Global modeling of the 14N216O line positions within the framework of the non-polyad model of effective Hamiltonian

Journal of Quantitative Spectroscopy & Radiative Transfer 231 (2019) 88–101

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Global modeling of the 14 N2 16 O line positions within the framework of the non-polyad model of effective Hamiltonian S.A. Tashkun a,b,∗ a

Laboratory of Theoretical Spectroscopy, V.E. Zuev Institute of Atmospheric Optics, Siberian Branch, Russian Academy of Sciences, Academician Zuev square 1, 634055 Tomsk, Russia b Climate and Environmental Physics Laboratory, Ural Federal University, 19, Mira av., 620002 Yekaterinburg, Russia

a r t i c l e

i n f o

Article history: Received 23 February 2019 Revised 17 April 2019 Accepted 17 April 2019 Available online 18 April 2019 Keywords: N2 O Rotation-vibration HITRAN Line positions Effective Hamiltonian

a b s t r a c t The global modeling of 14 N2 16 O line positions in the 0.8–14,917 cm−1 region has been performed using the non-polyad model of effective Hamiltonian. The effective Hamiltonian parameters were fitted to the measured line positions collected from the literature. The dimensionless weighted standard deviation of the fit is 1.71. The fitted set of 195 effective Hamiltonian parameters allowed reproduction 56,888 measured line positions with an RMS value of 0.006 cm−1 . Comparisons of the calculated line positions based on the fitted model with HITRAN2016 and a recent line list by Bertin et al. (J Quant Spectrosc Radiat Transf 2019;229:40–9) are presented and discussed. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction Nitrous oxide, N2 O, is a minor constituent of the Earth atmosphere but it plays an important role in atmospheric physics and chemistry. Being a greenhouse gas, it contributes to the atmospheric radiation balance. Through the atmospheric chemistry processes, N2 O participates in the ozone layer depletion. Also, nitrous oxide is one of the products of the burning of the organic fuels in the air [1]. For the calculation of the radiation balance inside an internal-combustion engine or jet propulsion, it is necessary to know the high-temperature spectrum of this molecule [2]. N2 O is one of the best-studied molecules concerning its molecular rovibrational spectra. Up to now, more than 60,0 0 0 measured high-resolution rovibrational transition positions in the ground electronic state of the principal isotopologue 14 N2 16 O have been published [3–66]. These measurements cover the 0.8–14,917 cm−1 spectral range and have measurement uncertainties ranging from 10−7 (3KHz) [13] to 0.05 cm−1 [4]. Most of the data come from laboratory Fourier transform and cavity ring-down measurements. The highest measured J value is 105 (the R 104 line of the ν 1 +ν 3 band [26]). The value of the highest energy level included in these data is 15,017 cm−1 (the ν 1 =0, ν 2 =0, ν 3 =7, J = 14 state [33]). This

∗ Corresponding author at: Laboratory of Theoretical Spectroscopy, V.E. Zuev Institute of Atmospheric Optics, Siberian Branch, Russian Academy of Sciences, Academician Zuev square 1, 634055 Tomsk, Russia. E-mail address: [email protected]

https://doi.org/10.1016/j.jqsrt.2019.04.023 0022-4073/© 2019 Elsevier Ltd. All rights reserved.

value is half of the ground electronic state dissociation energy, which is ∼30,300 cm−1 [67]. Theoretical modeling of these data is based on two approaches. The first approach uses the potential energy surface (PES) to model the Coulomb interaction of electrons with nuclei in the framework of the Bohr-Oppenheimer approximation. Energy levels are eigenvalues of the molecular Hamiltonian. The second uses the effective Hamiltonian approach. Ideally, a theoretical model based on a PES should describe experimental data with the accuracies compatible with experimental uncertainties. It is also desirable to model energy levels above 10,0 0 0 cm−1 with a root-mean-square error (RMS) ∼ 0.01 cm−1 or better. It will allow identifying new lines in laboratory spectra. Published PESs, however, do not allow these requirements to be met. The existing ab initio PESs provide residuals of the order 1 cm−1 even for the fundamental vibrational bands. Martin et al. [68] published an ab initio quartic force field which reproduces the fundamental experimental frequencies with an accuracy of several cm−1 . Császár [69] published an incomplete ab initio sextic force field, which reproduces these frequencies in the range 16–39 cm−1 . Wong and Bacslcay [70] presented an ab initio PES calculated by the CCSD(T) method. The PES was approximated as a Morse coordinate expansion up to sextic terms and allowed to reproduce 56 observed band origins up to 8,0 0 0 cm−1 with an RMS = 5.6 cm−1 . Nakamura and Kato [67] calculated an ab initio PES and presented a quantum calculation of 1647 vibrational bound states up to the dissociation threshold. They also analyzed the structure of the vibrational energy levels and found that

S.A. Tashkun / Journal of Quantitative Spectroscopy & Radiative Transfer 231 (2019) 88–101

N2 O cannot be identified simply as a regular or irregular system. Schröder et al. [71] published an equilibrium structure and the rovibrational energies obtained from a high-level ab initio PES. This surface reproduces harmonic vibrational frequencies with deviations within 0.6 cm−1 . The deviation between the calculated and experimental values of the rotational constant Be is about 0.002 cm−1 . Wang and Harcourt [72] presented a PES of the electronic ground state which has three minima. The global minimum has a linear N–N–O structure with C∞v symmetry. In addition, PES has other local minima: a less stable cyclic isomer with C2v symmetry and a less stable linear N–O–N isomer with D∞h symmetry. These minima have energies of 22,670 cm−1 and 38,720 cm−1 above the global minimum of N–N–O. Li and Varandas [73] reported a global, accurate double-many-body expansion PES for the electronic ground state. The topographic features of this PES confirm the existence of C2v and D∞h local minima previously reported in Ref. [72]. Several empirically adjusted PESs have also been published [74– 76]. Teffo and Chédin [74] constructed a PES expanded in terms of the mass-independent quasi-normal internal coordinates. Fortyseven expansion coefficients were fitted to 267 band centers, 319 inertia constants, and 333 centrifugal distortion constants belonging to six isotopic species 14 N2 16 O, 14 N15 N16 O, 15 N14 N16 O, 14 N2 18 O, 15 N 16 O, and 15 N 18 O. The vibrational band centers were fitted 2 2 with an RMS = 0.047 cm−1 . Based on the fitted PES, the molecular and spectroscopic parameters of 12 isotopic species were calculated to the fourth order of perturbation theory. The fitted PES reproduces about 60,0 0 0 14 N2 16 O measured line positions [3– 66] with an RMS = 0.689 cm−1 . The residuals for individual line positions exceed several wavenumbers. Yan et al. [75] reported an empirical PES for the electronic ground state. They used the Morsecosine expansion as an analytic representation of PES. Thirty-two PES expansion coefficients were fitted to 60 observed vibrational band origins up to 15,0 0 0 cm−1 , reported by Campargue et al. [33], with an RMS = 0.34 cm−1 . Zúñiga et al. [76] presented a set of generalized internal vibrational coordinates and an empirical Morsecosine PES. The PES parameters were fitted to the experimental vibrational frequencies by performing accurate variational calculations of highly excited bound vibrational states (up to 15,0 0 0 cm−1 ). Thirty-two PES expansion coefficients were fitted to a set of 70 observed vibrational frequencies with an RMS = 0.52 cm−1 . This PES is more accurate than the PES by Teffo and Chédin [74], which has an RMS = 3.72 cm−1 . Another theoretical approach was used in [77]. It is based on the effective polyad Hamiltonian model [78]. The parameters of the Hamiltonian were fitted to the 14 N2 16 O line positions in the range 0–9700 cm−1 . The input data file comprised 37,353 measured line positions collected from the literature. During the fitting, some lines perturbed by interpolyad resonance interactions were found and excluded from the fit. The fitted set of 138 parameters reproduced the inputted data with a dimensionless weighted standard deviation χ and an RMS of 4.06 and 0.0042 cm−1 , respectively. It was found that the number of lines perturbed by the interpolyad interactions increases with increasing vibrational and rotational excitations. This fact was also confirmed when this model was used to fit the line positions of other N2 O isotopologues [79,80]. Similar interpolyad perturbations were also found in the experimental FTS and CRDS spectra [40,47,49,50,62,65]. These perturbations make it ef f impossible to use HP for global modeling of N2 O line positions with a spectroscopic accuracy (0.001–0.01 cm−1 ). The goal of this work is to develop an effective Hamiltonian model capable of reproducing all known measured rotationvibration line positions in the range 0–15,0 0 0 cm−1 with the spectroscopic accuracy. One of the reasons for this study is to provide 14 N 16 O line positions and eigenstates for a future version of the 2

