Ethylene-1-13C (13C12CH4): First analysis of the ν2, ν3 and 2ν10 bands and re–analysis of the ν12 band and of the ground vibrational state

Journal of Quantitative Spectroscopy & Radiative Transfer 187 (2017) 403–413

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Ethylene-1-13C (13C12CH4): First analysis of the ν2, ν3 and 2 ν10 bands and re–analysis of the ν12 band and of the ground vibrational state O.N. Ulenikov a,n, O.V. Gromova a, E.S. Bekhtereva a, Yu.S. Aslapovskaya a, T.L. Tan b, C. Sydow c, C. Maul c, S. Bauerecker c a

Institute of Physics and Technology, National Research Tomsk Polytechnic University, Tomsk 634050, Russia Natural Sciences and Science Education, National Institute of Education, Nanyang Technological University, 1 Nanyang Walk, Singapore 637616, Singapore c Institut für Physikalische und Theoretische Chemie, Technische Universität Braunschweig, D–38106 Braunschweig, Germany b

art ic l e i nf o

a b s t r a c t

Article history: Received 13 August 2016 Received in revised form 10 October 2016 Accepted 10 October 2016 Available online 21 October 2016

High-resolution FTIR ro-vibrational spectra of the 13C12CH4 molecule in the region of 600–1700 cm
1, where the bands ν3, ν12 and ν2 are located, were recorded and analyzed with the Hamiltonian model. This model takes resonance interactions between these three bands as well as strong interactions with six neighboring bands, ν10, ν8, ν7, ν4, ν6, and 2ν10 into account. More than 3800 ro-vibrational transitions belonging to the bands ν3, ν12, ν2 and 2ν10 were assigned (for the first time for the ν2, ν3 and 2ν10 bands) with the maximum values of quantum numbers J max . /Kamax . equal to 22/8, 52/18, 30/11 and 27/12, respectively. On this basis, a set of 62 vibrational, rotational, centrifugal distortion and resonance interaction parameters was obtained from the weighted fit. These parameters reproduce 1562 initial “experimental” ro-vibrational energy levels obtained from unblended lines with the rms error drms = 2.6 × 10−4 cm−1. Furthermore, ground state parameters of the 13C12CH4 molecule were improved. & 2016 Elsevier Ltd. All rights reserved.

1. Introduction Ethylene (C2H4) is an important chemical species in many research fields such as chemistry, interstellar space, planetary nebulae, study of the atmospheres of the Earth and other planets, etc. Ethylene has been observed in the terrestrial atmosphere as a tropospheric pollutant, Refs. [1,2]. Beyond the earth's atmosphere, ethylene exists as a product of methane photochemistry in other planetary atmospheres, Refs. [3,4], and it has been observed in the atmospheres of the outer planets such as Jupiter, Saturn, and Neptune, Refs. [5–8], as well as in the atmosphere of Saturn's biggest moon Titan, Refs. [9–11]. As a consequence, the most abundant isotopic species 12C2H4 has been the subject of a number of studies involving line positions as well as line intensities (see, e.g., Refs. [12–14], and Refs. therein). On the other hand, the ethylene-1-13C isotopologue has seldom been discussed, Refs. [3,15–20], despite the fact that it is the most abundant isotopic impurity (2.2%) in normal samples (the 13C2H4 and C2H3D isotopic species are indeed about 100 times less abundant), and, as a consequence, one can expect to observe it in the spectra of the atmospheres of the outer planetary bodies (in particular, Titan) among the other 13C isotopologues as 13CH4, 13C12CH2, H13CN, 13CH3D, etc. (see, e.g., Refs. [21,22]). As mentioned in Ref. [16], knowledge of the highn

Corresponding author. E-mail address: [email protected] (O.N. Ulenikov).

http://dx.doi.org/10.1016/j.jqsrt.2016.10.009 0022-4073/& 2016 Elsevier Ltd. All rights reserved.

resolution ro-vibrational spectra of 13C12CH4 is very important for understanding the optical pumping process in ethylene. The present work on the 13C12CH4 is a continuation of our recent study of the high resolution spectra of ethylene and its different isotopologues in the infrared region, Refs. [23–33], with the 13C12CH4 molecule. As was mentioned above, 13C12CH4 was studied before only in a few articles, Refs. [15–20]. The first high resolution analysis of IR spectra of 13C12CH4 was made by De Vleeschouwer and co-authors in 1981, Ref. [15] where the ν5 + ν12 band (band center near 4503 cm
1) has been analyzed. One year later, the same authors reported the analysis of the 13C12CH4 IR spectra in the 10 μm spectral region where the ν7, ν8, ν4, ν6, and ν10 strongly interacting fundamentals are located. The next investigation of the transitions of the 13C12CH4 isotopologue was made 20 years later in Ref. [3] where 88 transitions of 13 12 C CH4 were recorded by infrared heterodyne spectroscopy, but not assigned. In 2010, one of us (see Ref. [17]) presented an analysis of the FTIR spectra of 13C12CH4 in the region of the ν12 band (1360– 1520 cm
1), which was considered to be isolated. Later the ν12 band was re-analyzed in Ref. [18] again as an isolated band. Also in 2010, Flaud with coauthors, Ref. [19], made an analysis of the FTIR spectra of 13 12 C CH4 in the 700–1190 cm
1 region (transitions belonging to the ν10, ν8, ν7 and ν4 bands have been assigned, and the ν6 band has been considered as “dark” band). In 2011, absolute line intensities and selfbroadened half-width coefficients of transitions were analyzed in the same spectral region and discussed in Ref. [20]. In this paper we present the results of a high-resolution FTIR

404

O.N. Ulenikov et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 187 (2017) 403–413

analysis of 13C12CH4 in the wave length spectral region, 1250– 1700 cm
1, where the bands ν3, ν12 and ν2 are located. In Section 2 we describe the conditions of our experiment. Description of the experimental spectra and results of the assignment of transitions are given in Section 3. Section 4 presents the theoretical background of the further analysis of experimental data: the use of Hamiltonian model and numerical estimation of the initial values of rotational and main resonance interaction parameters. It was found that the ground state parameters known in the literature do not allow us to correctly describe our experimental data. Therefore we re-analyzed the ground state parameters on the basis of the ground state combination differences (GSCD) derived from our experimental data. The obtained results of the ground vibrational state are presented in Section 5, whereas the outcome of the upper vibrational states analysis is discussed in Section 6.

