Global modeling of vibration-rotation spectra of the acetylene molecule

Journal of Quantitative Spectroscopy & Radiative Transfer 177 (2016) 59–74 Contents lists available at ScienceDirect Journal of Quantitative Spectro...

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Journal of Quantitative Spectroscopy & Radiative Transfer 177 (2016) 59–74

Contents lists available at ScienceDirect

Journal of Quantitative Spectroscopy & Radiative Transfer journal homepage: www.elsevier.com/locate/jqsrt

Global modeling of vibration-rotation spectra of the acetylene molecule O.M. Lyulin a,b, V.I. Perevalov a,n a Laboratory of Theoretical Spectroscopy, V.E. Zuev Institute of Atmospheric Optics, Siberian Branch, Russian Academy of Sciences, 1, Academician Zuev Square, 634055 Tomsk, Russia b National Research Tomsk State University, 36, Lenin av., 634050 Tomsk, Russia

a r t i c l e i n f o

abstract

Article history: Received 29 October 2015 Received in revised form 19 December 2015 Accepted 23 December 2015 Available online 13 January 2016

The global modeling of both line positions and intensities of the acetylene molecule in the 50–9900 cm 1 region has been performed using the effective operators approach. The parameters of the polyad model of effective Hamiltonian have been ﬁtted to the line positions collected from the literature. The used polyad model of effective Hamiltonian takes into account the centrifugal distortion, rotational and vibrational ℓ-doubling terms and both anharmonic and Coriolis resonance interaction operators arising due to the approximate relations between the harmonic frequencies: ω1 E ω3 E 5ω4 E 5ω5 and ω2 E 3ω4 E 3ω5. The dimensionless weighted standard deviation of the ﬁt is 2.8. The ﬁtted set of 237 effective Hamiltonian parameters allowed reproducing 24,991 measured line positions of 494 bands with a root mean squares deviation 0.0037 cm 1. The eigenfunctions of the effective Hamiltonian corresponding to the ﬁtted set of parameters were used to ﬁt the observed line intensities collected from the literature for 15 series of transitions: ΔP ¼ 0-13,15, where P¼ 5V1 þ 5V3 þ 3V2 þ V4 þ V5 is the polyad number (Vi are the principal vibrational quantum numbers). The ﬁtted sets of the effective dipole moment parameters reproduce the observed line intensities within their experimental uncertainties 2–20%. & 2016 Elsevier Ltd. All rights reserved.

Keywords: Acetylene Infrared Line position Line intensity Global modeling Effective Hamiltonian Effective dipole moment operator

1. Introduction The reference high temperature spectra of acetylene are of great importance for studying the atmospheres of carbon stars (see, for example, Ref. [1]) and exoplanets [2]. The respective spectra could be obtained experimentally or as a result of theoretical modeling. We know only two experimental papers [3,4] in which the laboratory high temperature spectra of acetylene were studied. The cited papers deal with the emission spectra of acetylene in the 3 mm rigion at temperatures from 870 K to 1455 K. The recorded spectra were theoretically modeled using the global effective rovibrational n

Corresponding author. Tel.: þ 7 3822491794; fax: þ 7 3822492086. E-mail address: [email protected] (V.I. Perevalov).

http://dx.doi.org/10.1016/j.jqsrt.2015.12.021 0022-4073/& 2016 Elsevier Ltd. All rights reserved.

Hamiltonian developed in Brussels [3]. To simulate the high temperature spectra, one has to know the effective dipole moment operator corresponding to the effective Hamiltonian used. The successful modeling of emission spectra of acetylene in 3 mm region performed in Ref. [4] was possible due to the fact that the absorption cross sections in this region are determined mostly by only one derivative of the dipole ∂μ moment function ∂q z [5]. For the simulation of high tem3 perature spectra in other spectral regions it is necessary to develop the effective dipole moment operator corresponding to used effective Hamiltonian. In this paper we present the results of the global modeling of both line positions and intensities performed using the global effective Hamiltonian and the corresponding dipole moment operator developed in Tomsk. These results will be used in the nearest future for the generation of high

60

O.M. Lyulin, V.I. Perevalov / Journal of Quantitative Spectroscopy & Radiative Transfer 177 (2016) 59–74

temperature database for acetylene molecule in the spectral region from 0.6 cm 1 to 10000 cm 1. The global effective Hamiltonian used in Tomsk was formulated in our paper [6] about 20 years ago. It is a polyad model which takes into account the rotational and vibrational ℓ-doubling terms, the anharmonic and Coriolis resonance interaction operators arrising due to the following approximate relations between the harmonic frequences [7]

ω1 ω3 5ω4 5ω5 ;

of the projections of the vibrational angular momentum on the molecular-ﬁxed axis z, connected with the double degenerate normal vibrations with the harmonic frequencies ωi. We use the Wang combinations of the basis functions V 1 V 2 V 3 V 4 V 5 ℓ4 ℓ5 J M K ε E 1 ¼ pﬃﬃﬃðV 1 V 2 V 3 V ℓ44 V ℓ55 J M K ¼ ℓ4 þℓ5 2 þ εjV 1 V 2 V 3 V 4 ℓ4 V 5 ℓ5 iJM K ¼ ℓ4 ℓ5 Þ; ð3Þ with

ω2 3ω4 3ω5 :

ð1Þ

To derive the above effective Hamiltonian we used the results obtained by Pliva [8], Hietanen [9], Huet et al. [10], Perevalov and Sulakshina [11], Abbouti Temsamani and Herman [12]. The preliminary results of the line position ﬁtting with this effective Hamiltonian have been published in our papers [13–15]. The effective dipole moment operator corresponding to our global effective Hamiltonian [6] has been developed in our papers [16,17]. We have performed the parameterization of the effective dipole moment matrix elements in the basis of the product of the eigenfunctions of the harmonic oscillators and the rigid symmetric top eigenfunctions. The suggested aproach for the line intensity calculations has been successfully used in the modeling of the observed high resolution spectra of acetylene in different spectral regions [17–24]. The C2H2 spectra predicted with our models of effective Hamiltonian and effective dipole moment operator have also been successfuly used for the analysis of the experimental spectra in different spectral regions [25–27]. The global modeling of the acetylene vibrational energies within the framework of the method of effective operators was performed in Brussels (see [12,28] and the references therein). Later this team has performed the successfull global modeling of C2H2 line positions using effective Hamiltonian which is very similar to that we used (see [3,29,30] and the references therein). The comparison of the results of two global C2H2 line position ﬁts is presented in Section 3.

ε ¼1 or 1 and

jV 1 V 2 V 3 V 4 V 5 00JMε ¼ 1i ¼ jV 1 V 2 V 3 V 04 V 05 4jJM0i:

Taking into account the symmetry properties of the basis functions each P block can be split into four isolated subblocks {P, q, ε}. Here ( g V 3 þ V 5 even q¼ ð5Þ u V 3 þ V 5 odd: The effective Hamiltonian can be presented by its matrix elements. For the sake of simplicity we present below the matrix elements of effective Hamiltonian in the multiplicative basis jV 1 V 2 V 3 V ℓ44 V ℓ55 JK⟩ ¼ V 1 V 2 V 3 V ℓ44 V ℓ55 ⟩JMK ¼ ℓ4 þ ℓ5 ⟩ but in our computer code the Wang combinations of the basis functions are used. In the case of free molecule, the magnetic quantum number M could be omitted. The diagonal on the principal vibrational quantum numbers matrix elements are given by the following equations: V 1 V 2 V 3 V 4 ℓ4 V 5 ℓ5 JK H ef f V 1 V 2 V 3 V 4 ℓ4 V 5 ℓ5 JK X X ωi V i þ ij X ij V i V j ¼ i

X Y ijl V i V j V l þ Y ab V i ℓa ℓb iab i X X þ Z VVVV þ Z ab V i V j ℓa ℓb ijlk ijlk i j l k ijab ij X X þ Z abcd ℓa ℓb ℓc ℓd þ W ijklm V i V j V k V l V m abcd ijklm X X X ab þ W ijk V i V j V k ℓa ℓb þ ðBe αi V i þ ij γ ij V i V j ijkab þ

þ

X

ab

g ab ℓa ℓb þ

X

γ ab ℓa ℓb þ ab

X

ijl

ε VVV þ ijk ijk i j k

½JðJ þ 1Þ K ðDe þ þ

Acetylene is a four atomic molecule which is linear in the ground electronic state and has D1h symmetry group. This molecule has ﬁve vibrational modes. Two of them ω4(Πg) and ω5(Πu) are degenerate bending modes, and the three þ þ others ω1(Σg ), ω2(Σg ) and ω3(Σu ), are nondegenerate stretching modes. Due to the approximate relations between the harmonic frequencies (1) the vibrational energy levels are grouped into clusters (polyads). A polyad can be numbered by a polyad number P ¼ 5V 1 þ 3V 2 þ 5V 3 þ V 4 þV 5 ;

ð2Þ

where Vi (i¼1, 2, 3, 4, 5) are the principal quantum numbers of the harmonic oscillators. Within the framework of the polyad model of effective Hamiltonian [6] its matrix in the basis of the the E eigenfunctions of harmonic oscillators product of V 1 V 2 V 3 V ℓ4 V ℓ5 jV 1 ijV 2 ijV 3 ijV 4 ℓ4 ijV 5 ℓ5 i and the rigid 5 4 symmetric top JMK is block-diagonal with respect to the polyad number P. Here ℓi ði ¼ 4; 5Þ are the quantum numbers

X ab

i

X 2

2. Effective Hamiltonian

ð4Þ

X i

X

βi V i þ

ε

ab V i ℓa ℓb Þ iab i

X

δ

VV ij ij i j

δab ℓa ℓb Þ½JðJ þ1Þ K 2 2 ;

ð6Þ

Vibrational ℓ-doubling D E V 1 V 2 V 3 V ℓ44 V ℓ55 JK H ef f V 1 V 2 V 3 V 4ℓ4 7 2 V 5ℓ5 8 2 JK X X X r i45 V i þ r VV þ r ab ℓ ℓ ¼ r 45 þ ij ij45 i j ab 45 a b i

þ r J45 JðJ þ1Þ þ r JJ45 ½JðJ þ 1Þ2

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðV 4 8ℓ4 ÞðV 4 7ℓ4 þ 2ÞðV 5 7ℓ5 ÞðV 5 8 ℓ5 þ 2Þ;

ð7Þ

Rotational ℓ-doubling E ℓt ef f ℓt 7 2 P P V t JK H V t JK 7 2 ¼ qt þ qti V i þ ij qtij V i V j i X J ab þ q ℓ ℓ þq JðJ þ1Þ a b t ab t

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðV t 8ℓt ÞðV t 7ℓt þ 2Þ½JðJ þ 1Þ KðK 71Þ½JðJ þ 1Þ ðK 71ÞðK 7 2Þ:

ð8Þ

Table 1 4 Δℓ5 ΔK List of the φΔℓ ΔV 1 ΔV 2 ΔV 3

ΔV1

ΔV2

ΔV 4 ΔV 5 ðV 1 ;

ΔV3

V 2 ; V 3 ; V 4 ; V 5 ; ℓ4 ; ℓ5 ; J; KÞ functions.