89

NOSD-296 (Nitrous Oxide Spectroscopic Databank) databank aimed at the atmospheric applications. This databank is assumed to have an intensity cutoff 10−30 cm/molecule at 296 K. Published hightemperature counterpart NOSD-10 0 0 [81] has an intensity cutoff 10−25 cm/molecule at 10 0 0 K and is based on the polyad effective Hamiltonian model [77]. The paper is organized as follows. The non-polyad effective Hamiltonian model is presented in Section 2. The input data file of the measured line positions is described in Section 3. In Section 4, we present the algorithm and results of the fitting of the model parameters to the measured line positions. In Section 5, we compare the results of our calculations with the HITRAN2016 database and with a recent CRDS line list in the region 5695 - 5910 cm−1 . Finally, a summary of the results obtained in this work is presented in Section 6. 2. Non-polyad model of effective Hamiltonian The effective Hamiltonian model Heff globally describing the rovibrational states of N2 O was presented and discussed in Refs. [74,78]. The model is based on a polyad structure following from the approximate relations between the harmonic frequencies

ω3 ≈ 2ω1 ≈ 4ω2 .

(1)

As a result, the vibrational states can be grouped into polyads with the pseudo quantum number P:

P = 2V1 + V2 + 4V3 ,

(2)

where V1 , V2 , and V3 are the principal vibrational quantum numbers associated with the ω1 stretch ( + ), ω2 bend (), and ω3 stretch ( + ) modes, respectively. Heff is a linear combination of all symmetry allowed products of the elementary vibrational (q, p) and rotational (Jx , Jy , Jz ) operators taken up to a given order of perturbation theory. The numerical coefficients of the combination are considered as adjustable parameters. The optimal values of the parameters are determined from fitting of the model to a set of observed line positions. Heff operator is presented by its matrix elements

V1 ,V2 ,l2 ,V3 ,J|H e f f |V1 +V1 ,V2 +V2 ,l2 +l2 ,V3 +V3 ,J in the basis of the harmonic oscillator and rigid symmetric top rotor eigenfunctions

|V1 , V2 , l2 , V3 , J = |V1 , V2 , l2 , V3 |J, K = l2 , where J is the total angular momentum quantum number, K is the quantum number of the projection of the total angular momentum on the molecular-fixed z-axis and l2 is the vibrational angular momentum quantum number: l2 = −V2 ,−V2 + 2, ..., V2 − 2, V2 . General expressions for Heff matrix elements up to the sixth order of perturbation theory are given below. Diagonal matrix element: (V1 = V2 = V3 = l2 = 0)

V1 ,V2 ,l2 ,V3 ,J|H e f f |V1 ,V2 , l2 , V3 , J    = ωiWi + xi jWiW j + yi jkWiW jWk i

+



ij

i jk

i jkm

+ l22 (xll +



yill Wi +

i





zi jkmWiW jWkWm +

+ J (J + 1 ) −

l22

 



ui jkmnWiW jWkWmWn

i jkmn

zi jll WiW j +

ij

Be +

+

i jk

ui jkll WiW jWk +zl l l l l22 )

i jk



αiWi +

i





εi jkWiW jWk +l22 (γll +

 ij

 i

γi jWiW j 

εill Wi )

90

S.A. Tashkun / Journal of Quantitative Spectroscopy & Radiative Transfer 231 (2019) 88–101



+ J (J + 1 ) − l22



+ J (J + 1 ) −

2 

( De +



βiWi +

i

3 l22

(He +





the block size, it is customary to introduce Wang-type basis functions:

ηi jWiW j +ηll l22 )

ij

φiWi ).

(3)

i

Diagonal l-doubling matrix element: (V1 = V2 = V3 = 0, l2 =±2)

V1 ,V2 ,l2 ,V3 ,J|H e f f |V1 ,V2 , l2 ± 2, V3 , J =



(V2 ± l2 + 2 )(V2 ∓ l2 )[J (J + 1 ) − l2 (l2 ± 1 )][J (J + 1 ) − (l2 ± 1 )(l2 ± 2 )]

× ( Le +



LiWi +

i



Li j WiW j +

ij





Li jkWiW j Wk

i jk



+ L J J ( J + 1 ) − ( l2 ± 1 ) ), 2

(4)

where Wi = Vi + gi /2, gi is the degeneracy of the i-th vibrational mode: gi = 1 for i = 1, 3 and gi = 2 for i = 2. In addition to the diagonal vibrational matrix elements, the model can take into account various types of resonance interactions. Each resonance is defined by a 3 component vector V = (V1 ,V2 ,V3 )and a numberP = 2V1 + V2 + 4V3 . The order of the resonance is |V1 | + |V2 | + |V3 | − 2. The resonances with P = 0 connect two vibrational states inside the polyad P. They are called intrapolyad resonances. Resonances with P > 0 are called interpolyad ones. They connect the vibrational states belonging to polyads P and P + P. Anharmonic resonance matrix element: (V2 =even, l2 =0)

V1 ,V2 ,l2 ,V3 ,J|H e f f |V1 +V1 ,V2 +V2 , l2 , V3 + V3 , J  V ,1 V /2,2 V ,1 = (V1 + d1 )[ 1 ] (V2 − l2 + d2 )[ 2 ] (V3 + d3 )[ 3 ] ×(Fe(V ) + + FJ

(V )



