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FTFIR-spectrum of the ground state of D2CO
Journal of Molecular Spectroscopy 228 (2004) 1–6 www.elsevier.com/locate/jms
FTFIR-spectrum of the ground state of D2CO Jarmo Lohilahti*, Veli-Matti Horneman Infrared Research Group, Department of Physical Sciences, University of Oulu, P.O. Box 3000, FIN-90014 Oulu, Finland Received 2 April 2004; in revised form 3 June 2004 Available online 23 July 2004
Abstract Ground state spectrum of the formaldehyde D2CO molecule in the range from 30 to 315 cm1 has been recorded by a Fourier transform infrared spectrometer. Besides usual DKa = 0 transitions also 2200 DKa = 2 and 60 DKa = 4 transitions have been observed. The quantum number limits of the assigned transitions are J = 5–59 and Ka = 0–20. The WatsonÕs A-type Hamiltonian in I r representation was fitted to the present and literature data, and the parameters up to eighth order were determined. The standard deviation of the literature and far infrared data are 3.17 kHz and 3.14 MHz, respectively. 2004 Elsevier Inc. All rights reserved. PACS: 33.20.Sn Keywords: Far infrared; FIR; Rotational spectrum; Ground state; Formaldehyde-d2; D2CO
1. Introduction Formaldehyde is a planar asymmetric top molecule. As a light, non-linear four atomic molecule it has been a prototype molecule of many ab initio calculations. For the theoretical calculations and comparison of the results of them, accurate parameters are needed. The D2CO isotopomer of the molecule is also possible medium for FIR laser [1]. In these devices the D2CO is excited by CO2 laser from ground state to v6 = 1 state, where FIR transition occurs. To assign the FIR-emission lines, the levels involved in laser operation must be known accurately. The newest ground state study of D2CO is presented by Bocquet et al. [2]. The most recent analysis of the v6 = 1 state, together with v3 = 1 and v4 = 1 states, has been presented by Perrin et al. [3]. Bocquet et al. recorded ground state spectrum of D2CO by several spectrometers and fitted the WatsonÕs A-type Hamiltonian to the assigned transitions as well as the lines found in the literature. The fit resulted in *
Corresponding author. Fax: +358-81-553-1287. E-mail address: Jarmo.Lohilahti@oulu.fi (J. Lohilahti).
0022-2852/$ - see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.jms.2004.06.010
all the parameters up to eight order except LJ, LK, and lK. Although the utilized spectrometers are accurate devices, continuous spectrum and observations of all the quantum numbers in the measured range were not recorded in [2]. The observed lower states of the transitions from [2] are presented in Fig. 1. Four different centimeter, millimeter, and submillimeterwave spectrometers in different frequency regions were used in [2]. The highest measured line 324,29 ‹ 314,28 is at 1853306.02 MHz 62 cm1. In the range from 40 to 62 cm1 37 lines were used. However, the ground state spectrum of D2CO extends over 60 wavenumber, see Figs. 2 and 3. With the setup of the Bruker IFS 120HR spectrometer authors were able to measure a good far infrared spectrum starting from 30 cm1. The lines of the D2CO below the limit had already been measured well by Bocquet et al. We did focus on to the higher wavenumber region where lines with high J and Ka values and also DKa „ 0 lines can be found. The aim of the present work is to improve the ground state parameters by enlarging the quantum number range and including DKa „ 0 data.
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J. Lohilahti, V.-M. Horneman / Journal of Molecular Spectroscopy 228 (2004) 1–6
Fig. 1. The lower states from [2] are presented in the figure on the left. The right-hand side figure consists the lower states observed in the present FIR measurements. The crosses, solid, and open circles present the lower states of R(0,1), R(2,1), and R(4,3) transitions, respectively.
Fig. 2. The present spectrum in the region of the highest transitions presented in [2]. Assigned lines in [2] and in the present analysis are marked with plus signs and open triangles, respectively. The spectrum is from the first measurement where the absorption was 27 Pa* 3.2 m.
