) (84) (48) (36) (15) (83) (41) (27)* (55) 1.68(35) (47)* (93) 1.676(59) (14) (63) (16) (57) (91) (89) (21) (61) 1.269(54) (19) (30) a Asterisks denote level sequences not connected to the rest by assigned transitions, See caption to Fig Fourier-transform far-infrared spectra and Ritz analysis 2.1. Experimental spectra Previously, CH 3 18 OH FIR spectra [25,27] had been recorded from 30 to 130 cm 1 at 2 Torr pressure and from 130 to 220 cm 1 at 2.8 Torr pressure at cm 1 resolution on the original Bomem Fourier-transform spectrometer at the National Research Council in Ottawa, employing a 15-cm cell at room temperature. In the present study, spectra were recorded from 15 to 350 cm 1 at cm 1 resolution and 400 mtorr pressure and from 330 to

6 J. Fisher et al. / Journal of Molecular Spectroscopy 245 (2007) Table 2 Asymmetry splitting coefficients for CH 18 3 OH substates of A torsional symmetry m t K S T U V (14) (60) (80) (33) (34) (79) (45) (17) (59) (66) (24) (15) (31) (16) (21) (59) (39) (34) (30) (30) (21) (31) (34) (48) (16) (53) cm 1 at cm 1 resolution and 250 mtorr pressure in eight transits of an 0.5-m absorption cell on the modified Bomem DA3.002 Fourier-transform spectrometer at NRC. Subsequently, further spectral recordings were obtained from 50 to 350 cm 1 at cm 1 resolution at two pressures of 165 and 740 mtorr in an 0.3-m path on the Bruker instrument at Justus-Liebig University in Giessen Assignment and analysis of the spectra Progressing from the initial core of assigned transitions from the earlier study [25,27], the Ritz program [16] was used to perform the majority of the CH 18 3 OH FIR assignments and determine an accurate set of energy level term values for the ground-state torsion rotation manifold. (Our Ritz databases of assigned transitions and CH 18 3 OH term values are available as supplementary data to the present work.) The Ritz program evaluates the term values involved in the assigned transitions directly from the Rydberg Ritz combination principle, i.e., it evaluates the set of term values f ~W i g, where ~W i ¼ E i =hc, which minimizes the expression v 2 ¼ X ð ~W i ~W j m ij Þ 2 ð1þ e 2 i;j ij where ~W i and ~W j are the term values of the ith and jth molecular levels, m ij is the experimental wavenumber of the corresponding transition, and e ij is its experimental accuracy. This procedure provides term values with accuracy of the order of 10 4 cm 1. This good accuracy is due both to the high experimental resolution and signalto-noise ratio of the Fourier-transform spectrometers and to the facts that (i) the liberal selection rules of the slightly asymmetric methanol molecule allow a given level to be accessed by many transitions, (ii) the program performs a simultaneous fit of all assigned transitions, and (iii) all possible closed transition loops are automatically checked. In general, the energy levels of methanol can be labeled by five quantum numbers, which we will denote in the present paper as (m,m t,k;j) T s. The quantum number m labels the small-amplitude vibrational state, but since only the ground vibrational state is investigated in the present work, m will be omitted here. The quantum number m t labels the torsional state, and K is the component of the total angular momentum J along the internal rotation axis. K has only non-negative values for A torsional symmetry, while it can have positive, or negative values in the case of E symmetry [18]. The label T s represents the symmetry species, A or E. Transitions can be labeled as ðv 0 t ; K0 ; J 0 Þ ðv 00 t ; K00 ; J 00 ÞA ðv 0 t ; K0 ; J 0 Þ ðv 00 t ; K00 ; J 00 ÞE where the primed quantities refer to the upper level of the transition and the double primed quantities refer to the lower level. The superscripts + or following the K values for A symmetry label the asymmetry-splitting components [19]. Level sequences, i.e., sequences of levels sharing all quantum numbers but J, will be represented by (T s,m t,k) or simply (m t,k) with the T s label being omitted when obvious from the context. Figs. 1 3 display schematically the level sequences and the transitions investigated in the present work for A- and E- symmetry. The E-symmetry transitions are separated according to the sign of K (K P 0, corresponding to E 1 -symmetry, and K 6 0, corresponding to E 2 -symmetry, according to another possible notation [17])(Figs. 2 and 3). So far, absorption transitions have been assigned, of which 7662 correspond to A-symmetry and to E-symmetry. The Ritz program has evaluated the energies of 1794 levels belonging to A symmetry and 2497 levels belonging to E symmetry. A useful way of approximating the term values of each level sequence (T s,m t,k) is via a Taylor series in powers of J(J + 1) according to the formula ~W ðt s ; v t ; K; JÞ ¼ X m a m ðt s ; v t ; KÞ½JðJ þ 1ÞŠ m The Taylor coefficients a m are given in Table 1 for A symmetry and in Table 4 for E symmetry. For low K, the term values of A-symmetry also have a resolvable contribution from asymmetry doubling D ~W asymm which, when expanded up to the third power in J(J + 1), has the form ð2þ ð3þ

7 12 J. Fisher et al. / Journal of Molecular Spectroscopy 245 (2007) 7 20 Table 3 Taylor expansion statistics for CH 18 3 OH substates of A torsional symmetry m t K J max J e e max r D ~W asymm ¼ ðj þ KÞ! ðj KÞ! fs þ JðJ þ 1ÞT þ½jðjþ1þš2 U þ½jðjþ1þš 3 Vg ð4þ The S, T, U, and V coefficients for the affected level sequences are reported in Table 2. The term values of A-symmetry affected by asymmetry doubling are thus approximated by ~W ða ; v t ; K; JÞ ¼ X m a m ða; v t ; KÞ½JðJ þ 1ÞŠ m 1 2 D ~W asymm Statistics of the Taylor expansions are shown in Table 3 for A symmetry, and in Table 5 for E symmetry. These tables report, for each level sequence, the quantum numbers identifying the sequence, the maximum fitted J value J max, the J value J e at which the maximum deviation of the Taylor fit from the experimental value is observed, the value of this maximum deviation e max, and the standard deviation r of the fit. In all of the tables, units of cm 1 are used for all quantities other than quantum numbers. The numbers within parentheses in Tables 1, 2, and 4 give the errors in units of the last digit. We present the Taylor coefficient tables here because they give insight into the physics of the molecule. However, whenever actual term values are needed, we recommend use of the original Ritz tables available in the supplementary data or on request from the authors. 3. Global analysis In support of the Orion molecular line surveys [1,2], there is a need for a comprehensive line list including information such as transition quantum number, lower state energy, and transition strength, thereby generating in turn a need for global Hamiltonian modeling. In this work, we have taken the torsional state limit as m t_max = 2, which corresponds to torsional energies ranging from 600 to 1400 cm 1 for K values from 0 to 15, well above the 370 cm 1 torsional potential barrier height. This limit represents a balance between quantum state coverage and the Boltzmann distribution of level populations to ensure a meaningful set of model parameters