 (V )  (V ) Fi Ui + Fi j UiU j + Fll(V ) l22 i

J (J + 1 ) −

l22



ij

),

(5)

where Ui = Vi + Vi + gi /2, di = gi , if Vi > 0 and 0 otherwise. V[m,d] is the factorial polynomial defined as V[0,d] = 1, 0[m,d] = 0, V[m,d] = V(V + d)(V + 2d)..., if m > 0 and V[m,d] = V(V − d)(V − 2d)..., if m < 0. The number of factors equals |m|. Matrix elements of anharmonic+l-type resonance interactions: (V2 =even, l2 =±2)

|V1 , V2 , |l2 |, V3 , J, C = (|V1 , V2 , l2 , V3 , J √ + (−1 )C−1 |V1 , V2 , −l2 , V3 , J )/ 2, l2 = 0, |V1 , V2 , 0, V3 , J, C = 1 = |V1 , V2 , 0, V3 , J, l2 = 0, where |l2 | = V2 ,V2 − 2, ..., 1(0) and C = 1 or 2. In the Wang basis, each J block is divided into two independent blocks. C = 1 (e-type) basis functions define the first block; C = 2 (f-type) ones define the second. The eigenvalues and eigenvectors (eigenstates) of the (J, C) block are labeled with three numbers (J, C, N), where the index N = 1,2,… ranks the eigenvalues in ascending order. This labeling provides a one-to-one correspondence between eigenstates and (J, C, N). The model of the effective Hamiltonian which includes only intrapolyad resonances is called the polyad model effective Hamiltoef f ef f nian HP . The main advantage of HP is that each (J, C) block is additionally divided into independent blocks defined by the polyad number P. Since the number of vibrational states belonging to a polyad P is finite and the size of each vibrational block is also finite, each (J, C, P) block has a finite size and can be built and diagonalized independently of other blocks. The eigenstates of the (J, C, P) block are unambiguously labeled with integers (J, C, P, n) where the index n ranks the eigenvalues of the (J, C, P) block in ascending order. It should be emphasized that all Coriolis resonances cannot be taken into account in this model, since all these resonances have V2 =odd and, therefore, P = 0. In all published papers on the global fittings of the line posief f tions of N2 O and its isotopologues the polyad model HP was used [77,79,80]. Unfortunately, the model cannot reproduce some of the measured bands with an accuracy compatible with measurement uncertainties due to the absence of interpolyad resonances in it. In this work, we propose a non-polyad model of the effective ef f Hamiltonian HNP to overcome this drawback. The model takes into account three types of interactions between the vibrational states|V1 , V2 , l2 , V3 , and |V1 + V1 , V2 + V2 , l2 + l2 , V3 + V3 . Interactions of the first type are the anharmonic resonances with V2 = even, l2 =0, and P=0,2. Interactions of the second type are the anharmonic+l2 -type resonances: V2 = even, l2 =± 2, and P=0. Interactions of the third type are the Coriolis resonances:

V1 ,V2 ,l2 ,V3 ,J|H e f f |V1 +V1 ,V2 +V2 , l2 ± 2s, V3 +V3 , J  = [J (J + 1 ) − l2 (l2 + s )][J (J + 1 ) − (l2 + s )(l2 + 2s )]  V ,1 V /2+1,2] × (V1 + d1 )[ 1 ] (V2 + sl2 + d2 )[ 2 (V2 − sl2 + d2 )[V2 /2−1,2] (V3 + d3 )[V3 ,1]    ×(FL(V ) + FLi(V )Ui + FLJ(V ) J (J + 1 ) − l22 ),

(6)

i

where s=± 1. Matrix elements of Coriolis resonance interactions: (V2 =odd, l2 =±1)

V1 , V2 , l2 , V3 , J|H |V1 + V1 , V2 + V2 , l2 + s, V3 + V3 , J  = J ( J + 1 ) − l2 ( l2 + s )  V ,1 n ,2 n −1,2] × (V1 + d1 )[ 1 ] (V2 + sl2 + d2 )[ 2 ] (V2 − sl2 + d2 )[ 2 (V3 + d3 )[V3 ,1]

  (V )  (V ) 1 × Ce(V ) + Ci Ui + Ci j UiU j + CK(V ) l2 (l2 + s ) + ef f



i

ij

+ CJ(V ) J (J + 1 ) − l2 (l2 + s ) −

2

 1 2

,

(7)

where n2 = V2 /2 + 1 if V2 > 0 and n2 = V2 /2 if V2 ≤ 0, and s=±1. In this basis the Heff matrix is divided into independent blocks, each block being defined by J quantum number. To further reduce

V2 = odd, l2 =± 1, P=1. There are ten different intrapolyad resonance interactions P=0 up to the sixth order. All of them ef f

were included in HP

. As for interpolyad resonances, there are 42 ef f

resonances with P=1,2 up to the sixth order. The HP ef f HNP ,

model is

a particular case of if we neglect all interpolyad resonances and remove parameters uijkmn , uijkll , ε ijk and Lijk from it. There are two other types of N2 O rovibrational levels labeling used in the literature. The spectroscopic community uses a set (V1 ,V2 ,l2 ,V3 ,J, C) to denote a rovibrational level. This labeling is adopted, in particular, for the HITRAN2016 [82] database. However, due to strong mixing between the normal modes V1 , V2 and V3 (Fermi resonance V1 =−1, V2 =2, V3 =0, and the anharmonic resonance V1 =−2, V2 =0, V3 =1) one-to-one labeling is possible only for low lying levels. Another type of labeling was proposed in Ref. [33]. A vibrational level is labeled by a set (P, l2 ,n), where the ranking index n, orders the levels of a cluster (P, l2 ) in ascend-

S.A. Tashkun / Journal of Quantitative Spectroscopy & Radiative Transfer 231 (2019) 88–101

91

Table 1 A summary of the sources of the observed line positions. reference

setupb

Nf

Vmin -Vmax g cm−1

min -max

Coles et al. [3] Herzberg et al. [4] Tetenbaum et al. [5] Burrus and Gordy [6] Lafferty and Lide [7] Pearson et al. [8] Amiot and Guelachvili [9] Farrenq and Dupre-Maquaire [10] Krell and Sams [11] Bogey [12] Casleton and Kukolich [13] Andreev et al. [14] Amiot and Guelachvili [15] Guelachvili [16] Jolma et al. [17] Pollock et al. [18] Brown and Toth [19] Wells et al. [20] Wells et al. [21] Toth [22,23] Hinz et al. [24] Zink et al. [25] Esplin et al. [26]a Vanek et al. [27] Vanek et al. [28] Vanek et al. [29] Yamada [30] Toth [31] Tan et al. [32] Campargue et al. [33], spectrum S1 Campargue et al. [33], spectrum S2 Campargue et al. [33] Tachikawa et al. [34] Campargue [35] Morino et al. [36] He et al. [37] Garnache et al. [38] Morino et al. [39] Toth [40] Barbe [41] Hippler and Quack [42] Oshika et al. [43] Weirauch et al. [44] Weirauch et al. [44] Bailly and Vervloet [45]; Bailly et al. [48] Campargue et al. [46] Bertseva et al. [47] Ding et al. [49] Bertseva et al. [50] Wang et al. [51] Herbin et al. [52] Drouin and Maiwald [53] Horneman [54] Liu et al. [55] Liu et al. [56] Liu et al. [57] Liu et al. [58] Milloud et al. [59] Toth [60] Yuan et al. [61] Lu et al. [62] Knabe et al. [63] Ting et al. [64] Ting et al. [64] Karlovets et al. [65] Werwein et al. [66]

MW PHOTO MW MW MW MW FTS FTS FTS MW MB MW FTS FTS FTS HET FTS HET HET FTS HET HET FTS HET HET HET FTS FTS FTS FTS FTS ICLAS HET ICLAS MW ICLAS ICLAS MW FTS FTS CRDS DLS FTS ICLAS FTS, emission ICLAS ICLAS ICLAS ICLAS FTS ICLAS MW FTS CRDS CRDS CRDS CRDS ICLAS FTS, CALCc OC CRDS EC-QCL DFG, NTHUd DFG, JPLe CRDS FTS