ΔV4

Nonaccidental resonance interactions 0 0 0 0

ΔV5

Δℓ4

Δℓ5

ΔK

0

74

84

0

Not

r4455

0

0

0

0

0

74

74

u55

0

0

0

0

0

74

82

72

u44/5

0

0

0

0

0

82

74

72

u4/55

2

0

0

0

K1/255

2

0

0

0

0

K1/244

Anharmonic resonance interactions 1 1 0 0 1

1

0

2

0

2

0

0

0

0

0

K11/33

0

0

0

2

2

0

0

0

K44/55

0

0

0

2

2

72

82

0

ℓ

0

1

1

1

1

71

81

0

K3/245

K44/55

1

0

1

1

1

71

81

0

K14/35

1

0

1

1

1

71

81

0

K15/34

1

1

2

2

0

0

0

0

K33/1244

1

1

2

0

2

0

0

0

K33/1255

0

1

1

1

3

71

81

0

K34/2555

2

0

2

2

2

0

0

0

K1144/3355

0

0

0

4

4

0

0

0

K4444/5555

ΔK¼ 7 2 resonance interactions 0 1 1 1

1

71

71

72

O3/245

0

2

0

72

72

O1/255

1

1

0

1

1

0

2

0

72

0

72

O1/244

1

0

1

1

1

71

71

72

O14/35

1

0

1

1

1

71

71

72

O15/34

0

0

0

2

2

0

72

72

ℓ5

Resonance Coriolis interactions 0 1 0 1

2

71

0

71

C2/455

0

71

0

71

Ca2/444

0

1

0

3

O44/55

ΔV 4 ΔV 5 ðV 1 ;

V 2 ; V 3 ; V 4 ; V 5 ; ℓ4 ; ℓ5 ; J; KÞ

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðV 4 7 ℓ4 þ 4ÞðV 4 7 ℓ4 þ 2ÞðV 4 8 ℓ4 ÞðV 4 8 ℓ4 2ÞðV 5 8 ℓ5 þ 4ÞðV 5 8 ℓ5 þ 2ÞðV 5 7 ℓ5 ÞðV 5 7 ℓ5 2Þ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 7 7 7 7 ðV 5 7 ℓ5 þ 4ÞðV 5 7 ℓ5 þ 2ÞðV 5 8 ℓ5 ÞðV 5 8 ℓ5 2Þf J;K f J;K 7 1 f J;K 7 2 f J;K 7 3 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 7 7 ðV 4 7 ℓ4 þ 4ÞðV 4 7 ℓ4 þ 2ÞðV 4 8 ℓ4 ÞðV 4 8 ℓ4 2ÞðV 5 7 ℓ5 ÞðV 5 8 ℓ5 þ 2Þf J;K f J;K 7 1 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ 7 7 ðV 4 7 ℓ4 ÞðV 4 8 ℓ4 þ 2ÞðV 5 7 ℓ5 þ 4ÞðV 5 7 ℓ5 þ 2ÞðV 5 8 ℓ5 ÞðV 5 8 ℓ5 2Þf J;K f J;K 7 1 rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðV 1 þ 1ÞV 2 V 25 ℓ25 rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðV 1 þ 1ÞV 2 V 24 ℓ24 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðV 1 1ÞV 1 ðV 3 þ 1ÞðV 3 þ 2Þ r ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ h i V 24 ℓ24 ðV 5 þ 2Þ2 ℓ25 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðV 4 8 ℓ4 ÞðV 4 8 ℓ4 2ÞðV 5 8 ℓ5 þ 2ÞðV 5 8 ℓ5 þ 4Þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ V 2 ðV 3 þ 1ÞðV 4 8 ℓ4 ÞðV 5 7 ℓ5 Þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ V 1 ðV 3 þ 1ÞðV 4 8 ℓ4 ÞðV 5 8 ℓ5 þ 2Þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ V 1 ðV 3 þ 1ÞðV 4 7 ℓ4 þ 2ÞðV 5 7 ℓ5 Þ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ V 1 V 2 ðV 3 þ 1ÞðV 3 þ 2Þ V 24 ℓ24 rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ V 1 V 2 ðV 3 þ 1ÞðV 3 þ 2Þ V 25 ℓ25 ﬃ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ h i ðV 2 þ 1ÞV 3 ðV 4 8 ℓ4 Þ ðV 5 þ 2Þ2 ℓ25 ðV 5 8 ℓ5 þ 4Þ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ h iﬃ V 1 ðV 1 1ÞðV 3 þ 1ÞðV 3 þ 2Þ V 24 ℓ24 ðV 5 þ 2Þ2 ℓ25 rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ h ih ih i V 24 ℓ24 ðV 4 2Þ2 ℓ24 ðV 5 þ 2Þ2 ℓ25 ðV 5 þ 4Þ2 ℓ25 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 7 7 V 2 ðV 3 þ 1ÞðV 4 8 ℓ4 ÞðV 5 8 ℓ5 Þf J;K f J;K 7 1 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 7 7 ðV 1 þ 1ÞV 2 ðV 5 8 ℓ5 ÞðV 5 8 ℓ5 2Þf J;K f J;K 7 1 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 7 7 ðV 1 þ 1ÞV 2 ðV 4 8 ℓ4 ÞðV 4 8 ℓ4 2Þf J;K f J;K 7 1 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 7 7 V 1 ðV 3 þ 1ÞðV 4 8 ℓ4 ÞðV 5 7 ℓ5 þ 2Þf J;K f J;K 7 1 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 7 7 V 1 ðV 3 þ 1ÞðV 4 7 ℓ4 þ 2ÞðV 5 8 ℓ5 Þf J;K f J;K 7 1 r ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 7 7 V 24 ℓ24 ðV 5 7 ℓ5 þ 4ÞðV 5 7 ℓ5 þ 2Þf J;K f J;K 7 1

O.M. Lyulin, V.I. Perevalov / Journal of Quantitative Spectroscopy & Radiative Transfer 177 (2016) 59–74

0

4 Δℓ5 ΔK φΔℓ ΔV 1 ΔV 2 ΔV 3

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ h i ﬃ 7 V 2 ðV 4 7 ℓ4 þ 2Þ ðV 5 þ 2Þ2 ℓ25 f J;K rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ h i ﬃ 7 V 2 ðV 4 7 ℓ4 þ 4Þ ðV 4 þ 2Þ2 ℓ24 f J;K 61

62

Table 1 (continued ) ΔV1

ΔV3

ΔV4

ΔV5

Δℓ4

Δℓ5

ΔK

Not

73

82

71

Cb2/444

1

0

71

71

C22/35

0

71

0

71

C22/14

2

1

0

71

71

Ca225/344

1

2

1

72

71

73

Cb225/344

1

0

5

2

71

0

71

C255/44444

1

1

2

1

72

81

71

C23/1445

0

1

0

3

0

0

2

1

0

1

2

0

1

0

2

1

0

2

0 1

4 Δℓ5 ΔK φΔℓ ΔV 1 ΔV 2 ΔV 3

ΔV 4 ΔV 5 ðV 1 ;

V 2 ; V 3 ; V 4 ; V 5 ; ℓ4 ; ℓ5 ; J; KÞ

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 7 V 2 ðV 4 7 ℓ4 þ 6ÞðV 4 7 ℓ4 þ 4ÞðV 4 7 ℓ4 þ 2ÞðV 5 7 ℓ5 ÞðV 5 8 ℓ5 þ 2Þf J;K qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 7 V 2 ðV 2 1ÞðV 3 þ 1ÞðV 5 7 ℓ5 þ 2Þf J;K qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 7 ðV 1 þ 1ÞV 2 ðV 2 1ÞðV 4 7 ℓ4 þ 2Þf J;K rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ h i 7 V 2 ðV 2 1ÞðV 3 þ 1Þ ðV 4 þ 2Þ2 ℓ24 ðV 5 8 ℓ5 Þf J;K qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 7 7 7 V 2 ðV 2 1ÞðV 3 þ 1ÞðV 4 7 ℓ4 þ 4ÞðV 4 7 ℓ4 þ 2ÞðV 5 8 ℓ5 Þf J;K f J;K 7 1 f J;K 7 2 rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ h ih i 7 V 2 ðV 4 7 ℓ4 þ 6Þ ðV 4 þ 4Þ2 ℓ24 ðV 4 þ 2Þ2 ℓ24 V 25 ℓ25 f J;K qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ 7 V 1 ðV 2 þ 1ÞðV 3 þ 1ÞðV 4 7 ℓ4 þ 2ÞðV 4 7 ℓ4 þ 4ÞðV 5 7 ℓ5 Þf J;K

f J;K ¼ J ðJ þ 1Þ K ðK 7 1Þ Not – notations for the principal parameter of the respective effective Hamiltonian matrix element.

O.M. Lyulin, V.I. Perevalov / Journal of Quantitative Spectroscopy & Radiative Transfer 177 (2016) 59–74

7

ΔV2

O.M. Lyulin, V.I. Perevalov / Journal of Quantitative Spectroscopy & Radiative Transfer 177 (2016) 59–74

The matrix elements of the resonance interaction operators (both anharmonic and Coriolis types) are given in the following way

63

3. Line position ﬁt The weighted ﬁt of the C2H2 effective Hamiltonian para-

⟨V 1 þ ΔV 1 V 2 þ ΔV 2 V 3 þ ΔV 3 V 4 þ ΔV 4 V 5 þ ΔV 5 ℓ4 þ Δℓ4 ℓ5 þ Δℓ5 J K þ ΔK H ef f V 1 V 2 V 3 V 4 V 5 ℓ4 ℓ5 J K⟩ (

X ΔℓΔK ΔV i ℓ4 Δℓ5 ΔK ℓΔK ¼ φΔ ð V ; V ; V ; V ; V ; ℓ ; ℓ ; J; K Þ RΔ RΔV Vi þ ΔV þ ΔV 1 ΔV 2 ΔV 3 ΔV 4 ΔV 5 1 2 3 4 5 4 5 i 2 i

X ab J X ΔℓΔK ΔV j ΔV i ΔℓΔK ΔℓΔK ΔℓΔK K 2 V þ þ R V þ þ R ℓ ℓ þ R J ð J þ 1 Þ þ R K ; a i j b Δ V Δ V Δ V Δ V ij ab ij 2 2 where the functions ℓ4 Δℓ5 ΔK φΔ ΔV 1 ΔV 2 ΔV 3 ΔV 4 ΔV 5 ðV 1 ; V 2 ; V 3 ; V 4 ; V 5 ; ℓ4 ; ℓ5 ; J; K Þ

for the resonance interactions considered in this work are presented in Table 1. In Eq. (9) we use simpliﬁed notations ℓΔK for the resonance interaction parameters RΔ : ΔV

ΔV ΔV 1 ; ΔV 2 ; ΔV 3 ; ΔV 4 ; ΔV 5 ; Δℓ- Δℓ4 ; Δℓ5 : Nonaccidental resonance interactions ðΔV ¼ 0; Δℓ a 0Þ are also presented by this equation.