intensities. The sample temperature was 296 ± 0.5 K . The final spectral resolution was mainly limited by Doppler broadening and resulted in 0.0023 cm
1 at 600 cm
1, 0.0033 cm
1 at 1200 cm
1 and 0.0050 cm
1 at 2000 cm
1. The pressure broadening was between 6 × 10−7 and 0.0006 cm
1 for the used sample pressures between 0.3 and 300 Pa. This means that it only has a minor contribution to the total line widths which were computed by the root mean square approximation of a convolution of Doppler, pressure and instrumental line widths and which is in accordance with the experimental results. The spectra were calibrated with N2O lines recorded at a (partial) N2O pressure of about 10 Pa with 200 and 230 scans at an optical resolution of 0.0021 cm
1. The mean divergence of measured N2O line positions (about 30 lines) from line positions published in the current HITRAN data base is around 2 × 10−4 cm
1 with a divergence of single lines below 4 × 10−4 . For optimization of data recording and line calibration we used data and procedures described in Refs. [34–36].

2. Experimental details The spectra of 13C12CH4 were recorded in the 600–1700 cm
1 region using the Bruker IFS 125 HR Michelson Fourier transform spectrometer located at the Nanyang Technological University in Singapore at an unapodized resolution of 0.0063 cm
1. The 13 12 C CH4 gas samples used in the experiments were supplied by Cambridge Isotope Laboratories in Massachusetts, USA and had a chemical purity better than 98%. All spectral measurements were carried out at an ambient temperature of about 296 ± 0.5 K with a Globar infrared source, a high-sensitivity liquid–nitrogen–cooled Hg–Cd–Te detector, KBr beamsplitter, and aperture size of 1.5 mm. A capacitance pressure gauge measured the vapor pressure to be 35, 100 and about 1000 Pa in the gas cell for the three spectra depicted in Table 1. A multiple-pass absorption cell with a total absorption length of 0.80 m was used. A total of 300, 720 and 1050 scans were co-added to produce the final spectra. The absorption lines of H2O were used to calibrate the 13C12CH4 spectra. In the Braunschweig Infrared Laboratory four spectra in the 600–1150 cm
1 region and two spectra in the 1000–2000 cm
1 region have been recorded using an IFS 120HR Fourier Transform infrared spectrometer (FTIR) combined with a stainless steel White cell with a base length of one meter and a maximum path– length of up to 50 m. A Globar IR source, a KBr beamsplitters, a mercury–cadmium–telluride (MCT) semiconductor detector and the sample, 13C12CH4 (gas) with a specified chemical purity of better than 99% purchased from Sigma Aldrich have been used. For detailed optical and recording parameters see Table 1; the optical resolution was 0.0021 cm
1 for spectra I to IV and 0.0025 cm
1 for spectra V and VI, the number of scans was between 330 and 1210 resulting in measuring durations between 15.8 and 48.5 hours, the optical path-length was 4 m for all spectra apart from spectrum VI (24 m) and the sample gas pressure was varied between 0.3 and 300 Pa to get lines with stronger and weaker line Table 1 Experimental setup for the regions 600 - 2000 cm
1 of the infrared spectrum of

3. Description of the spectra and assignment of transitions The survey spectra VI, VII and IX in the region of 600– 1700 cm
1 are shown in Fig. 1. As the spectra in the spectral region between 600 and 1250 cm
1 have been discussed in Refs. [19,20], we focused on the region between 1250 and 1700 cm
1. In this region one can see the ν12 band with clearly pronounced P-, Q-, and R-branches. The weak ν3 and ν2 bands can also be recognized at the left and right side of the ν12 band. Two bands, ν6 and 2ν10, which are located at the same region, are extremely weak and cannot be seen even in the strongest spectrum VI. The ν1 band of N2O is also visible in the central part of Fig. 1. Some small parts of the high–resolution spectra in the regions of the ν12, ν2, and ν3 bands are shown in the top parts of Figs. 2, 3, and 4, respectively. One can clearly see the pronounced regular structure of separate sets of transitions of these bands. The 13C12CH4 molecule is an asymmetric top of C2v symmetry with the asymmetry parameter κ = (2B − A − C ) /(A − C ) ≈ − 0.918 and with the symmetry isomorphic to the C2v point symmetry group. For convenience of the reader, the symmetry properties in 13 12 C CH4 are shown in Table 2 (reproduced from Ref. [19]): the list of irreducible representations and table of characters of the C2v symmetry group are presented in columns 1–5; symmetries of vibrational coordinates qλ , rotational operators, Jα , and of direction cosines, kzα , are shown in columns 6 and 7. In accordance with the selection rules for the asymmetric top molecules of the C2v symmetry (see, e.g., Refs. [37–42]), there are three types of vibrational bands which are allowed in absorption from the ground vibrational state: (1) the A1 ← A1 bands are the a− type ones, and the selection rules for them are ΔJ = 0, ± 1 and ΔKa= even, ΔKc = odd;

13 12

C CH4.

Spectr.

University

Region /cm
1

Resolution /cm
1

No. of scans

Source

Detector

Beamsplitter

Opt. pathlength/m

Aperture /mm

Temp. /°C

I II III IV V VI VII VIII IX

Braunschweig Braunschweig Braunschweig Braunschweig Braunschweig Braunschweig Singapore Singapore Singapore

600 - 1150 600 - 1150 600 - 1150 600 - 1150 1000 - 2000 1000 - 2000 600 - 1700 600 - 1700 600 - 1700

0.0021 0.0021 0.0021 0.0021 0.0025 0.0025 0.0063 0.0063 0.0063

330 600 350 370 1210 480 300 720 1050

Globar Globar Globar Globar Globar Globar Globar Globar Globar

MCT MCT MCT MCT MCT MCT MCT MCT MCT

KBr KBr KBr KBr KBr KBr KBr KBr KBr

4 4 4 4 4 24 0.8 0.8 0.8

1.5 1.5 1.5 1.5 1.3 1.3 1.3 1.3 1.3

23 23 23 23 23 23 23 23 23

7 7 7 7 7 7 7 7 7

0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5

Pressure /Pa

Calibr. gas

0.3 3 30 300 30 250 35 100 1000

N2O N2O N2O N2O N2O N2O H2O H2O H2O

O.N. Ulenikov et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 187 (2017) 403–413

405

(2) the B1 ← A1 bands are the b− type ones, and the selection rules for them are ΔJ = 0, ± 1 and ΔKa= odd, ΔKc = odd; (3) the B2 ← A1 bands are the c− type ones, and the selection rules for them are ΔJ = 0, ± 1 and ΔKa= odd, ΔKc = even.