1 59 1 15 3 35 3400 647 1847 4 1 90 3662 646 355 39 240 31 14 2282 26 9 3314 8 3 18 32 3211 244 2591 569 337 129 241 5 1 62 5 1401 79 8 544 2945 563 5076 956 1049 741 735 1446 1067 73 532 5096 2203 1191 1746 180 933 7 5910 24 44 175 2618 41

0.8 12,834.9 - 12,898.2 1.7 3.4 - 10.1 0.8 - 2.5 4.2 - 10.1 3584.8 - 8098.1 2098.5 - 2230.7 2267.1 - 2618.0 1.7 - 4.2 0.8 12.6 - 18.5 1831.7 - 3191.2 1118.1 - 1342.9 542.5 - 645.4 4341.1 - 4753.3 1132.0 - 4749.1 1104.8 - 1914.7 1257.3 - 1339.8 900.9 - 2392.5 1257.5 - 1335.0 1037.2 - 1084.6 1100.8 - 1360.0 1591.3 - 1672.7 50.1 - 50.2 896.9 - 989.7 20.1 - 47.6 1224.8 - 4664.3 554.0 - 619.4 6436.3 - 10,832.9 8837.6 10,826.0 11,819.4 - 14,916.6 897.0 - 1074.4 11,233.8 - 12,221.9 20.9 - 24.3 7788.5 9362.1 - 9419.8 20.9 - 24.3 3676.9 - 7795.2 903.8 - 1118.1 7783.5 - 7788.5 7515.5 - 7796.6 6436.3 - 10,441.6 10,694.2 - 12,141.2 2036.1 - 2266.3 10,084.0 - 12,021.1 9556.3 - 9972.7 8836.1 - 10,092.6 9057.9 - 9621.0 5313.7 - 8954.8 3900.8 - 4089.7 20.1 - 55.2 542.9 - 635.2 6001.8 - 6884.9 5906.3 - 6832.4 6764.9 - 7065.6 7647.5 - 7918.2 12,764.2 - 12,899.2 1741.9 - 5169.0 2271.2 - 2272.1 6949.8 - 7652.6 2189.3 - 2213.2 2121.2 - 2272.2 6.7 - 67.5 7915.8 - 8331.5 4418.2 - 4439.8

0.0033 30 0.0033 0.0067 - 0.033 0.002 - 0.0033 0.0 0 033 - 0.005 1.0 3.0 5.0 0.0 0 0334 0.0 0 0 013 0.0 0 03 - 0.00117 2.0 0.13 0.05 0.13 - 0.4 0.066 0.167 - 0.5 0.067 - 0.33 0.1 0.01 - 0.35 0.13 - 0.27 1.0 0.17 - 0.33 0.0067 0.1 - 0.33 0.1 0.05 - 5.0 0.08 5.0 5.0 20.0 0.0 0 033 15.0 0.0 0 067 1.0 5.0 0.0 0 067 0.03 - 1.0 0.03 1.0 2.0 5.0 5.0 0.4 10.0 10.0 5.0 5.0 3.0 1.0 0.0025 0.007 - 0.011 1.5 1.0 1.0 1.0 10.0 1.0 10.0 1.0 0.027 0.0 0 033 - 0.0032 0.0 0 01 - 0.0025 1.0 0.04 - 0.11

h

10−3 cm−1

Correction factor 1.0i 1.0 1.0 1.0 1.0 1.0 0.99999997 0.99999920 1.0 0 0 0 0 050 1.0 1.0 1.0 1.0 0.99999981 0.99999982 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.99999980 1.0 1.0 1.0 0.99999880 1.0 1.0 0 0 0 0 0 07 1.0 0 0 0 0320 1.0 0 0 0 030 0 1.0 1.0 1.0 1.0 1.0 0.99999830 1.0 1.0 0.99999991 1.0 1.0 0 0 0 0 040 0.99999970 1.0 1.0 1.0 0.99999900 0.99999960 0.99999990 1.0 1.0 1.0 0.99999999 0.99999996 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0 0 0 0 020 0.99999992

high-temperature measurements, T = 800 K. PHOTO - photographic spectrum; MW - microwave measurements; MB - maser beam measurements; HET - laser heterodyne measurements; FTS - Fourier transform spectroscopy; ICLAS - intracavity laser absorption spectroscopy; CRDS - cavity ring-down spectroscopy; OC - optical centrifuge measurements; DFG - difference frequency generation measurements; EC-QCL - external-cavity quantum cascade lasers measurements. c calculated data based on FTS measurements. d NTHU - measurements performed at National Tsing Hua University. e JPL - measurements performed at Jet Propulsion Laboratory. f N - the number of measured lines in a given source. g Vmin and Vmax - minimal and maximal values of the wavenumbers in a given source. h min and max - minimal and maximal values of the measurement uncertainties for a given source. i Correction factor was fixed to 1.0. a

b

92

S.A. Tashkun / Journal of Quantitative Spectroscopy & Radiative Transfer 231 (2019) 88–101

ing order. The reason for this labeling is based on the fact that P and l2 are considered good quantum numbers for states with small values of J, which are not strongly perturbed by interpolyad Coriolis interactions. In this work we will use (J, C, N) numbers to label ef f the energy levels of the HNP model. 3.

14 N 16 O 2

measured line positions data file

The input data file includes measured line positions collected from the literature [3–66]. The file contains more than 60,0 0 0 data that cover the 0.8–14,917 cm−1 spectral range. A summary of the data is given in Table 1. Column 1 contains references to data sources. The references are ordered according to the date of publication. Column2 indicates the setups used for measurements: PHOTO - photographic spectrum, MW - microwave measurements, MB - maser beam measurements, HET - laser heterodyne measurements, FTS - Fourier transform spectroscopy, ICLAS - intracavity laser absorption spectroscopy, CRDS - cavity ring-down spectroscopy, OC - optical centrifuge measurements, DFG - difference frequency generation measurements, and EC-QCL - external-cavity quantum cascade lasers measurements. The data file was supplemented with some calculated line positions taken from Toth’s line list SISAM.N2O [60] and labeled as CALC in the second column of Table 1. We assumed for these data an uncertainty 0.001cm−1 . The number of lines and spectrum ranges are shown in columns 3 and 4, respectively. The measurement uncertainties are given in column 5. Where possible, the uncertainties of the positions of individual lines were used. If information on individual uncertainties is not available, the averef f age uncertainty for the data source is given. Using the HNP model presented in Section 2, with parameters given in Ref. [77], we initially labeled each energy level with a triad (J, C, N). Spectroscopic labeling of these levels, taking from publications, was also placed in the data file. Since the spectra were recorded by various authors using various setups, the analysis of the internal consistency of the line positions from the point of view of assignments, possible measurement errors, misprints of published data, and recovering possible correction factors of different data sources relative to each other is an important part of data analysis. To perform such an analysis, we used the fundamental Rydberg-Ritz combination principle. The use of the principle, also known as term value method, has been proven as a useful tool for testing spectroscopic assignments [83,84]. In order to take into account possible correction factors, the Rydberg-Ritz principle was written as follows:

ck vkj←i = E j − Ei ,

(8)

where vkj←i is the measured line position belonging to the k-th source ck is the correction factor for this source, and Ei and Ej are the energies (term values) of the lower and upper states, respectively. Composite quantum number indexes i and j define the assignment of the measured transition vkj←i . In our case, i and j are sets of (J, C, N) numbers. The correction factor ck can be rewritten as ck = 1+δ k , where δ k is the offset of the k-th data source and Eq. (8) takes form

vkj←i = E j − Ei − δ k vkj←i .