for reliable extrapolation while giving wide coverage of observable transitions with strong to medium line strength The m t = 0, 1, and 2 dataset Our dataset for the CH 3 18 OH fitting included a total of MW, MMW, and FIR lines covering quantum number ranges m t 6 2 and J 6 30 for all identified K states of both A and E torsional symmetries. A group of 550 MW and MMW frequencies were collected from the available literature [20 24], while the Ritz analysis of the Fouriertransform spectra provided the FIR information. In the present global modeling, we were able to bring about 93% of the Ritz-assigned transitions into the fit (the remaining 7% of the Ritz assignments belong to transitions to the m t = 3 level). In our analysis, all of the transitions were critically evaluated for reliability and internal consis- ð5þ

8 Table 4 Taylor coefficients of J(J + 1) power-series expansions of Ritz term values for CH 3 18 OH substates of E torsional symmetry mt K a0 a a1 a2 a3 a4 a5 a6 a (16) (79) (17) (82) (12) (49) (73) (88) 2.86(38) (77) (71) (47) (19) (31) (12) (53) (51) 1.605(15) (17) (52) (71) 1.848(31) (59) (49) (12) (93) (18) (93) (18) 1.30(11) (31) (41) (27) (87) (22) 6.3(18) (65) (12) (12) (62) (55) (18) 1.97(19) (89) (21) (27) (17) (22) (65) 1.371(55) (20) (37) (36) (18) (98) (31) 6.8(270) (10) (19) (19) (92) (81) (20) 5.789(13) (34) (41) (19) (80) (94) (475) (12) (18) (14) (59) (19) (33) 4.98(15) (27) (20) (99) (36) 3.74(37) (16) (36) (42) (24) (19) (55) 8.7(46) (16) (27) (21) (15) (70) (22) (29) 1.679(11) (15) (28) (34) 1.518(11) (14) (33) (23) (40) (11) (18) 1.43(10) (26) (33) (21) (12) (18) 1.576(90) (20) (20) (62) (44) (88) (55) (51) 1.676(15) (18) (24) (91) (45) (28) (54) (37) (22) (41) (33) (12) (91) (13) 2.9(75) (21) (30) (22) (54) (88) 7.7(55) (17) (27) (21) (27) (22) (52) (57) (58) 1.937(17) (72) (15) 1.70(11) (40) (76) (80) (43) (12) (15) (59) (86) (57) (28) (67) (22) 8.9(24) (12) (31) (43) (31) (46) (13) 2.8(11) (41) (78) (79) (41) (10) (56) (20) (52) 2.485(38) (12) (17) (11) (98) (33) 1.324(29) (11) (18) (15) (61) (20) 1.960(18) (62) (10) (76) (38) (37) (81) (30) (28) (60) (15) (47) 4.760(44) (17) (32) (29) (36) (12) 2.14(12) (58) (14) (17) (11) (continued on next page) J. Fisher et al. / Journal of Molecular Spectroscopy 245 (2007)

9 Table 4 (continued) m t K a 0 a a 1 a 2 a 3 a 4 a 5 a 6 a (27) (46) 1.948(23) (47) (41) (40) (83) 2.343(54) (16) (22) (15) (11) (25) 2.49(19) (71) (13) (11) (35) (56) 8.98(30) (69) (71) (13) (62) (19) (22) 2.945(92) (18) (16) (100) (66) 1.449(13) (19) (12) 2.677(24) (10) (43) (11) (18) 1.08(11) (29) (40) (27) (25)* (35) 5.5(16) (30) (12)* (27) 2.10(20) (65) (94) (59) (15) 1.94(14) (55) (12) (14) (80) (46) (13) 1.49(11) (46) (95) (11) (60) (22) (49) 1.343(32) (82) (92) (16) (30) 1.522(16) (34) (31) (24) (12) (21) (81) 1.036(87) (39) (87) (10) (59) (22) (73) 1.543(64) (22) (36) (27) (29) (19) (66) (41) (23) (76) 1.579(75) (32) (69) (79) (46) (15) (37) 1.334(28) (90) (14) (10) (23) (43) 1.294(23) (44) (50) (12) 1.193(86) (27) (37) (23) (35) 1.430(17) (35) (31) (26) (28) (89) (11) (14) (19) 3.65(93) (20) (20) (18) (24) 1.89(12) (25) (26) (12) (80) 2.029(17) (13) (51) (11) (58) (17) (78) (23)* (19) (12)* (13) (14) (64) (36) (20) (37) (23) (45) (33) (30) (18) (93) (59) (91) (57) (18) (40) 1.359(25) (54) (14) (40) 1.461(33) (11) (16) (56) (28) (13) (19) (75) (11) (16) (84) (18) (81) (12) (52) 1.285(62) (28) (53) J. Fisher et al. / Journal of Molecular Spectroscopy 245 (2007) 7 20 a Asterisks denote level sequences not connected to the rest by assigned transitions, See caption to Fig. 1.

