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Reinvestigation of the ground and first torsional state of methylformate
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Journal of Molecular Spectroscopy 246 (2007) 158–166 www.elsevier.com/locate/jms
Reinvestigation of the ground and first torsional state of methylformate M. Carvajal b
a,*
, F. Willaert b, J. Demaison b, I. Kleiner
c
a Departamento de Fı´sica Aplicada, Facultad de Ciencias Experimentales, Universidad de Huelva, 21071 Huelva, Spain Laboratoire de Physique des Lasers, Atomes, et Mole´cules, UMR CNRS 8523, Universite´ de Lille I, F-59655 Villeneuve d’Ascq Ce´dex, France c Laboratoire Interuniversitaire des Syste`mes Atmosphe´riques, UMR CNRS 7583, Universite´ Paris 7 et Universite´ Paris 12, 61 av. Charles de Gaulle, F-94010 Cre´teil Ce´dex, France
Received 7 May 2007; in revised form 29 August 2007 Available online 20 September 2007
Abstract We have reinvestigated the laboratory spectrum for the methylformate HCOOCH3 molecule involving both the ground and first excited torsional states. We have fitted within almost experimental accuracy a data set for this molecule consisting of 3496 vt ¼ 0 and 774 vt ¼ 1 microwave lines, using the so-called ‘‘rho axis method’’ (RAM) and a model extended to include perturbation terms through eighth order. The previously published vt ¼ 0 and vt ¼ 1 microwave lines covering the J values up to 43 in the ground state and up to 18 in the first excited state have been extended by 434 new measurements from Lille in the 567–669 GHz spectral range, corresponding to transitions with J max ¼ 62, K max ¼ 22 in vt ¼ 0. The final fit requires only 49 parameters to achieve a unitless weighted standard deviation of 1.43 for a total of 4270 lines for the whole fit. This result represents an improvement over the previous fit which achieved a standard deviation of 1.96 for 3862 lines using 69 parameters. A calculation of the line strengths of torsion–rotation transitions up to J = 60 needed for the astronomical survey is also provided. 2007 Elsevier Inc. All rights reserved. Keywords: Methyl formate; Radioastronomy; Internal rotation; Microwave spectrum
1. Introduction Methyl formate (HCOOCH3) is one of the most important molecules in astrophysics and it has been detected in the hot cores of giant molecular clouds such as Orion KL and Sgr B2(N) and in comets [1–4]. The relatively large abundance of methylformate in hot core sources has led to the identification of an important number of interstellar lines (about 500 rotational lines belonging to the ground state were identified in the hot core regions where stars are forming and very recently 20 lines belonging to vt ¼ 1 were identified in Orion KL [3]). Even though methyl formate, HCOOCH3, is extremely abundant in numerous interstellar clouds [3,4] its mechanism of formation is not yet understood [5], requiring further studies.
*
Corresponding author. E-mail address: [email protected] (M. Carvajal).
0022-2852/$ - see front matter 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.jms.2007.08.009
Its rotational spectrum is dense because of a relatively small rotational A constant (almost 0.6 cm1), leading to the observation of very high J values (up to 70), because of the presence of two non-zero components of the dipole moment, leading to both a-type and b-type transitions (la ¼ 1:63 Debye and lb ¼ 0:68 Debye), and because of the existence of three states at low frequencies (the torsional mode at 130 cm1, the COC bending mode m12 at 318 cm1 and the out-of-plane bending mode m17 at 332 cm1), leading to the observation of rotational transitions within those levels populated at room temperature. These three low frequency vibrations can also in principle interact with each other, perturbing the frequency and intensity distribution, but no evidence for such perturbations has been observed in the range of the quantum numbers we study. To give an idea of the density of the spectrum, let us note that for a temperature of 296 K and with an arbitrary cutoff for the line strength of 104, more than 33 000 lines are predicted for the ground torsional
M. Carvajal et al. / Journal of Molecular Spectroscopy 246 (2007) 158–166
state vt ¼ 0 in the range 0–4000 GHz (with most of the transitions occurring in the range 0–1600 GHz). The methylformate spectrum is also complicated by the combination of a large amplitude motion, the torsion, with the large asymmetry splittings occurring in this fairly asymmetric near-prolate rotor ðj ¼ 0:78Þ. In particular, the internal rotation of the methyl group splits each rotational line into a doublet (characterized by the symmetry labels A and E). As the molecule is light and as the barrier to internal rotation is not so high, the internal rotation splittings are relatively large and the transition frequencies are extremely difficult to calculate with accuracy. For this reason, the identification of interstellar methyl formate was made possible only thanks to intense laboratory work, which has led to the assignment of more than 3000 lines up to 608 GHz in the vibrational ground state [6–14]. In the most recent studies, the rotational spectrum of the first excited torsional state was observed in the frequency range 7– 200 GHz [14]. Around 530 lines were also recorded up to 620 GHz for the vibrational ground state of 13C1-methyl formate [15]. For some time the 500–600 GHz spectral range has been of particular interest for astronomers because the radio telescopes which are in development (HERSCHEL, ALMA, SOFIA) will operate in this sub-millimeter wave range. For this reason, it is important to have accurate predictions for methyl formate in this spectral range. The challenge faced in the present study was to test the performance of the method when applied to the analysis of internal rotation phenomena at the especially high values of J and K quantum numbers one reaches in the 500– 600 GHz range. Although the present paper represents some improvement over past millimeter and microwave studies on methyl formate [6–14], the main effort was focused on getting higher J values, which are rather intense and of importance for astrophysical detection. The global fit achieved in the present work allows us to obtain a better determination of all the parameters compared to previous studies, especially the higher-order terms. Even though the microwave spectrum of methylformate is rather complex and dense as said above, this eight-atom molecule is rather small and well adapted to perform high level quantum chemical calculations. It can become another ‘‘test’’ model to validate ab initio [16] and Density Functional Theory calculations by comparing them with experimental results, in the same way as acetaldehyde CH3CHO or methanol CH3OH [17]. High-level ab initio calculations are now being performed in order to compare the barrier height obtained with the experimentally derived value. The equilibrium structure of methylformate which was never determined with high accuracy will also be determined in this future paper [18]. 2. Experimental details The submillimeterwave measurements were performed in Lille with a source-modulated spectrometer using
159
phase-stabilized backward wave oscillators working in the frequency range 300–700 GHz. The sources were Russian ISTOK backward-wave oscillators. They were phaselocked on the emission of a harmonic of a 2–20 GHz Hewlett-Packard (HP 83711A) synthetizer whose frequency was first multiplied by six and amplified by a Millitech active frequency multiplier. The intermediate frequency beat near 320 MHz was compared with the 32nd harmonic of the 10 MHz signal from a second HP synthetizer (HP 3325B). The mixer was a new planar Schottky diode optimized for the range 500–650 GHz and provided by the University of Virginia. It was placed in a parabolic structure. The large step frequency tuning of the source was obtained by changing the frequency of the 2–20 GHz synthetizer. The small step frequency scan is provided by means of a 10 MHz (9.75–10.25 MHz) synthesizer. The typical acquisition time was 0.1 s per point. A sine-wave frequency modulation of 5 kHz (modulation depth: 400 kHz) was applied to the 10 MHz and the signal was demodulated at twice this frequency. The circular absorption cell, 6 cm in diameter and 110 cm long, was made of stainless steel. The absorption was detected with a liquid helium cooled InSb bolometer (QMC). The measurements were made at room temperature and the pressure in the absorption cell was from 10 to 30 mTorr. Computer processing was used to improve the signal/noise ratio and to measure the frequency of the lines, whose accuracy is better than 50 kHz in most cases. 3. Theoretical model The Hamiltonian used in the present work is the socalled ‘‘RAM’’ (‘‘Rho Axis Method’’) internal-rotation Hamiltonian based on the work of Kirtman [19], Lees and Baker [20], and Herbst et al. [21]. Since rather complete descriptions of this method, which takes its name from the choice of axis system, has been presented in the literature [21–23], we will not repeat here such a general description. We also have applied this formalism and our code to a number of internal rotor molecules (see for example [24]). We will only emphasize here the various characteristics that the RAM program had to face for the present study. The threefold torsional potential barrier is rather high ðV 3 371 cm1 Þ but the value of the reduced height s = 4V3/9F (with F 5.49 cm1 being the internal rotation constant) is about 30, so torsional splittings reach values up to a few MHz in the torsional ground state. In addition, methyl formate has two small-amplitude vibrations [25], the COC bending mode m12 at 318 cm1 and the out-of-plane COC bending m17 mode at 332 cm1, which lie near the top of the barrier and therefore could cause some perturbations on the rotational–torsional energy levels under study in the present work. Third, methylformate is a fairly asymmetric near-prolate internal rotor molecule ðj ¼ 0:78Þ leading to a clustering of the transitions with the same Kc quantum number for high J, low Ka values. This large asymmetry, modeled by
160
M. Carvajal et al. / Journal of Molecular Spectroscopy 246 (2007) 158–166
the difference between the B and C constants multiplying the term ðP 2b P 2c Þ in the Hamiltonian, leads to the determination of a rather large number of torsionally dependent contributions to this term of the form P 2c ðP 2b P 2c Þ; cos 3cðP 2b P 2c Þ . . .. Finally, the value of the q parameter which represents the coupling term between internal rotation and global rotation in the kinetic energy operator F ðP c qP a Þ2 where F is the internal rotation constant, Pc is the internal angular momentum and Pa is the component of the global rotation angular momentum along the RAM a axis, is only 0.084. The consequence of such a small q value is to lengthen the period of the cosine function describing the torsional splittings as described in Eqs. (4) and (6) of Ref. [23]. For the F ðP c qKÞ2 sign choice of our program, the +Ka E-species levels belonging to even values of vt lie below the Ka levels, and the +Ka E species levels belonging to odd values of vt lie above the Ka levels for all values of jKaj from 1 to 18. This is the same labeling scheme as the one adopted for our previous acetic acid study [24]. In this reference, the connection with the more traditional J K a ;K c labeling scheme was also discussed.