2. Experimental The high-resolution far infrared spectrum of D2CO between 30 and 315 cm1 was measured by using the Bruker IFS 120HR Fourier transform spectrometer in the University of Oulu. High pressure mercury and Globar sources, a 6 mm Mylar beam splitter and a liquid helium cooled Si bolometer detector made by Infrared Laboratories were used. The bolometer was operating at 1.8 K by pumping out evaporated helium gas. The optical range was limited with a cold optical low pass filter with the cut-off frequency at 360 cm1. The sample was at room temperature in the White cell [4,5]. For getting better signal to noise ratio (S/N) the instrumental resolution due to the optical path difference was limited to 0.00136 cm1. The diameter of the Jacqui-
not stop was 3.15 mm. Due to this the aperture broadening was as low as 0.0005 cm1 at 70 cm1. However, a larger aperture could not be used. Doppler width of the molecule at 70 cm1 is about 0.00015 cm1. Together these line broadening terms gave the spectral resolution of about 0.00155 cm1 46 MHz in the measured spectrum. Because of the natural reasons a good signal to noise ratio (S/N) is difficult to achieve in FIR measurements. In the present measurement the peak to peak S/N is about 60 at 70 cm1 and it increases in going to the higher wavenumbers because black body radiation is higher there, see transmittance scales in Figs. 2–4. Three different spectra were collected. In the first absorption (27 Pa, 3.2 m) was adjusted in such a way that the lines around 60 cm1 had 90% absorption. Totally
J. Lohilahti, V.-M. Horneman / Journal of Molecular Spectroscopy 228 (2004) 1–6
3
Fig. 3. In the upper row there is presented the end of the R(2,1) serie from the spectrum recorded with Globar source and absorption 130 Pa* 9.6 m. The labels of the n symbol present the J and Ka values of the lower state. The R(0,1) transitions from the same lower levels are shown in the lower row. The lower row is from the first measurement with mercury lamp and absorption 27 Pa* 3.2 m. In the each small figure horizontal axis is wavenumbers in scale 0.2 cm1. The vertical scales are shown in the figures on the left and right for both rows.
Fig. 4. Beginning of the R(4,3) serie. The labels of the symbol present the J, Ka, and Kc values of the lower state. The spectrum was recorded with Globar source. In the each small figure horizontal axis is wavenumbers in scale 0.2 cm1. The vertical signal scales are shown in the figures on the left and right.
53 scans were coadded recording time being 10.5 h. For the second spectrum absorption was increased (130 Pa, 9.6 m) in order to get lines above 60 cm1. In higher absorption conditions 34 scans were recorded in 9 h. A somewhat weaker but more stabile Globar source was then utilized for the registration of 44 scans in 11.5 h. Peak positions were calculated using optimized center of gravity method [7]. With regard to this method the precision of the used peak positions is varying between 105 and 104 cm1. For example the precision of the reference lines shown in Fig. 2 is about 18 · 106 cm1. The calibration was first made with the aid of D2CO lines in the region 40–60 cm1 from [2]. However, because the S/N of the present spectrum is not very good in this region, the calibration factor k = mref/m was not well determined. The wavenumber precision in the region over 200 cm1 would have been degraded by the
calibration. On the other hand, the accuracy given for the D2CO lines in [2] is in the same order as the absolute accuracy of the new water lines [6]. The other advantage of these water lines is that they are lying in the upper region of the measured spectrum. The calibration with water is then rather an interpolation procedure than extrapolation, as it would be with the D2CO lines. Therefore the final calibration was made with the water lines. The accuracy of a used sample line at m0 in the spectrum is Dm00 ¼ m0 Dk þ kDm0 ;
ð1Þ
where Dk is the error of the calibration coefficient. It has a form Dk = Dmref/mcal + Dkstd, where Dmref = 30 · 106 cm1 is the mean accuracy of the lines in the calibration source [6], mcal is the average position of calibration lines in the present measurement, and Dkstd = 1.234 · 107 is the
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J. Lohilahti, V.-M. Horneman / Journal of Molecular Spectroscopy 228 (2004) 1–6
standard error of the calibration factor. The term Dmref/ mcal takes into account the absolute error of the reference source. Term Dm0 in Eq. (1) is the uncertainty to determine the line position in the spectrum. According to this the accuracy of the used lines is varying from 20 to 170 · 106 cm1. All three measurements were calibrated separately.
The D2CO molecule is an asymmetric top (j = 0.89) with C2v symmetry. The A-reduced WatsonÕs rotational Hamiltonian can be used, because the asymmetry parameter j for this molecule clearly deviates from prolate symmetric top limit (j = 1). The details of the Hamiltonian are given in Table 1. The Ir axis choice can be used, because the symmetry axis, dipole moment and greatest rotational axis are aligned along the CO-bond. Due to the axis choice and value of the asymmetry parameter, Ka is a good quantum number in describing the rotational states and transitions. The selection rules for the a-type transitions are: DK a ¼ 2m;
DKa = 0 DKa = 2 DKa = 4
Jmin
Jmax
K min a
K max a
Number of lines
15 5 10 r 81%
60 43 37 1.96r 94%
0 0 0 2.58r 97%
20 16 7 3.29r 98%
1992 1944 95
The standard deviation of the present data is r = 104 · 106 cm1. The lower part of the table shows the form of the obs calc distribution in percentages.