10 J. Fisher et al. / Journal of Molecular Spectroscopy 245 (2007) Table 5 Taylor expansion statistics for CH 18 3 OH substates of E torsional symmetry m t K J max J e e max r Table 5 (continued) m t K J max J e e max r tency using loop-sum and multiple-assignment check programs prior to fitting. Transitions that did not satisfy loop-sum relations or were multiply assigned were excluded. The uncertainties of all MW and MMW lines were taken as ±50 khz in the fit, except for a few unresolved K-doublet transitions which were given uncertainties of ±100 to ±200 khz. The uncertainties of the FIR transitions from the Fourier-transform spectra were grouped according to selection rule and torsional excitation energy, and were set to ±6 MHz ( cm 1 ) for m t =0 0and 1 0 transitions, ±9 MHz ( cm 1 ) for m t =1 1 and 2 1 transitions, and ±15 MHz ( cm 1 ) for m t =2 2 and 2 0 transitions. In certain cases we increased the uncertainties of lines known to be strongly overlapped, or of transitions expected to be very weak with K and J rotational quantum numbers changing in opposite direction. In the supplementary information for this paper, we have included a graphic representation of the range of our data in the form of adjacency matrices for the A and E MW/MMW and FT datasets. In these figures, each dot on the graph represents a transition, thus giving an informative visual display of all of the states included in the global modeling both for MW/MMW and FT accuracies.

11 16 J. Fisher et al. / Journal of Molecular Spectroscopy 245 (2007) 7 20 Table 6 Statistics of the dataset for the torsion rotation global fit to m t = 0, 1, 2 torsional states of CH 18 3 OH Overall Std. Dev. Unitless # data # of Params MW (Weight = 50 khz) RMS # Data FTFIR Weights RMS # Data Unitless MHz cm 1 MHz Unitless # cm m t = m t = m t =1 0 m t = m t = m t = m t =2 1 m t = m t = m t = Overall fit results and parameters Our global fit to the CH 3 18 OH data achieved convergence with an overall unitless weighted standard deviation of using 79 adjusted parameters. Detailed statistics are presented in Table 6. The MW and MMW lines were fitted with weighted RMS errors of for the 366 m t = 0 transitions, for the 154 m t = 1 transitions, and for the 30 m t = 2 transitions. The FTFIR lines were fitted with a weighted RMS error of Table 7 presents the parameters from the fit arranged according to increasing order of torsion rotation operator defined as n l + m, where l is the order of the torsional factor and m is the order of the rotational factor [32]. In Table 7, we did not attempt to compare our parameters with those of other workers, because the specific Hamiltonians used in the various studies almost all differ in some important aspect with significant effects on the resulting constants. However, the current set of parameters is comparable with similar fits using the same program for other methanol isotopomers [7 13], and the magnitudes of the lower order rotational and torsional parameters are consistent with the expected changes with isotopic mass. Out of the 79 reported fit parameters in Table 7, 15 have been used for the first time in our methanol isotopic global fits. In the early stages of the fitting, we had fixed k 6 and k 7 to ab initio values (k 6 = cm 1 and k 7 = cm 1 ) [33] because these two parameters are involved in an indeterminacy