Table 1 of [14]. We have thus decided to take some of them out of the fit. In addition, based on our fitting residuals, we have allocated an increased weight of 200 kHz, instead of the 100 kHz originally quoted by the authors, to all Oesterling data [12]. A noticeable improvement in our root-meansquare deviation was achieved by this change, since the unitless weighted standard deviation decreased from 1.52 to 1.43. We suggest that some Oesterling lines with observed-calculated deviations higher than exhibited by the other lines in the same series should be checked with a new experimental recording. The ‘‘best’’ fit allowed 49 parameters (much less than the 69 parameters used in [14]) to vary. The microwave rootmean-square deviations obtained are 94 kHz (for 3496 vt ¼ 0 lines) and 84 kHz (for 774 vt ¼ 1 lines), respectively, involving transitions with J 6 62, Ka 6 22 for the ground torsional state and J 6 17 and jKaj 6 7 for the vt ¼ 1 state. Separate root-mean-square deviations for the A and E species are 91 and 94 kHz for 2181 and 2089 lines respectively. Since we fit both species simultaneously, this demonstrates the similar quality of the fit for the two symmetry species. All the present standard deviations represent an improve-
4. Assignments and fit In the present paper, we present a combined analysis of (i) 2 lines published by Brown et al. [7] with a 3 kHz uncertainty, (ii) 51 lines published by Bauder [8] with a 20 kHz uncertainty, (iii) 97 lines published by Plummer et al. [10] with a 50 kHz uncertainty, (iv) 147 lines of the 148 originally fitted by Plummer et al. [11] with a 50 kHz uncertainty, (v) 44 lines from Demaison et al. [9] with a 50 kHz uncertainty, (vi) 2591 lines (of 2612) from Ogata et al. [14] also with a 50 kHz uncertainty and 7 lines from these authors with a 200 kHz uncertainty, (vii) 897 lines of the 908 originally published by Oesterling et al. [12] with a 200 kHz uncertainty and (viii) 434 new lines (of the 438 originally measured) from Lille, measured using broadband recording in various frequency intervals between 567 and 669 GHz with an uncertainty of 50 kHz. Starting from the same data set as in Ref. [14] which includes 3077 vt ¼ 0 lines and 785 vt ¼ 1 microwave transitions, we refit the data using our program [23,24]1 starting at low J and going up progressively. Some improvements were made in our program in order to fit transitions with higher J and K than was possible with the previous version of the code. During this process, we noticed that a number of previously published lines showed observed-calculated values which were much worse than the ones belonging to the rest of the same K sub-branch. Those lines correspond also to transitions which were given a weight of 250 kHz in 1 The source code for the fit, an example of input data file and a readme file are available at the web site (http://www.ifpan.edu.pl/~kisiel/introt/ introt.htm#belgi) managed by Dr. Zbigniew Kisiel. Extended versions of code are also available with one of the authors (I.K).
Table 1 Root-mean-square (rms) deviations from the global fita of transitions involving vt ¼ 0 and 1 torsional energy levels of methylformate (H12COOCH3) Number of parameters Number of lines rms of the 3496 MW vt ¼ 0 0 lines rms of the 774 MW vt ¼ 1 1 lines rms of the 2181 A symmetry lines rms of the 2089 E symmetry lines Sourceb Rangec (GHz)
vt, Jmax, Kmaxd Number of linese
49 4270 0.0940 MHz 0.0836 MHz 0.0906 MHz 0.0939 MHz Uncertaintiesf (MHz)
rmsg (MHz)
Brown around 1.6 0,1,1
2
0.003
0.0029
Bauder 8–58
0,40,10
51
0.020
0.0267
LILLE PluE PluA DeMa TYAM
0,62,22 0,30,15 0,40,18 0,28,12 0,40,17 1,18,7
434 147 97 44 2591
0.050h
0.0778
7 897
0.200i
0.1340
567–669 200–352 216–506 150–313 7–200
TYAM 7–200 Oest 250–510
0,43,18
a Parameter values are given in Table 2. Some examples of the observed minus calculated residuals are given in Table 3 for vt ¼ 0 lines and in Table 4 of the present paper for vt ¼ 1. b Sources of data: Brown – Ref. [7]; Bauder – Ref. [8]; LILLE data comes from present work; PluE – Ref. [11]; PluA – Ref. [10]; DeMa – Ref. [9]; TYAM – Ref. [14]; Oest – Ref. [12]. c Range containing the measurements in a given row. d vt state and maximum J and Ka for each group of measurements. e Number of MW lines in each uncertainty group. f One-sigma standard uncertainly in MHz used in the fit. g Root mean square deviation in MHz for each group. h In Ref. [9] the experimental uncertainty was assumed to be 30 kHz.
M. Carvajal et al. / Journal of Molecular Spectroscopy 246 (2007) 158–166
ment over those of Ref. [14], especially for the E species where the previous standard deviations were 134 and 163 kHz for vt ¼ 0 and 1, respectively. The overall quality of the fit is illustrated in Table 1, which gives the root-mean-square deviations for transitions grouped according to their measurement uncertainties (weight in the fit was proportional to reciprocal of squared uncertainty). Table 2 presents the values for the rotation–torsion parameters used in our model, which includes up to eighth order parameters, i.e. those with n = l + m = 8, where n is the total order of the operator, l is the order of the torsional part and m is the order of the rotational part, respectively, following the notation of our previous papers on the RAM method like Ref. [24]. In comparison with the previous work, only 49 parameters are needed to achieve nearly experimental accuracy. However because no direct information on the torsional vibrational frequency vt ¼ 1 0 is included in the fit, the correlation between the torsional parameters F, V3 and V6 is rather strong, increasing the true uncertainties of those parameters somewhat beyond the standard deviations given in Table 2. Tables 3 and 4 show some examples of the line assignments, the observed frequencies with the measurement uncertainties (in parentheses), the observed-calculated values, the references of the data sources, the line strengths for the transitions in the vt ¼ 0 and 1 torsional states, respectively, and the lower and upper state energies. The whole table of the fitted lines is available in the Supplementary Data. Despite our efforts, some lines from the new measurements show somewhat large observed calculated values, but at the present time it is unclear whether the discrepancies occur because of some missing high order parameters or because some of lines which present large observed calculated values, are causing the parameters to shift. Indeed some previously measured MW lines show inadequate combination differences, i.e. violate a ‘‘closed combination differences loop criterion’’ similar to that described in Ref. [20]. Indeed of all the ‘‘loops’’ checked, 12% of them (involving 1747 energy levels) show combination differences exceeding about 0.4 MHz. As a test of quality of our parameters it is useful to compare the rotational constants among references. Hence a transformation of our rotational constants from the RHO axis system (RAM) to the principal axis system (PAM) was carried out. In Table 5 both the RAM rotational constants obtained in the fitting (see Table 2) in MHz and their transformed values in the principal axis system are presented. The 3 · 3 matrix of RAM rotational constants (ARAM, BRAM, CRAM, plus the off-diagonal inertia constant Dab) was diagonalized using the values of those parameters from Table 2. This transformation corresponds to a rotation of the RAM axis system (where the a-axis lies along the direction of the q vector) into the PAM axis system. This rotation is about the out-of-plane c-axis, which stays the same in the two axis systems. The angle ðhRAM Þ
161
Table 2 Torsion–rotation parameters needed for the global fit of transitions involving vt ¼ 0 and vt ¼ 1 torsional energy levels of methylformate (H12COOCH3) nlma Operatorb 220 211 202
440 431 422
413
404
642
624
633
606
826 844
(1 cos 3c)/2 P 2c Pc Pa P 2a P 2b P 2c ðP a P b þ P b P a Þ P 4c (1 cos 6c)/2 P 3c P a P 2c P 2 2P 2c ðP 2b P 2c Þ sin 3cðP a P c þ P c P a Þ (1 cos 3c) P2 ð1 cos 3cÞP 2a ð1 cos 3cÞðP 2b P 2c Þ ð1 cos 3cÞðP a P b þ P b P a Þ P 2c P 2a P 2c ðP a P b þ P b P a Þ P cP aP 2 P c P 3a P c fP a ; ðP 2b P 2c Þg P c fP 2a ; P b g P4 P 2 P 2a P 4a 2P 2 ðP 2b P 2c Þ fP 2a ; ðP 2b P 2c Þg ðP 3a P b þ P b P 3a Þ (1 cos 6c) P2 ð1 cos 6cÞðP 2b P 2c Þ 2P 4c ðP 2b P 2c Þ (1 cos 3c) P4 ð1 cos 3cÞðP 2b P 2c ÞP 2 ð1 cos 3cÞfP 2a ; ðP 2b P 2c Þg 2P 2c ðP 2b P 2c ÞP 2 ð1 cos 3cÞðP a P b þ P b P a ÞP 2 ð1 cos 3cÞðP 3a P b þ P b P 3a Þ ð1 cos 3cÞP 2a P 2 P 3c P 2 P a P 3c P 3a P 3c fP a ; ðP 2b P 2c Þg P 3c fP 2a ; P b g P6 P 4 P 2a P 2 P 4a P 6a ð1 cos 3cÞðP 2b P 2c ÞP 4 2P 4c ðP 2b P 2c ÞP 2
Parameter Present workc V3 F q ARAM BRAM CRAM Dab k4 V6 k3 Gv c1 Dac Fv k5 c2 dab k2 Dab Lv k1 c4 dab DJ DJK DK dJ dK DabK Nv c11 c3 fv c2J c2K c1J dabJ dabK k5J k3J k3K c12 ddab HJ HJK HKJ HK c2JJ c3J
370.924(113) 5.49038(129) 0.08427127(723) 0.5884101(188) 0.3082455(179) 0.17711843(416) 0.1649794(162) 0.0004368(184) 23.9018(636) 0.00012758(711) 0.000002709(432) 0.000018117(264) 0.0068896(540) 0.0025827(184) 0.0112949(386) 0.0012608(253) 0.0063031(176) 0.00002837(166) 0.000008874(434) 0.000003932(110) 0.000000596(279) 0.0000001100(561) 0.000010141(145) 0.00000042854(455) 0.0000019285(527) 0.0000036534(594) 0.00000017990(227) 0.00000028824(900) 0.0000020747(108) 0.0000507(127) 0.0014751(202) 0.00000045962(750) 0.00000009957(441) 0.00000005483(445) 0.00000024458(404) 0.0000000017386(211) 0.00000012488(883) 0.00000019649(625) 0.0000005853(125) 0.00000007061(198) 0.00000008230(461) 0.00000006946(110) 0.000000061998(808) 0.000000000000333(35) 0.000000000015998(570) 0.00000000007620(187) 0.00000000009098(281) 0.000000000001746(201) 0.000000000029116(514)
a
Notation of Ref. [24]; n = l + m, where n is the total order of the operator, l is the order of the torsional part and m is the order of the rotational part. b Notation of Ref. [24]. {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 qP a Þ2 þ ARAM P 2a . c Values of the parameters from the present fit for vt ¼ 0 and 1. All values are in cm1, except for q which is unitless. Statistical uncertainties are given in parentheses in units of the last quoted digit.