3. Spectrum and assignment
DJ ¼ 0; 1;
Table 2 The data set of the present measurement
DK c ¼ ð2n þ 1Þ;
ð2Þ
where m and n may obtain values 0, 1, 2, etc. For the most intense transitions changes in quantum numbers are DJ = +1 and DKa = 0 and DKc = ±1, i.e., R(0,1) in other notation. We started with the assignments in [2]. Further DKa = 0 lines were assigned iteratively by comparing the measured and predicted positions. The process was straightforward, because the parameters given in [2] are good already. A computer program was used to calculate the energy levels utilizing existing literature and FIR-data and parameters. The same program was let to calculate line positions and strengths of the weaker lines with DKa „ 0. It was rather easy task to collect the series of the R(2,1) and R(4,3) transitions down to the noise level. Some of the weak series are shown in Figs. 3 and 4. The detailed structure of the complete data set used in this work is given in Table 2 and the data are presented graphically in Fig. 1. The output file of the fitting program is available electronically from the journal
archives. A line can be present more than once in the fit, if the line is composed of more than one transition for example from asymmetric doublets. Also the line can be taken from different recordings.
4. Analysis The uncertainty of the ground state lines measured by different centimeter, millimeter, submillimeterwave, and FTIR-spectrometers is varying. In the analysis weighting of the lines has an essential role. Each datum was weighted with square of the reciprocal error of the data. The lines presented in [2] are given with their errors which vary in rather wide range, from 1 to 5000 kHz, because the centimeter, millimeter, and submillimeterwave spectrometers have different precisions. According to the discussion given in the Experimental section, the accuracy of 5 MHz can be safely given for the FIR lines. In the final fitting pure, unblended lines were accepted into the FIR-data set. Coincidence, incompletely splitted components of asymmetric doublets and otherwise impure lines were removed from the fit. Blended lines were noticed during the assignment process and finally during the fit as irregular obs calc values. The parameters of the present and previous analysis [2] are presented in Table 3. Differences in the rotational and centrifugal parameters are not significant and deviations are within two or three standard deviations. Also the standard deviations are approximately in the same order of magnitude. The reason for this is that the rotational constants and centrifugal parameters are determined by the transitions between levels with low J and
Table 1 Hamiltonian operator 1 1 1 H ¼ ðB þ CÞJ2 þ A þ ðB þ CÞ J 2z þ ðB CÞJ 2xy DK J 4z DJK J 2z J2 DJ J4 dK fJ 2z ; J 2xy g dJ J 2xy J2 þ H K J 6z þ H KJ J 4z J2 þ H JK J 2z J4 þ H J J6 2 2 2 þ hK fJ 4z ; J 2xy g þ hKJ fJ 2z ; J 2xy gJ2 þ hJ J 2xy J4 þ LK J 8z þ LKKJ J 6z J2 þ LJK J 4z J4 þ LJJK J 2z J6 þ LJ J8 þ lK fJ 6z J 2xy g þ lKJ fJ 4z ; J 2xy gJ2 þ lJK fJ 2z ; J 2xy gJ4 þ 2lJ J 2xy J6 with: J 2xy ¼ J 2x J 2y and fA; Bg ¼ AB þ BA
J. Lohilahti, V.-M. Horneman / Journal of Molecular Spectroscopy 228 (2004) 1–6 Table 3 Molecular constants of the ground state Ref. [2]
This work
A (MHz) B (MHz) C (MHz)
141653.5586(19) 32283.56502(43) 26185.31545(38)
141653.5494(16) 32283.56403(30) 26185.31517(28)
DJ (kHz) DJK (kHz) DK (kHz) dJ (kHz) dK (kHz)
52.7908(11) 620.286(10) 4484.99(11) 11.45602(12) 522.7875(51)
52.78633(60) 620.2122(59) 4484.879(49) 11.455142(74) 522.7613(42)
HJ (Hz) HJK (Hz) HKJ (Hz) HK (Hz) hJ (Hz) hJK (Hz) hK (Hz)
0.16303(71) 16.804(24) 34.9(13) 524.2(25) 0.07029(23) 8.786(12) 308.18(34)
0.15456(20) 16.543(13) 34.827(66) 522.40(44) 0.06860(12) 8.667(10) 309.86(28)