ð9Þ

meters has been performed to the observed line positions collected from the literature [5, 19–23,25–27,29–52] using the computer code GLOFILM (Global Fitting of Linear Molecules). This computer code has been elaborated for solving the inverse problem by the least-squares method. The ﬁle of the input data contained 28,972 line positions belonging to 494 bands lying in the 50–9900 cm 1 region. The description of the input data together with the results of the ﬁt is given in Table 2. The weighting has been performed using the reciprocal values of

Table 2 Experimental data and summary of the line position ﬁt. Reference

Setupa

Nﬁtb

Nexclc

Uncertaintyd (cm 1)

νmin–νmaxe (cm 1)

RMSf (10 3 cm 1)

Jacquemart et al. [49] Amyay et al. [51] Kabbadj et al. [33] Gomez et al. [50] Gomez et al. [48] Jacquemart et al. [41] Jacquemart et al. [42] Jacquemart et al. [44] D’Cunha et al. [34] Vander Auwera et al. [5] Rinsland et al. [31] Sarma et al. [37] D’Cunha et al. [32] Lyulin et al. [21] Lyulin et al. [22] Lyulin et al. [22] Henningsen and Sørensen [39] Kou et al. [35] Keppler et al. [38] Mandin et al. [43] Lyulin et al. [19] Lyulin et al. [20] Tran et al. [45] Lyulin et al. [23] Robert et al. [29] Jacquemart et al. [46] Lyulin et al. [25] Lyulin et al. [26] Amyay et al. [30] Vander Auwera et al. [40] Jacquemart et al. [47] Lyulin et al. [27] Twagirayezu [52]

FTS FTS FTS FTS FTS FTS FTS FTS FTS FTS FTS FTS FTS FTS FTS FTS FTS FTS FTS FTS FTS FTS FTS FTS CRDS FTS CRDS CRDS CRDS FTS FTS FTS FC SDSDS

116 1089 2588 464 403 152 463 245 336 465 1357 995 653 441 439 360 106 647 1744 424 155 667 502 89 1401 217 1777 2649 1105 574 313 1920 135

1 0 141 3 11 0 23 0 36 69 432 111 159 3 82 267 0 174 269 0 12 16 44 22 370 16 382 292 437 52 6 551 0

0.0005 0.0002–0.01 0.0002 0.0005 0.0002 0.0002 0.0002 0.0002 0.0004 0.0006 0.01 0.0005 0.001 0.0002 0.001 0.0005 0.0001 0.0003 0.001 0.0005–0.001 0.0005–0.005 0.0002–0.005 0.0005 0.001 0.002 0.0005 0.003–0.005 0.002–0.003 0.0006–0.009 0.0003–0.006 0.0005–0.001 0.001–0.003 0.000001f

65.9–191.3 62.5–222.6 51.9–1439.7 1153.2–1419.9 1229.6–1451.2 1860.4–2255.0 1810.1–2235.1 2515.5–2752.4 2589.2–2759.7 2584.0–3364.1 3139.8–3398.9 3170.9–3953.6 3989.1–4190.2 4423.2–4786.4 5041.8–5566.5 5704.8–6062.6 6459.1–6664.2 6367.9–6734.5 5704.8–6861.9 3183.1–3375.3 3181.9–3327.0 3768.4–4208.5 6298.7–6853.6 6299.6–6665.8 5885.2–6998.3 7042.8–7471.0 7129.2–7917.9 5744.9–6421.1 5688.2–7080.8 7062.1–9877.2 7676.4–9887.9 7000.9–7499.0 6448.3–6564.2

0.1 0.5 0.6 0.7 0.5 0.5 0.5 0.4 0.7 0.7 4.7 2.2 0.9 1.2 4.0 1.0 0.5 0.7 1.7 1.7 4.2 0.5 1.1 1.6 3.0 0.6 6.8 3.7 12.1 2.7 3.3 4.1 1.2

Notes: a FTS – Fourier transform spectroscopy; CRDS – cavity ring down spectroscopy; FC SDSDS – frequency comb referenced sub-Dopler saturation dip spectroscopy. b Nﬁt – number of lines of a given source included into the ﬁts. c Nexcl– number of lines of a given source excluded from the ﬁt. d νmin and νmax – minimal and maximal values of the wavenumbers in a given source. e RMS deviation of the ﬁt for a given source. f In the ﬁt for the weighting we used value 0.0002 cm 1.

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O.M. Lyulin, V.I. Perevalov / Journal of Quantitative Spectroscopy & Radiative Transfer 177 (2016) 59–74

60 50

Mean deviation RMS

Deviation, 10-3 cm-1

40 30 20 10 0 -10 -20 -30 -40 0

2000

4000

6000

8000

10000

-1

Band center, cm

Fig. 1. Mean deviations and RMS residuals deﬁned for a given band versus a band center.

the uncertainties of the line positions reported by the authors of the respective data sources. In a number of cases, the reported uncertainties were substituted by our estimated values, ﬁrst of all, because the authors of the publications give usually the uncertainties for the strong and unblended lines while other reported lines have considerably larger uncertainties for the line positions. In the process of ﬁtting we have removed 3981 lines from the ﬁle of the input data. These lines are outliers. Usually they are very weak. The respective residuals are randomly distributed versus the rotational quantum number for a given band. If we saw any smooth dependance of the residuals versus the rotational quantum number J, the corresponding lines were preserved in the ﬁle of the input data. In the cases of Refs. [25–27], we had the real recorded spectra. The selection has been done as a result of the analysis of these spectra. To ensure a good extrapolation ability of the derived set of the effective Hamiltonian parameters, in the ﬁnal ﬁt we have retained only parameters with standard deviation less

Table 3 Effective Hamiltonian parameters (in cm 1).

ω1 ω2 ω3 ω4 ω5 X11 X12 X13 X14 X15 X22 X23 X24 X25 X33 X34 X35 X44 X45 X55 g44 g45 g55 Y115 Y122 Y123 Y124 Y125 Y134 Y135 Y144 Y145 Y155 Y44 1 Y45 1 Y55 1 Y222 Y223 Y224 Y225 Y234 Y235 Y244 Y245

Notation

Value

3398.4749(8)a 1981.702(2) 3316.247(1) 608.9941(3) 729.2224(1) 25.5147(7) 11.216(2) 104.5600(6) 14.024(1) 10.916(2) 7.394(2) 5.6756(9) 12.498(1) 1.4466(6) 28.270(2) 9.287(2) 8.972(1) 3.0998(3) 2.4316(3) 2.4335(1) 0.7347(3) 6.5328(1) 3.53576(9) 0.0887(9) 0.141(1) 0.547(2) 0.114(1) 0.149(1) 0.219(2) 0.248(2) 0.1861(5) 0.067(1) 0.0494(3) 0.143(1) 0.260(1) 0.0301(1) 0.0084(4) 0.0461(8) 0.2553(9) 0.0845(6) 0.0424(7) 0.0741(6) 0.3975(8) 0.0898(3)

Y445 Y455 Y555 Y44 4 Y45 4 Y44 5 45 Z11 Z55 11 Z1244 Z1245 Z45 12 Z1445 Z1555 Z44 14 Z45 14 Z55 14 Z45 15 Z2444 Z2445 Z44 24 Z44 25 Z2244 Z2245 Z44 22 Z3444 Z3455 Z45 34 Z55 34 Z44 35 Z45 35 Z55 35 Z3344 Z3345 Z45 33 Z4444 Z4445 Z4455 Z4555 Z44 44 Z45 44 Z55 45 Z45 55 Z55 55 Z4445

Notation 0.1104(1) 0.0557(3) 0.00900(2) 0.0011(3) 0.0754(2) 0.00475(9) 0.0173(5) 0.0105(1) 0.0214(5) 0.0343(6) 0.0339(4) 0.0604(4) 0.00110(7) 0.0070(4) 0.0332(4) 0.0152(4) 0.0273(4) 0.0094(2) 0.0096(2) 0.00767(8) 0.0096(2) 0.0175(4) 0.0118(3) 0.0114(2) 0.0055(2) 0.0461(6) 0.0624(8) 0.0455(8) 0.051(1) 0.113(1) 0.0051(1) 0.0101(3) 0.0368(9) 0.0329(9) 0.004922(8) 0.00522(3) 0. 00947(3) 0.0038(1) 0.00241(7) 0.00192(5) 0.00074(3) 0.00031(3) 0.00078(2) 0.00190(5)

Be α1 α2 α3 α4 α5 γ11 γ12 γ13 γ14 γ15 γ22 γ23 γ24 γ25 γ33 γ34 γ35 γ44 γ45 γ55 γ44 γ45 γ55 ε344 ε455 ε45 4 ε44 5 ε45 5 ε55 5 De β1 β2 β3 β4 β5 δ33 δ35 δ44 δ45

Value 1.1766434(4) 0.68397(4) 10 2 0.61732(4) 10 2 0.58486(6) 10 2 0.13582(2) 10 2 0.22252(2) 10 2 0.365(3) 10 4 0.641(5) 10 4 1.824(3) 10 4 1.151(2) 10 4 0.249(2) 10 4 0.087(3) 10 4 0.839(5) 10 4 0.763(2) 10 4 0.560(1) 10 4 0.343(4) 10 4 0.478(4) 10 4 0.282(4) 10 4 0.0556(9) 10 4 0.211(2) 10 4 0.2605(6) 10 4 0.6279(9) 10 4 2.184(2) 10 4 1.081(1) 10 4 0.34(2) 10 5 0.162(6) 10 5 0.411(9) 10 5 0.534(9) 10 5 0.33(1) 10 5 0.164(5) 10 5 0.16222(2) 10 5 0.144(1) 10 7 0.052(2) 10 7 0.062(5) 10 7 0.348(1) 10 7 0.2524(6) 10 7 0.60(3) 10 8 0.52(4) 10 8 0.060(7) 10 8 0.50(1) 10 8

O.M. Lyulin, V.I. Perevalov / Journal of Quantitative Spectroscopy & Radiative Transfer 177 (2016) 59–74

65

Table 3 (continued ) Notation Y255 44 Y2 55 Y2 Y333 Y334 Y344 44 Y3 45 Y3 Y444

0.0624(1) 0.1317(4) 0.0267(1) 0.3945(4) 0.189(1) 0.0509(5) 0.0597(2) 0.079(2) 0.05137(6)

Value W22444 W22555 W24444 W25555 W34444 W35555 W44 444 W45555 W55 555

Notation

Value

0.00231(7) 0.00068(4) 0.00030(1) 0.00022(1) 0.00060(2) 0.000289(9) 0.000221(4) 0.00024(2) 0.000046(4)

Resonance interaction parameters ΔV Δℓb

Notation

0 0 0 0 0 2-2 r45 r145 r245 r345 r445 r545 J r45 r4445 r44 45 0 0 0 0 0 2 0 q4 q41 q42 q43 q44 q45 J q4 q423 0 0 0 0 0 0 2 q5 q51 q52 q54 q55 J q5 q515 0 0 0 2-2 0 0 K44/55 (K44/55)2 (K44/55)3 (K44/55)J (K44/55)14 (K44/55)23 0 0 0 2-2 2-2 ℓ K 44=55 ðℓ K 44=55 Þ1 ðℓ K 44=55 Þ5 0000004 0 0 0 0 0 4-2 0 0 0 0 0-2 4 0 0 0 2-2 0 2 0 1-1 1 1 1-1