between the RHO a-axis and the principal a-axis can be obtained from the following expression:
162
Table 3 Assignmentsa, observed valuesb, residuals from the fitc, line strengthsd, sourcese, and energy levelsf for selected methylformate microwave transitions within the vt = 0 state J0
K0a
K 0c
P 0a
v00t
J00
K 00a
K 00c
P 00
Obs. Freq. (Unc) (MHz)b
Obs calc (MHz)c
Line Str.d
Refe
E 0 (cm1)f
E00 (cm1)
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
6 7 8 9 10 11 12 13 14 15 21 22 23 24 25 26 27 28 32 38 39 40 41 42 43 49 50 51 53 56 57 58 46 47 48 50 52 53 54 56 57 58 59 60 46 47 48 50
5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 17 17 17 17 17 2 2 2 2 2 2 2 18 18 18 18
1 2 3 4 5 6 7 8 9 10 16 17 18 19 20 21 22 23 27 33 34 35 36 37 38 44 45 46 48 51 52 53 29 30 31 33 35 51 52 54 55 56 57 58 28 29 30 32
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
5 6 7 8 9 10 11 12 13 14 20 21 22 23 24 25 26 27 31 37 38 39 40 41 42 48 49 50 52 55 56 57 45 46 47 49 51 52 53 55 56 57 58 59 45 46 47 49
5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 17 17 17 17 17 2 2 2 2 2 2 2 18 18 18 18
0 1 2 3 4 5 6 7 8 9 15 16 17 18 19 20 21 22 26 32 33 34 35 36 37 43 44 45 47 50 51 52 28 29 30 32 34 50 51 53 54 55 56 57 27 28 29 31
73665.582(50) 86029.804 98435.693(50) 110890.210(50) 123403.403(50) 135988.491(50) 148664.544(50) 161458.217(50) 174406.161(50) 187555.025(50) 271532.798(50) 285542.584(200) 299123.858(200) 312165.784(200) 324596.208(200) 336373.838(200) 347494.169(200) 358008.265(200) 396951.238(200) 457227.937(50) 467585.380(200) 477981.080(200) 488403.977(200) 498846.142(200) 509301.582(200) 572154.497(50) 582636.171(50) 593117.395(50) 614077.171(50) 645505.446(50) 655977.286(50) 666446.913(50) 569096.503(50) 581719.679(50) 594361.029(50) 619701.062(50) 645124.212(50) 584920.937(50) 595437.262(50) 616460.517(50) 626967.354(50) 637470.721(50) 647970.750(50) 658467.283(50) 568292.658(50) 580851.692(50) 593425.076(50) 618616.760(50)
0.154 0.397 0.114 0.050 0.034 0.002 0.021 0.012 0.008 0.010 0.041 0.037 0.018 0.038 0.136 0.014 0.198 0.130 0.025 0.114 0.028 0.029 0.075 0.187 0.092 0.029 0.087 0.030 0.047 0.098 0.031 0.172 0.021 0.023 0.080 0.097 0.231 0.061 0.006 0.045 0.020 0.050 0.059 0.102 0.004 0.047 0.040 0.009
0.487436E + 01 0.911594E + 01* 0.129621E + 02 0.165448E + 02 0.199432E + 02 0.232075E + 02 0.263712E + 02 0.294577E + 02 0.324834E + 02 0.354611E + 02 0.529188E + 02 0.557931E + 02 0.586151E + 02 0.613648E + 02 0.640328E + 02 0.666206E + 02 0.691424E + 02 0.716248E + 02 0.817250E + 02 0.976085E + 02 0.100286E + 03 0.102969E + 03 0.105658E + 03 0.108351E + 03 0.111049E + 03 0.127357E + 03 0.130090E + 03 0.132799E + 03 0.138077E + 03 0.147876E + 03 0.152558E + 03 0.157239E + 03 0.106137E + 03 0.109201E + 03 0.112252E + 03 0.118320E + 03 0.124348E + 03 0.159264E + 03 0.162342E + 03 0.168479E + 03 0.171567E + 03 0.174665E + 03 0.177762E + 03 0.180853E + 03 0.104078E + 03 0.107184E + 03 0.110273E + 03 0.116419E + 03
TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM PluA Oest Oest Oest Oest Oest Oest Oest Oest PluA Oest Oest Oest Oest Oest NEW NEW NEW NEW NEW NEW NEW NEW NEW NEW NEW NEW NEW NEW NEW NEW NEW NEW NEW NEW NEW NEW NEW
91.0294 93.8991 97.1825 100.8814 104.9977 109.5338 114.4927 119.8784 125.6960 131.9521 179.1828 188.7075 198.6852 209.0980 219.9253 231.1455 242.7367 254.6786 305.7165 392.1303 407.7273 423.6710 439.9624 456.6022 473.5906 582.8575 602.2921 622.0764 662.6937 726.2407 748.1217 770.3520 648.2667 667.6708 687.4966 728.4156 771.0293 618.0133 637.8749 678.6502 699.5635 720.8273 742.4413 764.4054 664.1024 683.4775 703.2720 744.1213
88.5722 91.0294 93.8991 97.1825 100.8814 104.9977 109.5338 114.4927 119.8784 125.6960 170.1255 179.1828 188.7075 198.6852 209.0980 219.9253 231.1455 242.7367 292.4756 376.8789 392.1303 407.7273 423.6710 439.9624 456.6022 563.7725 582.8575 602.2921 642.2103 704.7090 726.2407 748.1217 629.2837 648.2667 667.6708 707.7446 749.5103 598.5024 618.0133 658.0873 678.6502 699.5635 720.8273 742.4413 645.1461 664.1024 683.4775 723.4864
M. Carvajal et al. / Journal of Molecular Spectroscopy 246 (2007) 158–166
v0t
33 27 28 29 31 27 28 30 32 27 29 31 26 28 30 18 19 19 19 19 20 20 20 20 21 21 21 22 22 22 51 45 46 47 49 46 47 49 51 47 49 51 47 49 51 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 34 28 29 30 32 28 29 31 33 28 30 32 27 29 31 18 19 19 19 19 20 20 20 20 21 21 21 22 22 22
a
52 46 47 48 50 47 48 50 52 48 50 52 48 50 52 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Upper and lower state quantum numbers are indicated by 0 and 00 , respectively. Torsion–rotation levels of A species, only given in the first part of the table, have a ‘‘parity’’ label; levels of E species, given in the second part of the table, have a signed Ka value [21]. b Observed vt = 0 microwave transitions in MHz, with estimated uncertainties in parentheses. c Observed minus calculated values in MHz. If followed by an ‘*’, the line was excluded from the fit. d Calculated line strengths [23]. e The references indicate, respectively: Brow: Ref. [7]; Baud: Ref. [8]; PluA, PluE: Ref. [10,11]; Oest: Ref. [12];DeMa: Ref. [9]; TYAM: Ref. [13,14]; NEW: present results. f Upper and lower state energy (cm1).
765.1770 661.9641 680.8979 700.2489 740.2042 698.6595 717.9910 757.9027 799.4818 736.6867 776.5624 818.1001 756.3277 796.1735 837.6767 786.6543 680.8979 700.2489 720.0175 760.8094 717.9910 737.7386 778.4837 820.8974 756.4170 797.1234 839.4928 776.0435 816.7178 859.0503 NEW NEW NEW NEW NEW NEW NEW NEW NEW NEW NEW NEW NEW NEW NEW 0.122518E + 03 0.101902E + 03 0.105052E + 03 0.108186E + 03 0.114411E + 03 0.102804E + 03 0.105983E + 03 0.112291E + 03 0.118542E + 03 0.103667E + 03 0.110064E + 03 0.116396E + 03 0.101239E + 03 0.107728E + 03 0.114145E + 03 0.114 0.005 0.019 0.017 0.088 0.003 0.101 0.003 0.017 0.093 0.011 0.078 0.129 0.281 0.425 643872.491(50) 567619.486(50) 580128.726(50) 592649.412(50) 617726.870(50) 579542.320(50) 592019.011(50) 617001.594(50) 642024.619(50) 591498.047(50) 616401.761(50) 641337.599(50) 591065.894(50) 615903.391(50) 640766.508(50)
E 0 (cm1)f Line Str.d Obs calc (MHz)c K 0c K0a J0 v0t
Table 3 (continued)
P 0a
v00t
J00
K 00a
K 00c
P 00
Obs. Freq. (Unc) (MHz)b
Refe
E00 (cm1)
M. Carvajal et al. / Journal of Molecular Spectroscopy 246 (2007) 158–166
tanð2hRAM Þ ¼
2Dab ðA BÞ
163
ð1Þ
From the values of the rotational constants A, B and Dab from Table 2, hRAM ¼ 24:83 degrees. In Table 5 we also corroborate our result with an ab initio calculation (MP2/VTZ) of the principal axis rotational constants from the molecular structure. The angles in degrees between the principal axis (a,b,c) and the methyl top axis (i) obtained from our fitting and calculated by means of MP2/VTZ are also compared in Table 5. 5. Intensity calculation For astrophysical detections, line intensities are of course very important. In the case of the methylformate molecule, we need to take into account the internal rotation splittings. In the same way that the Hamiltonian was used in the calculation and fit of the line positions, the calculation of the line strengths was also carried out in the RHO axis system. For this purpose, the components of the dipole moment must be expressed in this non-principal reference system. The approach and the equations used here are given in detail in Ref. [23]. To obtain the line strengths in the RAM system, the dipole moment components must be transformed by a rotation of the principal-axis dipole moment components: la cosðhRAM Þ sinðhRAM Þ la ¼ ð2Þ sinðhRAM Þ cosðhRAM Þ lb PAM lb RAM Our intensity calculations will be based on the assumption that the microwave intensities are only driven by the permanent electric dipole moments [23], without any dependence on the torsional angle and other vibrational coordinates. In the case of methyl formate, the permanent electric dipole moment components have the values la ¼ þ1:63 D and lb ¼ þ0:68 D in the principal axis system [8]. The direction in which the dipole moment was found to lie at an angle of 39.4(2) from the C@O bond (Fig. 1, [6]). In the ‘‘chemical’’ convention we have adopted, la and lb have positive signs with respect to the chosen principal axes (i.e. the dipole moment is pointing from the positive charges to the negative charges of the molecule), a convention which we also used for acetic acid [24]. The RAM dipole moment components obtained by Eq. (2) are given in Table 6. The line strengths for all the assigned transitions were obtained and are presented in Tables 3 and 4 for the torsional ground and first excited states, respectively. A comparison between our calculated line strengths and those calculated by Ogata et al. [14] shows that they are in general in very good agreement. One cautionary comment should be made on the calculated intensities here. For the A species transitions certain selection rules on the parity (corresponding to A1 $ A2 overall selection rules) must be obeyed. However as a consequence of carrying out the calculations without factorizing the A block of the Hamil-
164
Table 4 Assignmentsa, observed valuesb, residuals from the fitc, line strengthsd, sourcese, and energy levelsf for selected methylformate microwave transitions within the vt = 1 state J0
K0a
K 0c
P 0a
v00t
J00
K00a
K 00c
P 00
Obs. Freq. (Unc) (MHz)b
Obs calc (MHz)c
Line Str.d
Refe
E 0 (cm1)f
E00 (cm1)
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
3 4 5 6 7 8 9 10 11 12 13 14 15 16 8 9 10 11 12 13 14 15 8 9 10 11 12 13 14 15 5 6 7 8 9 10 11 12 13 14 15 16 17 8 9 10 11 12
0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 1 1 1 1 1 1 1 1 1 1 1 1 1 7 7 7 7 7
3 4 5 6 7 8 9 10 11 12 13 14 15 16 3 4 5 6 7 8 9 10 2 3 4 5 6 7 8 9 4 5 6 7 8 9 10 11 12 13 14 15 16 2 3 4 5 6
+ + + + + + + + + + + + + + + + + + + + + +
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
2 3 4 5 6 7 8 9 10 11 12 13 14 15 8 9 10 11 12 13 14 15 8 9 10 11 12 13 14 15 4 5 6 7 8 9 10 11 12 13 14 15 16 8 9 10 11 12
0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 1 1 1 1 1 1 1 1 1 1 1 1 1 6 6 6 6 6
2 3 4 5 6 7 8 9 10 11 12 13 14 15 4 5 6 7 8 9 10 11 3 4 5 6 7 8 9 10 3 4 5 6 7 8 9 10 11 12 13 14 15 3 4 5 6 7
+ + + + + + + + + + + + + + + + + + + + + +