relation together with F and V 6 [34]. However, through careful trial and error tests we found that it was not necessary or helpful to fix these terms and, indeed, we were able to improve our standard deviation after the k 6 and k 7 terms were removed in the final stages of fitting. In the late stages of the fitting, we noticed D ab parameter was poorly determined which was also the case for other methanol global fits [8,9,11,13]. Though the D ab is a lower operator in torsion and rotational angular momentum, we decided to remove this parameter from our final fit Near degeneracies and couplings In Fig. 4, we plot J-reduced energies as a function of K for A and E symmetries, where the solid line is for A symmetry and the dashed and dotted lines are for +K and K E symmetry. For the m t = 0 ground torsional state, the small torsional splittings are hardly visible at the energy scale of Fig. 4, and the overall curves simply follow the quadratic K-rotation energy dependence. For the m t =1 and 2 excited torsional states, however, the larger torsional splittings significantly alter the quadratic K-rotation behavior and lead to accidental near degeneracies between certain torsion rotation states that can lead to significant coupling and perturbation. Possible sets of coupled levels that can be seen from the J-reduced energy diagram are: (K,m t ) = {(0, 0), (0, ±1), (0,±2)} E levels, {(0,9), (1,5)} A levels, etc. In several cases, the interacting state wavefunctions are sufficiently mixed that forbidden perturbationinduced transitions are observed in the spectrum. Because the effective J-rotational constants are state dependent, near-degenerate K-energy stacks can display J-localized level crossings as J increases, resulting in energy shifts that cannot be modeled by J(J + 1) power series fits alone. However, in our global analysis, the Hamiltonian includes off-diagonal torsional, K-rotational and J-rotational matrix elements that account for the interactions, and the perturbations in all of these coupled systems have been successfully modeled CH 3 18 OH database up to 3 THz To assist Orion survey studies, the CH 3 18 OH global fit parameters from the present work have been used for a general prediction of torsion rotation transition information for m t_max =3,K max = 15, and J max = 35. Along with the calculated transition frequencies, uncertainty estimates were also determined for each transition from the variance covariance matrix of the least squares analyses as described in Ref. [35]. Lower state energies are given, referenced to the K =0A m t = 0 level which is about 127 cm 1 above the bottom of the torsional potential well as used in Ref. [36]. For calculation of the line strengths, which involve the square of the product of a dipole moment component with the torsional overlap and rotational matrix elements, the dipole moments were expanded as a truncated Fourier series l i = l i0 + l i3 Æ 1 cos3cæ (i = a, b, and l c 0). The experimental dipole moments [37] were given as effective