Fig. 1. Survey spectrum of 13C12CH4 in the region of 600–1700 cm
1. Experimental conditions: room temperature for all spectra; absorption path length, number of scans and sample pressure are 24 m, 480 and 250 Pa for spectrum VI, 0.8 m, 300 and 35 Pa for spectrum VII, and 0.8 m, 1050 and about 1000 Pa for spectrum IX. Centers of the bands studied earlier are marked in black, centers of the ν3, ν2 and 2ν10 bands studied in this paper for the first time, are marked in red in the electronic version.

For this reason, as it follows from Table 2, all four studied bands, ν2(A1), ν3(A1), ν12(A1) and 2ν10(A1), can be identified as a− type bands. The assignment of the transitions was made with the ground state combination differences method. Here, the rotational energies of the ground vibrational state have been calculated with two sets of parameters from Refs. [18] and [19] (for convenience of the reader, they are reproduced in columns 2 and 3 of Table 3 from Refs. [18] and [19]). As the result of the assignment, more than 3800 transitions with the maximum values of upper quantum numbers J max. = 52 and Kamax. = 18 have been assigned to the ν12 band. Additionally, for the first time we were able to assign more than 910 and 480 transitions to the weak bands ν2 and ν3 and 14 transitions to the very weak 2ν10 band (for more details, see statistical information in Table 4). It is interesting to compare the strengths of the bands, which are shown in Fig. 1 and already have been studied earlier, with the

Fig. 2. A small part of the high-resolution spectrum of the 13C12CH4 molecule in the region of the Q –branch of the ν12 band (upper trace). Experimental conditions correspond to the spectrum VIII (see Table 1, for details). Lines belonging to the clusters of the Q Q Ka(J )− type are marked on the top part of Fig. 2. The middle part of Fig. 2 shows a detailed structure of one separate cluster, Q Q 5(J ) (lines of this cluster are marked by black dots). One can see splittings for J¼ 15 and 16. A few lines belonging to the ν12 band are marked by open circles. The bottom trace is the simulated spectrum (see text, for details).

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O.N. Ulenikov et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 187 (2017) 403–413

Fig. 3. A small part of the high resolution spectrum of the 13C12CH4 molecule in the region of the Q –branch of the ν2 band. Experimental conditions correspond to the spectrum VI (see Table 1, for details). Clusters Q Q Ka(J ) ( Ka = 3, …, 9) are marked by dark and open circles, dark triangles, dark and open stars, dark and open dark and open squares, respectively. The bottom trace is the simulated spectrum (see text, for details).

Fig. 4. A small part of the high resolution spectrum of the 13C12CH4 molecule in the region of the Q –branch of the ν2 band (upper trace). Experimental conditions correspond to the spectrum VI (see Table 1, for details). The bottom trace is the simulated spectrum (see text, for details). Clusters Q Q 5(J ) , Q Q 6(J ) and Q Q 7(J ) are marked by dark circles, open circles, and dark triangles, respectively. Some unassigned lines in the upper trace (they are marked by dark stars) belong, probably, to the “hot” ν10 + ν12 − ν10 band.

Table 2 Symmetry types and characters of irreducible representations of the C2v group (application to 13C12CH4). Repr. 1

E 2

C2 3

σv(xz ) 4

σv(yz ) 5

Vibr. 6

q1, q2, q3, q11, q12

Rot.(Ir) 7

A1

1

1

1

1

A2

1

1


1


1

q4 ,

Jz, kzz

B1

1


1

1


1

q5, q6, q9, q10

Jy, kzy

B2

1


1


1

1

q7, q8

Jx, kzx

strengths of the bands discussed in the present paper. To realize this, the first five lines of Table 5 show the values of band intensities (in cm
1/(molecule cm
1)) at 296 K for the ν10, ν8, ν7, ν4 and ν6 bands (reproduced from Table 3 of Ref. [20]). The next three lines of Table 5 present corresponding values which were estimated in the present paper from comparative analysis of line strengths of the same name transitions of the ν8 and ν7 bands, on the one hand, and of the ν12, ν3 and ν2 bands, on the other hand. In this case, we estimated ratios of the main effective dipole moment parameters from line strengths of pairs of the same name transitions (8–10 such pairs for any pair of absorption

O.N. Ulenikov et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 187 (2017) 403–413

Table 3 Spectroscopic parameters of the ground vibrational state of the (in cm
1)a.

C CH4 molecule

Ref. [18] 2

Ref. [19] 3

This work 4

This work/Ref. [19]b 5

A B C

4.86518 0.9760945 0.810882 1.599

4.864699 0.976094 0.810887 0.86728

4.8647231(29) 0.97609212(39) 0.81088756(52) 0.87174(88)

4.8641651(28) 0.97609399(58) 0.81088904(48) 0.86825(39)

ΔJK × 105

0.9982

1.00483

1.00438(63)

1.00427(81)

ΔJ × 105

0.140099

0.14014

0.139910(37)

δK × 10

1.079

0.9792

6

0.262 208.0

ΔK × 104

5

δJ × 10

8

HK × 10

Band 1

Band intensity 2

ν10a

0.328 × 10−18

ν8a

0.633 × 10−18

ν7a

0.1116 × 10−16

ν4a

0.141 × 10−18

0.140372(88)

ν6a

0.230 × 10−19

0.9825(27)

0.9931(46)

ν3b

(0.7 ± 0.2) × 10−19

0.26426

0.26325(36)

0.26351(43)

ν12b

(1.6 ± 0.3) × 10−18

0.6196

1.060(89)

0.647(39)

ν2b

(0.9 ± 0.2) × 10−19

HKJ × 10


0.424

HJK × 109

0.1845

9

HJ × 1011 2.9

8

hK × 10

0.0593(65)


0.50(33)

a

0.220(91)

b

0.2501

0.342(70)

0.346

0.55(30)

hKJ × 109

0.1138

0.117(33)

0.178(48)

11

0.1098

0.130(22)

0.180(34)

hJ × 10

LK × 1010


0.169(37)

LJ × 1015

0.480(89)

4800.8

rms × 104 a b

5.6

2.1

2.4

Table 4 Statistical information for the studied bands of the max

a

Band

Center/cm

1

2

3

Kamax 4

5

ν12 , [18] ν12 , twe ν2 , tw ν3 , tw

1439.3461 1439.3461 1606.0945 1336.8380 1659.9060

41 52 30 22 27

14 18 11 8 12

1602 2410 910 480 14

2ν10 f tw

J

Ntr

Nl

b

6

950 395 211 6

From Ref. [20]. This work.