35781.717(50) 47157.063(50) 58157.430(50) 68840.033(50) 79324.821(50) 89731.695(50) 100136.911(50) 110571.569(50) 121040.237(50) 131536.624(50) 142052.800(50) 152581.782(50) 163118.722(50) 173660.281(50) 152606.026(50) 152443.185(50) 152213.504(50) 151905.159(50) 151511.582(50) 151033.416(50) 150483.224(50) 149890.033(50) 180586.240(50) 180509.008(50) 180391.481(50) 180223.247(50) 179993.047(50) 179689.693(50) 179301.511(50) 178818.702(50) 63888.606(50) 76203.676(50) 88220.742(50) 99869.136(50) 111094.105(50) 121901.608(50) 132378.695(50) 142664.666(50) 152891.340(50) 163142.587(50) 173452.699(50) 183824.379(50) 194247.050(50) 181831.461(50) 181706.050(50) 181532.761(50) 181300.551(50) 180997.053(50)
0.024 0.035 0.048 0.049 0.073 0.073 0.067 0.114 0.044 0.029 0.065 0.087 0.148 0.249 0.031 0.281 0.018 0.031 0.047 0.020 0.024 0.052 0.099 0.082 0.028 0.041 0.074 0.007 0.032 0.070 0.022 0.036 0.031 0.090 0.049 0.078 0.059 0.039 0.044 0.020 0.039 0.081 0.120 0.087 0.035 0.067 0.032 0.100
0.788670E + 01 0.104706E + 02 0.130376E + 02 0.156075E + 02 0.181969E + 02 0.208090E + 02 0.234381E + 02 0.260779E + 02 0.287241E + 02 0.313746E + 02 0.340285E + 02 0.366853E + 02 0.393450E + 02 0.420074E + 02 0.126452E + 01 0.162707E + 01 0.197763E + 01 0.232142E + 01 0.266209E + 01 0.300222E + 01 0.334359E + 01 0.368716E + 01 0.894613E + 00 0.128657E + 01 0.165778E + 01 0.201520E + 01 0.236369E + 01 0.270675E + 01 0.304702E + 01 0.338657E + 01 0.127149E + 02 0.153992E + 02 0.180415E + 02 0.206475E + 02 0.232217E + 02 0.257801E + 02 0.283460E + 02 0.309356E + 02 0.335508E + 02 0.361853E + 02 0.388319E + 02 0.414859E + 02 0.441445E + 02 0.879032E + 00 0.126522E + 01 0.163179E + 01 0.198566E + 01 0.233167E + 01
TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM
204.2199 205.7929 207.7328 210.0290 212.6750 215.6682 219.0084 222.6967 226.7341 231.1217 235.8601 240.9497 246.3907 252.1834 233.0920 236.7482 240.8143 245.2915 250.1811 255.4846 261.2034 267.3390 239.1158 242.7693 246.8315 251.3031 256.1851 261.4784 267.1842 273.3038 208.1580 210.6999 213.6426 216.9739 220.6796 224.7458 229.1615 233.9203 239.0202 244.4620 250.2478 256.3795 262.8589 238.5992 242.2659 246.3428 250.8308 255.7307
203.0263 204.2199 205.7929 207.7328 210.0290 212.6750 215.6682 219.0084 222.6967 226.7341 231.1217 235.8601 240.9497 246.3907 228.0017 231.6632 235.7370 240.2245 245.1273 250.4467 256.1838 262.3393 233.0920 236.7482 240.8143 245.2915 250.1811 255.4846 261.2034 267.3390 206.0269 208.1580 210.6999 213.6426 216.9739 220.6796 224.7458 229.1615 233.9203 239.0202 244.4620 250.2478 256.3795 232.5340 236.2049 240.2875 244.7832 249.6933
M. Carvajal et al. / Journal of Molecular Spectroscopy 246 (2007) 158–166
v0t
8 9 10 11 6 6 6 6 13 14 15 16 1 1 1 1 7 8 9 10 7 7 7 7
a
13 14 15 16 1 1 1 1
Upper and lower state quantum numbers are indicated by 0 and 00 , respectively. Torsion–rotation levels of A species, only given in the first part of the table, have a ‘‘parity’’ label; levels of E species, given in the second part of the table, have a signed Ka value [21]. b Observed vt = 1 microwave transitions in MHz, with estimated uncertainties in parentheses. c Observed minus calculated values in MHz. If followed by an ‘*’, the line was excluded from the fit. d Calculated Line Strengths [23]. e The references indicate, respectively: Brow: Ref. [7]; Baud: Ref. [8]; PluA, PluE: Ref. [10,11]; Oest: Ref. [12]; DeMa: Ref. [9]; TYAM: Ref. [13,14]; NEW: present results. f Upper and lower state energy (cm1).
255.0194 260.7632 266.9265 273.5114 261.0438 266.7712 272.9141 279.4739 TYAM TYAM TYAM TYAM 180607.773(50) 180116.664(50) 179504.871(50) 178751.035(50)
0.021 0.026 0.025 0.079
0.267339E + 01 0.301355E + 01 0.335428E + 01 0.369712E + 01
E 0 (cm1)f K00a J00 K 0c K0a J0 v0t
Table 4 (continued)
P 0a
v00t
K 00c
P 00
Obs. Freq. (Unc) (MHz)b
Obs calc (MHz)c
Line Str.d
Refe
E00 (cm1)
M. Carvajal et al. / Journal of Molecular Spectroscopy 246 (2007) 158–166
165
Table 5 Rotational constants in the RHO axis system (RAM) and in the principal axis system (PAM)
A(MHz) B(MHz) C(MHz) Dab (MHz) <(i,a) <(i,b) <(i,c) hRAM
RAMa
PAMb
PAMc
17640.0910 9240.9676 5309.8769 4945.9580
19939.5304 6954.4483 5309.8769 0.0
19848.5032 7006.1251 5351.3017 0.0
52.989 37.011 90.000
58.568 31.432 90.000
24.83d
Angles between the principal axis and the methyl top axis. a Rotational constants obtained in our work for RAM-axis system (see Table 2). b Rotational constants of our work transformed to the Principal Axis System and angles in degrees between the principal axis (a,b,c) and the methyl top axis (i). c Calculation of the principal axis rotational constants from the molecular structure (MP2/VTZ) and of the angles in degrees between the principal axis (a,b,c) and the methyl top axis (i). d The angle hRAM between the a-principal axis and the a-RAM axis is given in degrees and obtained from Eq. (1) with the parameters ARAM, BRAM, CRAM, and Dab of Table 2.
Fig. 1. The chemical structure, principal axes and direction of the dipole moment of methyl formate [6]. The a–b plane is a plane of symmetry.
Table 6 Dipole moment components in Debye in the principal axis system (PAM) and in the RHO axis system (RAM)
la lb a
RAM
PAMa
+1.765 0.067
+1.63 +0.68
Experimental dipole moment from Ref. [8].
tonian matrix into A1 and A2 submatrices, the parity of the numerically generated eigenvector is not well defined when K-type doublets remain degenerate. Intensities for such cases were obtained by calculating both the formally parity-allowed and parity-forbidden components of a given
166
M. Carvajal et al. / Journal of Molecular Spectroscopy 246 (2007) 158–166
K-type doublet transition, and then ascribing all the calculated intensity to the two allowed transitions. The transition energies and line strengths have been predicted for a number of lines within the torsional ground state and the first excited state (around 33100 lines for vt ¼ 0 and 38280 lines for vt ¼ 1 at a temperature of 296 K and with a cutoff of 104 for the line strength. The prediction involves transition energies up to 4000 GHz and J 6 60, Ka 6 60, with most of the transitions falling in the range 0–1600 GHz. These lines should be useful for astrophysicists in the identification of new spectral lines in the interstellar space.
6. Conclusion A new global analysis of the rotational levels in the torsional ground and first excited states for the parent methyl formate has been carried out by using the RHO Axis Model. In this study around 400 new submillimeter lines provided by the group of Lille were assigned and included in a joint fit with the existing data in the microwave, millimeter and submillimeter spectral region. An analysis of our rotational constants shows that they are in agreement with other spectroscopic parameters of different authors. Due to the importance of this molecule for the remote sensing of the interstellar medium we have calculated not only the frequencies but also the relative intensities of the available data and we have predicted them for nearly 71 380 lines (at 296 K and with a cutoff for the line strength of 104) within vt ¼ 0 and vt ¼ 1. In the future, a new recording of some available data lines would be valuable in order to obtain those lines with better accuracy and to refine the rotational parameters. Also a recording of high resolution far infrared data in the 130 cm1 region where the torsional fundamental vt ¼ 1 0 band (and overtones) absorb would be important to refine the torsional parameters and make them uncorrelated.
Acknowledgments I.K. would like to thank the ‘‘Institut du De´veloppement et des Ressources en Informatique Scientifique’’ IDRIS, for the allocation of computer time. M.C. thanks the CNRS for financial support (project CERC3). I.K. and J.D. thank the PEPCO-NEI cooperation project (MENESR, France). The authors would also like to thank Dr. J.T. Hougen for numerous discussions during the work, V.V. Ilyushin for providing us a combination differences
of the experimental data, and M.L. Senent for numerous useful comments about the work. Appendix A. Supplementary data Supplementary data for this article are available on ScienceDirect (www.sciencedirect.com) and as part of the Ohio State University Molecular Spectroscopy Archives (http://msa.lib.ohio-state.edu/jmsa_hp.htm). References [1] A. Nummelin, P. Bergman, A. Hjalmarson, P. Friberg, W.M. Irwine, T.J. Millar, S. Saito, Astrophys. J. Suppl. Ser. 128 (2000) 213–243. [2] G.A. Blake, E.C. Sutton, G.R. Masson, T.G. Philips, Astrophys. J. Suppl. Ser. 60 (1986) 357–374. [3] K. Kobayashi, K. Ogata, S. Tsunekawa, S. Takano, Astrophys. J. 657 (2007) L17–L19. [4] F.J. Lovas, NIST Recommended rest frequencies for observed interstellar molecular microwave transitions (Gaithersburg, NIST). 2004. [5] A. Horn, H. Møllendal, O. Sekiguchi, E. Uggerud, H. Roberts, E. Herbst, A.A. Viggiano, T.D. Fridgen, Astrophys. J. 611 (2004) 605–614. [6] R.F. Curl, J. Chem. Phys. 30 (1959) 1529–1536. [7] R.D. Brown, J.G. Crofts, F.F. Gardner, P.D. Godfrey, B.J. Robinson, J.B. Whiteoak, Astrophys. J. 197 (1975) L29–L31. [8] A. Bauder, J. Phys. Chem. Ref. Data 8 (1979) 583–618. [9] J. Demaison, D. Boucher, A. Dubrulle, B.P. Van Eijck, J. Mol. Spectrosc. 102 (1983) 260–263. [10] G.M. Plummer, G.A. Blake, E. Herbst, F.C. De Lucia, Astrophys. J. Suppl. 55 (1984) 633–656. [11] G.M. Plummer, E. Herbst, F.C. De Lucia, G.A. Blake, Astrophys. J. Suppl. 60 (1986) 949–961. [12] L.C. Oesterling, S. Albert, F.C. De Lucia, K.V.L.N. Sastry, E. Herbst, Astrophys. J. 521 (1999) 255–260. [13] Y. Karakawa, K. Oka, H. Odashima, K. Takagi, S. Tsunekawa, J. Mol. Spectrosc. 210 (2001) 196–212. [14] K. Ogata, H. Odashima, K. Takagi, S. Tsunekawa, J. Mol. Spectrosc. 225 (2004) 14–32. [15] F. Willaert, H. Møllendal, E. Alekseev, M. Carvajal, I. Kleiner, J. Demaison, J. Mol. Struct. 795 (2006) 4–8. ´ nguez-Go´mez, [16] M.L. Senent, M. Villa, F.J. Mele´ndez, R. Domı Astrophys. J. 627 (2005) 567–576. [17] L-H. Xu, R.M. Lees, J.T. Hougen, Chem. Phys. 110 (1999) 3835–3841. [18] J. Demaison, A.G. Csa´sza´r, E. Czinki, J. Lie´vin, in preparation. [19] B. Kirtman, J. Chem. Phys. 37 (1962) 2516–2539. [20] R.M. Lees, J.G. Baker, J. Chem. Phys. 48 (1968) 5299–5318. [21] E. Herbst, J.K. Messer, F.C. De Lucia, P. Helminger, J. Mol. Spectrosc. 108 (1984) 42–57. [22] C.C. Lin, J.D. Swalen, Rev. Mod. Phys. 31 (1959) 841–892. [23] J.T. Hougen, I. Kleiner, M. Godefroid, J. Mol. Spectrosc. 163 (1994) 559–586. [24] V.V. Ilyushyn, E.A. Alekseev, S.F. Dyubko, I. Kleiner, J. Mol. Spectrosc. 220 (2003) 170–186. [25] T. Shimanouchi, Tables of Molecular Vibrational Frequencies Consolidated, vol. 1, National Bureau of Standards, 1972, pp. 1–160.