LJJK (mHz) LJK (mHz) LKKJ (mHz) LK (mHz) lJ (mHz) lJK (mHz) lKJ (mHz) lK (mHz)
0.937(17) 0.94(12) 12.42(82)
0.3627(70) 13.66(25) 45.74(79) 100.2(13) 0.000193(35) 0.1394(36) 5.92(17) 181.6(36)
a
0.00291(13) 0.115(10) 17.58(33)
5
errors were reduced 50% from the corresponding values in [2]. The inclusion of the LK- and lK-parameters changes further the other L-parameters. Finally, contribution of the FIR lines to the ground state parameters was tested in following way. First DKa = 0 lines were fitted together with literature data. This had no significant effect to the values and accuracies of B- and C-rotational constants and purely from J depended quartic and sextic centrifugal distortion parameters. The values of the K depended parameters did not change significantly but the accuracies worsen. The most obvious this decrease was detected in standard deviations of the A-, DK-, and HK-parameters. Already in this stage the values and accuracies of the octic parameters improved. The improvement of octic parameters is a consequence of the high J and K values of DKa = 0 data, see Table 2. Secondly 2039 DKa = 2 and DKa = 4 lines were added. Following normal distribution law the addition data gave the better standard error of all the parameters. Moreover, the accuracy of K depended parameters were improved more than the accuracy of J depended parameters. This proves how vital the DKa „ 2 transitions are in determination of A-, DK-, etc., parameters.
Errors given in parentheses are 1 SD in units of last digit guoted.
5. Discussion Ka quantum numbers. The low J and Ka transitions are measured accurately in [2] by the centimeter, millimeter, and submillimeterwave spectrometers and the addition of the high J and Ka transitions measured by the FTIR-spectrometer does not have effect to the low order terms. By inspecting of the H- and L-parameters in Table 3 one can see more differences between previous and present analysis. The values of the H-parameters are being changed about 1%, but diminution of standard errors in the present result is significant. Thus the sextic parameters were fairly well determined in [2], but yet the extension of the present data set makes them still more reliable. The extension of the quantum number region affects most obviously to the L-parameters. In the present analysis all octic parameters except LJ have been determined. Initially this parameter was also determined, but in order to get better overall statistic in the final fit, the impure lines were excluded from the fit. Among the impure lines were also high J transitions and therefore omission of the impure lines degraded the error of the LJ-parameter and finally LJ was constrained to zero. Comparison with the present and previous L values of [2] shows big differences which are due to different data set and existence of LK- and lK-parameters. The fit was also performed with the same data, but without LK- and lK-parameters. The other L-parameters differed then few ten per cent and their
It has been shown in this paper, that the ground state spectrum of D2CO molecule extends to wide region. The usual R(0,1) end at about 100 cm1. The weaker R(2,1) and R(4,3) transitions have been observed in the regions 50–290 and 90–210 cm1, respectively. The reason for observing lines in such wide region are following. First of all the A-rotational constant is relatively big, so that R(2,1) and R(4,3) can be found at higher frequencies. Second the dipole moment of formaldehyde has relatively large value too [8,9]. In that sense formaldehyde molecule can be compared with water molecule, which has also large rotational constants, dipole moment, and very wide rotational spectrum [10–13].
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] M. Inguscio, G. Moruzzi, K.M. Evenson, D.A. Jennings, J. Appl. Phys. 80 (1986) R161–R191.
