Analysis of the high-resolution water spectrum up to the Second Triad and to J=30

Journal of Molecular Spectroscopy 303 (2014) 36–41

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Analysis of the high-resolution water spectrum up to the Second Triad and to J ¼ 30 L.H. Coudert a,⇑, Marie-Aline Martin-Drumel b,c,1, Olivier Pirali b,c a Laboratoire Inter-universitaire des Systèmes Atmosphériques, UMR 7583 du CNRS, Universités Paris Est Créteil et Paris Diderot, 61 Avenue du Général de Gaulle, 94010 Créteil Cedex, France b SOLEIL Synchrotron, L’Orme des Merisiers, Saint-Aubin, 91192 Gif-Sur-Yvette, France c Institut des Sciences Moléculaires (ISMO), CNRS – Université Paris XI, Bât. 210, 91405 Orsay Cedex, France

a r t i c l e

i n f o

Article history: Received 12 March 2014 In revised form 7 July 2014 Available online 17 July 2014 Keywords: High temperature water FIR spectroscopy Anomalous centrifugal distortion Radio-frequency discharge Bending–Rotation approach Line position analysis Line strength analysis

a b s t r a c t We report high temperature spectroscopic measurements of water vapor carried out in the far infrared domain. The new data set contains numerous transitions characterized by a J-value larger than 30 and allows us to reach rotational levels of water for which information was either unavailable or inaccurate. This new data set, along with previously published microwave, far infrared, and infrared measurements, is fitted using a modified version of the Bending–Rotation approach and allows us to perform the first analysis of the high-resolution spectrum of water vapor up to J ¼ 30 for the ground and (0 1 0) states, up to J ¼ 27 for the First Triad States, and up to J ¼ 19 for the Second Triad States. The results of the analysis are used to build a spectroscopic database supporting high-resolution investigations performed at temperatures as high as 1500 K. Ó 2014 Elsevier Inc. All rights reserved.

1. Introduction Modeling the high-resolution spectrum of the water molecule is theoretically challenging as this molecule is among the most difficult to deal with from the spectroscopic point of view. Water is a light triatomic molecule displaying a strong coupling between overall rotation and its three vibrational modes due to the lightness of the hydrogen atoms and the shortness of the OH bonds. The vibrational mode the most coupled to the overall rotation is the large amplitude bending m2 mode giving rise to the so-called anomalous centrifugal distortion [1]. Water is probably the molecule for which the largest number of theoretical approaches aimed at accounting for its high-resolution spectroscopic data have been developed. Approaches based on the rotational Hamiltonian of a semi-rigid molecule were first used [2] but, although accurate, did not allow us to account for the energy of rotational levels with J > 15. To reach higher J-values, modified rotational Hamiltonians were introduced as in the Generating Function [3] and the Euler Hamiltonian [4] approaches. Because they are not based on a physical model and as they lack Dv 2 > 0 ⇑ Corresponding author. E-mail address: [email protected] (L.H. Coudert). Present address: I. Physikalisches Institut, Universität zu Köln, 50937 Köln, Germany. 1

http://dx.doi.org/10.1016/j.jms.2014.07.003 0022-2852/Ó 2014 Elsevier Inc. All rights reserved.

matrix elements, these approaches tend to require a large number of spectroscopic parameters. First principle calculations based on the exact Hamiltonian of the molecule and relying on a tridimensional potential energy surface retrieved by ab initio calculations were developed more recently and make it possible to reach high J-values and high lying vibrational states. Such calculations are very satisfactory from the physical point of view but still lack spectroscopic accuracy. For instance, with the calculations of Refs. [5,6], the experimental energies reported in Ref. [7] for rotational levels with J ¼ 15 belonging to the low lying (0 2 0) vibrational state are reproduced with root mean square (RMS) values of 0.072 and 0.187 cm1, respectively, which do not compare well with the largest uncertainty value of the experimental energies, 0.005 cm1. The Bending–Rotation Hamiltonian, introduced some time ago, is an effective approach in which the overall rotation and the bending m2 mode are treated together. This approach was first applied to the ground [8] and (0 1 0) vibrational states [9–11] and later extended to the vibrational states of the First [7,12] and Second [13,14] Triads. Although this approach only allows us to deal with a limited number of vibrational states, it is nonetheless very accurate. It is more accurate than first principle calculations [5,6] since the experimental energies of the reduced set of rotational levels mentioned above were reproduced in a previous investigation [12] with an RMS value of 0.003 cm1, well within the largest uncertainty value of the experimental energies. The Bending–Rotation