Journal of Molecular Spectroscopy 246 (2007) 158–166 www.elsevier.com/locate/jms
Reinvestigation of the ground and first torsional state of methylformate M. Carvajal b
a,*
, F. Willaert b, J. Demaison b, I. Kleiner
c
a Departamento de Fı´sica Aplicada, Facultad de Ciencias Experimentales, Universidad de Huelva, 21071 Huelva, Spain Laboratoire de Physique des Lasers, Atomes, et Mole´cules, UMR CNRS 8523, Universite´ de Lille I, F-59655 Villeneuve d’Ascq Ce´dex, France c Laboratoire Interuniversitaire des Syste`mes Atmosphe´riques, UMR CNRS 7583, Universite´ Paris 7 et Universite´ Paris 12, 61 av. Charles de Gaulle, F-94010 Cre´teil Ce´dex, France
Received 7 May 2007; in revised form 29 August 2007 Available online 20 September 2007
Abstract We have reinvestigated the laboratory spectrum for the methylformate HCOOCH3 molecule involving both the ground and first excited torsional states. We have fitted within almost experimental accuracy a data set for this molecule consisting of 3496 vt ¼ 0 and 774 vt ¼ 1 microwave lines, using the so-called ‘‘rho axis method’’ (RAM) and a model extended to include perturbation terms through eighth order. The previously published vt ¼ 0 and vt ¼ 1 microwave lines covering the J values up to 43 in the ground state and up to 18 in the first excited state have been extended by 434 new measurements from Lille in the 567–669 GHz spectral range, corresponding to transitions with J max ¼ 62, K max ¼ 22 in vt ¼ 0. The final fit requires only 49 parameters to achieve a unitless weighted standard deviation of 1.43 for a total of 4270 lines for the whole fit. This result represents an improvement over the previous fit which achieved a standard deviation of 1.96 for 3862 lines using 69 parameters. A calculation of the line strengths of torsion–rotation transitions up to J = 60 needed for the astronomical survey is also provided. 2007 Elsevier Inc. All rights reserved. Keywords: Methyl formate; Radioastronomy; Internal rotation; Microwave spectrum
1. Introduction Methyl formate (HCOOCH3) is one of the most important molecules in astrophysics and it has been detected in the hot cores of giant molecular clouds such as Orion KL and Sgr B2(N) and in comets [1–4]. The relatively large abundance of methylformate in hot core sources has led to the identification of an important number of interstellar lines (about 500 rotational lines belonging to the ground state were identified in the hot core regions where stars are forming and very recently 20 lines belonging to vt ¼ 1 were identified in Orion KL [3]). Even though methyl formate, HCOOCH3, is extremely abundant in numerous interstellar clouds [3,4] its mechanism of formation is not yet understood [5], requiring further studies.
*
Corresponding author. E-mail address: [email protected] (M. Carvajal).
0022-2852/$ - see front matter 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.jms.2007.08.009
Its rotational spectrum is dense because of a relatively small rotational A constant (almost 0.6 cm1), leading to the observation of very high J values (up to 70), because of the presence of two non-zero components of the dipole moment, leading to both a-type and b-type transitions (la ¼ 1:63 Debye and lb ¼ 0:68 Debye), and because of the existence of three states at low frequencies (the torsional mode at 130 cm1, the COC bending mode m12 at 318 cm1 and the out-of-plane bending mode m17 at 332 cm1), leading to the observation of rotational transitions within those levels populated at room temperature. These three low frequency vibrations can also in principle interact with each other, perturbing the frequency and intensity distribution, but no evidence for such perturbations has been observed in the range of the quantum numbers we study. To give an idea of the density of the spectrum, let us note that for a temperature of 296 K and with an arbitrary cutoff for the line strength of 104, more than 33 000 lines are predicted for the ground torsional
M. Carvajal et al. / Journal of Molecular Spectroscopy 246 (2007) 158–166
state vt ¼ 0 in the range 0–4000 GHz (with most of the transitions occurring in the range 0–1600 GHz). The methylformate spectrum is also complicated by the combination of a large amplitude motion, the torsion, with the large asymmetry splittings occurring in this fairly asymmetric near-prolate rotor ðj ¼ 0:78Þ. In particular, the internal rotation of the methyl group splits each rotational line into a doublet (characterized by the symmetry labels A and E). As the molecule is light and as the barrier to internal rotation is not so high, the internal rotation splittings are relatively large and the transition frequencies are extremely difficult to calculate with accuracy. For this reason, the identification of interstellar methyl formate was made possible only thanks to intense laboratory work, which has led to the assignment of more than 3000 lines up to 608 GHz in the vibrational ground state [6–14]. In the most recent studies, the rotational spectrum of the first excited torsional state was observed in the frequency range 7– 200 GHz [14]. Around 530 lines were also recorded up to 620 GHz for the vibrational ground state of 13C1-methyl formate [15]. For some time the 500–600 GHz spectral range has been of particular interest for astronomers because the radio telescopes which are in development (HERSCHEL, ALMA, SOFIA) will operate in this sub-millimeter wave range. For this reason, it is important to have accurate predictions for methyl formate in this spectral range. The challenge faced in the present study was to test the performance of the method when applied to the analysis of internal rotation phenomena at the especially high values of J and K quantum numbers one reaches in the 500– 600 GHz range. Although the present paper represents some improvement over past millimeter and microwave studies on methyl formate [6–14], the main effort was focused on getting higher J values, which are rather intense and of importance for astrophysical detection. The global fit achieved in the present work allows us to obtain a better determination of all the parameters compared to previous studies, especially the higher-order terms. Even though the microwave spectrum of methylformate is rather complex and dense as said above, this eight-atom molecule is rather small and well adapted to perform high level quantum chemical calculations. It can become another ‘‘test’’ model to validate ab initio [16] and Density Functional Theory calculations by comparing them with experimental results, in the same way as acetaldehyde CH3CHO or methanol CH3OH [17]. High-level ab initio calculations are now being performed in order to compare the barrier height obtained with the experimentally derived value. The equilibrium structure of methylformate which was never determined with high accuracy will also be determined in this future paper [18]. 2. Experimental details The submillimeterwave measurements were performed in Lille with a source-modulated spectrometer using
159
phase-stabilized backward wave oscillators working in the frequency range 300–700 GHz. The sources were Russian ISTOK backward-wave oscillators. They were phaselocked on the emission of a harmonic of a 2–20 GHz Hewlett-Packard (HP 83711A) synthetizer whose frequency was first multiplied by six and amplified by a Millitech active frequency multiplier. The intermediate frequency beat near 320 MHz was compared with the 32nd harmonic of the 10 MHz signal from a second HP synthetizer (HP 3325B). The mixer was a new planar Schottky diode optimized for the range 500–650 GHz and provided by the University of Virginia. It was placed in a parabolic structure. The large step frequency tuning of the source was obtained by changing the frequency of the 2–20 GHz synthetizer. The small step frequency scan is provided by means of a 10 MHz (9.75–10.25 MHz) synthesizer. The typical acquisition time was 0.1 s per point. A sine-wave frequency modulation of 5 kHz (modulation depth: 400 kHz) was applied to the 10 MHz and the signal was demodulated at twice this frequency. The circular absorption cell, 6 cm in diameter and 110 cm long, was made of stainless steel. The absorption was detected with a liquid helium cooled InSb bolometer (QMC). The measurements were made at room temperature and the pressure in the absorption cell was from 10 to 30 mTorr. Computer processing was used to improve the signal/noise ratio and to measure the frequency of the lines, whose accuracy is better than 50 kHz in most cases. 3. Theoretical model The Hamiltonian used in the present work is the socalled ‘‘RAM’’ (‘‘Rho Axis Method’’) internal-rotation Hamiltonian based on the work of Kirtman [19], Lees and Baker [20], and Herbst et al. [21]. Since rather complete descriptions of this method, which takes its name from the choice of axis system, has been presented in the literature [21–23], we will not repeat here such a general description. We also have applied this formalism and our code to a number of internal rotor molecules (see for example [24]). We will only emphasize here the various characteristics that the RAM program had to face for the present study. The threefold torsional potential barrier is rather high ðV 3 371 cm1 Þ but the value of the reduced height s = 4V3/9F (with F 5.49 cm1 being the internal rotation constant) is about 30, so torsional splittings reach values up to a few MHz in the torsional ground state. In addition, methyl formate has two small-amplitude vibrations [25], the COC bending mode m12 at 318 cm1 and the out-of-plane COC bending m17 mode at 332 cm1, which lie near the top of the barrier and therefore could cause some perturbations on the rotational–torsional energy levels under study in the present work. Third, methylformate is a fairly asymmetric near-prolate internal rotor molecule ðj ¼ 0:78Þ leading to a clustering of the transitions with the same Kc quantum number for high J, low Ka values. This large asymmetry, modeled by
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M. Carvajal et al. / Journal of Molecular Spectroscopy 246 (2007) 158–166
the difference between the B and C constants multiplying the term ðP 2b P 2c Þ in the Hamiltonian, leads to the determination of a rather large number of torsionally dependent contributions to this term of the form P 2c ðP 2b P 2c Þ; cos 3cðP 2b P 2c Þ . . .. Finally, the value of the q parameter which represents the coupling term between internal rotation and global rotation in the kinetic energy operator F ðP c qP a Þ2 where F is the internal rotation constant, Pc is the internal angular momentum and Pa is the component of the global rotation angular momentum along the RAM a axis, is only 0.084. The consequence of such a small q value is to lengthen the period of the cosine function describing the torsional splittings as described in Eqs. (4) and (6) of Ref. [23]. For the F ðP c qKÞ2 sign choice of our program, the +Ka E-species levels belonging to even values of vt lie below the Ka levels, and the +Ka E species levels belonging to odd values of vt lie above the Ka levels for all values of jKaj from 1 to 18. This is the same labeling scheme as the one adopted for our previous acetic acid study [24]. In this reference, the connection with the more traditional J K a ;K c labeling scheme was also discussed.