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[2] R. Bocquet, J. Demaison, J. Cosleou, A. Friedrich, L. Margueles, S. Macholl, H. Ma¨der, M.M. Beaky, G. Winnewisser, J. Mol. Spectrosc. 195 (1999) 345–355. [3] A. Perrin, J.-M. Flaud, A. Predoi-Cross, M. Winnewisser, B.P. Winnewisser, G. Mellau, M. Lock, J. Mol. Spectrosc. 187 (1998) 61–69. [4] T. Ahonen, P. Karhu, V.-M. Horneman. An optimized Whitetype gas cell for the Bruker IFS 120 high resolution FTIR spectrometer. 15th Colloquium on High Resolution Molecular Spectroscopy. Glasgow, Scotland. 8–13 September, 1997. [5] Available form:. [6] V.-M. Horneman, R. Anttila, J. Pietila¨, Seppo Alanko, M. Koivusaari, Transferring the high accuracy of the 10 mm CO2
[7] [8] [9] [10] [11] [12] [13]
laser bands to far infrared region with internal calibration of CS2, to be published. V.-M. Horneman, Acta Univ. Oulu A 239 (1992) 127. K. Tanaka, Y. Nakahara, M. Yamaguchi, T. Tanaka, J. Opt. Soc. Am. B 4 (1987) 1181–1187. B. Fabricant, D. Krieger, J.S. Muenter, J. Chem. Phys. 67 (1977) 1576–1586. J.K. Messer, F.C. De Lucia, P. Helminger, Int. J. Infrared Millimeter Waves 4 (1983) 505–539. J. Kauppinen, T. Ka¨rkka¨inen, E. Kyro¨, J. Mol. Spectrosc. 71 (1978) 15–45. F. Winther, J. Mol. Spectrosc. 65 (1977) 405–419. R. Paso, V.-M. Horneman, JOSA B 12 (1995) 1813–1838.
FTFIR-spectrum of the ground state of D2CO Jarmo Lohilahti*, Veli-Matti Horneman Infrared Research Group, Department of Physical Sciences, University of Oulu, P.O. Box 3000, FIN-90014 Oulu, Finland Received 2 April 2004; in revised form 3 June 2004 Available online 23 July 2004
Abstract Ground state spectrum of the formaldehyde D2CO molecule in the range from 30 to 315 cm1 has been recorded by a Fourier transform infrared spectrometer. Besides usual DKa = 0 transitions also 2200 DKa = 2 and 60 DKa = 4 transitions have been observed. The quantum number limits of the assigned transitions are J = 5–59 and Ka = 0–20. The WatsonÕs A-type Hamiltonian in I r representation was fitted to the present and literature data, and the parameters up to eighth order were determined. The standard deviation of the literature and far infrared data are 3.17 kHz and 3.14 MHz, respectively. 2004 Elsevier Inc. All rights reserved. PACS: 33.20.Sn Keywords: Far infrared; FIR; Rotational spectrum; Ground state; Formaldehyde-d2; D2CO
1. Introduction Formaldehyde is a planar asymmetric top molecule. As a light, non-linear four atomic molecule it has been a prototype molecule of many ab initio calculations. For the theoretical calculations and comparison of the results of them, accurate parameters are needed. The D2CO isotopomer of the molecule is also possible medium for FIR laser [1]. In these devices the D2CO is excited by CO2 laser from ground state to v6 = 1 state, where FIR transition occurs. To assign the FIR-emission lines, the levels involved in laser operation must be known accurately. The newest ground state study of D2CO is presented by Bocquet et al. [2]. The most recent analysis of the v6 = 1 state, together with v3 = 1 and v4 = 1 states, has been presented by Perrin et al. [3]. Bocquet et al. recorded ground state spectrum of D2CO by several spectrometers and fitted the WatsonÕs A-type Hamiltonian to the assigned transitions as well as the lines found in the literature. The fit resulted in *
Corresponding author. Fax: +358-81-553-1287. E-mail address: Jarmo.Lohilahti@oulu.fi (J. Lohilahti).
0022-2852/$ - see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.jms.2004.06.010
all the parameters up to eight order except LJ, LK, and lK. Although the utilized spectrometers are accurate devices, continuous spectrum and observations of all the quantum numbers in the measured range were not recorded in [2]. The observed lower states of the transitions from [2] are presented in Fig. 1. Four different centimeter, millimeter, and submillimeterwave spectrometers in different frequency regions were used in [2]. The highest measured line 324,29 ‹ 314,28 is at 1853306.02 MHz 62 cm1. In the range from 40 to 62 cm1 37 lines were used. However, the ground state spectrum of D2CO extends over 60 wavenumber, see Figs. 2 and 3. With the setup of the Bruker IFS 120HR spectrometer authors were able to measure a good far infrared spectrum starting from 30 cm1. The lines of the D2CO below the limit had already been measured well by Bocquet et al. We did focus on to the higher wavenumber region where lines with high J and Ka values and also DKa „ 0 lines can be found. The aim of the present work is to improve the ground state parameters by enlarging the quantum number range and including DKa „ 0 data.