(9)

Eq. (9) with different indexes i, j and k constitutes an overdetermined system of linear equations. The quantities δ k are set to zero or fixed values for some reference sources, and they are considered unknown for other sources. For the absolute line position measurements (MB, MW, HET), all δ k are set to zero. They are considered as reference sources. The assigned data file of measured line positions can be represented as a set of graphs, where energy

levels are considered as vertices and transitions are considered as edges. On the graph, every two energy levels are connected by a chain of transitions. Such graphs we call trees. In Ref. [84] they are called spectroscopic networks. If the number Ntra of the transitions of a tree is greater than Nlev +Noffset , where Nlev is the number of energy levels of the tree and Noffset is the number of unknown offsets, Eq. (9) can be solved in the least-squares sense to determine the experimental energies Ei and the correction factors ck . If Ntree < Nlev + Noffset , then the system is undetermined and the tree is called floating. More information can be found in Ref. [84]. Since the line position data file includes transitions with different measurement uncertainties ε ranging from 3 KHz for the MB measurements [13] to several 10−2 cm−1 for ICLAS [33,35,46,47,59], and photographic [4] measurements, the corresponding observed line positions should be weighted. Ideally, for each measured line position the corresponding measurement uncertainty should be given. In practice, this is done only for a small part of the measurements, as a rule, for high precision data measured by the MB, MW and HET techniques. Toth [22,23,31,40] for his FTS measurements used the number of decimal places after the decimal point as an estimate of ε . For most FTS, CRDS, and ICLAS measurements, the authors provide only an estimate for ε , valid for unblended lines, lines that do not have low signal-to-noise ratio, well-resolved lines, et cetera. Given all this, we use the following algorithm for assigning uncertainties to measurements. If uncertainty is explicitly reported, we use it. For Toth’s FTS measured data we used the number of decimals after the decimal point as an estimate of ε . For Toth’s calculated data taken from the SISAM.N2O line list, we took ε =0.001 cm−1 . Finally, if there is no information about uncertainty, we use an estimated value based on the type of setup and the publication date. Solving of Eq. (9) is equivalent to minimizing the dimensionless expression

χRIT Z =

   { (1 + δ k )vkj←i − (E j − Ei ) /ε kj←i } Ntra − Nlev − No f f set

,

(10)

where the summation runs over all transitions included in the tree, and ε kj←i is the measurement uncertainty of the ν kj←i line position. As a rule, the system (9) consists of tens of thousands of equations (the number of observed line positions) and thousands of unknowns (the number of energy levels and the number of offsets). However, the matrix of the system is extremely sparse: each row contains no more than three nonzero entries. That fact enables us to use an efficient algorithm [85] for sparse linear least squares. To solve the system (9) in the least squares sense, we developed a set of Fortran-90 procedures called “RITZ." The correction factors found from the Ritz analysis are in a good agreement with those found earlier. For example, our correction factors 1.0 0 0 0 030 0 and 1.0 0 0 0 0320 found for spectra S1 and S2 [33], are close to the factors 1.0 0 0 0 033961 and 1.0 0 0 0 03757 reported in [44]. The value of the dimensionless standard deviation of the solution χ RITZ provides a measure of the overall consistency of the observed wavenumbers in the data file. Values of the individual dimensionless residuals

r kj←i =

ck vkj←i − (E j − Ei )

ε kj←i

(11)

allow us to detect outliers, i.e., misprints, misassignments or poorly measured wavenumbers. It should be emphasized that the value of χ RITZ does not depend on any Hamiltonian model and is calculated only from the first principles of quantum mechanics. This approach, which can be viewed as a generalization of the well-known method of combination differences, is especially useful for a data file with many lines originating from hot transitions.

S.A. Tashkun / Journal of Quantitative Spectroscopy & Radiative Transfer 231 (2019) 88–101 Table 2 ef f model. Resonance interactions used in the HNP

where

#

Type of resonance

V1

V2

V3

P

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Anharmonic (Fermi) Anharmonic Anharmonic Anharmonic Anharmonic Anharmonic Anharmonic Anharmonic Anharmonic Coriolis Coriolis Coriolis Coriolis Coriolis Coriolis Coriolis Coriolis Anharmonic Anharmonic Anharmonic

−1 −2 −1 0 −2 −4 −3 −3 −2 −1 −1 −2 0 −2 2 −3 0 1 −1 −5

2 0 −2 −4 4 0 2 −2 −4 −1 3 1 −3 −1 −3 −1 −5 −4 −4 0

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

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

r j←i =

measurement uncertainty. Summation over n passes through all measured line positions. The calculated line position vcalc of a tranj→i sition between the upper state Ej and the lower state Ei is defined as vcalc = E j − Ei . The composite indices j and i are actually sets of j→i three numbers (J, C, N) as discussed in Section 2. Whenn  m, a value of χ ∼ 1 means that the model is capable of reproducing the measured line positions with an accuracy comparable to measurement uncertainties. ef f Although HNP depends on the parameters x linearly, its eigenvalues Ei (x) are nonlinear functions of x. Therefore, minimizing the dimensionless standard deviation (14) is an iterative process that requires knowledge of the derivatives ∂ Ei /∂ xα , i = 1, 2, …, n and α =1, 2, …, m. Their values are given by the Hellmann-Feynman theorem:

ef f

The HNP matrix is block-diagonal with respect to the rotational quantum number J and the Wang parity C (C = 1 for energy levels of type ’e’ and C = 2 for energy levels of type ’f’). Each (J, C) block of ef f ef f HNP can be constructed and diagonalized independently. For HP , each (J,C) block, in turn, is block-diagonal with respect the polyad ef f number P. Each (P, J, C) block has a finite size. Thus, the HP model has no problems with the convergence and basis truncation. Due ef f to interpolyad resonance interactions, each (J, C) block of the HNP model has infinite size. To process it using a computer, we must truncate it and check the convergence properties of its eigenvalef f ues. Our input data file includes data up to P = 28, while the HNP model was built up to P = 30. We checked that this ensures that all highly excited energy levels converge better than 0.001 cm−1 . ef f The HNP model is a linear combination of the structural matrices Hα defined by Eqs. (3)-(7) with the numerical parameters x1 ,x2 ,..., xn :

xα Hα .