Table 1 of [14]. We have thus decided to take some of them out of the fit. In addition, based on our fitting residuals, we have allocated an increased weight of 200 kHz, instead of the 100 kHz originally quoted by the authors, to all Oesterling data [12]. A noticeable improvement in our root-meansquare deviation was achieved by this change, since the unitless weighted standard deviation decreased from 1.52 to 1.43. We suggest that some Oesterling lines with observed-calculated deviations higher than exhibited by the other lines in the same series should be checked with a new experimental recording. The ‘‘best’’ fit allowed 49 parameters (much less than the 69 parameters used in [14]) to vary. The microwave rootmean-square deviations obtained are 94 kHz (for 3496 vt ¼ 0 lines) and 84 kHz (for 774 vt ¼ 1 lines), respectively, involving transitions with J 6 62, Ka 6 22 for the ground torsional state and J 6 17 and jKaj 6 7 for the vt ¼ 1 state. Separate root-mean-square deviations for the A and E species are 91 and 94 kHz for 2181 and 2089 lines respectively. Since we fit both species simultaneously, this demonstrates the similar quality of the fit for the two symmetry species. All the present standard deviations represent an improve-
4. Assignments and fit In the present paper, we present a combined analysis of (i) 2 lines published by Brown et al. [7] with a 3 kHz uncertainty, (ii) 51 lines published by Bauder [8] with a 20 kHz uncertainty, (iii) 97 lines published by Plummer et al. [10] with a 50 kHz uncertainty, (iv) 147 lines of the 148 originally fitted by Plummer et al. [11] with a 50 kHz uncertainty, (v) 44 lines from Demaison et al. [9] with a 50 kHz uncertainty, (vi) 2591 lines (of 2612) from Ogata et al. [14] also with a 50 kHz uncertainty and 7 lines from these authors with a 200 kHz uncertainty, (vii) 897 lines of the 908 originally published by Oesterling et al. [12] with a 200 kHz uncertainty and (viii) 434 new lines (of the 438 originally measured) from Lille, measured using broadband recording in various frequency intervals between 567 and 669 GHz with an uncertainty of 50 kHz. Starting from the same data set as in Ref. [14] which includes 3077 vt ¼ 0 lines and 785 vt ¼ 1 microwave transitions, we refit the data using our program [23,24]1 starting at low J and going up progressively. Some improvements were made in our program in order to fit transitions with higher J and K than was possible with the previous version of the code. During this process, we noticed that a number of previously published lines showed observed-calculated values which were much worse than the ones belonging to the rest of the same K sub-branch. Those lines correspond also to transitions which were given a weight of 250 kHz in 1 The source code for the fit, an example of input data file and a readme file are available at the web site (http://www.ifpan.edu.pl/~kisiel/introt/ introt.htm#belgi) managed by Dr. Zbigniew Kisiel. Extended versions of code are also available with one of the authors (I.K).
Table 1 Root-mean-square (rms) deviations from the global fita of transitions involving vt ¼ 0 and 1 torsional energy levels of methylformate (H12COOCH3) Number of parameters Number of lines rms of the 3496 MW vt ¼ 0 0 lines rms of the 774 MW vt ¼ 1 1 lines rms of the 2181 A symmetry lines rms of the 2089 E symmetry lines Sourceb Rangec (GHz)
vt, Jmax, Kmaxd Number of linese
49 4270 0.0940 MHz 0.0836 MHz 0.0906 MHz 0.0939 MHz Uncertaintiesf (MHz)
rmsg (MHz)
Brown around 1.6 0,1,1
2
0.003
0.0029
Bauder 8–58
0,40,10
51
0.020
0.0267
LILLE PluE PluA DeMa TYAM
0,62,22 0,30,15 0,40,18 0,28,12 0,40,17 1,18,7
434 147 97 44 2591
0.050h
0.0778
7 897
0.200i
0.1340
567–669 200–352 216–506 150–313 7–200
TYAM 7–200 Oest 250–510
0,43,18
a Parameter values are given in Table 2. Some examples of the observed minus calculated residuals are given in Table 3 for vt ¼ 0 lines and in Table 4 of the present paper for vt ¼ 1. b Sources of data: Brown – Ref. [7]; Bauder – Ref. [8]; LILLE data comes from present work; PluE – Ref. [11]; PluA – Ref. [10]; DeMa – Ref. [9]; TYAM – Ref. [14]; Oest – Ref. [12]. c Range containing the measurements in a given row. d vt state and maximum J and Ka for each group of measurements. e Number of MW lines in each uncertainty group. f One-sigma standard uncertainly in MHz used in the fit. g Root mean square deviation in MHz for each group. h In Ref. [9] the experimental uncertainty was assumed to be 30 kHz.
M. Carvajal et al. / Journal of Molecular Spectroscopy 246 (2007) 158–166
ment over those of Ref. [14], especially for the E species where the previous standard deviations were 134 and 163 kHz for vt ¼ 0 and 1, respectively. The overall quality of the fit is illustrated in Table 1, which gives the root-mean-square deviations for transitions grouped according to their measurement uncertainties (weight in the fit was proportional to reciprocal of squared uncertainty). Table 2 presents the values for the rotation–torsion parameters used in our model, which includes up to eighth order parameters, i.e. those with n = l + m = 8, where n is the total order of the operator, l is the order of the torsional part and m is the order of the rotational part, respectively, following the notation of our previous papers on the RAM method like Ref. [24]. In comparison with the previous work, only 49 parameters are needed to achieve nearly experimental accuracy. However because no direct information on the torsional vibrational frequency vt ¼ 1 0 is included in the fit, the correlation between the torsional parameters F, V3 and V6 is rather strong, increasing the true uncertainties of those parameters somewhat beyond the standard deviations given in Table 2. Tables 3 and 4 show some examples of the line assignments, the observed frequencies with the measurement uncertainties (in parentheses), the observed-calculated values, the references of the data sources, the line strengths for the transitions in the vt ¼ 0 and 1 torsional states, respectively, and the lower and upper state energies. The whole table of the fitted lines is available in the Supplementary Data. Despite our efforts, some lines from the new measurements show somewhat large observed calculated values, but at the present time it is unclear whether the discrepancies occur because of some missing high order parameters or because some of lines which present large observed calculated values, are causing the parameters to shift. Indeed some previously measured MW lines show inadequate combination differences, i.e. violate a ‘‘closed combination differences loop criterion’’ similar to that described in Ref. [20]. Indeed of all the ‘‘loops’’ checked, 12% of them (involving 1747 energy levels) show combination differences exceeding about 0.4 MHz. As a test of quality of our parameters it is useful to compare the rotational constants among references. Hence a transformation of our rotational constants from the RHO axis system (RAM) to the principal axis system (PAM) was carried out. In Table 5 both the RAM rotational constants obtained in the fitting (see Table 2) in MHz and their transformed values in the principal axis system are presented. The 3 · 3 matrix of RAM rotational constants (ARAM, BRAM, CRAM, plus the off-diagonal inertia constant Dab) was diagonalized using the values of those parameters from Table 2. This transformation corresponds to a rotation of the RAM axis system (where the a-axis lies along the direction of the q vector) into the PAM axis system. This rotation is about the out-of-plane c-axis, which stays the same in the two axis systems. The angle ðhRAM Þ
161
Table 2 Torsion–rotation parameters needed for the global fit of transitions involving vt ¼ 0 and vt ¼ 1 torsional energy levels of methylformate (H12COOCH3) nlma Operatorb 220 211 202
440 431 422
413
404
642
624
633
606
826 844
(1 cos 3c)/2 P 2c Pc Pa P 2a P 2b P 2c ðP a P b þ P b P a Þ P 4c (1 cos 6c)/2 P 3c P a P 2c P 2 2P 2c ðP 2b P 2c Þ sin 3cðP a P c þ P c P a Þ (1 cos 3c) P2 ð1 cos 3cÞP 2a ð1 cos 3cÞðP 2b P 2c Þ ð1 cos 3cÞðP a P b þ P b P a Þ P 2c P 2a P 2c ðP a P b þ P b P a Þ P cP aP 2 P c P 3a P c fP a ; ðP 2b P 2c Þg P c fP 2a ; P b g P4 P 2 P 2a P 4a 2P 2 ðP 2b P 2c Þ fP 2a ; ðP 2b P 2c Þg ðP 3a P b þ P b P 3a Þ (1 cos 6c) P2 ð1 cos 6cÞðP 2b P 2c Þ 2P 4c ðP 2b P 2c Þ (1 cos 3c) P4 ð1 cos 3cÞðP 2b P 2c ÞP 2 ð1 cos 3cÞfP 2a ; ðP 2b P 2c Þg 2P 2c ðP 2b P 2c ÞP 2 ð1 cos 3cÞðP a P b þ P b P a ÞP 2 ð1 cos 3cÞðP 3a P b þ P b P 3a Þ ð1 cos 3cÞP 2a P 2 P 3c P 2 P a P 3c P 3a P 3c fP a ; ðP 2b P 2c Þg P 3c fP 2a ; P b g P6 P 4 P 2a P 2 P 4a P 6a ð1 cos 3cÞðP 2b P 2c ÞP 4 2P 4c ðP 2b P 2c ÞP 2
Parameter Present workc V3 F q ARAM BRAM CRAM Dab k4 V6 k3 Gv c1 Dac Fv k5 c2 dab k2 Dab Lv k1 c4 dab DJ DJK DK dJ dK DabK Nv c11 c3 fv c2J c2K c1J dabJ dabK k5J k3J k3K c12 ddab HJ HJK HKJ HK c2JJ c3J
370.924(113) 5.49038(129) 0.08427127(723) 0.5884101(188) 0.3082455(179) 0.17711843(416) 0.1649794(162) 0.0004368(184) 23.9018(636) 0.00012758(711) 0.000002709(432) 0.000018117(264) 0.0068896(540) 0.0025827(184) 0.0112949(386) 0.0012608(253) 0.0063031(176) 0.00002837(166) 0.000008874(434) 0.000003932(110) 0.000000596(279) 0.0000001100(561) 0.000010141(145) 0.00000042854(455) 0.0000019285(527) 0.0000036534(594) 0.00000017990(227) 0.00000028824(900) 0.0000020747(108) 0.0000507(127) 0.0014751(202) 0.00000045962(750) 0.00000009957(441) 0.00000005483(445) 0.00000024458(404) 0.0000000017386(211) 0.00000012488(883) 0.00000019649(625) 0.0000005853(125) 0.00000007061(198) 0.00000008230(461) 0.00000006946(110) 0.000000061998(808) 0.000000000000333(35) 0.000000000015998(570) 0.00000000007620(187) 0.00000000009098(281) 0.000000000001746(201) 0.000000000029116(514)
a
Notation of Ref. [24]; n = l + m, where n is the total order of the operator, l is the order of the torsional part and m is the order of the rotational part. b Notation of Ref. [24]. {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 qP a Þ2 þ ARAM P 2a . c Values of the parameters from the present fit for vt ¼ 0 and 1. All values are in cm1, except for q which is unitless. Statistical uncertainties are given in parentheses in units of the last quoted digit.