12 Table 7 The 79 torsion rotation parameters (in cm 1 ) in the global fit to m t = 0, 1, and 2 torsional states of 18 O methanol and comparison with CH 3 OH, 13 CH 3 OH, CD 3 OH, CH 3 OD, and CD 3 OD Term Operator b Parameter b Methanol global fit parameters c Order nlm a CH 3 OH (Present work) CH 3 OH (Ref. [8]) 13 CH 3 OH (Ref. [9]) CD 3 OH (Ref. [10]) CH 3 OD (Ref. [11]) CD 3 OD (Ref. [13]) 220 (1 cos3c)/2 V (45) (7) (10) (2) (1) (7) P 2 c F (11) (2) (9) (5) (1) (2) 211 PcPa q d (11) d (1) d (4) d (7) d (2) d (1) d 202 P 2 a A (37) (2) (3) (3) (5) (4) P 2 b B (25) (2) (5) (6) (4) (3) P 2 c C (24) (3) (5) (7) (4) (3) {Pa,Pb} Dab (36) (4) (5) (2) (3) (2) 440 P 4 c k (23) (1) (1) (6) (6) (1) 10 3 (1 cos6c)/2 V (26) 1.60(5) (4) (5) (4) 2.191(7) 431 P 3 c P a k (81) (3) (4) (3) (2) (5) P 2 c P 2 Gv (41) (4) (6) (2) (2) (1) 10 5 P 2 c P 2 a k (11) (4) (5) (4) (2) (7) Pc 2 ðp 2 b P 2 c Þ c (16) (1) (3) (2) (1) (1) 10 5 P 2 c fp a; P bg Dab 1.9(1) (2) (9) (8) 10 4 sin3c{pa,pc} Dac (40) (1) (2) (3) (2) (2) 10 2 sin3c{p b,p c } D bc 0.569(20) (1) (3) (2) (3) (2) 10 3 (1 cos3c)p 2 Fv (40) (8) (8) (3) (9) (2) 10 3 ð1 cos 3cÞP 2 a k (52) (1) (1) (2) (5) (9) 10 2 ð1 cos 3cÞðP 2 b P 2 c Þ c (12) (3) (5) (2) (8) (2) 10 5 (1 cos3c){p a,p b } d ab (17) (2) (5) (2) (5) (5) PcPaP 2 Lv (72) (9) (1) (8) (6) (4) 10 4 P cp 3 a k (11) (1) (3) (3) (2) (4) 10 2 P cfp a; ðp 2 b P 2 c Þg c (15) (1) (3) (2) (1) (1) 10 4 P cðp 2 a P b þ P bp 2 a Þ d ab 1.292(21) (2) (3) (3) (1) (1) P 4 DJ (94) (5) (1) (4) (2) (1) 10 6 P 2 P 2 a DJK (44) (2) (3) (7) (8) (5) 10 5 P 4 a D K (89) (1) (5) (3) (1) (1) P 2ðP 2 b P 2 c Þ dj 5.319(11) (2) (2) (2) (3) (3) 10 8 fp 2 a ; ðp 2 b P 2 c Þg dk (87) (7) (3) (8) (2) (1) 10 5 {Pa,Pb}P 2 DabJ 1.26(6) (1) (6) (4) 10 7 fp 3 a ; P bg D abk 1.256(21) (1) (9) (3) (4) (4) (1 cos9c)/2 V (10) 1.0(2) 1.037(fixed) P 6 c k4b (25) (2) (2) (9) (1) P 5 c P a k 3B 4.032(11) (7) (10) (4) (5) P 4 c P 2 M v 8.065(27) (8) (1) (3) (1) 10 8 P 4 c P 2 a K (19) (1) (2) (7) (9) (1) P 4 c ðp 2 b P 2 c Þ c (32) 10 7 P 4 c fp a; P bg DD ab 0.996(33) (fixed) 7.054(7) 10 9 (1 cos6c)p 2 Nv 5.438(44) (3) (3) (6) (2) 10 6 ð1 cos 6cÞP 2 a K (41) (4) (4) (4) (1) (8) 10 4 ð1 cos 6cÞðP 2 b P 2 c Þ c (22) (3) (5) (4) (1) (1) 10 5 (1 cos6c){p a,p b } dd ab 2.664(28) (5) (1) (2) (9) 10 4 (continued on next page) J. Fisher et al. / Journal of Molecular Spectroscopy 245 (2007)

13 Table 7 (continued) Term Operator b Parameter b Methanol global fit parameters c Order nlm a CH3 18 OH (Present work) CH3OH (Ref. [8]) 13 CH3OH (Ref. [9]) CD3OH (Ref. [10]) CH3OD (Ref. [11]) CD3OD (Ref. [13]) 633 P 3 c P ap 2 k3j (79) (2) (5) (1) (1) 10 7 P 3 c P 3 a K (18) (9) (2) (6) (8) (1) 10 5 P 3 c fp a; ðp 2 b P 2 c Þg c (67) (8) (3) 10 9 P 3 c fp 2 a ; P bg dd ab 0.952(31) (fixed) {(1 cos3c), PaP 2 Pc} K6J 1.2(4) 10 5 fð1 cos 3cÞ; P 3 a P cg K6K 2.797(55) (6) (fixed) 2.3(2) (9) P 2 c P 4 g v 0.727(10) (1) (3) (1) (1) (2) 10 9 P 2 c P 2 a P 2 k 2J (96) (2) (6) (2) (3) 10 7 P 2 c P 4 a k2k (10) (3) (1) (2) (4) P 2 c P 2 ðp 2 b P 2 c Þ c5 2.22(16) (fixed) 2P 2 c fp 2 a ; ðp 2 b P 2 c Þg c (18) (38) (1) 10 8 (1 cos3c)p 4 f v 6.707(57) (7) (1) (5) (3) (2) 10 9 ð1 cos 3cÞP 2 a P 2 k5j 4.501(29) (2) (3) (7) (5) (3) 10 7 ð1 cos 3cÞP 4 a fk 4.526(90) (1) (3) (3) (1) (3) 10 4 ð1 cos 3cÞðP 2 b P 2 c ÞP 2 c 2J 0.814(17) (8) (1) 10 7 ð1 cos 3cÞfP 2 a ; ðp 2 b P 2 c Þg c (37) (fixed) sin3c{pa,pc}p 2 D acj 1.651(17) P c P a P 4 l v (90) (1) (3) (9) (2) 10 9 P cp 3 a P 2 k v (56) (2) (3) (8) (3) 10 7 P cp 5 a lk (32) (fixed) 8.91(4) (1) 10 5 P cp 2 fp a; ðp 2 b P 2 c Þg c7 1.57(11) (fixed) P cfp 3 a ; ðp 2 b P 2 c Þg c 7K 0.31(3) P 6 H J 1.160(64) (fixed) 0.29(4) P 4 P 2 a HJK 