(12)

α =1

The line position vj → i (x) of the transition between the upper and lower energy states is defined as

v j→i (x ) = E j (x ) − Ei (x ),

(13) ef f

where Ei (x) and Ej (x) are the eigenvalues of HNP (x ). The fitting procedure aimed to minimize the dimensionless standard deviation defined according to the usual formula

χ (x ) =

n 1  2 r j→i , n−m j→i

(15)

vcalc are the observed and calculated line position, and δ obs is the j→i j→i

4. Least squares fitting

n 

vobs − vcalc j←i j←i ε j←i

is the weighted dimensionless residual, n is the number of fitted line positions, m is the number of adjusted parameters, vobs and j→i

The final data file included more than 61 thousand entries. The highest polyad number P was 28 (the 7v3 band [33]), the highest rotational quantum number J was 105 (the P 105 line of the v3 band) [26]. Ritz analysis revealed 678 entries with anomalously large RITZ residuals (11). They were removed from the data file along with 329 entries which were labeled by experimenters as poorly measured. Statistically significant correction factors different from one were found for 20 sources. They are listed in the last column of Table 1. The transition frequencies of the data file were adjusted for these factors.

ef f HNP (x ) =

93

(14)

∂ Ei = ti (x )|Hα |ti (x ), ∂ xα where |ti (x) is the eigenvector of HNP (x ) corresponding to the eigenvalue Ei (x). The upper and lower state ranking numbers N and N  assigned to observed transitions depend on x. At the beginning of the fitef f ting process, they were set using the HP parameters from [77]. During iterations for each line position vobs , the ranking number j→i Nj of the upper state is adjusted by minimizing the value of the expression |vobs − (EJ j ,C j ,N j − EJi ,Ci ,Ni )|. Without this procedure, the j←i iterative process sometimes fails to converge. The task of minimizing the dimensionless standard deviation (14) for the nonlinear model (12),(13) belongs to the class of optimization problems, known as the least-squares fitting of parameterized Hamiltonians to energy levels. A common feature of this class is the presence of multiple local minima in a parameter space [86]. This makes finding the global minimum a difficult task. We started from the set of parameters given in Ref. [77]. Our goal is to find the deepest minimum, where the values of the optimal parameters should, if possible, be consistent with the orders arising from perturbation theory. To do this, we must explore the parameter space, performing fittings from different starting points and using different sets of adjustable parameters. Since the dimension of our parameter space is very large (more than 100), the trial and error process is very time consuming. We stopped it after finding a set of 195 parameters that reproduces 56,888 measured line positions in the 0.8–14,917 cm−1 spectral range with χ = 1.71. Also, the values of the fitted parameters are, in general, consistent with the orders of perturbation theory, the weighted residuals (15) are scattered around zero, and there are no noticeable outliers. However, it should be emphasized that there is no guarantee that we found the global minimum. The resonance interactions included in the fief f nal HNP model are listed in Table 2. Parameter values, along with their standard errors are given in the Supplementary materials. The fitted model reproduces most of the observed positions with residuals close to their experimental uncertainties. To illustrate this, we plotted the values of the weighted residuals (15) versus the measurement uncertainties ε j ← i in Fig. 1. It is seen that most of the residuals are scattered around zero. Before the fitting process, the FTS data from Ref. [33] were removed from the input data file. The rationale is they were revised in Ref. [44], where more accurate line positions for them were reported. Some of the bands from Ref. [33], measured using the ICLAS technique, were ef f

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Fig. 1. Values of the weighted residuals (15) versus the measurement uncertainties.

Table 3 Summary of the ICLAS cold bands measured in Ref. [35]. #

N2 O band in the cluster notation

Spectral range cm−1

smax cm/molecule

H2 O band

Mean residual cm−1

Standard deviation cm−1

1 2 3 4 5 6

(20,0,12) - (0,0,1) (20,0,15) - (0,0,1) (20,0,19) - (0,0,1) (20,0,21) - (0,0,1) (20,0,23) - (0,0,1) (22,0,11) - (0,0,1)

11,233 - 11,283 11,301 - 11,357 11,512 - 11,558 11,604 - 11,641 11,697 - 11,721 12,172 - 12,222

2.1e-24 4.7e-25 1.8e-26 1.7e-25 6.4e-25 4.8e-23

0 03–0 0 0 0 03–0 0 0 131–0 0 0 131–0 0 0 131–0 0 0 211–0 0 0

0.032 0.033 0.081 0.129 0.132 −0.004

0.014 0.013 0.027 0.015 0.027 0.030

also removed, as they were measured in Ref. [59] more accurately. During the fitting, we also deleted a few hundred data with unacceptably large weighted residuals. These data could not be verified by Ritz analysis. The choice and significance of the parameters, along with their values, was analyzed using the singular numbers sα and right eigenvectors Vα of the dimensionless m × n Jacobi matrix J:

J j→i,α

  ∂v j→i ∂ E j ∂ Ei xα = = − . δ j→i ∂ xα δ j→i ∂ xα ∂ xα xα

(16)

 The linear combinations nβ =1 cαβ xβ of the fitted parameters x, corresponding to large singular numbers are well determined from the fitted data. Here cαβ are the components of the right eigenvector Vα , which corresponds to the singular number sα . These combinations are important for the model and are called ’stiff’ combinations [87]. In contrast, the linear combinations corresponding to small singular numbers have little effect on the behavior of the model. These unimportant parameter combinations are referred to as ’sloppy.’ The singular numbers of the Jacobi matrix (16) are plotted in Fig. 2 in descending order. The first 30 singular numbers cover the range 105 - 1012 . Linear ef f combinations of the HNP parameters corresponding these numbers have dominant contributions from the following parameters: ω1 , ω2 , ω3 , Be , α 3 , Fe(−1,2,0) , x13 , x33 , x11 , x12 , Fe(−2,0,1) , x23 , x22 , α 2 , F2(−1,2,0 ) , F1(−2,0,1 ) , Fe(−1,−2,1 ) , F1(−1,2,0 ) , De , Le , y222 , y111 , FJ(−1,2,0 ) , (−2,0,1 ) F3(−2,0,1 ) , z1133 , y133 , y113 , F13 , y223 , F3(−1,2,0 ) , y2ll , F3(−1,−2,1 ) , (−1,−2,1 ) y123 , F2(−2,0,1 ) , Fe(0,−4,1 ) , y3ll , F2(−1,−2,1 ) , F1(−1,−2,1 ) , F11 . These

parameters are involved in stiff combinations and determine the core of the model. On the other hand, the parameters entering in sloppy combinations are responsible for less important features of the model’s behavior. In general, it is reasonable to expect that the most important parameters, in accordance with perturbation theory, enter the stiff combinations whereas the least important ones enter the sloppy combinations. However, sometimes a parameter which normally is considered as an unimportant and which should be small has an anomalously large value and enters into a stiff combination. For example, the parameter z1133 is included in the stiff combination corresponding to the singular number 4.764 × 105 . It has a value −0.081, which is typical for the more important parameters yijk . Attempts to remove z1133 from the fit or constrain it to some reasonable value led to a significant increase in the fitted χ value. Perhaps this situation is similar to that discussed in [87], where it was stated that if a model has many sloppy combinations, then the parameters have a compensating effect on the collective behavior of the model. Any sloppy parameter can vary widely, and the model could still fit the data. We found that the singular value decomposition of the Jacobi matrix provides a convenient tool for analyzing the relative importance of the fitting parameters of the model. The dimensionless weighted standard deviation of the fit is 1.71. The fitted set of 195 parameters made it possible to reproduce the measured line positions with an RMS = 0.006 cm−1 . Of the almost 57,0 0 0 line positions used in the fit, only 96 lines have the dimensionless residuals larger than 10. As for usual residuals r = vobs - vcalc , 9424 lines have |r| < 10−4 cm−1 , 14,675 lines have |r| <

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95

Fig. 2. The singular numbers of the dimensionless Jacobi matrix of the fitted model.

Fig. 3. Residuals vobs - vcalc and the measurement uncertainties for the region 970 0–15,0 0 0 cm−1 .