2
J. Lohilahti, V.-M. Horneman / Journal of Molecular Spectroscopy 228 (2004) 1–6
Fig. 1. The lower states from [2] are presented in the figure on the left. The right-hand side figure consists the lower states observed in the present FIR measurements. The crosses, solid, and open circles present the lower states of R(0,1), R(2,1), and R(4,3) transitions, respectively.
Fig. 2. The present spectrum in the region of the highest transitions presented in [2]. Assigned lines in [2] and in the present analysis are marked with plus signs and open triangles, respectively. The spectrum is from the first measurement where the absorption was 27 Pa* 3.2 m.
2. Experimental The high-resolution far infrared spectrum of D2CO between 30 and 315 cm1 was measured by using the Bruker IFS 120HR Fourier transform spectrometer in the University of Oulu. High pressure mercury and Globar sources, a 6 mm Mylar beam splitter and a liquid helium cooled Si bolometer detector made by Infrared Laboratories were used. The bolometer was operating at 1.8 K by pumping out evaporated helium gas. The optical range was limited with a cold optical low pass filter with the cut-off frequency at 360 cm1. The sample was at room temperature in the White cell [4,5]. For getting better signal to noise ratio (S/N) the instrumental resolution due to the optical path difference was limited to 0.00136 cm1. The diameter of the Jacqui-
not stop was 3.15 mm. Due to this the aperture broadening was as low as 0.0005 cm1 at 70 cm1. However, a larger aperture could not be used. Doppler width of the molecule at 70 cm1 is about 0.00015 cm1. Together these line broadening terms gave the spectral resolution of about 0.00155 cm1 46 MHz in the measured spectrum. Because of the natural reasons a good signal to noise ratio (S/N) is difficult to achieve in FIR measurements. In the present measurement the peak to peak S/N is about 60 at 70 cm1 and it increases in going to the higher wavenumbers because black body radiation is higher there, see transmittance scales in Figs. 2–4. Three different spectra were collected. In the first absorption (27 Pa, 3.2 m) was adjusted in such a way that the lines around 60 cm1 had 90% absorption. Totally
J. Lohilahti, V.-M. Horneman / Journal of Molecular Spectroscopy 228 (2004) 1–6
3
Fig. 3. In the upper row there is presented the end of the R(2,1) serie from the spectrum recorded with Globar source and absorption 130 Pa* 9.6 m. The labels of the n symbol present the J and Ka values of the lower state. The R(0,1) transitions from the same lower levels are shown in the lower row. The lower row is from the first measurement with mercury lamp and absorption 27 Pa* 3.2 m. In the each small figure horizontal axis is wavenumbers in scale 0.2 cm1. The vertical scales are shown in the figures on the left and right for both rows.
Fig. 4. Beginning of the R(4,3) serie. The labels of the symbol present the J, Ka, and Kc values of the lower state. The spectrum was recorded with Globar source. In the each small figure horizontal axis is wavenumbers in scale 0.2 cm1. The vertical signal scales are shown in the figures on the left and right.
53 scans were coadded recording time being 10.5 h. For the second spectrum absorption was increased (130 Pa, 9.6 m) in order to get lines above 60 cm1. In higher absorption conditions 34 scans were recorded in 9 h. A somewhat weaker but more stabile Globar source was then utilized for the registration of 44 scans in 11.5 h. Peak positions were calculated using optimized center of gravity method [7]. With regard to this method the precision of the used peak positions is varying between 105 and 104 cm1. For example the precision of the reference lines shown in Fig. 2 is about 18 · 106 cm1. The calibration was first made with the aid of D2CO lines in the region 40–60 cm1 from [2]. However, because the S/N of the present spectrum is not very good in this region, the calibration factor k = mref/m was not well determined. The wavenumber precision in the region over 200 cm1 would have been degraded by the
calibration. On the other hand, the accuracy given for the D2CO lines in [2] is in the same order as the absolute accuracy of the new water lines [6]. The other advantage of these water lines is that they are lying in the upper region of the measured spectrum. The calibration with water is then rather an interpolation procedure than extrapolation, as it would be with the D2CO lines. Therefore the final calibration was made with the water lines. The accuracy of a used sample line at m0 in the spectrum is Dm00 ¼ m0 Dk þ kDm0 ;
ð1Þ
where Dk is the error of the calibration coefficient. It has a form Dk = Dmref/mcal + Dkstd, where Dmref = 30 · 106 cm1 is the mean accuracy of the lines in the calibration source [6], mcal is the average position of calibration lines in the present measurement, and Dkstd = 1.234 · 107 is the
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J. Lohilahti, V.-M. Horneman / Journal of Molecular Spectroscopy 228 (2004) 1–6
standard error of the calibration factor. The term Dmref/ mcal takes into account the absolute error of the reference source. Term Dm0 in Eq. (1) is the uncertainty to determine the line position in the spectrum. According to this the accuracy of the used lines is varying from 20 to 170 · 106 cm1. All three measurements were calibrated separately.