3.082(38) (6) (2) (4) P 4 a P 2 HKJ (14) (1) (7) (2) (8) (7) 10 7 P 6 a H K (50) (2) (6) (2) (1) (2) 10 6 P 2 fp 2 a ; ðp 2 b P 2 c Þg hjk 1.32(10) (fixed) 880 P 8 c k 4BB (91) P 7 c P a k 3BB 6.707(53) P 6 c P 2 k 4BJ 2.52(26) P 6 c P 2 a k 4BK 1.854(14) 10 6 (1 cos9c){pa,pb} V 9ab 2.417(41) P 5 c P ap 2 k 3BJ 0.508(48) P 5 c P 3 a k 3BK 2.848(19) P 4 c P 2 a P 2 K 1J 2.52(22) P 4 c P 4 a K 1K 2.628(16) 10 6 ð1 cos 6cÞfP 2 a ; ðp 2 b P 2 c Þg c 11K 3.60(45) 10 8 ð1 cos 6cÞfP 3 a ; P bg dd abk 1.242(65) P 3 c P 3 a P 2 k 3JK (86) P 2 c P 6 a k 2KK 4.501(25) P cp 7 a l kk 5.974(32) J. Fisher et al. / Journal of Molecular Spectroscopy 245 (2007) 7 20 a Order of the Hamiltonian term in the notation of Ref. [32]: n = l + m, where n is the total order of the operator, l is the order of the torsional factor, and m is the order of the rotational factor. b Notation of Ref. [7]. {A,B} AB + BA. The product of the parameter and operator from a given row yields the term actually used in the vibration rotation torsion Hamiltonian, except for F, q, and A, which occur in the Hamiltonian in the form F ðp c þ qp aþ 2 þ AP 2 a. Parameters marked with an asterisk and shown in red are used in a methanol global fit for the first time. c Parameter uncertainties are given in parentheses, and represent one standard deviation in the last digit. d q is unitless.

14 J. Fisher et al. / Journal of Molecular Spectroscopy 245 (2007) dipole moments for each of m t = 0, 1, 2, 3. The Fourier expansion coefficients were then determined as the least squares solution to the over determined system of equations Al f = l e, where l e is the column vector containing the effective dipole moments for m t =0,1,2,3,l f is the column vector containing the Fourier expansion coefficients, and the matrix A is defined as A i1 1and A i2 hm t ¼ i 1; r; J; Kj1 cos 3cjm t ¼ i 1; r; J; Ki for i = 1, 2, 3, 4. There is some variation of the expectation values Æm t, r, J, Kj1 cos3cjm t, r, J, Kæ with r and K, so the values used were the average of the K =0, r = 0, and K =0,r = 1 values. The intensities were calculated in four ways: (i) l a0, l a3, l b0, l b3 were set to the values determined by the above method; (ii) l a0 and l b0 were set to the average (over m t ) of the experimental l a and l b ; (iii) l a0 and l b0 were set to the same values as (i), but l a3 and l b3 set to 0; and (iv) l a0 and l b0 were set to the experimental l a and l b corresponding to m t =0. Where MW/MMW transitions have been reported, the original measurements are also listed. For the FTFIR transitions, frequencies were converted from cm 1 to MHz using the speed of light (1 cm 1 = MHz). The complete line list is available on request from one of us (lxu@unbsj.ca). 4. Discussion and conclusions In this work, FTFIR spectra of CH 3 18 OH were measured from 15 to 470 cm 1 at the National Research Council of Canada and Justus-Liebig University, Giessen, and have been analyzed with the Ritz program [16] based on the Ritz combination principle. A total of transitions have been assigned covering torsional states m t =0, 1, 2, and 3. Among them, 7662 are of A torsional symmetry, and are of E torsional symmetry. State dependent coefficients in J(J + 1) power series expansions of the transition wavenumbers have been obtained for the assigned subbands, together with asymmetry splitting coefficients