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L.H. Coudert et al. / Journal of Molecular Spectroscopy 303 (2014) 36–41

approach also allows us to reach microwave accuracy since it has been used recently to fit the microwave and terahertz spectra of water [12] up to the First Triad. Recently two new sets of measurements have been reported for the water molecule. The high-resolution far infrared (FIR) spectrum of water was recorded at a temperature of nearly 1800 K [12] and 32 microwave and terahertz transitions within the Second Triad [15] were observed for the first time. The former measurements extend the body of available high-resolution transitions to higher J-values while the latter ones provide us with very accurate spectroscopic information about Second Triad states. These new data also pose a challenge for theoretical model aimed at accounting for the high-resolution spectrum of water. In this paper a line position analysis of a large body of highresolution data pertaining to the water molecule is performed using a slightly modified version of the Bending–Rotation approach. The maximum value of J is 30, which is a higher value than in previous similar investigations [7,12–14]. The fitted data set includes the FIR transition measured in Ref. [12], new FIR transitions measured in the present work with the radio-frequency discharge setup of this reference, the newly reported microwave transitions [15], and previously published data. The results of the line position analysis are used to build a spectroscopic database for hot water vapor. This paper has five remaining sections. In Section 2 the new FIR spectroscopic data are presented. In Section 3 the modified version of the Bending–Rotation approach is introduced. Section 4 deals with the line position analysis. In Section 5, a line strength analysis is performed and the database for hot water vapor is built. Section 6 is the discussion.

tions, the agreement was satisfactory as the RMS value of the difference between the observed wavenumber and that calculated with the MARVEL database [18] was 0.003 cm1 which compares well with the experimental uncertainty of the new FIR transitions [12,19]. However, for 100 transitions, the residual values were too large and ranged from 0.005 to 0.111 cm1. Table 1 shows the 10 transitions displaying the largest residuals. 3. Modified Bending–Rotation approach The Bending–Rotation approach as applied to interacting vibrational states with one quantum of energy in either stretching mode was introduced in Lanquetin et al. [7]. An effective Bending– Rotation Hamiltonian was built in Section 3.D of this reference by adding Bending–Rotation distortion operators to the exact Hamiltonian of the molecule. Bending Hamiltonians, including the contribution from these distortion operators, were then obtained for each stretching state and for given values of J and k, as indicated in Eq. (37) of Lanquetin et al. [7]. Bending energies were afterwards computed diagonalizing these bending Hamiltonians. In Coudert et al. [14], a change was made to the Bending–Rotation approach and involved accounting rigorously for the coupling term between the v ¼ 0 and v ¼ 1 stretching states. In the present modified approach, the effects of the Bending– Rotation distortion operators are taken into account in a perturbative manner. They are not included in the bending Hamiltonians, but their effects are evaluated after computing bending energies. This change leads to a more stable least squares fitting of the line positions and to an increased accuracy. 4. Line position analysis

2. Experimental The FIR transitions reported in this work were measured using an experimental setup based on emission spectroscopy of hot water vapor produced in a radio-frequency (RF) discharge plasma experiment. This experiment has previously been described in details in Ref. [16]. Briefly, a continuous flow of water vapor going through a Pyrex cell is excited by an RF discharge up to 1000 W. The intense light emission is collected by an off-axis parabolic mirror and focused onto the entrance iris of the Fourier-Transform (FT) Bruker 125HR interferometer located on the AILES beamline of SOLEIL synchrotron [17]. Two FIR emission spectra of water vapor have been recorded in the frequency range 50–700 cm1 with this experiment using the following conditions of pressure, resolution, and discharge power: 10.5 mTorr, 0.004 cm1, 900 W, and > 15 mTorr, 0.010 cm1, 1000 W. In the previous investigation [12], assignments of these two spectra were reported for water transitions involving rotational levels with J 6 27 and up to the First Triad. In the present work, transitions involving rotational levels with J 6 38 and up to the Second Triad are reported and consist of 6757 transitions including 6273 b-type rotational transitions within a given state, 255 a-type transitions connecting the (0 2 0) and (0 0 1) states and the (1 0 0) and (0 0 1) states, and 229 b-type transitions connecting the (0 2 0) and (1 0 0) states and the (0 1 0) and (0 2 0) states. The list of assigned transitions can be retrieved from the Supplementary Material and includes the transitions reported previously [12]. The new FIR transitions involve rotational levels for which experimental energies were either unavailable or inaccurate. The upper and lower levels of 4819 unambiguously assigned unblended transitions, displaying a satisfactory residual in the line position analysis of Section 4, were searched for in the MARVEL database of Tennyson et al. [18]. For two transitions, the upper or lower levels could not be found. For the remaining 4817 transi-