between the RHO a-axis and the principal a-axis can be obtained from the following expression:
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Table 3 Assignmentsa, observed valuesb, residuals from the fitc, line strengthsd, sourcese, and energy levelsf for selected methylformate microwave transitions within the vt = 0 state J0
K0a
K 0c
P 0a
v00t
J00
K 00a
K 00c
P 00
Obs. Freq. (Unc) (MHz)b
Obs calc (MHz)c
Line Str.d
Refe
E 0 (cm1)f
E00 (cm1)
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
6 7 8 9 10 11 12 13 14 15 21 22 23 24 25 26 27 28 32 38 39 40 41 42 43 49 50 51 53 56 57 58 46 47 48 50 52 53 54 56 57 58 59 60 46 47 48 50
5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 17 17 17 17 17 2 2 2 2 2 2 2 18 18 18 18
1 2 3 4 5 6 7 8 9 10 16 17 18 19 20 21 22 23 27 33 34 35 36 37 38 44 45 46 48 51 52 53 29 30 31 33 35 51 52 54 55 56 57 58 28 29 30 32
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
5 6 7 8 9 10 11 12 13 14 20 21 22 23 24 25 26 27 31 37 38 39 40 41 42 48 49 50 52 55 56 57 45 46 47 49 51 52 53 55 56 57 58 59 45 46 47 49
5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 17 17 17 17 17 2 2 2 2 2 2 2 18 18 18 18
0 1 2 3 4 5 6 7 8 9 15 16 17 18 19 20 21 22 26 32 33 34 35 36 37 43 44 45 47 50 51 52 28 29 30 32 34 50 51 53 54 55 56 57 27 28 29 31
73665.582(50) 86029.804 98435.693(50) 110890.210(50) 123403.403(50) 135988.491(50) 148664.544(50) 161458.217(50) 174406.161(50) 187555.025(50) 271532.798(50) 285542.584(200) 299123.858(200) 312165.784(200) 324596.208(200) 336373.838(200) 347494.169(200) 358008.265(200) 396951.238(200) 457227.937(50) 467585.380(200) 477981.080(200) 488403.977(200) 498846.142(200) 509301.582(200) 572154.497(50) 582636.171(50) 593117.395(50) 614077.171(50) 645505.446(50) 655977.286(50) 666446.913(50) 569096.503(50) 581719.679(50) 594361.029(50) 619701.062(50) 645124.212(50) 584920.937(50) 595437.262(50) 616460.517(50) 626967.354(50) 637470.721(50) 647970.750(50) 658467.283(50) 568292.658(50) 580851.692(50) 593425.076(50) 618616.760(50)
0.154 0.397 0.114 0.050 0.034 0.002 0.021 0.012 0.008 0.010 0.041 0.037 0.018 0.038 0.136 0.014 0.198 0.130 0.025 0.114 0.028 0.029 0.075 0.187 0.092 0.029 0.087 0.030 0.047 0.098 0.031 0.172 0.021 0.023 0.080 0.097 0.231 0.061 0.006 0.045 0.020 0.050 0.059 0.102 0.004 0.047 0.040 0.009
0.487436E + 01 0.911594E + 01* 0.129621E + 02 0.165448E + 02 0.199432E + 02 0.232075E + 02 0.263712E + 02 0.294577E + 02 0.324834E + 02 0.354611E + 02 0.529188E + 02 0.557931E + 02 0.586151E + 02 0.613648E + 02 0.640328E + 02 0.666206E + 02 0.691424E + 02 0.716248E + 02 0.817250E + 02 0.976085E + 02 0.100286E + 03 0.102969E + 03 0.105658E + 03 0.108351E + 03 0.111049E + 03 0.127357E + 03 0.130090E + 03 0.132799E + 03 0.138077E + 03 0.147876E + 03 0.152558E + 03 0.157239E + 03 0.106137E + 03 0.109201E + 03 0.112252E + 03 0.118320E + 03 0.124348E + 03 0.159264E + 03 0.162342E + 03 0.168479E + 03 0.171567E + 03 0.174665E + 03 0.177762E + 03 0.180853E + 03 0.104078E + 03 0.107184E + 03 0.110273E + 03 0.116419E + 03
TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM PluA Oest Oest Oest Oest Oest Oest Oest Oest PluA Oest Oest Oest Oest Oest NEW NEW NEW NEW NEW NEW NEW NEW NEW NEW NEW NEW NEW NEW NEW NEW NEW NEW NEW NEW NEW NEW NEW
91.0294 93.8991 97.1825 100.8814 104.9977 109.5338 114.4927 119.8784 125.6960 131.9521 179.1828 188.7075 198.6852 209.0980 219.9253 231.1455 242.7367 254.6786 305.7165 392.1303 407.7273 423.6710 439.9624 456.6022 473.5906 582.8575 602.2921 622.0764 662.6937 726.2407 748.1217 770.3520 648.2667 667.6708 687.4966 728.4156 771.0293 618.0133 637.8749 678.6502 699.5635 720.8273 742.4413 764.4054 664.1024 683.4775 703.2720 744.1213
88.5722 91.0294 93.8991 97.1825 100.8814 104.9977 109.5338 114.4927 119.8784 125.6960 170.1255 179.1828 188.7075 198.6852 209.0980 219.9253 231.1455 242.7367 292.4756 376.8789 392.1303 407.7273 423.6710 439.9624 456.6022 563.7725 582.8575 602.2921 642.2103 704.7090 726.2407 748.1217 629.2837 648.2667 667.6708 707.7446 749.5103 598.5024 618.0133 658.0873 678.6502 699.5635 720.8273 742.4413 645.1461 664.1024 683.4775 723.4864
M. Carvajal et al. / Journal of Molecular Spectroscopy 246 (2007) 158–166
v0t
33 27 28 29 31 27 28 30 32 27 29 31 26 28 30 18 19 19 19 19 20 20 20 20 21 21 21 22 22 22 51 45 46 47 49 46 47 49 51 47 49 51 47 49 51 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 34 28 29 30 32 28 29 31 33 28 30 32 27 29 31 18 19 19 19 19 20 20 20 20 21 21 21 22 22 22
a
52 46 47 48 50 47 48 50 52 48 50 52 48 50 52 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Upper and lower state quantum numbers are indicated by 0 and 00 , respectively. Torsion–rotation levels of A species, only given in the first part of the table, have a ‘‘parity’’ label; levels of E species, given in the second part of the table, have a signed Ka value [21]. b Observed vt = 0 microwave transitions in MHz, with estimated uncertainties in parentheses. c Observed minus calculated values in MHz. If followed by an ‘*’, the line was excluded from the fit. d Calculated line strengths [23]. e The references indicate, respectively: Brow: Ref. [7]; Baud: Ref. [8]; PluA, PluE: Ref. [10,11]; Oest: Ref. [12];DeMa: Ref. [9]; TYAM: Ref. [13,14]; NEW: present results. f Upper and lower state energy (cm1).
765.1770 661.9641 680.8979 700.2489 740.2042 698.6595 717.9910 757.9027 799.4818 736.6867 776.5624 818.1001 756.3277 796.1735 837.6767 786.6543 680.8979 700.2489 720.0175 760.8094 717.9910 737.7386 778.4837 820.8974 756.4170 797.1234 839.4928 776.0435 816.7178 859.0503 NEW NEW NEW NEW NEW NEW NEW NEW NEW NEW NEW NEW NEW NEW NEW 0.122518E + 03 0.101902E + 03 0.105052E + 03 0.108186E + 03 0.114411E + 03 0.102804E + 03 0.105983E + 03 0.112291E + 03 0.118542E + 03 0.103667E + 03 0.110064E + 03 0.116396E + 03 0.101239E + 03 0.107728E + 03 0.114145E + 03 0.114 0.005 0.019 0.017 0.088 0.003 0.101 0.003 0.017 0.093 0.011 0.078 0.129 0.281 0.425 643872.491(50) 567619.486(50) 580128.726(50) 592649.412(50) 617726.870(50) 579542.320(50) 592019.011(50) 617001.594(50) 642024.619(50) 591498.047(50) 616401.761(50) 641337.599(50) 591065.894(50) 615903.391(50) 640766.508(50)
E 0 (cm1)f Line Str.d Obs calc (MHz)c K 0c K0a J0 v0t
Table 3 (continued)
P 0a
v00t
J00
K 00a
K 00c
P 00
Obs. Freq. (Unc) (MHz)b
Refe
E00 (cm1)
M. Carvajal et al. / Journal of Molecular Spectroscopy 246 (2007) 158–166
tanð2hRAM Þ ¼
2Dab ðA BÞ
163
ð1Þ
From the values of the rotational constants A, B and Dab from Table 2, hRAM ¼ 24:83 degrees. In Table 5 we also corroborate our result with an ab initio calculation (MP2/VTZ) of the principal axis rotational constants from the molecular structure. The angles in degrees between the principal axis (a,b,c) and the methyl top axis (i) obtained from our fitting and calculated by means of MP2/VTZ are also compared in Table 5. 5. Intensity calculation For astrophysical detections, line intensities are of course very important. In the case of the methylformate molecule, we need to take into account the internal rotation splittings. In the same way that the Hamiltonian was used in the calculation and fit of the line positions, the calculation of the line strengths was also carried out in the RHO axis system. For this purpose, the components of the dipole moment must be expressed in this non-principal reference system. The approach and the equations used here are given in detail in Ref. [23]. To obtain the line strengths in the RAM system, the dipole moment components must be transformed by a rotation of the principal-axis dipole moment components: la cosðhRAM Þ sinðhRAM Þ la ¼ ð2Þ sinðhRAM Þ cosðhRAM Þ lb PAM lb RAM Our intensity calculations will be based on the assumption that the microwave intensities are only driven by the permanent electric dipole moments [23], without any dependence on the torsional angle and other vibrational coordinates. In the case of methyl formate, the permanent electric dipole moment components have the values la ¼ þ1:63 D and lb ¼ þ0:68 D in the principal axis system [8]. The direction in which the dipole moment was found to lie at an angle of 39.4(2) from the C@O bond (Fig. 1, [6]). In the ‘‘chemical’’ convention we have adopted, la and lb have positive signs with respect to the chosen principal axes (i.e. the dipole moment is pointing from the positive charges to the negative charges of the molecule), a convention which we also used for acetic acid [24]. The RAM dipole moment components obtained by Eq. (2) are given in Table 6. The line strengths for all the assigned transitions were obtained and are presented in Tables 3 and 4 for the torsional ground and first excited states, respectively. A comparison between our calculated line strengths and those calculated by Ogata et al. [14] shows that they are in general in very good agreement. One cautionary comment should be made on the calculated intensities here. For the A species transitions certain selection rules on the parity (corresponding to A1 $ A2 overall selection rules) must be obeyed. However as a consequence of carrying out the calculations without factorizing the A block of the Hamil-
164
Table 4 Assignmentsa, observed valuesb, residuals from the fitc, line strengthsd, sourcese, and energy levelsf for selected methylformate microwave transitions within the vt = 1 state J0
K0a
K 0c
P 0a
v00t
J00
K00a
K 00c
P 00
Obs. Freq. (Unc) (MHz)b
Obs calc (MHz)c
Line Str.d
Refe
E 0 (cm1)f
E00 (cm1)
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
3 4 5 6 7 8 9 10 11 12 13 14 15 16 8 9 10 11 12 13 14 15 8 9 10 11 12 13 14 15 5 6 7 8 9 10 11 12 13 14 15 16 17 8 9 10 11 12