0 1-1 1 1 1 1

ðℓ K 44=55 ÞJ u55 u44/5 u55/4 ℓ5 O44/55 K3/245 (K3/245)1 (K3/245)2 (K3/245)3 (K3/245)4 (K3/245)5 (K3/245)J (K3/245)44 (K3/245)45 (K3/245)55 O3/245 (O3/245)5

Value

ΔV Δℓb

Notation

Value

1.58859(5) 0.0405(5) 0.02616(4) 0.0748(5) 0.00610(8) 0.02408(6) 0. 4849(2) 10 4 0.129(2) 10 2 0.253(3) 10 2 0.131064(4) 10 2 0. 386(9) 10 5 0. 665(9) 10 5 0. 68(1) 10 5 0. 276(3) 10 5 1. 644(5) 10 5 1.015(3) 10 8 0. 55(2) 10 5 0.116497(4) 10 2 1.02(1) 10 5 0. 165(4) 10 5 2.561(3) 10 5 0. 979(3) 10 5 0. 985(3) 10 8 0. 067(7) 10 5 2.120(1) 0.0587(7) 0.112(2) 0. 397(5) 10 4 0.0367(4) 0.069(2) 0.6002(5)

1 1 0 0 2 0 0

2 0-2 2-2 0 0

K1/255 (K1/255)1 (K1/255)2 (K1/255)3 (K1/255)4 (K1/255)J O1/255 K1/244 (K1/244)1 (K1/244)3 (K1/244)4 O1/244 K14/35 (K14/35)1 (K14/35)2 (K14/35)3 (K14/35)J O14/35 K15/34 (K15/34)1 (K15/34)2 (K15/34)3 (K15/34)5 (K15/34)J O15/34 K11/33 (K11/33)2 (K11/33)4 (K11/33)5 (K11/33)J K1144/3355

1.807(1) 0.260(2) 0.0150(6) 0.300(5) 0.0806(5) 0. 115(3) 10 4 0.060(2) 10 4 2.585(5) 0.557(5) 0.456(2) 0.0833(7) 0.117(3) 10 4 4.148(2) 0.263(3) 0.137(1) 0.507(3) 0. 405(2) 10 4 0.163(1) 10 4 4.453(7) 0.847(4) 0.072(2) 0.310(4) 0.0334(4) 0. 433(5) 10 4 0.115(3) 10 4 25.7476(8) 0.074(3) 0.111(2) 0.043(1) 0. 585(3) 10 4 0.065(1)

0.1471(7) 0.00362(7) 0.217(2) 10 4 2.11(8) 10 10 0.78(1) 10 6 0.33(2) 10 6 0.173(3) 10 4 2.3582(4) 0.062(1) 0.0279(2) 0.0675(9) 0.01937(8) 0.0273(1) 0. 3125(5) 10 4 0.00102(2) 0.00161(8) 0.00073(5) 0.210(1) 10 4 0.0173(8) 10 4

1 1 0 0 2 0 2 1 1 0 2 0 0 0

1 1 0 2 0 2 0 1 0-1 1-1 1-1

1 0-1 1-1 1 1 1 0-1-1 1 1-1

1 0-1-1 1 1 1 2 0-2 0 0 0 0

0-1 1 1-3 1-1

K34/2555

0.0394(8)

-1-1 2-2 0 0 0

K33/1244

0.199(2)

1 -1 2 0-2 0 0

K33/1255

0 0 0 4-4 0 0 0 0 0 0 0 4-4 0-1 0 1 2 1 0 0-1 0 3 0 1 0

0-1 0 3 0 3-2 0-2 1 0 1 0 1 1-2 0 1 0 1 0

0-2 1 2-1 0 1 0-2 1 2-1 2 1 0-1 0 5-2 1 0 1-1-1 2 1 2-1

K4444/5555 r44/55 C2/455 (C2/455)5 Ca2/444 (Ca2/444)1 (Ca2/444)4 (Ca2/444)K Cb2/444 C22/35 C22/14 (C22/14)1 (C22/14)4 Ca225/344 Cb225/344 C255/44444 C23/1445

0.591(5) 0.02671(4) 0.000730(3) 0.21(1) 10 2 0. 034(2) 10 2 0.373(7) 10 2 0.144(3) 10 2 0.022(1) 10 2 0. 0124(4) 10 2 0.0247(9) 10 2 0.191(5) 10 2 0.877(8) 10 2 0.551(6) 10 2 0.415(3) 10 2 0.184(3) 10 2 0.0113(5) 10 4 0.74(5) 10 4 0.050(2) 10 2

Notations for the resonance interaction parameters presented in this table are deﬁned in Table 1 and they are linked to those given by Eq. (9) in the Δℓ4 Δℓ5 ΔK ¼ 0 ΔK ¼ 0 ΔK ¼ 1 ΔℓΔK ¼ 2 4 Δℓ5 ΔK a 0 following way: RΔℓ - K ΔV or ℓ K ΔV , RΔℓ - C Δℓ -OΔℓ - r, RΔℓ - u, the superscripts J and K outside the ΔV , RΔV ΔV , RΔV ¼ 0 ΔV ΔV ΔV ¼ 0 parentheses are used for the parameters describing the rotational dependence of an interaction parameter, the numerical subscripts outside the parentheses are used for the parameters describing the dependence of an interaction parameter on the principal vibrational quantum numbers. a Between parentheses, the conﬁdence interval (1 SD) is in the units of the last quoted digit. b ΔV Δℓ - Δv1Δv2Δv3Δv4Δv5Δℓ4Δℓ5.

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Table 4 Results of the global ﬁts of C2H2 line positions. Fits

Wavenumber range, cm 1

This work 50–9900 Brussels [30] 50–8900

Number of ﬁtted/excluded line Number of adjusted positions parameters

RMS, 10 3 cm 1 Weighted standard deviation

24991/3981 18415

3.65

237 396

2.84 1.07

Table 5 Comparison of the effective Hamiltonian parameters for C2H2 molecule obtained in this work to those from Ref. [30]. Vibrational

ω1 ω2 ω3 ω4 ω5 X11 X12 X13 X14 X15 X22 X23 X24 X25 X33 X34 X35 X44 X45 X55 g44 g45 g55 r45 r145 r245 r345 r445 r545 J r45

Rotational

TW

Ref. [30]

3398.4749(8) 1981.702(2) 3316.247(1) 608.9941(3) 729.2224(1) 25.5147(7) 11.216(2) 104.5600(7) 14.024(1) 10.916(2) 7.394(2) 5.6756(9) 12.498(1) 1.4466(6) 28.270(2) 9.287(2) 8.972(1) 3.0998(3) 2.4316(3) 2.4335(1) 0.7347(3) 6.5328(1) 3.53576(9) 1.58859(5) 0.0405(5) 0.02616(4) 0.0748(5) 0.00610(8) 0.02408(6) 0.4849(2) 10 4

3401.153(21) 1981.68287(8) 3313.204(21) 608.9901(7) 729.2068(3) 28.041(32) 11.229(2) 104.5551(20) 13.971(7) 10.749(5) 7.36696(7) 4.920(31) 12.502(1) 1.412(3) 25.125(32) 9.377(6) 9.144(6) 3.1095(6) 2.4208(4) 2.4222(3) 0.7420(8) 6.5403(5) 3.5396(3) 1.55970(1) 0.0316(3) 0.02201(2) 0.063(1) 0.0097(1) 0.0138(2) 0.4903(1) 10 4

Be α1 α2 α3 α4 α5 De β1 β2 β3 β4 β5 q4 q41 q42 q43 q44 q45 J q4 q5 q51 q52 q53 q54 q55 J q5

Resonance interaction parameters

TW

Ref. [30]

1.1766434(4) 0.68397(4) 10 2 0.61732(4) 10 2 0.58486(6) 10 2 0.13582(2) 10 2 0.22252(2) 10 2 0.16222(3) 10 5 0.144(1) 10 7 0.052(2) 10 7 0.062(5) 10 7 0.348(1) 10 7 0.2524(6) 10 7 0.131064(4) 10 2 0.386(9) 10 5 0. 704(9) 10 5 0. 665(9) 10 5 0. 276(3) 10 5 1.644(5) 10 5 1.015(3) 10 8 0.116497(4) 10 2 1.02(1) 10 5 0. 165(4) 10 5 2.561(3) 10 5 -0. 979(3) 10 5 0. 985(3) 10 8

1.17665(7) 0.68433(2) 10 2 0.61741(2) 10 2 0.58568(7) 10 2 0.135737(6) 10 2 0.222770(7) 10 2 0.162677(9) 10 5 0.1479(7) 10 7 0.049(3) 10 7 0.146(3) 10 7 0.3380(3) 10 7 0.2607(3) 10 7 0.130769(1) 10 2 -0. 742(8) 10 5 -0. 651(4) 10 5 -0. 898(2) 10 5 0. 230(2) 10 5 1.712(2) 10 5 0.987(1) 10 8 0.117462(1) 10 2 0.77(1) 10 5 0.064(2) 10 5 0.08(1) 10 5 2.734(5) 10 5 1.099(4) 10 5 0. 969(1) 10 8

K44/55 K3/245 K1/255 K1/244 K14/35 K15/34 K11/33 C2/455 Ca2/444 C22/35 C22/14

TW

Ref. [30]

2.120(1) 2.3582(4) 1.807(1) 2.585(5) 4.148(2) 4.453(7) 25.7476(8) 0.21(1) 10 2 0.373(7) 10 2 0.191(5) 10 2 0.877(8) 10 2

2.123(1) 2.28754(1) 1.918(2) 2.798(4) 4.064(2) 4.629(4) 30.139(9) 0.146(8) 10 2 0.181(3) 10 2 0.106(3) 10 2 0.188(3) 10 2

Note: The effective Hamiltonian parameters are given in cm 1. Some parameters from Ref. [30] are recalculated to the notations of the present work. Contrary to Ref. [30] we do not use the numerical factors in the expressions for the nondiagonal matrix elements, hence (r45)our ¼ 1/4(r45 )Ref. [30], (q4)our ¼1/4(q4)Ref. [30], (q5)our ¼1/4(q5)Ref. [30], (K44/55)our ¼ 1/4(K44/55)Ref. [30], (K3/245)our ¼1/8(K3/245)Ref. [30], (C)our ¼ 1/4 (C)Ref. [30], etc.