In the line position analysis, rovibrational energies were calculated using the Bending–Rotation approach as outlined in Section 3. Data were introduced in a weighted least-squares fitting program in which each data point was given a weight equal to the square of the inverse of its experimental uncertainty. The data set consists of three different types of data: rotational energy levels, FIR and infrared (IR) lines, and microwave transitions. The number of data is 24 461 and the unitless standard deviation of the fit is 1.2. Unresolved K-type doublets were treated as in Section 4 of Ref. [10]. The analysis results are presented below for each data set with the help of several quantities including STD, the unitless number introduced in Section 3.2.1 of Ref. [14].

Table 1 FIR linesa displaying large residuals with the MARVEL database [18]. J

Ka

Kc

v

J

Ka

Kc

v

rb

O–Cc

26 21 27 19 27 29 22 27 26 29

11 7 13 19 11 4 15 7 11 7

15 15 15 0 16 25 7 21 15 23

(0 0 0) (0 2 0) (0 0 0) (0 2 0) (0 0 0) (0 1 0) (1 0 0) (0 1 0) (0 0 0) (0 0 0)

26 21 27 19 26 29 21 26 26 28

10 6 12 18 12 3 14 6 8 6

16 16 16 1 15 26 8 20 18 22

(0 0 0) (0 2 0) (0 0 0) (0 2 0) (0 0 0) (0 1 0) (0 2 0) (0 1 0) (0 0 0) (0 0 0)

241.144 258.958 288.328 307.655 330.104 406.711 479.124 588.510 591.281 604.753

53 97 67 27 36 58 111 31 67 27

a Unblended FIR transitions, unambiguously assigned and satisfactorily fitted in Section 4, are listed. They are identified with their rotational and vibrational quantum numbers and with their observed wavenumber r. b The experimental wavenumber is given in cm1. Its experimental uncertainty is on the order of 0.001 cm1. c O–C is the observed minus calculated residual, in 103 cm1, calculated with experimental levels extracted from the MARVEL database [18]. The 10 transitions listed display the largest residuals.

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L.H. Coudert et al. / Journal of Molecular Spectroscopy 303 (2014) 36–41

For the ground and (0 1 0) states, the experimental energies of Lanquetin et al. [7] were taken up to J ¼ 25 and 21, respectively, and those of Tennyson et al. [18] for 26 6 J 6 30 and 22 6 J 6 30, respectively. For the First Triad states, the experimental energies of Lanquetin et al. [7] were taken up to J ¼ 20 and those of Tennyson et al. [18] for 21 6 J 6 26. At last, for the Second Triad states, the experimental energies of Coudert et al. [13] were taken up to J ¼ 10 and those of Tennyson et al. [18] for 11 6 J 6 18. The energy uncertainties given by these authors were in many cases found to be underestimated as inclusion of their experimental energies led to a large increase of the unitless standard deviation of the fit. In order to obtain a meaningful fit, energy uncertainties were modified and some levels were excluded from the analysis as they displayed an experimental energy differing by more than 0.06 cm1 from the calculated one. The results for the 4291 experimental rotational energies are given in Table 2 which lists the number of levels, the maximum J-value, RMS, and STD for each vibrational state. This table should be compared with Tables 1 and 3 of Ref. [13] and Table 4 of Ref. [14]. Calculated energies for the levels considered in the analysis can be found in the Supplementary Material. The fitted FIR and IR transitions include the new FIR transitions described in Section 2, IR transitions measured at room temperature [20–30], and FIR and IR transitions measured using hot water vapor spectroscopy [13,31–38]. The analysis was restricted to J ¼ 30 for transitions involving the ground and (0 1 0) states, to J ¼ 27 (19) for transitions involving the First (Second) Triad states. For the 5511 new FIR transitions, an experimental uncertainty of 0.8 or 1:4  103 cm1 was taken depending on the spectrum used [12]; a larger value was taken in some cases for weak lines. The results of the analysis are given in Table 3 which lists the number of transitions, maximum J- and K a -values, RMS, and STD for each type of band. This table emphasizes that the STD values are satisfactory and close to 1 up to the First Triad. Slightly less satisfactory values arise for b-type transitions within Second Triad States. The observed minus calculated table for the new FIR transitions can be found in the Supplementary Material. Table 4 summarizes the results obtained in the analysis for previously reported FIR and IR data sets. In addition to the same information as in Table 3, this