5 × 10−4 cm−1 , 9336 lines have |r| < 10−3 cm−1 , 17,556 lines have |r| < 5 × 10−3 cm−1 , and 5827 have |r| > 10−2 cm−1 . The J values of the R branch for the (17,1,6) - (0,0,1) band, reported in Ref. [33] and labeled as “extra lines” were reassigned from J to J-1 to be consistent with the measured values of the same band reported in Ref. [40]. The observed data in Ref. [77] covered the range 0 - 9700 cm−1 . In our fit, we used data above 9700 cm−1 from the ICLAS [33,35,44,46,47,49,59], FTS [44] and photographic [4] measurements. The measurement uncertainties of the ICLAS data are be-

tween 0.005 and 0.020 cm−1 . The total number of fitted data was 3308. Residuals vobs - vcalc together with measurement uncertainties for the range 970 0–15,0 0 0 cm−1 were plotted in Fig. 3. Fifty percent of all data have the weighted residuals (15) within ±2. However, there are 14 data whose weighted residuals exceed 10. They are located in the region 11,500–11,750 cm−1 where the differences vobs - vcalc reach 0.18 cm−1 . All data in this region were obtained from ICLAS measurements [35], where 241 line positions of six cold bands were observed in the range 11,20 0–12,30 0 cm−1 . The measurement uncertainty of these data is 0.015 cm−1 . The

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Fig. 4. Residuals vcalc - vHITRAN plotted versus line position.

Fig. 5. An enlarged version of Fig. 4 where the y-axis has limits of ±0.01 cm−1 .

band indices, cluster labeling (P,l2 ,n) and spectral ranges are given in Table 3 in columns 1, 2, and 3, respectively. The line positions were calibrated using the atmospheric water absorption lines from HITRAN1992 [88]. We give the intensity of the strongest water line that falls within the band’s range, and its vibrational labeling in column 4 and 5, respectively. These data were taken from HITRAN2016. Mean values and standard deviations of the residuals vobs - vcalc for each band are listed in the last two columns. For bands 3, 4 and 5, the

mean residuals are about 0.1 cm−1 , and the strong water lines belong to the 131–0 0 0 band. For bands 1, 2, and 6, the mean residuals are about 0.03 cm−1 , which is consistent with the χ value of the fit. The strong water lines that fall within the ranges of these bands belong to the 0 03–0 0 0 and 211–0 0 0 bands. The line positions of the strong lines of these bands in HITRAN1992 and HITRAN2016 coincide within 0.002 cm−1 , while the line positions of the strong lines of the 131–0 0 0 band differ to 1.0 cm−1 . We believe that some lines of the 131–0 0 0 band given in HITRAN1992 are un-

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97

Fig. 6. Residuals vcalc - vHITRAN plotted versus line intensity.

Fig. 7. An enlarged version of Fig. 6 where the y-axis has limits of ±0.01 cm−1 .

reliable, and bands 3, 4, 5 need to be recalibrated using more accurate HITRAN2016 data. 5. Comparison with other line lists HITRAN2016 [82] 14 N2 16 O line positions comprise 33,074 entries and cover the spectral range 0.8 - 7797 cm−1 . More than 32,0 0 0 entries originate from Toth’s SISAM.N2O line list [60]. Refs. [89–92] were used as sources of other 912 entries. Toth’s data have an intensity cutoff 2.0 × 10−25 cm/molecule. The minimum in-

tensity is 1.23 × 10−28 cm/molecule (the P 98 line of the 01101– 0 0 0 01 band). The HITRAN error code for most data is ierr=4, which means that the estimated uncertainty of HITRAN data is between 0.0 0 01 and 0.001 cm−1 . Using the fitted parameters, we calculated the file with the line positions, which were then compared with HITRAN2016. We were able to assign all data, except for 36. Most of the unassigned lines belong to the 0441–0440 band. The residuals vcalc - vHITRAN are plotted in Fig. 3. Out of 33,074 HITRAN2016 data, the residuals of 13,955 data are within ±0.0 0 01 cm−1 , the residuals of 16,677

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Fig. 8. Residuals vcalc - vHITRAN for the 04401–03301 band.

Fig. 9. Residuals vHEFF - vobs plotted versus line position.

data are within ±0.001 cm−1 , and the residuals of 2248 data are within ±0.01 cm−1 . There are only 159 data that have residuals greater than 0.1 cm−1 in absolute value. As follows from Fig. 4, all these data originate from the bands with large v2 values. An enlarged version of Fig. 4 is presented in Fig. 5 where the y-axis is limited to ±0.01 cm−1 . It is instructive also to plot residuals versus HITRAN intensities. These plots are shown in Figs. 6 and 7 with the same y-axis limits as in Figs. 4 and 5. It is seen that all outliers are located in the 10−23 - 10−25 cm/molecule intensity interval.

The Toth data [60] are mainly calculated values based on fitted spectroscopic constants. He used the following expression to model the energy levels Ei and Ej , which enter in Eq. (13)

Ev (J ) = Gv + Bv [J (J + 1 )] − Dv [J (J + 1 )] + Hv [J (J + 1 )] 2

+ L v [J ( J + 1 ) ] , 4

3

(17)

where Gv is the vibrational term value, Bv is the rotational constant, Dv , Hv , and Lv are the centrifugal distortion constants, J is the angular momentum quantum number. Values of these constants

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were fitted to the measured line positions. If a resonance interaction perturbs a band, the fit leads to unacceptably large residuals for some lines. In this case, the measured values were placed in SISAM.N2O instead of the calculated values for these lines. A lineby-line comparison of the HITRAN2016 data and our calculation is placed in the Supplementary material. To understand the reason for the significant discrepancies between our calculations and the HITRAN data for the bands with large values of the v2 quantum number, consider the 04401–03301 band. The HITRAN data of this band were based on FTS measurements by Toth [31]. The spectroscopic constantsG04401 , B04401 , D04401 , H04401 , and L04401 of the 04,401 state were obtained from 15 measured lines of the 04411–04401 band [31]. These measurements covered data from J = 6 to J = 31. Similarly, the spectroscopic constants G03301 , B03301 , D03301 , and H03301 of the 03301 state were obtained in the same work from 172 measured line positions of the 04411–03301, 04412–03301, 07301–03301, 13311–03301, 15301–03301, 23301– 03301, and 23302–03301 bands in the interval from J = 3 to J = 50. The fitted constants were used to calculate the line positions of the 04401–03301 band from J = 3 to J = 45 using Eqs. (13),(17). This J interval is divided into three parts. The interpolation part is from J = 6 to J = 31 and two extrapolation parts, one is from J = 5 to J = 3 and another is from J = 32 to J = 45. In addition to the data from Toth, our fitted data also included the measurements of the 04411– 04401 band reported in Ref. [45]. The maximal J values for the P and R branches were 47 and 44, respectively. The values of residuals vcalc - vHITRAN were depicted in Fig. 8 as a function of the number m defined as usual: m=-J for the P branch and m = J + 1 for the R branch. The intervals of interpolation and extrapolation are also shown. The residuals are within ±0.0022 cm−1 in the interpolation interval. Outside this interval, the residuals grow rapidly, reaching 0.45 cm−1 for the R 44 line. This behavior is typical when one uses the polynomial expression (17) for extrapolations at high J values. The danger of such extrapolations is discussed in Refs. [93,94]. ef f To test the predictive capabilities of the fitted HNP model, we used the most recent measurements [95]. In this work, high sensitivity CRDS of N2 O in the 5695 - 5910 cm−1 range was carried out. The number of the measured 14 N2 16 O line positions was 2188. The number of bands was 29. The estimated uncertainty on the line center determination of unblended lines is ∼0.001 cm−1 . For each measured line, the authors provided the prediction based on ef f the HP model [77]. We also made similar predictions with our ef f