than 20% of the parameter value. The ﬁt reached the value 2.84 for the weighted standard deviation. The ﬁtted set of the 237 effective Hamiltonian parameters allowed reproducing 24,991 measured line positions with the root mean squares (RMS) deviation of 0.0037 cm 1. The mean deviations and RMS deviations deﬁned for a given band are plotted in Fig. 1 versus band center. As one can see from this ﬁgure, we have large residuals in the region between 5800 cm 1 and 8000 cm 1 where very weak bands recorded with CRDS spectrometer are situated. The accuracy of the line position determination for very weak bands is very low because of blending with considerably stronger bands. It should be emphasized that the largest residuals are for the bands which are represented by a few lines of one of the branches. These lines are not supported

by the associated combination differences. The set of the effective Hamiltonian parameters is given in Table 3. The comparison of the results of our ﬁt to that published earlier [30] is given in Table 4. Compared to Ref. [30] in our ﬁt we used additional input line positions obtained recently as a result of the analyses of the CRDS spectra in the 5851–6341 cm 1 [26] and 7244–7918 cm 1 [25] regions and the FTS spectra in the 7000–7500 cm 1 region [27]. We also extended the ﬁtting range from 50–8900 cm 1, as it was in the case of Ref. [30], up to 50–9900 cm 1. In our ﬁt we used a considerably smaller amount of adjustable parameters but we introduced several additional higher order resonance anharmonic and Coriolis interaction matrix elements. The lower order parameters obtained in two ﬁts are in good agreement, see Table 5. The

O.M. Lyulin, V.I. Perevalov / Journal of Quantitative Spectroscopy & Radiative Transfer 177 (2016) 59–74

67

Table 6 Comparison of some ﬁtted and transformed effective Hamiltonian parameters to those from Ref. [30]. Parameter

ω1

ω3

X11

X33

X13

K11/33

TW TW (transformed) Ref. [30]

3398.475 3401.225 3401.153

3316.247 3313.497 3313.204

25.515 28.265 28.041

28.270 25.520 25.125

104.56 104.56 104.555

102.990 120.556 120.556

exception is the case of the resonance Coriolis interaction constants. As it has been discussed in our paper dealing with the effective Hamiltonian for the CO2 molecule [53] because of different possible phase choices all signs of the Coriolis interaction parameters in the effective Hamiltonian can be changed simultaneously. Some higher order parameters of the effective Hamiltonian differ considerably in two ﬁts. There are several reasons for this. First of all, the sets of the input data for ﬁtting the effective Hamiltonian parameters differ considerably. Then the expansion of the effective Hamiltonian is truncated at different orders of the perturbation theory in these two ﬁts. But the main reason is the ambiguity of an effective Hamiltonian when it includes the resonance interaction terms in the explicit form. In Table 6 we give the comparison of the x11, x33, x13 and K11/33 parameters for which there exists the following approximate relation [54] 1 1 x11 x33 x13 K 11=33 : 4 4

ð10Þ

The Eq. (10) is written in the notations of Ref. [30]. As one can see from this table these relations are quite sensible for both sets of the effective Hamiltonian parameters. Moreover using the unitary transformation of the effective Hamiltonian ef f H~ ¼ eiS H ef f e iS

ð11Þ

with the generator of the form iS ¼ S11=33 ðða1þ Þ2 a23 ða3þ Þ2 a21 Þ;

ð12Þ

aiþ

and ai are the creation and anihilation operators where of the vibrational quantum with the harmonic frequency ωi and S11=33 is the real parameter of the generator, we can match two sets of the parameters. Indeed this transformation contributes to the parameters under discussion as follows

ω~ 1 ¼ ω1 þ S11=33 K 11=33 ;

ð13Þ

ω~ 3 ¼ ω3 S11=33 K 11=33 ;

ð14Þ

x~ 11 ¼ x11 þS11=33 K 11=33 ;

ð15Þ

x~ 33 ¼ x33 S11=33 K 11=33 ;

ð16Þ

K~ 11=33 ¼ K 11=33 þ8ðω3 ω1 ÞS11=33 :

ð17Þ

Tilde is used for the transformed parameters. Choosing the parameter S11=33 of the generator (11) in such a way to convert the value of our interaction parameter K 11=33 to the value from

Ref. [30] we contribute to the parameters ω1 ; ω3 ; x11 and x33 : As one can see from Table 6 two sets of the parameters after this transformation are very close to each other. In other words, Eq. (10) could not be used for the veriﬁcation of a ﬁtted set of the effective Hamiltonian parameters.

4. Effective dipole moment matrix elements The line intensities within the framework of the method of effective operators could be calculated using the eigenfunctions of the effective Hamiltonian f Ψ efPNJq ε¼

X 5V 1 þ 3V 2 þ 5V 3 þ V 4 þ V 5 ¼ P

J

C

V 1 V 2 V 3 V 4 V 5 ℓ4 ℓ5 V 1 V 2 V 3 V 4 V 5 ℓ 4 ℓ 5 J K ε

PNqε

ℓ4 ℓ5

ð18Þ J

V 1 V 2 V 3 V 4 V 5 ℓ4 ℓ5 C PNqε

where is the expansion coefﬁcient of an eigenfunction, P is the polyad number and N is the ranking index of an eigenvalue in a (P, q, ε, J) block. In this case, one has to use an effective dipole moment operator instead of the dipole moment operator. The former is obtained from the dipole moment operator by the same unitary transformation as the effective Hamiltonian from the vibration-rotation Hamiltonian. The application of this approach to the C2H2 line intensity calculations is presented in detail in our papers [16,17]. The intensity SP 0 N0 J 0 q0 ε0 ’PNJqε expressed as cm/molecule at temperature T (K) of an absorption line corresponding to the transition P 0 N0 J 0 q0 ε0 ’PNJqε is related to the vibration–rotation transition dipole moment squared W P 0 N0 J 0 q0 ε0 ’PNJqε by the well known equation: SP0 N0 J 0 q0 ε0 ’PNJqε

3 ν 0 00 0 0 hcE ¼ 8π Cg PNJqε P N J Qq ðεT’PNJqε exp kTPNJqε Þ 3hc

hcνP 0 N0 J 0 q0 ε0 ’PNJqε W P 0 N0 J 0 q0 ε0 ’PNJqε ; 1 exp kT

ð19Þ

where EPNJqε is the energy of the lower state, νP0 N0 J 0 q0 ε0 ’PNJqε is the wavenumber of a transition, Q (T) is the total internal partition function at temperature T, C is the isotopic abundance and g PNJqε is the nuclear spin statistical weight, c is the speed of the light in vacuum, h is the Planck constant and k is the Boltzmann constant. Within the framework of the method of effective operators the transition dipole moment squared can be presented by the following equation [16,17]

68

O.M. Lyulin, V.I. Perevalov / Journal of Quantitative Spectroscopy & Radiative Transfer 177 (2016) 59–74

X W P 0 N0 J 0 q0 ε0 ’PNJqε ¼ ð2J þ1Þ 5V þ3V þ 5V þ V þ V ¼ P 1 5 2 3 4 ℓ4 ℓ5

X

J

5ΔV 1 þ3ΔV 2 þ 5ΔV 3 þ ΔV 4 þ ΔV 5 ¼ ΔP

V 1 V 2 V 3 V 4 V 5 ℓ4 ℓ5 C PNqε

Δℓ4 Δℓ5

V 1 þ ΔV 1 V 2 þ ΔV 2 V 3 þΔV 3 V 4 þ ΔV 4 V 5 þ ΔV 5 ℓ4 þΔℓ4 ℓ5 þ Δℓ5 Δℓ4 Δℓ5 M P 0 N0 q0 ε0 ΔV qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Δℓ4 Δℓ5 f ΔV ðV; ℓ4 ; ℓ5 Þ 1 þ δℓ4 ; 0 δℓ5 ; 0 þδℓ4 þ Δℓ4 ; 0 δℓ5 þ Δℓ5 ; 0 2δℓ4 ; 0 δℓ5 ; 0 δℓ4 þ Δℓ4 ; 0 δℓ5 þ Δℓ5 ; 0 0

J C

92 8 > !> > > = < X ΔVΔℓ4 Δℓ5 X ΔVΔℓ4 Δℓ5 Δℓ Δℓ 5 4 ΔV κ a ð2ℓt þ Δℓt Þ þ F Vi þ ðJ; K Þ ΦΔJΔK ðJ; KÞ 1 þ ΔJ ΔK > > i t > > t i ; :

Because of the utilization of the Wang combinations of the basis functions in some cases Eq. (20) has additional terms (see Ref. [17] for details). In this equation δi;j is the Kronecker symbol. The functions ΦΔJ ΔK ðJ; K Þ for ΔK ¼ 0; 71 are the Clebsh–Gordan coefﬁcients: ΦΔJ ΔK ðJ; K Þ ¼ 1 ΔK JK J þ ΔJ K þ ΔK ; ð21Þ for ΔK ¼ 72 these functions are given in Refs. [16,17] and Δℓ Δℓ the vibrational functions f ΔV4 5 ðV; ℓ4 ; ℓ5 Þ can be calculated using the equations presented in Appendix of Ref. 4 Δℓ5 ðJ; K Þ are [17]. The Herman-Wallis type functions ΔV F Δℓ ΔJΔK also given in Refs. [16,17] and are repeated below in a

ð20Þ

simpliﬁed form:

ΔK ¼ 71 Q-branch ΔV F Δℓ4 Δℓ5 ðJ; K Þ ¼ bΔV Δℓ4 Δℓ5 J ΔJ ΔK

1 ΔV Δℓ4 Δℓ5 þ dJQ K ΔK þ J ðJ þ 1Þ; 2

ð22Þ P- and R-branches ΔV F Δℓ4 Δℓ5 ðJ; KÞ ¼ bΔV Δℓ4 Δℓ5 m þdΔV Δℓ4 Δℓ5 m2 ; J J ΔJ ΔK

ð23Þ

Table 7 Summary of the line intensity ﬁts. Series ΔP

Number of linesa

Jmax

Experimental uncertainty

χ

RMS (%)

5% Ref. [49] 5% Refs. [55, 56] 5% Refs. [48,50,59] 2% Refs. [57,58] 5% Refs. [41,42] 5% Ref. [44] 5% Ref. [43,60] 2% Ref. [5] 5–20% Ref. [19] 5% Ref. [20] 5% Ref. [21] 5–10% Ref. [22] 5–10% Ref. [22] 5–20% Ref. [26] 2% Ref. [61] 5% Ref. [45] 10% Ref. [23] 5–20% Ref. [26] 3–10% Ref. [27,46] 3–10% Ref. [47] 3–10% Ref. [47] 7–10 % Ref. [47]

0.86 0.82 0.60

4.27 3.62 2.87

2 4 6

0.7 0.46 0.57

3.14 1.36 2.70

18 8 6

0.61 0.65 0.64 0.4

3.03 2.59 4.34 6.73

10 11 15 37

0.69

5.05

26

0.73 0.47 0.67 0.48

4.58 1.36 2.06 3.35

18 2 3 4

0 1 2

117(117) 431(432) 989(1033)

1 5 16

30 38 45

3 4 5

497(519) 245(245) 645(769)

12 5 16

35 25 41

6 7 8 9

656(674) 428(444) 283(382) 1570(2468)

8 8 7 35

43 29 29 37

10

1191(1404)

34

37

11 12 13 15

1226(2544) 38(38) 91(91) 186(190)

24 1 2 4

35 23 31 33

a b

nb

Number of bands

Number of ﬁtted line intensities (number of observed line intensities). n is the number of adjusted parameters.

O.M. Lyulin, V.I. Perevalov / Journal of Quantitative Spectroscopy & Radiative Transfer 177 (2016) 59–74

where m ¼ J; J þ 1 for P- and R-branches, respectively.