(v8 = 1), (v7 = 1), (v6 = 1) and (v4 = 1). Consequently, for the theoretical analysis of the assigned transitions we used the model of the effective Hamiltonian which takes into account resonance interactions between all nine mentioned vibrational bands. In this case, the states (v10 = 1), (v8 = 1), (v7 = 1), (v6 = 1) and (v4 = 1) were considered as the “dark” states (i.e., corresponding experimental bands were not analyzed in our study, because they have been discussed, in detail, in Ref. [19]; however, the influence of the states (v10 = 1), (v8 = 1), (v7 = 1), (v6 = 1), (v4 = 1) on the studied vibrational states have been taken into account in the fit procedure). Such a model was discussed in the literature on spectroscopy many times and has the form of an operator matrix (see, e.g., Refs. [43–45]):

Values in parentheses are 1s standard errors. See text, for details.


1

Table 5 Band intensities (in cm
1/(molecule cm
2)) at 296 K for the ν10, ν8, ν7, ν4, ν6, ν3, ν12 and ν2 bands of 13 12 C CH4.

13 12

Parameter 1

407

13 12

C CH4 molecule.

H vib . − rot . =

m1c

m2c

m3c

drmsd

7

8

9

10

5.7 11.4 15.0 0.0

3.0 2.24 3.40 3.42 1.29

77.7 68.5 66.5 100.0

16.6 20.1 18.5 0.0

a

Ntr is the number of assigned transitions. Nl is the number of obtained upper-state energies. Here mi ¼ ni /Nl× 100% (i¼ 1, 2, 3); n1, n2, and n3 are the numbers of upper-state energies for which the differences δ = E exp − E calc satisfy the conditions δ ≤ 2  10
4 cm
1, 2  10
4 cm
1 <δ ≤ 4× 10
4 cm
1, and δ > 4 × 10−4 cm−1, respectively. d In 10
4 cm
1. e The “tw” means “this work”. f For the 2ν10 band only transitions with the value of quantum number Ka = 14 were assigned without doubts. b c

bands) and, on this basis, estimated numerically intensity of the reference band (ν8 and/or ν7) and intensities of the bands ν12, ν3 and ν2. The intensity of the 2ν10 band was not estimated, because we obtained from the experimental data only a few of information about this band.

4. Hamiltonian model and estimation of spectroscopic parameters of the upper vibrational states 4.1. Effective hamiltonian As shown in the preliminary analysis, a correct description of rovibrational energy values of the (v12 = 1), (v3 = 1) and (v2 = 1) vibrational states is possible only if one considers the closely located (v10 = 2) state and, what is more important, the states (v10 = 1),

∑ |v〉〈v˜|H vv˜.

(1)

v, v˜

In accordance with the above said, nine vibrational states are taken into account in the operator matrix, Eq. (1): |v〉, |v∼〉 ¼1, 2, 3,…,8, 9; |1〉 = (v3 = 1, A1), |2〉 = (v12 = 1, A1), |3〉 = (v2 = 1, A1), |4〉 = (v10 = 2, A1). |5〉 = (v8 = 1, B2), |6〉 = (v7 = 1, B2), |7〉 = (v4 = 1, A2 ), |8〉 = (v10 = 1, B1), and |9〉 = (v6 = 1, B1). As the consequence, the resonance interactions of four types should be taken into account in Eq. (1) (see below). As to the diagonal blocks, H vv , of the operator, Eq. (1), they describe the unperturbed rotational structure of the vibrational states |v〉 and have a form of the reduced Watson's Hamiltonian in the A –reduction and Ir representation, Refs. [46,47]:

[

Hvv = E v + Av −

1 v 1 1 2 (B + C v) Jz2 + (Bv + C v) J 2 + (Bv − C v) Jxy 2 2 2

]

[

]+ − 2δ vJ J 2Jxy2 v v v 2 v 4 2 Jz J + H vJK Jz2J 4 + H vJ J 6 + [Jxy + HKv Jz6 + HKJ , hK Jz4 + hJK J 2Jz2 + hJ J 4 ]+ 2 − ΔKv Jz4 − ΔvJK Jz2 J 2 − ΔvJ J 4 − δKv Jz2 , Jxy

v Jz6J 2 + LvJK Jz4J 4 + LKv Jz8 + LKKJ

[

v

v

v

v

2 v Jz2J 6 + LvJ J 8 + Jxy , lK Jz6 + lKJ J 2Jz4 + l JK J 4Jz2 + l J J 6 + LKJJ

]+ + …,

(2)

where Jα ( α = x, y, z ) are the components of the angular momentum operator defined in the molecule-fixed coordinate system; 2 Jxy = Jx2 − Jy2; [… , …] denotes anticommutator; Av, Bv, and Cv are the effective rotational constants connected with the vibrational states (v ), and the other parameters are the different order centrifugal distortion coefficients. We may distinguish between four types of coupling operators ˜ H vv , (v ≠ v˜ ), corresponding to the four different types of resonance interactions which can occur in a set of vibrational states, A1, A2, B1, and B2, of the C2v asymmetric top molecules. If the product ˜ Γ = Γ v ⊗ Γ v of the symmetries species of the states v and v˜ is

408

O.N. Ulenikov et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 187 (2017) 403–413

Table 6 Spectroscopic parameters of some vibrational states of the

C CH4 molecule (in cm
1).