Table 2 Analysis resultsa for experimental energy levels. State

N

J

RMSb

STDc

(0 0 0) (0 1 0) (0 2 0) (1 0 0) (0 0 1) (0 3 0) (1 1 0) (0 1 1)

791 751 573 562 605 325 343 341

30 30 26 26 26 18 18 18

8.7 12.6 13.3 10.7 9.1 11.4 11.2 5.2

1.1 1.5 1.4 1.3 1.1 1.2 1.2 0.6

a Number of fitted experimental levels N, maximum J-value, RMS, and STD are given for each vibrational state. b RMS is the root mean square deviation of the observed minus calculated residuals in 103 cm1. c STD is the unitless number defined in Ref. [14].

Table 3 Analysis results for the new FIR transitions.a State/Band

N

J

Ka

RMSb

STDc

(0 0 0) (0 1 0) (0 2 0) (1 0 0) (0 0 1) (0 3 0) (1 1 0) (0 1 1) (0 1 0) (0 2 0) (0 2 0) (1 0 0)

898 765 723 735 835 385 362 415 6 181 73 133

30 30 27 27 27 19 19 19 19 23 22 21

21 18 19 17 17 12 15 16 12 18 20 8

1.0 1.3 1.4 1.5 1.6 1.5 1.7 1.6 0.9 2.1 3.0 1.0

1.0 1.2 1.4 1.3 1.2 1.8 1.8 1.6 1.2 1.1 2.0 0.7

(0 2 0) (1 0 0) (0 0 1) (0 0 1)

a Number of transitions N, maximum J- and K a -values, RMS, and STD values are given for transitions within the state or belonging to the band in the column headed State/Band. This data set is described in Section 2. b RMS is the root mean square deviation of the observed minus calculated residuals in 103 cm1. c STD is the unitless number defined in Ref. [14].

Table 4 Analysis resultsa for previously reported FIR and IR data sets. References

Statesb

N

nc

J

Ka

RMSd

STDe

Johns [20] Paso and Horneman [21] Kauppinen et al. [22] Mandin et al. [39] Dana et al. [40] Toth [23] Toth [24] Toth [25] Mikhailenko et al. [27] Esplin et al. [31] Polyansky et al. [32–35] Mikhailenko et al. [26] Toth [28] Coudert et al. [13] Zobov et al. [36] Coheur et al. [38] Horneman et al. [30] Toth [29] Zobov et al. [37]

(0 0 0) (0 0 0) (0 0 0) (0 0 0), (0 0 0), (0 0 0), (0 0 0), (0 1 0), (0 0 0), (0 0 0), All All (0 0 0), All (0 0 0), (0 0 0), (0 0 0) (0 0 0), All

59 228 18 8 42 1632 2204 236 915 597 1572 503 164 1485 90 26 109 1540 2481

1 4 1 1 1 5 3 0 25 2 39 21 2 25 14 4 5 17 29

12 17 13 24 19 20 19 12 18 23 25 18 14 21 8 8 17 17 28

7 11 10 11 15 14 11 7 11 13 25 11 9 15 7 7 10 8 23

0.2 0.2 0.2 6.6 5.4 0.3 0.5 0.3 0.7 3.0 4.8 1.6 1.4 0.8 1.8 7.6 0.0 0.4 9.6

1.1 0.9 1.8 1.3 1.1 0.4 0.9 0.9 1.1 0.9 0.5 2.0 1.0 0.9 0.8 1.2 1.3 0.7 1.0

(0 1 0) (0 1 0) (0 1 0) (0 1 0), (1 0 0), (0 0 1) (0 2 0) (0 2 0) (0 1 0), (0 2 0), (0 3 0)