ΔK ¼ 0; Δℓ4 ¼ 0; Δℓ5 ¼ 0 4 Δℓ5 The function ΔV F Δℓ ΔJΔK ¼ 0 ðJ; K Þ for P- and R-branches is given by the same Eq. (23). It should be noted that Qbranch is absent for ℓ4 ¼ ℓ5 ¼ 0 in this case. For ℓ4 a0 and 4 Δℓ5 ℓ5 a0 the function ΔV F Δℓ ΔJ ¼ 0 ΔK ¼ 0 ðJ; K Þ is given by the second term of Eq. (22).

4.1. Speciﬁc matrix elements Compared to the linear triatomic molecules, the linear tetratomic molecules have speciﬁc matrix elements with ΔK ¼ 0; Δℓ4 a 0; Δℓ5 a 0 [16,17]. Two cases have to be considered: (i) K ¼0, (ii) K a0. (i) K ¼0 4 Δℓ5 In this case, the function ΔV F Δℓ ΔJΔK ¼ 0 ðJ; K Þ is given by Eq. (23) for P- and R-branches. The Q-branch appears due to activation by the vibration–rotation interactions. The corresponding term is one order of magnitude smaller than those responsible for the P- and Rbranches. In this case the expression within the braces of Eq. (20) has to be substituted by the expression ΔV Δℓ4 Δℓ5

bJQ

1þ

X

κ i ΔV

Δℓ4 Δℓ5

Vi þ

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ X ΔV Δℓ Δℓ 4 5 at ð2ℓt þ Δℓt Þ JðJ þ 1Þ: t

i

ð24Þ (ii) K a0 4 Δℓ5 In this case, the function ΔV F Δℓ ΔJΔK ¼ 0 ðJ; K Þ is again given by Eq. (23) for the P- and R-branches, but for the Q-branch one has ! ΔV bΔℓ4 Δℓ5 JQ ΔV ΔV þ dJ F ΔJ ΔK ¼ 0 ðJ; K Þ ¼ J ðJ þ1Þ: ð25Þ K

ΔV Δℓ Δℓ

It should be emphasized that the bJQ 4 5 parameter ΔV Δℓ4 Δℓ5 has the same order of magnitude as the bJ

parameter. Therefore, in this particular case, a very strong 4 Δℓ5 dependence of the ΔV F Δℓ ΔJΔK ¼ 0 ðJ; K Þ function on the angular momentum quantum number J is observed. The parameters of the matrix elements of the effective 4 Δℓ5 4 −Δℓ5 dipole moment operator M Δℓ ¼ M −Δℓ , ΔV ΔV Δℓ4 Δℓ5 ΔV −Δℓ4 −Δℓ5 κΔV ¼ κ ð i ¼ 1; 2; 3; 4; 5 Þ, i i Δℓ4 Δℓ5 −Δℓ4 −Δℓ5 aΔV ¼ −aΔV ðt ¼ 4; 5Þ, t t ΔV Δℓ4 Δℓ5 ΔV −Δℓ4 −Δℓ5 ΔV Δℓ Δℓ5 ΔV −Δℓ4 −Δℓ5 bJ ¼ bJ , bJQ 4 ¼ −bJQ , ΔV Δℓ4 Δℓ5 ΔV−Δℓ4 −Δℓ5 ΔV Δℓ Δℓ5 ΔV−Δℓ4 −Δℓ5 dJ ¼ dJ , dJQ 4 ¼ dJQ involved in Eq. (20) and Eqs. (22)–(25) describe simultaneously the line intensities of the hot and cold bands belonging to the same ΔP series.

5. Line intensity ﬁts Using the approach described above, we performed the least-squares ﬁttings of the line intensities for ﬁfteen series of transitions: ΔP¼0–13 and 15 covering the 50–9900 cm 1 spectral range. The observed line intensities were gathered from the literature [5,19–23,26,27, 41–50,55–61]. The ﬁttings were performed separately for each spectral region deﬁned by the value of ΔP. The V 1 V 2 V 3 V 4 V 5 ℓ4 ℓ5 expansion coefﬁcients J C PNqε of the eigenfunctions were obtained from the global ﬁt of the effective Hamiltonian parameters to the observed line positions (see Section 2). The partition function Q ðTÞ¼414.03 at 296 K is taken from Ref. [62]. The summary of all ﬁts performed in this work is presented in Table 7. In all cases, the obtained sets of the effective dipole moment parameters reproduce the measured line intensities within their experimental uncertainties. In Fig. 2 we present the plot of the residuals of the line intensity ﬁts. The aim of the ﬁtting procedure is to minimize the value of the dimensionless weighted standard deviation, χ. The weighting has been performed using the published values of the relative uncertainties of the line intensity measurements. We use the same deﬁnitions of the dimensionless weighted standard deviation, the root mean

Mesured intensities

1E-16 1E-18 1E-20 1E-22 1E-24 1E-26 1E-28 Obs.-Calc.(%)

1E-30 40 20 0 -20 0

2000

4000

69

6000

8000

10000

Wavenumber, cm-1 Fig. 2. Bottom panel: residuals of the line intensity ﬁts. Upper panel: intensity diagram of the lines involved into the line intensity ﬁts.

70

O.M. Lyulin, V.I. Perevalov / Journal of Quantitative Spectroscopy & Radiative Transfer 177 (2016) 59–74

Table 8 Effective dipole moment parameters. Parametera ΔP¼ 0 M bJQ ΔP¼ 1 M κ5 bJ dJ ΔP¼ 2 M κ4 a4 bJ M M ΔP¼ 3 M κ4 bJ dJ M a4 bJ dJ dJQ M κ4 bJ dJ dJQ M M M M ΔP¼ 4 M κ4 bJ M bJ M bJ M ΔP¼ 5 M κ4 κ5 bJ M M ΔP¼ 6 M κ4 bJ M bJ M bJ dJQ M M ΔP¼ 7 M κ4 bJ M κ4 bJ M

ΔV4

ΔV5

Δℓ4

Δℓ5

Valueb

0 0

1 1

1 1

1 1

1 1

2.562(5) 10 2 0.247(7) 10 2

0 0 0 0

0 0 0 0

0 0 0 0

1 1 1 1

0 0 0 0

1 1 1 1

0 0 0 0 0 0

0 0 0 0 0 1

0 0 0 0 0 0

1 1 1 1 1 0

1 1 1 1 1 1

1 1 1 1 1 0

1 1 1 1 1 1

2.692(1) 10 2 0.0281(7) 0.0048(1) 0.131(2) 10 2 0.371(2) 10 4 1.074(2) 10 2

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 1 1 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 2 2 2 2 2 2 2 2 2 2 1 1 0 0

3 3 3 3 1 1 1 1 1 1 1 1 1 1 1 1 3 3

0 0 0 0 2 2 2 2 2 0 0 0 0 0 1 1 2 2

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3

5.989(9) 10 4 0.022(3) 0.181(6) 10 2 0.19(3) 10 4 2.63(2) 10 4 0.197(5) 1.28(2) 10 2 5.9(1) 10 4 4.3(4) 10 4 6.03(1) 10 4 0.54(2) 0.47(1) 10 2 2.98(8) 10 4 1.0(1) 10 4 0.1884(5) 10 2 9.52(3) 10 4 0.207(7) 10 4 6.3(4) 10 6

0 0 0 0 0 1 1 0

1 1 1 0 0 0 0 0

0 0 0 1 1 0 0 0

0 0 0 1 1 0 0 3

1 1 1 0 0 1 1 1

0 0 0 1 1 0 0 1

1 1 1 0 0 1 1 1

0.2401(2) 10 2 0.194(4) 0.553(8) 10 2 1.0464(8) 10 2 0.360(6) 10 2 1.129(1) 10 2 0.401(8) 10 2 1.637(2) 10 4

0 0 0 0 0 1

0 0 0 0 1 0

1 1 1 1 0 0

0 0 0 0 1 1

0 0 0 0 1 1

0 0 0 0 1 1

0 0 0 0 1 1

9.050(4) 10 2 0.028(1) 0.012(1) 0.161(2) 10 2 4.2(1) 10 4 3.7(5) 10 4

1 1 1 0 0 0 0 0 0 0

0 0 0 0 0 1 1 1 1 0

0 0 0 1 1 0 0 0 0 1

0 0 0 1 1 0 0 0 2 1

1 1 1 0 0 3 3 3 1 2

0 0 0 1 1 0 0 0 2 1

1 1 1 0 0 1 1 1 1 2

0.4939(5) 10 2 0.018(2) 0.698(4) 10 2 0.4390(6) 10 2 0.856(5) 10 2 3.345(4) 10 4 0.768(7) 10 2 1.36(5) 10 4 1.09(2) 10 4 1.4(1) 10 4

1 1 1 0 0 0 0

0 0 0 0 0 0 0

0 0 0 1 1 1 1

1 1 1 2 2 2 0

1 1 1 0 0 0 2

1 1 1 0 0 0 0

1 1 1 0 0 0 0

3.898(5) 10 4 0.085(3) 0.499(7) 10 2 5.575(6) 10 4 0.034(2) -0.256(6) 10 2 4.843(5) 10 4

ΔV1

ΔV2

ΔV3

0 0

0 0

0 0 0 0

0.1579(2) 0.013(2) 0.072 (5) 10 2 0.47(3) 10 4

O.M. Lyulin, V.I. Perevalov / Journal of Quantitative Spectroscopy & Radiative Transfer 177 (2016) 59–74

71

Table 8 (continued ) Parametera

ΔV1

ΔV2

ΔV3

ΔV4

ΔV5

Δℓ4

Δℓ5

Valueb

bJ M κ4 bJ ΔP¼ 8 M bJ M bJ M bJ M M dJ dJQ M M bJ M bJ ΔP¼ 9 M κ4 bJ M M M κ4 bJ M bJ M κ4 κ5 a5 bJ M M bJ M M M bJ M M bJ M κ4 M bJ M bJ M κ5 a4 a5 bJ M ΔP¼ 10 M κ4 κ5 bJ M M κ5 bJ dJ M M

0 0 0 0

0 1 1 1

1 0 0 0

0 3 3 3

2 1 1 1

0 1 1 1

0 1 1 1

0.522(7) 10 2 0.137(1) 10 4 0.23(2) 0.686) 10 2

0 0 0 0 0 0 0 0 0 0 1 0 0 1 1

1 1 2 2 1 1 1 0 0 0 0 1 1 0 0

1 1 0 0 0 0 0 1 1 1 0 0 0 0 0

0 0 1 1 4 4 4 3 3 3 2 2 2 0 0

0 0 1 1 1 1 1 0 0 0 1 3 3 3 3

0 0 1 1 2 2 0 1 1 1 0 2 2 0 0

0 0 1 1 1 1 1 0 0 0 1 1 1 1 1

2.12(1) 10 4 0.65(3) 10 2 0.953(3) 10 4 0.32(2) 10 2 9.7(6) 10 6 1.4(1) 10 2 0.12(1) 10 4 3.2(3) 10 6 0.24(3) 10 2 0.35(5) 10 2 1.319(8) 10 4 0.142(2) 10 4 0.84(7) 10 2 0.230(1) 10 4 0.20(4) 10 2