13 12

Parameter

(v10 = 1)

1

This work 2

Ref. [19]. 3

This work 4

Ref. [19]. 5

This work 6

Ref. [19]. 7

This work 8

Ref. [19]. 9

This work 10

Ref. [18]. 11

825.0 4.8647231 0.97609212 0.81088756 0.87174

825.40602 4.867733 0.9730521 0.8094999 0.780064

933.0 4.8647231 0.97609212 0.81088756 0.87174

932.19572 4.81899 0.97043737 0.81163415 0.550837

947.0 4.8647231 0.97609212 0.81088756 0.87174

947.44452 4.8826261 0.97054678 0.812155603 0.991121

1027.19(86) 4.7826(31) 0.97609212 0.81088756 0.87174

1025.6976 4.4433180 0.9766767 0.81088927 0.636671

1439.338012(33) 4.869626(18) 0.9769391(18) 0.8092053(19) 0.87174

1439.34612 4.9251533 0.98224541 0.80937507 1.8607

ΔKJ × 105

1.00438

1.08818

1.00438

7.14575

1.00438

1.10520

1.00438

1.23091

1.00438

0.8624

ΔJ × 105

0.139910

0.1330042

0.139910

0.1362014

0.139910

0.13506146

0.139910

0.1405140

0.13838(42)

0.146566

E A B C

ΔK × 104

(v8 = 1)

(v7 = 1)

(v12 = 1)

(v4 = 1)

δK × 105

0.9825

0.9792

0.9825

0.49264

0.9825

0.772643

0.9825

0.9792

0.9825

1.3780

δJ × 106

0.26325

0.140086

0.26325

0.238574

0.26325

0.2380449

0.26325

0.266477

0.26325

0.29660

HK × 108

1.060

0.6196

1.060

0.6196

1.060

0.6196

1.060

0.6196

1.060

209.184


0.424

HKJ × 109 0.0593

HJK × 109


0.424

0.1845

0.0593

0.1845

0.2501

HJ × 1011

0.0593

0.2501

0.346

hK × 108


0.424


0.424

0.1845

0.0593

0.1845

0.2501

0.346

0.0593

0.2501

0.346

0.346

3.0866

hJK × 109

0.117

0.1138

0.117

0.1138

0.117

0.1138

0.117

0.1138

0.117

hJ × 1011

0.130

0.1098

0.130

0.1098

0.130

0.1098

0.130

0.1098

0.130


0.169

10

LK × 10


0.169

Parameter

(v6 = 1)

1

This work 12


0.169


0.169


0.169

(v3 = 1)

(v2 = 1)

(v10 = 2)

Ref[19]. 13

This work 14

This work 15

This work 16

1219.79(67) 4.8637(31) 0.97805(19) 0.81172(20) 0.87174

1215.0 4.864699 0.976094 0.806596 0.86728

1336.84572(62) 4.874285(19) 0.9748841(50) 0.8086702(21) 0.91856(89)

1606.094207(44) 4.8520978(58) 0.96939675(86) 0.80627838(58) 0.87174

1659.906(48) 4.853618(90) 0.946451(29) 0.8109513(91) 0.87174

1.00438

1.00483

1.00438

1.00438

1.00438

ΔJ × 10

0.139910

0.13514

0.139910

0.13624(37)

0.139910

δK × 106

0.9825

0.9792

0.9825

0.9825

0.9825

0.26325

0.26426

0.26325

0.26325

0.26325

1.060

0.6196

1.060

1.060

1.060

0.0593

0.0593

0.0593

E A B C

ΔK × 104

ΔKJ × 105 6

7

δJ × 10

HK × 107


0.424

9

HKJ × 10

0.0593

HJK × 1011

0.1845 0.2501

12

HJ × 10

0.346

hK × 109 0.117

0.1138

0.117

0.117

0.117

12

0.130

0.1098

0.130

0.130

0.130

11


0.169


0.169


0.169


0.169

12

hJK × 10

hJ × 10

LK × 10 a

Values in parentheses are 1σ statistical confidence intervals (in last digits). Parameters presented without confidence intervals have been constrained to values estimated theoretically (see text for details).

equal to A1 (i.e., Γ v = Γ v ), then the states v and v˜ are connected by an anharmonic resonance interaction, and the corresponding interaction operator has the form, Ref. [48] ˜

Hvv˜ = +

vv˜

vv˜

F0 +

(

F xy

Jx2

vv˜

−

F K Jz2 Jy2

+

)+

vv˜

vv˜

2

FJ J +

vv˜

F KK Jz4

+

vv˜

F KJ Jz2J2

+

vv˜

4

F JJ J + …

˜ F Kxy⎡⎣ Jz2, Jx2 − Jy2 ⎤⎦ + vvF JxyJ2 Jx2 − Jy2 + … (3)

(

)

(

)

When Γ = B2, a b–type Coriolis interaction of the following type is possible, Ref. [50]: (1) (1) (2) (2) Hvv˜ = iJxHvv ˜ + Hvv˜ iJx + [Jy , Jz ]Hvv˜ + Hvv˜ [Jy , Jz ] +

⎡ iJ , J2 − J2 ⎤H (3) + H (3)⎡ iJ , J2 − J2 ⎤ + … ⎣ x x vv˜ ⎣ x y ⎦ vv˜ x y ⎦

(

)

(

)

(5)

If Γ = B1, then the following c –type Coriolis interaction is allowed, Ref. [49]:

Finally, if the product is Γ = A2, then the states v and v˜ are connected by an a –type Coriolis resonance interaction of the form, Ref. [50]:

(1) (1) (2) (2) Hvv˜ = iJyHvv ˜ + Hvv˜ iJy + [Jx , Jz ]Hvv˜ + Hvv˜ [Jx , Jz ]

(1) (2) (2) Hvv˜ = iJzHvv ˜ + [Jx , Jy ]Hvv˜ + Hvv˜ [Jx , Jy ]

2 2 ⎤ (3) (3)⎡ + ⎡⎣ iJy, Jx2 − Jy2 ⎤⎦Hvv ˜ + Hvv˜ ⎣ iJy, Jx − Jy ⎦ + …

(

)

(

)

(4)

2 2 ⎤ (3) (3)⎡ + ⎡⎣ iJz , Jx2 − Jy2 ⎤⎦Hvv ˜ + Hvv˜ ⎣ iJz , Jx − Jy ⎦ + …

(

)

(

)

(6)

O.N. Ulenikov et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 187 (2017) 403–413

Table 7 Some sets of resonance interaction parameters of Parameter

C CH4a.