0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 1 1 1 1 1 1 1 1 1 1 1 1 1 7 7 7 7 7
3 4 5 6 7 8 9 10 11 12 13 14 15 16 3 4 5 6 7 8 9 10 2 3 4 5 6 7 8 9 4 5 6 7 8 9 10 11 12 13 14 15 16 2 3 4 5 6
+ + + + + + + + + + + + + + + + + + + + + +
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
2 3 4 5 6 7 8 9 10 11 12 13 14 15 8 9 10 11 12 13 14 15 8 9 10 11 12 13 14 15 4 5 6 7 8 9 10 11 12 13 14 15 16 8 9 10 11 12
0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 1 1 1 1 1 1 1 1 1 1 1 1 1 6 6 6 6 6
2 3 4 5 6 7 8 9 10 11 12 13 14 15 4 5 6 7 8 9 10 11 3 4 5 6 7 8 9 10 3 4 5 6 7 8 9 10 11 12 13 14 15 3 4 5 6 7
+ + + + + + + + + + + + + + + + + + + + + +
35781.717(50) 47157.063(50) 58157.430(50) 68840.033(50) 79324.821(50) 89731.695(50) 100136.911(50) 110571.569(50) 121040.237(50) 131536.624(50) 142052.800(50) 152581.782(50) 163118.722(50) 173660.281(50) 152606.026(50) 152443.185(50) 152213.504(50) 151905.159(50) 151511.582(50) 151033.416(50) 150483.224(50) 149890.033(50) 180586.240(50) 180509.008(50) 180391.481(50) 180223.247(50) 179993.047(50) 179689.693(50) 179301.511(50) 178818.702(50) 63888.606(50) 76203.676(50) 88220.742(50) 99869.136(50) 111094.105(50) 121901.608(50) 132378.695(50) 142664.666(50) 152891.340(50) 163142.587(50) 173452.699(50) 183824.379(50) 194247.050(50) 181831.461(50) 181706.050(50) 181532.761(50) 181300.551(50) 180997.053(50)
0.024 0.035 0.048 0.049 0.073 0.073 0.067 0.114 0.044 0.029 0.065 0.087 0.148 0.249 0.031 0.281 0.018 0.031 0.047 0.020 0.024 0.052 0.099 0.082 0.028 0.041 0.074 0.007 0.032 0.070 0.022 0.036 0.031 0.090 0.049 0.078 0.059 0.039 0.044 0.020 0.039 0.081 0.120 0.087 0.035 0.067 0.032 0.100
0.788670E + 01 0.104706E + 02 0.130376E + 02 0.156075E + 02 0.181969E + 02 0.208090E + 02 0.234381E + 02 0.260779E + 02 0.287241E + 02 0.313746E + 02 0.340285E + 02 0.366853E + 02 0.393450E + 02 0.420074E + 02 0.126452E + 01 0.162707E + 01 0.197763E + 01 0.232142E + 01 0.266209E + 01 0.300222E + 01 0.334359E + 01 0.368716E + 01 0.894613E + 00 0.128657E + 01 0.165778E + 01 0.201520E + 01 0.236369E + 01 0.270675E + 01 0.304702E + 01 0.338657E + 01 0.127149E + 02 0.153992E + 02 0.180415E + 02 0.206475E + 02 0.232217E + 02 0.257801E + 02 0.283460E + 02 0.309356E + 02 0.335508E + 02 0.361853E + 02 0.388319E + 02 0.414859E + 02 0.441445E + 02 0.879032E + 00 0.126522E + 01 0.163179E + 01 0.198566E + 01 0.233167E + 01
TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM TYAM
204.2199 205.7929 207.7328 210.0290 212.6750 215.6682 219.0084 222.6967 226.7341 231.1217 235.8601 240.9497 246.3907 252.1834 233.0920 236.7482 240.8143 245.2915 250.1811 255.4846 261.2034 267.3390 239.1158 242.7693 246.8315 251.3031 256.1851 261.4784 267.1842 273.3038 208.1580 210.6999 213.6426 216.9739 220.6796 224.7458 229.1615 233.9203 239.0202 244.4620 250.2478 256.3795 262.8589 238.5992 242.2659 246.3428 250.8308 255.7307
203.0263 204.2199 205.7929 207.7328 210.0290 212.6750 215.6682 219.0084 222.6967 226.7341 231.1217 235.8601 240.9497 246.3907 228.0017 231.6632 235.7370 240.2245 245.1273 250.4467 256.1838 262.3393 233.0920 236.7482 240.8143 245.2915 250.1811 255.4846 261.2034 267.3390 206.0269 208.1580 210.6999 213.6426 216.9739 220.6796 224.7458 229.1615 233.9203 239.0202 244.4620 250.2478 256.3795 232.5340 236.2049 240.2875 244.7832 249.6933
M. Carvajal et al. / Journal of Molecular Spectroscopy 246 (2007) 158–166
v0t
8 9 10 11 6 6 6 6 13 14 15 16 1 1 1 1 7 8 9 10 7 7 7 7
a
13 14 15 16 1 1 1 1
Upper and lower state quantum numbers are indicated by 0 and 00 , respectively. Torsion–rotation levels of A species, only given in the first part of the table, have a ‘‘parity’’ label; levels of E species, given in the second part of the table, have a signed Ka value [21]. b Observed vt = 1 microwave transitions in MHz, with estimated uncertainties in parentheses. c Observed minus calculated values in MHz. If followed by an ‘*’, the line was excluded from the fit. d Calculated Line Strengths [23]. e The references indicate, respectively: Brow: Ref. [7]; Baud: Ref. [8]; PluA, PluE: Ref. [10,11]; Oest: Ref. [12]; DeMa: Ref. [9]; TYAM: Ref. [13,14]; NEW: present results. f Upper and lower state energy (cm1).
255.0194 260.7632 266.9265 273.5114 261.0438 266.7712 272.9141 279.4739 TYAM TYAM TYAM TYAM 180607.773(50) 180116.664(50) 179504.871(50) 178751.035(50)
0.021 0.026 0.025 0.079
0.267339E + 01 0.301355E + 01 0.335428E + 01 0.369712E + 01
E 0 (cm1)f K00a J00 K 0c K0a J0 v0t
Table 4 (continued)
P 0a
v00t
K 00c
P 00
Obs. Freq. (Unc) (MHz)b
Obs calc (MHz)c
Line Str.d
Refe
E00 (cm1)
M. Carvajal et al. / Journal of Molecular Spectroscopy 246 (2007) 158–166
165
Table 5 Rotational constants in the RHO axis system (RAM) and in the principal axis system (PAM)
A(MHz) B(MHz) C(MHz) Dab (MHz) <(i,a) <(i,b) <(i,c) hRAM
RAMa
PAMb
PAMc
17640.0910 9240.9676 5309.8769 4945.9580
19939.5304 6954.4483 5309.8769 0.0
19848.5032 7006.1251 5351.3017 0.0
52.989 37.011 90.000
58.568 31.432 90.000
24.83d
Angles between the principal axis and the methyl top axis. a Rotational constants obtained in our work for RAM-axis system (see Table 2). b Rotational constants of our work transformed to the Principal Axis System and angles in degrees between the principal axis (a,b,c) and the methyl top axis (i). c Calculation of the principal axis rotational constants from the molecular structure (MP2/VTZ) and of the angles in degrees between the principal axis (a,b,c) and the methyl top axis (i). d The angle hRAM between the a-principal axis and the a-RAM axis is given in degrees and obtained from Eq. (1) with the parameters ARAM, BRAM, CRAM, and Dab of Table 2.
Fig. 1. The chemical structure, principal axes and direction of the dipole moment of methyl formate [6]. The a–b plane is a plane of symmetry.
Table 6 Dipole moment components in Debye in the principal axis system (PAM) and in the RHO axis system (RAM)
la lb a
RAM
PAMa
+1.765 0.067
+1.63 +0.68
Experimental dipole moment from Ref. [8].
tonian matrix into A1 and A2 submatrices, the parity of the numerically generated eigenvector is not well defined when K-type doublets remain degenerate. Intensities for such cases were obtained by calculating both the formally parity-allowed and parity-forbidden components of a given
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M. Carvajal et al. / Journal of Molecular Spectroscopy 246 (2007) 158–166
K-type doublet transition, and then ascribing all the calculated intensity to the two allowed transitions. The transition energies and line strengths have been predicted for a number of lines within the torsional ground state and the first excited state (around 33100 lines for vt ¼ 0 and 38280 lines for vt ¼ 1 at a temperature of 296 K and with a cutoff of 104 for the line strength. The prediction involves transition energies up to 4000 GHz and J 6 60, Ka 6 60, with most of the transitions falling in the range 0–1600 GHz. These lines should be useful for astrophysicists in the identification of new spectral lines in the interstellar space.
6. Conclusion A new global analysis of the rotational levels in the torsional ground and first excited states for the parent methyl formate has been carried out by using the RHO Axis Model. In this study around 400 new submillimeter lines provided by the group of Lille were assigned and included in a joint fit with the existing data in the microwave, millimeter and submillimeter spectral region. An analysis of our rotational constants shows that they are in agreement with other spectroscopic parameters of different authors. Due to the importance of this molecule for the remote sensing of the interstellar medium we have calculated not only the frequencies but also the relative intensities of the available data and we have predicted them for nearly 71 380 lines (at 296 K and with a cutoff for the line strength of 104) within vt ¼ 0 and vt ¼ 1. In the future, a new recording of some available data lines would be valuable in order to obtain those lines with better accuracy and to refine the rotational parameters. Also a recording of high resolution far infrared data in the 130 cm1 region where the torsional fundamental vt ¼ 1 0 band (and overtones) absorb would be important to refine the torsional parameters and make them uncorrelated.
Acknowledgments I.K. would like to thank the ‘‘Institut du De´veloppement et des Ressources en Informatique Scientifique’’ IDRIS, for the allocation of computer time. M.C. thanks the CNRS for financial support (project CERC3). I.K. and J.D. thank the PEPCO-NEI cooperation project (MENESR, France). The authors would also like to thank Dr. J.T. Hougen for numerous discussions during the work, V.V. Ilyushin for providing us a combination differences
of the experimental data, and M.L. Senent for numerous useful comments about the work. Appendix A. Supplementary data Supplementary data for this article are available on ScienceDirect (www.sciencedirect.com) and as part of the Ohio State University Molecular Spectroscopy Archives (http://msa.lib.ohio-state.edu/jmsa_hp.htm). References [1] A. Nummelin, P. Bergman, A. Hjalmarson, P. Friberg, W.M. Irwine, T.J. Millar, S. Saito, Astrophys. J. Suppl. Ser. 128 (2000) 213–243. [2] G.A. Blake, E.C. Sutton, G.R. Masson, T.G. Philips, Astrophys. J. Suppl. Ser. 60 (1986) 357–374. [3] K. Kobayashi, K. Ogata, S. Tsunekawa, S. Takano, Astrophys. J. 657 (2007) L17–L19. [4] F.J. Lovas, NIST Recommended rest frequencies for observed interstellar molecular microwave transitions (Gaithersburg, NIST).