The D2CO molecule is an asymmetric top (j = 0.89) with C2v symmetry. The A-reduced WatsonÕs rotational Hamiltonian can be used, because the asymmetry parameter j for this molecule clearly deviates from prolate symmetric top limit (j = 1). The details of the Hamiltonian are given in Table 1. The Ir axis choice can be used, because the symmetry axis, dipole moment and greatest rotational axis are aligned along the CO-bond. Due to the axis choice and value of the asymmetry parameter, Ka is a good quantum number in describing the rotational states and transitions. The selection rules for the a-type transitions are: DK a ¼ 2m;
DKa = 0 DKa = 2 DKa = 4
Jmin
Jmax
K min a
K max a
Number of lines
15 5 10 r 81%
60 43 37 1.96r 94%
0 0 0 2.58r 97%
20 16 7 3.29r 98%
1992 1944 95
The standard deviation of the present data is r = 104 · 106 cm1. The lower part of the table shows the form of the obs calc distribution in percentages.
3. Spectrum and assignment
DJ ¼ 0; 1;
Table 2 The data set of the present measurement
DK c ¼ ð2n þ 1Þ;
ð2Þ
where m and n may obtain values 0, 1, 2, etc. For the most intense transitions changes in quantum numbers are DJ = +1 and DKa = 0 and DKc = ±1, i.e., R(0,1) in other notation. We started with the assignments in [2]. Further DKa = 0 lines were assigned iteratively by comparing the measured and predicted positions. The process was straightforward, because the parameters given in [2] are good already. A computer program was used to calculate the energy levels utilizing existing literature and FIR-data and parameters. The same program was let to calculate line positions and strengths of the weaker lines with DKa „ 0. It was rather easy task to collect the series of the R(2,1) and R(4,3) transitions down to the noise level. Some of the weak series are shown in Figs. 3 and 4. The detailed structure of the complete data set used in this work is given in Table 2 and the data are presented graphically in Fig. 1. The output file of the fitting program is available electronically from the journal
archives. A line can be present more than once in the fit, if the line is composed of more than one transition for example from asymmetric doublets. Also the line can be taken from different recordings.
4. Analysis The uncertainty of the ground state lines measured by different centimeter, millimeter, submillimeterwave, and FTIR-spectrometers is varying. In the analysis weighting of the lines has an essential role. Each datum was weighted with square of the reciprocal error of the data. The lines presented in [2] are given with their errors which vary in rather wide range, from 1 to 5000 kHz, because the centimeter, millimeter, and submillimeterwave spectrometers have different precisions. According to the discussion given in the Experimental section, the accuracy of 5 MHz can be safely given for the FIR lines. In the final fitting pure, unblended lines were accepted into the FIR-data set. Coincidence, incompletely splitted components of asymmetric doublets and otherwise impure lines were removed from the fit. Blended lines were noticed during the assignment process and finally during the fit as irregular obs calc values. The parameters of the present and previous analysis [2] are presented in Table 3. Differences in the rotational and centrifugal parameters are not significant and deviations are within two or three standard deviations. Also the standard deviations are approximately in the same order of magnitude. The reason for this is that the rotational constants and centrifugal parameters are determined by the transitions between levels with low J and
Table 1 Hamiltonian operator 1 1 1 H ¼ ðB þ CÞJ2 þ A þ ðB þ CÞ J 2z þ ðB CÞJ 2xy DK J 4z DJK J 2z J2 DJ J4 dK fJ 2z ; J 2xy g dJ J 2xy J2 þ H K J 6z þ H KJ J 4z J2 þ H JK J 2z J4 þ H J J6 2 2 2 þ hK fJ 4z ; J 2xy g þ hKJ fJ 2z ; J 2xy gJ2 þ hJ J 2xy J4 þ LK J 8z þ LKKJ J 6z J2 þ LJK J 4z J4 þ LJJK J 2z J6 þ LJ J8 þ lK fJ 6z J 2xy g þ lKJ fJ 4z ; J 2xy gJ2 þ lJK fJ 2z ; J 2xy gJ4 þ 2lJ J 2xy J6 with: J 2xy ¼ J 2x J 2y and fA; Bg ¼ AB þ BA
J. Lohilahti, V.-M. Horneman / Journal of Molecular Spectroscopy 228 (2004) 1–6 Table 3 Molecular constants of the ground state Ref. [2]
This work
A (MHz) B (MHz) C (MHz)
141653.5586(19) 32283.56502(43) 26185.31545(38)