13 12

Value

1,2

F0

409

Parameter

0.901(92)

1,2

F K × 102

Value

Parameter

Value


0.603(86)

1,2

0.1933(78)

F J × 102


0.4038(91)

0.444(18)

1,2

2,3


0.241(15)

2,3


0.3181(76)

2,3

0.644(27)

2,3


0.930(34)

2,3

0.194(14)

2,3

0.756(27)


0.175(31)

2,4

1,2

F xy × 10

3

F K × 10

F xy × 103

1,4

F xy × 10

2,4

2,4

5

F xyK × 10

F J × 102

F xyK × 105

F JJ × 107

F xyJ × 107

0.332(24)

2


0.1170(85)

F K × 10

2,4

F J × 103

F xy × 102

F xyK × 10

0.7321(17)

3,4


0.4436(64)

3,4

0.3152(12)

3,4


0.338(15)

3,4

1,5 1


0.80

1,5 1 CK

× 103


0.2973(18)

2,5 1

0.06

2,7 1 CK

× 104


0.573(85)

2,7 1 CJ ×

1,9 1 CK × 103 1,9 2 Cxz × 102

0.361(29)

1,9 1 CKK

2,9 1 CK

0.292(53)

3,4

3,4

F K × 10 6

F JJ × 10

3,4

F xy × 10

3,4

F xyKK × 106

2

C ≡ (2Bζ x )1,5 C ≡ (2Bζ x )2,5 C ≡ (2Bζ x )3,5 x 1,6

0.04

C ≡ (2Bζ x )2,6

1.61

C ≡ (2Bζ )

2,6 1

3,6 1


0.07

5,6

3.00(82)

C ≡ (2Bζ x )3,6

F0

C ≡ (2Aζ z )1,7

4.77

z 2,7

C ≡ (2Aζ )

2,7 1 CKK 3,7 1

× 10

6

0.14304(68)

5

F xyK × 10


0.816(49)

3,4

6


0.396(14)

F xyKKK × 109

0.581(51)

F KKK × 10

F KKJ × 10 F xyJ × 10

0.1161(11)

104

0.544(19)

× 105

0.3149(71)

0.14

C ≡ (2Aζ z )3,7 y 2,8

0.11

C ≡ (2Cζ y )3,8

0.10

C ≡ (2Cζ )

3,8 1


0.42

5,8 1

C ≡ (2Cζ y )5,8

6,8 1

y 6,8

7,8 1

x 7,8


4.27

C ≡ (2Cζ )


1.68

C ≡ (2Bζ )

0.58

1,9 1

C ≡ (2Cζ y )1,9

1,9 1 CKJ

F KK × 103

3,4


0.381(27)

7

2,8 1

3,4

0.2526(41)


0.11

1,7 1

2,7 1

0.3447(12) 6

F J × 102


1.40

3,5 1

1,6 1


0.1928(36)

0.6823(92)

5


0.1812(67)

× 106

2,9 1

0.12

3,9 1


0.23

C ≡ (2Cζ y )2,9 C ≡ (2Cζ y )3,9

0.358(33)

× 104

4.24

5,9 1

C ≡ (2Aζ z )5,9

6,9 1

z 6,9

0.52

7,9 1

x 7,9


0.05

C ≡ (2Aζ )

C ≡ (2Bζ )

a Values in parentheses are 1 σ standard errors. Value of the parameters presented without standard errors has been estimated theoretically (see text for details) and were not varied in the fit procedure.

(i ) The operators Hvv ˜ , i = 1, 2, 3, … in Eqs. (4)–(6) have the form:

(i ) Hvv ˜ =

1 vv˜ i C + 2 +

vv˜ i 2 C K Jz

vv˜ i C KKK Jz6

+

+

1 vv˜ i 2 CJ J + 2

vv˜ i C KKJ Jz4J2

+

vv˜ i C KK Jz4

vv˜ i C KJJ Jz2J 4

+

+

vv˜ i C KJ Jz2J2

+

1 vv˜ i 4 C JJ J 2

1 vv˜ i 6 C JJJ J + … 2

(7)

(1) The initial values of unperturbed vibrational energies have been estimated on the basis of results of the isotopic substitution theory, (see, e.g., Refs. [51–53]). To realize this, transformation coefficients, lNαλ , of the 13C12CH4 species have been calculated from the set of equations (for details, see Refs. [51,52])

ανμων′ 2 = 4.2. Estimation of spectroscopic parameters Here we briefly discuss the problem of numerical estimation of spectroscopic parameters of vibrational states considered in the present study. This is important because the “inverse spectroscopic problem” (fit) of nine interacting vibrational states belongs to the class of so-called uncorrect inverse problems (in our case, there are numerous correlations between parameters of the diagonal and non-diagonal blocks). The mostly suitable way to understand and overcome the problem is a numerical estimation of maximum possible numbers of spectroscopic parameters of the effective Hamiltonian, Eq. (1). In our study:

∑ Aλμωλ2ανλ, λ

Aλμ = δλμ −

∑ Nα

(m N − mN ) ′ lNαλlNαμ, mN ′

(8)

(9)

and

l Nγλ = ′

⎛ m ⎞1/2 N ⎟ lNαμ(α−1)μλ. ⎝mN⎠ ′

∑ Kαγe ⎜ αμ

(10)

All the values in the right hand sides of Eqs. (8)–(10) are known values of the “parent”, 12C2H4, molecule, or can be obtained during solving of Eqs. (8)–(10). In turn, the

410

O.N. Ulenikov et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 187 (2017) 403–413

JK K a c

Fig. 5. Plots of dependency of some differences Δ J′ Ka′ Kc′ =

(exp .) JKa Kc δ J′ K ′ K ′ a c

−

(calc .) JKa Kc δ J′ K ′ K ′ a c

between experimental and calculated values of some sets of ground state combination

differences on the value of quantum number J. The curve I a corresponds to the values ΔJJ +0 2J 0 J + 2 (the values (exp .)δ JJ +02J 0 J + 2 have been calculated with the parameters from Ref. Kc [19]). The curves II a - V a correspond to the values ΔJJ 810K ′Kc , ΔJJ +101 K8′Kc , ΔJJ +8 1K10 , and ΔJJ +101 K12′ Kc , respectively (corresponding values ′ c

c

c

c

(exp.) JKa Kc δ J′ K ′ K ′ a c

have been calculated with the

parameters from Ref. [18]). Curves on the bottom part of the figure correspond to results obtained on the base of our ground state parameters from column 4 of Table 3.

Table 8 Transitions assigned to the 2ν10 band of Uppera J

13 12

C CH4.

Lowera Ka

Kc

J′

1

Line position

Ka′ 2

Kc′

Transm.