for subbands of A symmetry for which K-doubling was resolved. We were able to bring more than 93% of the FTFIR transitions in the first three torsional states m t = 0, 1, and 2 combined with 550 previously published MMW and MW measurements into a global analysis employing a Hamiltonian and program used successfully before for a variety of methanol isotopomers. The global fitting of this dataset was accomplished with an overall weighted standard deviation of 1.072, i.e., essentially to within mean experimental uncertainty. The torsion rotation model employed 79 adjustable parameters and included terms up to 8th order in the torsion rotation operators, some of which were introduced for the first time. A CH 3 18 OH database has been compiled up to 3 THz using the global ð6þ fit parameters from this work together with the derived permanent dipole moment components and their variation in torsion from Ref. [37]. The database and the results of the fit in the form of the full dataset together with the least squares (m obs m calc ) residuals have been deposited as supplementary information in the journal archive. A limiting factor in the current global analysis is the relatively narrow range of transitions measured to high accuracy with MW and MMW techniques. While the quantum state coverage of the Fourier-transform FIR data is much greater, the FT uncertainties are also much larger, ranging from a few MHz to tens of MHz. Thus, new MMW frequency measurements for CH 3 18 OH are needed in order to improve our fits and our predictive ability. As well, a broader body of precision measurements would help to better define those high order parameters that are not necessarily well determined in the current fit. Acknowledgments The authors thank Drs. I. Kleiner and M. Godefroid for making their acetaldehyde (CH 3 CHO) internal-rotation global-fitting program available for this work, and Dr. Jon T. Hougen for his continuing interest in the project. L.H.X. and R.M.L. thank the Natural Sciences and Engineering Research Council of Canada for financial support of this research program. J. Fisher and G. Paciga are grateful for NSERC-USRA (Undergraduate Summer Research Award) support. Appendix A. Supplementary data Supplementary CH 3 18 OH Line list: Microwave and Terahertz transitions of CH 3 18 OH in order of frequency. An asterisk after a measured frequency indicates that the transition was not included in the fit. Supplementary data for this article are available on ScienceDirect ( and as part of the Ohio State University Molecular Spectroscopy Archives ( lib.ohio-state.edu/jmsa_hp.htm). References [1] S. Leurini, P. Schilke, K.M. Menten, D.R. Flower, J.T. Pottage, Li-Hong Xu, Astron. Astrophys. 422 (2004) [2] Private communication, from Jose Cernicharo, of the Instituto de Estructura de la Materia in Madrid, Spain. [3] Herschel HIFI website. Available from: < divisions/lea/hifi/>. [4] SOFIA website. Available from: < [5] ALMA website. Available from: < [6] F.F. Gardner, J.B. Whiteoak, J. Reynolds, W.L. Peters, T.B.H. Kuiper, Mon. Not. Roy. Astron. Soc. 240 (1989) 35. [7] Li-Hong Xu, J.T. Hougen, J. Mol. Spectrosc. 169 (1995) [8] Li-Hong Xu, J.T. Hougen, J. Mol. Spectrosc. 173 (1995) [9] Li-Hong Xu, M.S. Walsh, R.M. Lees, J. Mol. Spectrosc. 179 (1996) [10] M.S. Walsh, Li-Hong Xu, R.M. Lees, J. Mol. Spectrosc. 188 (1998)