141653.5494(16) 32283.56403(30) 26185.31517(28)
DJ (kHz) DJK (kHz) DK (kHz) dJ (kHz) dK (kHz)
52.7908(11) 620.286(10) 4484.99(11) 11.45602(12) 522.7875(51)
52.78633(60) 620.2122(59) 4484.879(49) 11.455142(74) 522.7613(42)
HJ (Hz) HJK (Hz) HKJ (Hz) HK (Hz) hJ (Hz) hJK (Hz) hK (Hz)
0.16303(71) 16.804(24) 34.9(13) 524.2(25) 0.07029(23) 8.786(12) 308.18(34)
0.15456(20) 16.543(13) 34.827(66) 522.40(44) 0.06860(12) 8.667(10) 309.86(28)
LJJK (mHz) LJK (mHz) LKKJ (mHz) LK (mHz) lJ (mHz) lJK (mHz) lKJ (mHz) lK (mHz)
0.937(17) 0.94(12) 12.42(82)
0.3627(70) 13.66(25) 45.74(79) 100.2(13) 0.000193(35) 0.1394(36) 5.92(17) 181.6(36)
a
0.00291(13) 0.115(10) 17.58(33)
5
errors were reduced 50% from the corresponding values in [2]. The inclusion of the LK- and lK-parameters changes further the other L-parameters. Finally, contribution of the FIR lines to the ground state parameters was tested in following way. First DKa = 0 lines were fitted together with literature data. This had no significant effect to the values and accuracies of B- and C-rotational constants and purely from J depended quartic and sextic centrifugal distortion parameters. The values of the K depended parameters did not change significantly but the accuracies worsen. The most obvious this decrease was detected in standard deviations of the A-, DK-, and HK-parameters. Already in this stage the values and accuracies of the octic parameters improved. The improvement of octic parameters is a consequence of the high J and K values of DKa = 0 data, see Table 2. Secondly 2039 DKa = 2 and DKa = 4 lines were added. Following normal distribution law the addition data gave the better standard error of all the parameters. Moreover, the accuracy of K depended parameters were improved more than the accuracy of J depended parameters. This proves how vital the DKa „ 2 transitions are in determination of A-, DK-, etc., parameters.
Errors given in parentheses are 1 SD in units of last digit guoted.
5. Discussion Ka quantum numbers. The low J and Ka transitions are measured accurately in [2] by the centimeter, millimeter, and submillimeterwave spectrometers and the addition of the high J and Ka transitions measured by the FTIR-spectrometer does not have effect to the low order terms. By inspecting of the H- and L-parameters in Table 3 one can see more differences between previous and present analysis. The values of the H-parameters are being changed about 1%, but diminution of standard errors in the present result is significant. Thus the sextic parameters were fairly well determined in [2], but yet the extension of the present data set makes them still more reliable. The extension of the quantum number region affects most obviously to the L-parameters. In the present analysis all octic parameters except LJ have been determined. Initially this parameter was also determined, but in order to get better overall statistic in the final fit, the impure lines were excluded from the fit. Among the impure lines were also high J transitions and therefore omission of the impure lines degraded the error of the LJ-parameter and finally LJ was constrained to zero. Comparison with the present and previous L values of [2] shows big differences which are due to different data set and existence of LK- and lK-parameters. The fit was also performed with the same data, but without LK- and lK-parameters. The other L-parameters differed then few ten per cent and their
It has been shown in this paper, that the ground state spectrum of D2CO molecule extends to wide region. The usual R(0,1) end at about 100 cm1. The weaker R(2,1) and R(4,3) transitions have been observed in the regions 50–290 and 90–210 cm1, respectively. The reason for observing lines in such wide region are following. First of all the A-rotational constant is relatively big, so that R(2,1) and R(4,3) can be found at higher frequencies. Second the dipole moment of formaldehyde has relatively large value too [8,9]. In that sense formaldehyde molecule can be compared with water molecule, which has also large rotational constants, dipole moment, and very wide rotational spectrum [10–13].
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).
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