Spectrumb

Upper energy

Mean value

δc

cm
1

in %

(cm
1)

(cm
1)

(10
4 cm
1)

3

4

5

6

7

8

25

14

d

24 26

14 14

d d

1496.5708 1405.6312

68.5 78.0

IX IX

2806.5621 2806.5625

2806.5623


1

26

14

d

25 27 27

14 14 16

d d d

1497.8344 1403.3301 1168.0208

80.7 76.2 90.5

VI VI VI

2852.4047 2852.4042 2852.4039

2852.4042

0

27

14

d

26 27

14 14

d d

1499.1254 1450.9822

60.5 69.9

IX IX

2900.0567 2900.0563

2900.0565

2

a

The notation [JKa d] means the couple of degenerated states [JKa Kc = J − Ka] and [JKa Kc = J − Ka + 1]. See Table 1. δ = (E exp . − E calc .) (in units of 10
4 cm
1) is the difference between the experimental value of upper energy from column 7 and corresponding value calculated with the parameters from Tables 6 and 7. b c

knowledge of the transformation coefficients, lNαλ , allows one to estimate shifts of band centers and line positions under isotopic substitution, Ref. [53]. Results of numerical estimations are presented in Table 6. (2) Because the cubic force field parameters of the ethylene molecule are unknown, it is difficult to correctly estimate values of effective rotational parameters even on the basis of the isotopic substitution theory. So, we have used one of the statements of the vibration-rotation theory (see, e.g., Refs. [54,55]) which says that the values of both rotational, and of different centrifugal distortion parameters of the excited vibrational states of a “normal” molecule cannot be strongly changed from the values of corresponding parameters of the ground vibrational state. For that reason, the initial values of all rotational and centrifugal distortion parameters of all

excited vibrational states have been taken to be equal to the values of corresponding parameters of the ground vibrational state (see Table 6). (3) The initial values of the main Coriolis interaction parameters, vv˜ 1 C , have been estimated with the simple formulas, Refs. [55,56], α ζλμ =

∑ (lNβλlNγμ − lNγλlNβμ) N

(11)

and vλvμ 1

α C = 2Bαζλμ ,

(12)

where lNαλ are discussed above transformation coefficients of the 13C12CH4 molecule; Bz ≡ A , Bx ≡ B , and By ≡ C rotational

O.N. Ulenikov et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 187 (2017) 403–413

411

the parameters from Ref. [19]). The curves II a - V a correspond to Kc the values ΔJJ 810K ′Kc , ΔJJ +101 K8′Kc , ΔJJ +8 1K 10 , and ΔJJ +101 K12′ Kc , respectively c c c′ c (corresponding values (exp.)δ JJ′ KK′aKK′ c have been calculated with the a c parameters from Ref. [18]). As one can see from Fig. 5, the values of JK K differences, Δ J′ Ka′ Kc′ may reach the values from 40 × 10−4 cm−1 up to a c
1 3.4 cm . At the same time, the accuracy of our experiment is about 2 × 10−4 cm−1. It means that the ground state parameters of both papers (Refs. [18] and [19]) can be improved. To understand and improve the ground state parameters, we constructed more than 3400 ground state combination differences JK K of the (exp.)δ J′ Ka′ Kc -type ( J max. = 50, Kamax. = 12, ΔJ = 0, ± 1, ± 2, a c ΔKa = 0, ± 2). As all three studied bands, ν12, ν2 and ν3, are bands

Fig. 6. The vibrational energy levels of 13C12CH4 in the region of 800–1700 cm
1. Resonance interactions which have been taken into account in the present study are marked by thick solid lines (Fermi-interactions), dash lines ( B –type Coriolis interactions between vibrational states of the symmetry A1 and B2, or A2 and B1), thin solid lines ( A –type Coriolis interactions between vibrational states of the symmetry A1 and A2, or B1 and B2), and dot lines (–type Coriolis interactions between vibrational states of the symmetry A1 and B1, or A2 and B2), respectively.

parameters; and index α in right hand side of Eq. (12) is determined by the symmetry of the vibrational states vλ and vμ . Corresponding estimated values are shown in Table 7.

5. Re-analysis of the ground vibrational state As shown in the assignment of transitions of the experimental spectra, the rotational structure of the ground vibrational state is not totally correct both in Ref. [18], and in Ref. [19] and may be improved. To illustrate this, Fig. 5 presents some plots of dependency of differences JK K δ J′ Kaa′ Kcc′

= E JK

a Kc

JK K

Δ J′ Kaa′ Kcc′ =

(exp .) JKa Kc δ J′ Ka′ Kc′

−

(calc .) JKa Kc δ J′ Ka′ Kc′

(here

− E J′ Ka′ Kc′ ) between experimental and calculated va-

lues of some sets of the ground state combination differences on the value of quantum number J. The curve I a corresponds to the values ΔJJ +0 2J 0 J + 2 (the values (exp .)δ JJ +02J 0 J + 2 have been calculated with

JK K of the a –type, again only (exp.)δ J′ Ka′ Kc′ ( Ka′ = Ka ) can be constructed. a c For that reason, we additionally analyzed the c –type bands ν7 and ν8, and the b− type band ν10. This gave us the possibility to construct more than 1500 ground state combination differences (exp.) JKa Kc δ J′ Ka′ Kc′ with the values (Ka′ − Ka ) = ± 2. Then the constructed ground state combination differences were used as input data in the weighed fit of parameters of the ground vibrational state. Results of the fit are shown in column 4 of Table 3. As one can see, the HKJ, HJ and hK parameters are absent in column 3. They were omitted from consideration because absolute values of these parameters obtained from the fit were less than their 1σ standard errors. It is useful to compare the results obtained in the present study with corresponding results from Ref. [18] and [19] which are shown in columns 2 and 3 of Table 3. One can see that the values of the lower order parameters in all three cases are in close agreement to each other. However, the set of parameters derived in the present study reproduces the initial ground state combination differences, obtained from infrared transitions in this paper, with the rms –deviation of 2.1 × 10−4 cm−1 which is close to the experimental accuracy of our FTIR data (see also bottom part of Fig. 5). At the same time, the drms values for the analyzed GSCD, which have been calculated with the parameters from columns 2 and 3, are 0.48 cm
1 and 5.6 × 10−4 cm−1, respectively. It is of interest to compare the values of parameters in columns 3 and 4 of Table 3. One can see that the values of all parameters, with the exception of HK, HJK, and hJ, are close to each other. At the same time, the values of the mentioned three parameters differ considerably from each other. We believe that the reason of this discrepancy is in the larger number of the ground state combination differences which have been used as the initial data in our analysis (the last statement can be illustrated, e.g., by the comparison of curves I a and I b on Fig. 5: it is clear that the increasing of the value ΔJJ0+J 20J + 2 on the curve I a with the increasing of the value J is the consequence of the absence of corresponding ground state combination differences in Ref. [19]). Consequently, an influence of the initial experimental data on the values of

Fig. 7. Observed - calculated term values and fit statistics for the studied vibrational states of

13 12

C CH4.