0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 2 2 2 0 0 0 0 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 2 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 0 0 0 0 1 1 1 1 1 1

1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 2 2 2 2 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0

1 1 1 1 1 2 2 2 2 2 1 1 1 1 1 1 0 0 2 2 0 0 2 0 0 0 0 1 1 3 3 0 0 0 0 0 2

0 0 0 0 2 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1 1 1 4 4 3 3 3 3 1 1 1 1 1 1 1 1

1 1 1 1 1 2 2 2 0 0 1 1 1 1 1 1 0 0 2 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0

0 0 0 2 0 1 1 1 1 1 0 0 0 0 0 2 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1

1.321(3) 10 4 0.29(1) 0.41(2) 10 2 0.06(1) 10 4 0.442(8) 10 4 0.331(1) 10 4 0.83(2) 0.47(2) 10 2 0.180(3) 10 4 0.72(7) 10 2 5.60(3) 10 4 0.028(3) 0.247(4) 0.142(3) 0.555(9) 10 2 0.81(1) 10 4 6.281(9) 10 4 0.51(2) 10 2 0.101(1) 10 4 0.262(5) 10 4 1.603(6) 10 4 0.69(3) 10 2 8.2(2) 10 6 0.993(6) 10 6 1.14(4) 10 2 1.84(1) 10 6 1.9(1) 0.92(1) 10 6 1.7(1) 10 2 0.1003(6) 10 4 0.57(5) 10 2 8.0(7) 10 6 0.18(3) 0.99(2) 0.20(2) 5.3(5) 10 2 0.407(8) 10 4

1 1 1 1 1 1 1 1 1 0 0

0 0 0 0 0 1 1 1 1 1 1

1 1 1 1 1 0 0 0 0 1 1

0 0 0 0 0 1 1 1 1 2 0

0 0 0 0 0 1 1 1 1 0 2

0 0 0 0 2 1 1 1 1 0 0

0 0 0 0 2 1 1 1 1 0 0

1.0636(9) 10 2 0.1186(8) 0.031(1) 0.023(2) 10 2 1.96(9) 10 4 7.797(7) 10 4 0.077(2) 0.029(3) 10 2 0.52(2) 10 4 2.066(7) 10 4 2.096(6) 10 4

72

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Table 8 (continued ) Parametera

ΔV1

ΔV2

ΔV3

ΔV4

ΔV5

Δℓ4

Δℓ5

Valueb

dJ M M M κ4 κ5 a4 a5 M dJ M κ5 M κ4 a4 ΔP¼ 11 M κ4 bJ M M M κ4 M M κ4 κ5 bJ M M M κ4 κ5 bJ ΔP¼ 12 M bJ ΔP¼ 13 M M dJ ΔP¼ 15 M κ4 M M

0 2 2 0 0 0 0 0 0 0 0 0 0 0 0

1 0 0 0 0 0 0 0 0 0 2 2 2 2 2

1 0 0 2 2 2 2 2 2 2 0 0 0 0 0

0 1 1 1 1 1 1 1 1 1 3 3 1 1 1

2 1 1 1 1 1 1 1 1 1 1 1 3 3 3

0 1 1 1 1 1 1 1 1 1 1 1 1 1 1

0 1 1 1 1 1 1 1 1 1 1 1 1 1 1

0.78(8) 10 4 1.12(2) 10 4 4.354(6) 10 4 0.64(3) 10 4 0.50(5) 0.32(2) 0.21(2) 0.08(1) 0.1407(2) 10 2 0.28(4) 10 4 0.1146(6) 10 4 1.03(5) 1.06(2) 10 4 0.347(9) 0.680(6)

0 0 0 0 1 0 0 0 1 1 1 1 1 1 2 2 2 2

0 0 0 1 1 2 2 3 0 0 0 0 1 1 0 0 0 0

2 2 2 1 0 1 1 0 1 1 1 1 0 0 0 0 0 0

0 0 0 1 2 0 0 1 1 1 1 1 0 2 0 0 0 0

1 1 1 2 1 0 0 1 0 0 0 0 3 1 1 1 1 1

0 0 0 1 0 0 0 1 1 1 1 1 0 2 0 0 0 0

1 1 1 0 1 0 0 1 0 0 0 0 1 1 1 1 1 1

3.370(4) 10 4 0.028(6) 0.996(7) 10 2 1.5(3) 10 6 2.4(3) 10 6 4.43(1) 10 4 0.043(8) 3.9(2) 10 6 3.334(4) 10 4 0.040(5) 0.033(6) 1.138(9) 10 2 3.1(1) 10 6 0.9(2) 10 6 4.121(3) 10 4 0.044(2) 0.022(2) 0.593(4) 10 2

1 1

0 0

1 1

2 2

0 0

0 0

0 0

1.237(2) 10 4 0.11(1) 10 2

1 1 1

1 2 2

1 0 0

0 1 1

0 1 1

0 1 1

0 1 1

6.756(8) 10 4 0.869(1) 10 4 0.84(4) 10 4

0 0 1 2

0 0 1 0

3 3 1 1

0 0 2 0

0 0 0 0

0 0 0 0

0 0 0 0

5.62(1) 10 4 0.029(5) 0.455(9) 10 4 2.60(2) 10 4

a b

Parameters M are given in Debye while the other parameters are dimensionless. Conﬁdence intervals (1 SD, in unit of the last quoted digit) are given between parenthesis.

squares of the residuals (RMS), and mean residual (MR) for a given band as in our paper [17]. For all ﬁts the ﬁtted sets of the effective dipole moment parameters are presented in Table 8.

6. Discussion and conclusions In this paper we present the results of the global modeling of the high resolution spectra of the acetylene molecule in the 50–9900 cm 1 region. The modeling has been performed within the framework of the method of effective operators. The polyad model of the effective Hamiltonian has been used. We reached nearly experimental accuracy of the reproduction of the observed line

positions. The line intensities are reproduced within their experimental uncertainties. We have found that the polyad model of the effective Hamiltonian works quite well in the region of the vibration–rotation energy levels from 0 to 11,000 cm 1. The same conclusion has been done by the Brussels team [30] which has performed the global ﬁtting of the line positions of this molecule in the wavenumber range 50–8900 cm 1. The obtained sets of the effective Hamiltonian and the effective dipole moment parameters will be used in the nearest future for the generation of the high temperature database for the acetylene molecule. Compared to Ref. [30] in the global ﬁt of the line positions we extended the wavenumber region from 50– 8900 cm 1 to 50–9900 cm 1 and considerably enlarged the number of the input line positions. Nevertheless we

O.M. Lyulin, V.I. Perevalov / Journal of Quantitative Spectroscopy & Radiative Transfer 177 (2016) 59–74

use a considerably smaller amount of the adjustable effective Hamiltonian parameters. The weighted standard deviation of the ﬁt is slightly better in Ref. [30] than in our case. To ensure good extrapolation abilities of the effective Hamiltonian parameters we decided to restrict the number of the higher-order parameters. The accuracy of the line position reproduction we reached is quite enough for the high temperature applications, in which the low resolution spectra are usually used. We also performed the global modeling of the line intensities in the 50– 9900 cm 1 wavenumber region that was not done by the Brussels team.

Acknowledgment O.M. Lyulin acknowledges the support in the frame of the program of development of competitiveness of the National Research Tomsk State University (Russia).

References [1] Ridgway ST, Hall DNB, Kleinmann SG, Weinberger DA, Wojslaw RS. Circumstellar acetylene in the infrared spectrum of IRCþ 101 216. Nature 1976;264:345. [2] Tinetti G, Beaulieu JP, Henning T, Meyer M, Micela G, Ribas I, et al. EChO: exoplanet characterisation observatory. Exp Astron 2012;34: 311–53. [3] Amyay B, Robert S, Herman M, Fayt A, Raghavendra B, Moudens A, et al. Vibration–rotation pattern in acetylene. II. Introduction of Coriolis coupling in the global model and analysis of emission spectra of hot acetylene around 3 mm. J Chem Phys 2009;131: 114301. [4] Moudens A, Georges R, Benidar A, Amyay B, Herman M, Fayt A, et al. Emission spectroscopy from optically thick laboratory acetylene samples at high temperatures. J Quant Spectrosc Radiat Transf 2011;112:540–9. [5] Vander Auwera J, Hurtmans D, Carleer M, Herman M. The ν3 fundamental in C2H2. J Mol Spectrosc 1993;157:337–57. [6] Perevalov VI, Lobodenko EI, Teffo JL. Reduced effective Hamiltonian for global ﬁtting of C2H2 rovibrational lines. In: Proceedings of XIIth symposium and school on high-resolutions molecular spectroscopy. St. Petersburg (Russian Federation). SPIE, vol. 3090, 1997. p. 143–9. [7] Kellman ME, Chen G. Approximate constants of motion and energy transfer pathways in highly excited acetylene. J Chem Phys 1991;95: 8671–2. [8] Pliva J. Molecular constants for the bending modes of acetylene 12 C2H2. J Mol Spectrosc 1972;44:165–82. [9] Hietanen J. ℓ-Resonance effects in the hot bands 3ν5 2ν5, (ν4 þ2ν5) (ν4 þ ν5) and (2ν4 þ ν5) 2ν4 of acetylene. Mol Phys 1983;49:1029–38. [10] Huet TR, Herman M, Johns JWC. The bending vibrational levels in þ C2D2 (X~ 1 Σg ). J Chem Phys 1991;94:3407–14. [11] Perevalov VI, Sulakshina ON. Reduced effective vibrationalrotational Hamiltonian for bending vibrational levels of acetylene molecule. In: Proceedings of XIth symposium and school on highresolutions molecular spectroscopy. Moscow (Russian Federation). SPIE, vol. 2205, 1993. p. 182–7. [12] Abbouti Temsamani M, Herman M. The vibrational energy levels in acetylene 12C2H2: Towards a regular pattern at higher energies. J Chem Phys 1995;102:6371–84. [13] Lyulin OM, Perevalov VI, Tashkun SA, Teffo JL. Global ﬁtting of the vibrational-rotational line positions of acetylene molecule. In: Proceedings of XIIIth symposium and school on high-resolutions molecular spectroscopy. Tomsk (Russian Federation). SPIE, vol. 4063, 2000. p. 126–33. [14] Lyulin OM, Perevalov VI, Teffo JL. Global ﬁtting of the vibrationalrotational line positions of acetylene molecule in the far and middle infrared regions. In: Proceedings of XIVth symposium on highresolutions molecular spectroscopy. Krasnoyarsk (Russian Federation). SPIE, vol. 5311, 2004. p. 134–43.