HNP model. Residuals vHEFF - vobs are plotted in Fig. 9. It can be seen that our model correctly takes into account all intrapolyad ef f and interpolyad perturbations. The HNP model reproduces all these data with a mean residual −0.0 0 0 06 cm−1 and an RMS = 0.0023 cm−1 . In particular, the line positions of the Coriolis interacting bands 0821i-0220i and 5110i-0220i, i = 1, 2, were predicted with ef f an accuracy close to the measurement uncertainty. The HP residuals of these bands, located in the 5690 - 5730 cm−1 spectral ef f range, are between −0.075 and 0.2 cm−1 , while the HNP residuals −1 are smaller, within ±0.004 cm . There is only one line, located at 5873.208 cm−1 , for which both models give a residual ∼0.031 cm−1 . This line is assigned as the Q 3 line of the 31112–01101 band. We consider it an outlier whose line position is poorly measured or reported. 6. Conclusions The 14 N2 16 O line positions collected in the literature has been satisfactorily fitted using the non-polyad model of effective Hamiltonian. The dimensionless weighted standard deviation of the fit is 1.71. This work significantly improves and expands the previous analysis of 14 N2 16 O line positions, which used the polyad model of

99

effective Hamiltonian [77]. A comparison of our calculations with the HITRAN2016 data showed that most of the data are reliable, and the stated line position accuracy 0.0 0 01 - 0.001 cm−1 is valid for ∼93% of them. The model has excellent predictive capabilities, which was demonstrated in the simulation of the CRDS measurements in the region 5695 - 5910 cm−1 [95]. Acknowledgments The author acknowledges Drs. V.I. Perevalov and J.L. Teffo for useful discussions. S. Reshetnik is acknowledged for preparing the initial version of the data file. Supplementary materials Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.jqsrt.2019.04.023. References [1] Liu H, Gibbs BM. Modelling of NO and N2 O emissions from biomass-fired circulating fluidized bed combustors. Fuel 2002;81:271–80. [2] Scherson Y, Lohner K, Cantwell B, Kenny T. Small-scale planar nitrous oxide monopropellant thruster for “green” propulsion and power generation. In: Proceedings of the 46th AIAA/ASME/SAE/ASEE joint propulsion conference and exhibit. AIAA2010-6828, Nashville, TN; 2010. [3] Coles DK, Elyash ES, Gorman JG. Microwave absorption spectra of N2 O. Phys Rev 1949;76:973. [4] Herzberg G, Herzberg L. Rotation-vibration spectra of diatomic and simple polyatomic molecules with long absorbing paths VI The spectrum of nitrous oxide (N2O) below 1.2 μm. J Chem Phys 1950;18:1551–61. [5] Tetenbaum SJ. Six-millimeter spectra of OCS and N2 O. Phys Rev 1952;88:772–4. [6] Burrus CA, Gordy W. Millimeter and submillimeter wave spectroscopy. Phys Rev 1956;101:599–602. [7] Lafferty WJ, Lide DR Jr. Rotational constants of excited vibrational states of 14 N2 16 O. J Mol Spectrosc 1964;14:407–8. [8] Pearson R, Sullivan T, Frenkel L. Microwave spectrum and molecular parameters for 14 N2 16 O. J Mol Spectrosc 1970;34:440–9. [9] Amiot C, Guelachvili G. Vibration-rotation bands of 14 N2 16 O: 1.2 μm-3.3 μm region. J Mol Spectrosc 1974;51:475–91. [10] Farrenq R, Dupre-Maquaire J. Vibrational luminescence of N2 O excited by dc discharge. Rotation-vibration constants. J Mol Spectrosc 1974;49:268–79. [11] Krell JM, Sams RL. Vibration-rotation bands of nitrous oxide: 4.1 μm region. J Mol Spectrosc 1974;51:492–507. [12] Bogey M. Microwave absorption spectroscopy in the v3 states of OCS and N2 O through energy transfer from N2 ∗ . J Phys B: Atom Mol Phys 1975;8:1934–8. [13] Casleton KH, Kukolich SG. Beam maser measurements of hyperfine structure in 14 N2 O. J Chem Phys 1975;62:2696–9. [14] Andreev BA, Burenin AV, Karyakin EN, Krupnov AF, Shapin SM. Submillimeter wave spectrum and molecular constants of N2 O. J Mol Spectrosc 1976;62:125–48. [15] Amiot C, Guelachvili G. Extension of the 106 samples Fourier spectrometry to the indium antimonide region: vibration-rotation bands of 14 N2 16 O: 3.3-5.5μm region. J Mol Spectrosc 1976;59:171–90. [16] Guelachvili G. Absolute N2 O wavenumbers between 1118 and 1343 cm−1 by Fourier transform spectroscopy. Can J Phys 1982;60:1334–47. [17] Jolma K, Kauppinen J, Horneman VM. Vibration-rotation spectrum of N2 O in the region of the lowest fundamental v2 . J Mol Spectrosc 1983;101:278–84. [18] Pollock CR, Petersen FR, Jennings DA, Wells JS. Absolute frequency measurements of the 0 0 02-0 0 0 0, 20 01-0 0 0 0, and 1201-0 0 0 0 bands of N2 O by heterodyne spectroscopy. J Mol Spectrosc 1984;107:62–71. [19] Brown LR, Toth RA. Comparison of the frequencies of NH3 , CO2 , H2 O, N2 O, CO, and CH4 as infrared calibration standards. J Opt Soc Am B 1985;2:842–56. [20] Wells JS, Jennings DA, Hinz A, Murray JS. Heterodyne frequency measurements on N2O at 5.3 and 9.0 μm. J Opt Soc Am B 1985;2:857–61. [21] Wells JS, Hinz A. Heterodyne frequency measurements on N2 O between 1257 and 1340 cm−1 . J Mol Spectrosc 1985;114:84–96. [22] Toth RA. Frequencies of N2 O in the 1100- to 1440-cm−1 region. J Opt Soc Am B 1986;3:1263–81. [23] Toth RA. N2 O vibration-rotation parameters derived from measurements in the 900-1090- and 1580-2380-cm−1 regions. J Opt Soc Am B 1987;4:357–74. [24] Hinz A, Wells JS, Maki AG. Heterodyne measurements of hot bands and isotopic transitions of N2 O near 7.8 μm. Z Phys D 1987;5:351–8. [25] Zink LR, Wells JS, Maki AG. Heterodyne frequency measurements on N2 O near 1060 cm−1 . J Mol Spectrosc 1987;123:426–33. [26] Esplin MP, Barowy WM, Huppi RJ, Vanasse GA. High-resolution fourier spectroscopy of nitrous oxide at elevated temperatures. Mikrochim. Acta [Wien] 1988 II:403-7. [27] Vanek MD, Schneider M, Wells JS, Maki AG. Heterodyne measurements on N2 O near 1635 cm−1 . J Mol Spectrosc 1989;134:154–8.

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