412

O.N. Ulenikov et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 187 (2017) 403–413

parameters is redistributed between different high-order centrifugal distortion coefficients. To illustrate this effect, column 5 of Table 3 shows the results of the fit of the same set of parameters, as in column 3, with our initial GSCD data. One can see that the values of high order centrifugal distortion coefficients in column 5 also differ considerably from the values of parameters in column 3. From comparison of the rms –values in columns 4 and 5 one can see that they are comparable. However, we prefer to use parameters from column 4 in the further analysis because (a) the rms –value in column 4 is about 10 % less than the rms –value in column 5, and (b) the number of parameters in column 4 is one less than in column 5.

6. Ro-vibrational energy levels and determination of spectroscopic parameters The new ground state parameters obtained in Section 5 were used then for calculation of ground state rotational energy levels which, in turn, were used in the re-assignments of transitions in the recorded FTIR spectra. The final list of the more than 3800 assigned transitions is presented in the Supplementary material. From these transitions we obtained 1562 upper ro-vibrational energy levels (for details, see statistical information in Table 4; list of transitions assigned to the band 2ν10 is presented in Table 8) which were used then as input data in a weighted least square fit with the aim to determine rotational, centrifugal distortion, and resonance interaction parameters for the set of states (v3 = 1), (v12 = 1), (v2 = 1), and (v10 = 2). Results of the fit are presented in columns 8, 10, 12, 14, 15, and 16 of Table 6 and in Table 7 (values in parentheses are 1σ statistical confidence intervals). As the analysis shows, correct description of the experimental data is possible only if strong resonance interactions of the studied states with the states (v4 = 1), (v6 = 1), (v7 = 1), (v8 = 1) and (v10 = 1) are taken into account (see Fig. 6, where the scheme of vibrational energy terms and resonance interactions in the studied region are shown). Therefore, some parameters of these vibrational states were also fitted (see Table 6). Parameters presented without confidence intervals have been constrained to the values estimated theoretically or equal to the values of corresponding parameters of the ground vibrational state. One can see that the values of parameters in columns 8, 10, 12, 14, 15, and 16 correlate very well with the values of corresponding parameters of the ground vibrational state from column 4 of Table 3. Also for comparison, columns 3, 5, 7, 9, 11, and 13 of Table 6 present values of parameters of preceding studies from Refs. [19] and [18]. Because, as was discussed above, diagonal block's parameters can differ from corresponding parameters of the ground vibrational state only slightly, we only varied band centers, rotational and the most important of the Δ–, δ –centrifugal distortion coefficients in the fit. To achieve a satisfactory correspondence between theoretical and experimental results, the number of varied resonance interaction parameters was larger than usually used in analogous fits. In this case, if a value of varied parameter was less or comparable with its 1σ statistical confidence interval, such parameter was omitted or constrained to the theoretically estimated value. The set of parameters obtained from the fit which is presented in Tables 6 and 7 reproduces the values of initial 1562 FTIR energy levels with the rms –deviation equal to 2.6 × 10−4 cm−1 (for more details, see statistical information in Table 4). To illustrate the quality of the fit, column 8 of Tables 8 and column 9 of Supplementary Material show the values of differences δ (in units of 10
4 cm
1) between experimental and calculated values of ro-vibration energy levels and/or line positions. It can be also interesting to compare the results obtained in the present study with the results of an analogous study of the ν12 band in Ref. [18] (one can find such comparison in Table 4). In particular, one can see that 1.5 times more number of experimental transitions/energy levels

is described by the number of varied parameters which is comparable with the preceding study. To give the reader an opportunity to estimate the quality of the results, Fig. 7 shows the fit residuals for ro-vibrational energies as a function of the quantum number J for studied vibrational states which demonstrates good agreement between the experimental and calculated results. As one more illustration of the correctness of the results obtained in the present study, the simulated spectra of the studied bands were constructed (see the bottom part of Figs. 2, 3, and 4). Calculation schemes from Refs. [57–59] were used in the estimation of the line strengths (in this case, only one main dipole moment parameter and Doppler profile of the lines were used in estimation of relative line strengths values). One can see that using of even such simple model allow one to describe peculiarities of spectra with more than satisfactory accuracy.

7. Conclusion We recorded and analyzed the high resolution ro-vibrational spectra of the 13C12CH4 molecule in the region of 600 – 1700 cm
1, where the bands ν3, ν12, ν2 and 2ν10 are located. For the ν12 band, more than 2400 transitions with the maximum values of quantum numbers J max. = 52 and /Kamax. = 18 were assigned. More than 1400 transitions were assigned for the first time to the bands ν2, ν3 and 2ν10. The ground vibrational state was re-analyzed on the basis of our new experimental data. The improved set of ground state parameters was obtained, and used for determination of upper ro-vibrational energy levels and spectroscopic parameters. The latter were fitted in the Hamiltonian model which takes into account resonance interactions both between four analyzed vibrational states, (v3 = 1), (v12 = 1), (v2 = 1), (v10 = 2), and between these four states and additional five vibrational states, (v10 = 1), (v8 = 1), (v7 = 1), (v4 = 1), (v6 = 1), which interact with four studied ones. The set of 62 parameters obtained from the fit reproduces values of 1562 initial “experimental” ro-vibrational energy levels (positions of more than 3800 experimentally recorded and assigned transitions) with the rms error drms = 2.6 × 10−4 cm−1. Ground state parameters of the 13C12CH4 molecule were improved, as well.

Acknowledgments This work was funded by the Volkswagen Foundation, Germany (grant 90239). Part of the work was supported by the project “Leading Russian Research Universities” (Project FTI-24/2016 of the Tomsk Polytechnic University, Russia), by the DAAD grant N1.710.2016, and by the Deutsche Forschungsdemeinschaft (grants BA 2176/3-2, BA 2176/4-1, BA 2176/4-2, and BA 2176/5-1).

Appendix A. Supplementary data Supplementary data associated with this article can be found in the online version at http://dx.doi.org/10.1016/j.jqsrt.2016.10.009.

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