73

[15] Perevalov VI, Tashkun SA, Lyulin OM, Teffo JL. Global modelling of high-resolution spectra of linear molecules CO2, N2O and C2H2. In: Proceedings of the NATO advanced research workshop on remote sensing of the atmosphere for environmental security. Rabat (Morocco), 2005. p. 139–59. [16] Perevalov VI, Lyulin OM, Teffo JL. Global treatment of line intensities of vibrational-rotational transitions of acetylene molecule. Approach and design formulas. Atmos Ocean Opt 2001;14:730–8. [17] Perevalov VI, Lyulin OM, Jacquemart D, Claveau C, Teffo JL, Dana V, Mandin JY, Valentin A. Global ﬁtting of line intensities of acetylene molecule in the infrared using the effective operator approach. J Mol Spectrosc 2003;218:180–9. [18] Lyulin OM, Perevalov VI. Line intensities of vibration-rotational transitions of acetylene molecule in the 1.5-μm region. Atmos Ocean Opt 2004;17:485–8. [19] Lyulin OM, Perevalov VI, Mandin JY, Dana V, Jacquemart D, RégaliaJarlot L, et al. Line intensities of acetylene in the 3-μm region: new measurements of weak hot bands and global ﬁtting. J Quant Spectrosc Radiat Transf 2006;97:81–98. [20] Lyulin OM, Perevalov VI, Mandin JY, Dana V, Gueye F, Thomas X, et al. Line intensities of acetylene: measurements in the 2.5-μm spectral region and global modeling in the ΔP ¼ 4 and 6 series. J Quant Spectrosc Radiat Transf 2007;103:496–523. [21] Lyulin OM, Perevalov VI, Gueye F, Mandin JY, Dana V, Thomas X, et al. Line positions and intensities of acetylene in the 2.2-μm region. J Quant Spectrosc Radiat Transf 2007;104:133–54. [22] Lyulin OM, Jacquemart D, Lacome N, Perevalov VI, Mandin JY. Line parameters of acetylene in the 1.9 and 1.7 μm spectral regions. J Quant Spectrosc Radiat Transf 2008;109:1856–74. [23] Lyulin OM, Perevalov VI, Tran H, Mandin JY, Dana V, Régalia-Jarlot L, et al. Line intensities of acetylene: new measurements in the 1.5μm spectral region and global approach in the ΔP¼ 10 series. J Quant Spectrosc Radiat Transf 2009;110:1815–24. [24] Lyulin OM, Perevalov VI. Effective dipole moment parameters of 12 C2H2 for the 100, 7.7, 1.4, 1.3, 1.2 and 1.0 μm regions. J Mol Spectrosc 2011;266:75–80. [25] Lyulin OM, Campargue A, Mondelain D, Kassi S. The absorption spectrum of acetylene by CRDS between 7244 and 7918 cm 1. J Quant Spectrosc Radiat Transf 2013;130:327–34. [26] Lyulin OM, Mondelain D, Beguier S, Kassi S, Vander Auwera J, Campargue A. High-sensitivity CRDS absorption spectrum of acetylene between 5851 and 6341 cm 1. Mol Phys 2014;112:2433–44. [27] Lyulin OM, Vander Auwera J, Campargue A. The Fourier transform absorption spectrum of acetylene between 7000 and 7500 cm 1. J Quant Spectrosc Radiat Transf 2015;160:85–93. [28] El Idrissi MI, Liévin J, Campargue A, Herman M. The vibrational energy pattern in acetylene (IV): updated global vibrational constants for 12C2H2. J Chem Phys 1999;110:2074–86. [29] Robert S, Herman M, Fayt A, Campargue A, Kassi S, Liu A, Wang L, Di Lonardo G, Fusina L. Acetylene, 12C2H2: new CRDS data and global vibration-rotation analysis up to 8600 cm 1. Mol Phys 2008;106: 2581–605. [30] Amyay B, Herman M, Fayt A, Campargue A, Kassi S. Acetylene, 12 C2H2: reﬁned analysis of CRDS spectra around 1.52 μm. J Mol Spectrosc 2011;267:80–91. [31] Rinsland CP, Baldacci A, Rao KN. Acetylene bands observed in carbon stars: a laboratory study and an illustrative example of its application to IRC þ10216. Astrophys J Suppl Ser 1982;49:487–513. [32] D'Cunha R, Sarma YA, Guelachvili G, Farrenq R, Kou Q, Malathy Devi V, et al. Analysis of the high-resolution spectrum of acetylene in the 2.4 mm region. J Mol Spectrosc 1991;148:213–25. [33] Kabbadj Y, Herman M, Di Lonardo G, Fusina L, Johns JWC. The bending energy levels of C2H2. J Mol Spectrosc 1991;150:535–65. [34] D'Cunha R, Sarma YA, Job VA, Guelachvili G, Rao KN. Fermi coupling and l-type resonance effect in the hot bands of acetylene: the 2650 cm 1 region. J Mol Spectrosc 1993;157:358–68. [35] Kou Q, Guelachvili G, Abbouti Temsamani M, Herman M. The absorption spectrum of C2H2 around ν1 þ ν3: energy standards in the 1.5 mm region and vibrational clustering. Can J Phys 1994;72:1241–50. [36] Sarma YA, D’Cunha R, Gelachvili G, Farrenq R, Rao KN. Strech-bend levels of acetylene: Analysis of the hot bands in the 3800 cm 1 region. J Mol Spectrosc 1995;173:561–73. [37] Sarma YA, D’Cunha R, Gelachvili G, Farrenq R, Malathy Devi V, Benner DC, et al. Strech-bend levels of acetylene: analysis of the hot bands in the 3300 cm 1 region. J Mol Spectrosc 1995;173:574–84. [38] Keppler KA, Mellau G, Klee S, Winnewisser BP, Winnewisser M, Pliva J, et al. Precision measurements of acetylene spectra at 1.4–1.7 mm recorded with 352.5-m pathlength. J Mol Spectrosc 1996;175: 411–20.

74

O.M. Lyulin, V.I. Perevalov / Journal of Quantitative Spectroscopy & Radiative Transfer 177 (2016) 59–74

[39] Henningsen J, Sørensen GO. On modeling the overtone bands of acetylene in the 1500 nm region. Abstracts of the Seventeenth Colloquium on High Resolution Molecular Spectroscopy. University of Nijmegen, 9–13 September 2001, poster L17. p. 267. [40] Vander Auwera J, El Hachtouki, Brown L. Absolute line wavenumbers in the near infrared: 12C2H2 and 12C16O2. Mol Phys 2002;100:3563–76. [41] Jacquemart D, Mandin JY, Dana V, Régalia-Jarlot L, Thomas X, Von der Heyden P. Multispectrum ﬁtting measurements of line parameters for 5-μm cold bands of acetylene. J Quant Spectrosc Radiat Transf 2002;75:397–422. [42] Jacquemart D, Mandin JY, Dana V, Régalia-Jarlot L, Plateaux JJ, Décatoire D, Rothman LS. The spectrum of acetylene in the 5-mm region from new line-parameter measurements. J Quant Spectrosc Radiat Transf 2003;76:237–67. [43] Mandin JY, Jacquemart D, Dana V, Régalia-Jarlot L, Barbe A. Line intensities of acetylene at 3 μm. J Quant Spectrosc Radiat Transf 2005;92:239–60. [44] Jacquemart D, Lacome N, Mandin JY, Dana V, Lyulin OM, Perevalov VI. Multispectrum ﬁtting of line parameters for 12C2H2 in the 3.8-μm spectral region. J Quant Spectrosc Radiat Transf 2007;103:478–95. [45] Tran H, Mandin JY, Dana V, Régalia-Jarlot L, Thomas X, Von der Heyden P. Line intensities in the 1.5-μm spectral region of acetylene. J Quant Spectrosc Radiat Transf 2007;108:342–62. [46] Jacquemart D, Lacome N, Mandin JY, Dana V, Tran H, Gueye FK, et al. The IR spectrum of 12C2H2: line intensity measurements in the 1.4μm region and update of the databases. J Quant Spectrosc Radiat Transf 2009;110:717–32. [47] Jacquemart D, Lacome N, Mandin JY. Line intensities of 12C2H2 in the 1.3, 1.2 and 1 μm regions. J Quant Spectrosc Radiat Transf 2009;110:733–42. [48] Gomez L, Jacquemart D, Lacome N, Mandin JY. Line intensities of 12 C2H2 in the 7.7 μm spectral region. J Quant Spectrosc Radiat Transf 2009;110:2102–14. [49] Jacquemart D, Gomez L, Lacome N, Mandin JY, Pirali, Roy P. Measurements of absolute line intensities in the ν5–ν4 band of 12C2H2 using SOLEIL synchrotron far infrared AILES beamline. J Quant Spectrosc Radiat Transf 2010;111:1223–33. [50] Gomez L, Jacquemart D, Lacome N, Mandin JY. New line intensity measurements for 12C2H2 around 7.7 μm and HITRAN format line

[51]

[52]

[53]

[54]

[55] [56]

[57]

[58]

[59]

[60]

[61] [62]

list for applications. J Quant Spectrosc Radiat Transf 2010;111: 2256–64. Amyay B, Herman M, Fayt A, Fusina L, Predoi-Cross A. High resolution FTIR investigation of 12C2H2 in the FIR spectral range using synchrotron radiation. Chem Phys Lett 2010;491:17–9. Twagirayezu S, Cich MJ, Sears TJ, McRaven CP, Hall GE. Frequencycomb referenced spectroscopy of ν4- and ν5-excited hot bands in the 1.5 mm spectrum of C2H2. J Mol Spectrosc 2015;316:64–71. Teffo JL, Lyulin OM, Perevalov VI, Lobodenko EI. Application of the effective operator approach to the calculation of 12C16O2 line intensities. J Mol Spectrosc 1998;187:28–41. Lehmann KK. On the relation of Child and Lawton's harmonically coupled anharmonic–oscillator model and Darling–Dennison coupling. J Chem Phys 1983;79:1098. Mandin JY, Dana V, Claveau C. Line intensities in the ν5 band of acetylene 12C2H2. J Quant Spectrosc Radiat Transf 2000;67:429–46. Jacquemart D, Claveau C, Mandin JY, Dana V. Line intensities of hot bands in the 13.6 μm spectral region of acetylene 12C2H2. J Quant Spectrosc Radiat Transf 2001;69:81–101. Vander Auwera J. Absolute intensities measurements in the ν4 þ ν5 band of 12C2H2: analysis of Herman-Wallis effects and forbidden transitions. J Mol Spectrosc 2000;201:143–50. Lepere M, Blanquet G, Walrand J, Bouanich JP, Herman M, Vander Auwera J. Self-broadening coefﬁcients and absolute line intensities in the ν4 þ ν5 band of acetylene. J Mol Spectrosc 2007;242:25–30. Podolske JR, Lawenstein M, Varanasi P. Diode laser line strength measurements of the (ν4 þ ν5)0 band of 12C2H2. J Mol Spectrosc 1984;107:241–9. Jacquemart D, Mandin JY, Dana V, Claveau C, Vander Auwera J, Herman M, Rothman LS, Régalia-Jarlot L, Barbe A. The IR acetylene spectrum in HITRAN: update and new results. J Quant Spectrosc Radiat Transf 2003;82:363–82. Hachtouki El, Vander Auwera J. Absolute line intensities in acetylene: the1.5 μm region. J Mol Spectrosc 2002;216:355–62. Fischer J, Gamache RR, Goldman A, Rothman LS, Perrin A. Total internal partition sums for molecular species in the 2000 edition of the HITRAN database. J Quant Spectrosc Radiat Transf 2003;82: 401–12.

Global modeling of vibration-rotation spectra of the acetylene molecule

Global modeling of vibration-rotation spectra of the acetylene molecule

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