(0 3 0), (1 1 0), (0 1 1) (0 1 0), (0 2 0), (1 0 0), (0 0 1) (0 1 0), (0 2 0), (1 0 0), (0 0 1) (0 1 0), (0 3 0), (1 1 0), (0 1 1)

a Number of transitions N, number of excluded transitions, states involved, maximum J- and K a -values, RMS, and STD values are given for the data set reported in the investigation in the Reference column. b The states involved are given in this column. All stands for the eight vibrational states up to the Second Triad. c n is the number of transitions excluded from the analysis disregarding those removed because available in another reference. d RMS is the root mean square deviation of the observed minus calculated residuals in 103 cm1. e STD is the unitless number defined in Ref. [14].

L.H. Coudert et al. / Journal of Molecular Spectroscopy 303 (2014) 36–41 Table 5 Analysis results for the microwave transitionsa. State/Band

N

J

Ka

RMSb

STDc

(0 0 0) (0 1 0) (0 2 0) (1 0 0) (0 0 1) (0 3 0) (1 1 0) (0 1 1) (0 2 0) (0 2 0) (1 0 0)

252 172 47 46 41 6 15 15 4 4 24

18 14 11 14 9 4 6 6 9 6 10

11 8 7 5 5 2 2 3 5 6 7

0.2 0.3 1.7 0.4 0.5 1.9 0.6 1.1 1.5 0.7 0.4

1.4 1.7 8.3 2.1 2.3 8.1 2.6 4.8 0.6 4.3 1.6

(1 0 0) (0 0 1) (0 0 1)

a Number of transitions N, maximum J- and K a -values, RMS, and STD values are given for microwave transitions within the state or belonging to the band in the column headed State/Band. b RMS is the root mean square deviation of the observed minus calculated residuals in MHz. c STD is the unitless number defined in Ref. [14].

B0e V 01

Valueb 37.35557(5) 5.15(3)

Parametera B2e V 20

Valuec 36.5502(7) 3771.34(3)

V 02

19454.9(2)

V 21

586.8(5)

V 03

683. (4)

V 22

19135. (4)

V 04

3147. (4)

V 23

1288. (21)

V 05

4115. (10)

V 24

3960. (68)

V 06

1510. (16)

V 25

3929. (83)

V 07

1602. (13)

V 26

V 08

509. (31)

V 27 V 28

B1e V 10 V 11 V 12 V 13 V 14 V 15 V 16 V 17 V 18

712. (250) 776.730047c 1731.10318c

36.665(3) 3666.83(4)

B01 e

1.9806(30)

B02 e

1.993(12)

B12 e

0.61614(2)

te

0.294883558c

150.7(6) 19128. (5) 231. (34) 3702. (73) 829. (340) 5405. (260)

which was set to 0.01 cm1. 29 transitions of Zobov et al. [37] were excluded from the analysis as they displayed residuals that were too large. Among those, 12 are listed by the authors as transitions having an unconfirmed assignments [37]. The FIR transitions reported by Horneman et al. [30] also provide a test for the present model as they are characterized by very small uncertainties ranging from 105 to 0:12  103 cm1. As emphasized by Table 4, their wavenumbers could be reproduced with a satisfactory STD of 1.3. The microwave data set consists of the transitions reported recently by Yu et al. [15] and those reported previously by Yu et al. [12]. Table 5 gives the results of the analysis for the microwave transitions. The largest STD values are for transitions within the (0 2 0) and (0 3 0) states. The number of varied spectroscopic parameters is 399. Only parameters which correspond to the lowest powers of rotational operators in the theoretical model [14] are reported. They are given in Table 6 which should be compared to Table 6 of our previous investigation [14]. We can see that the largest changes are for high-order potential energy terms in ðt  t e Þn with n > 4. 5. Line strength analysis

Table 6 Effective Hamiltonian parameters. Parametera

39

8181. (960) 1533.799 10c

a

Parameters are defined in Eqs. [32]-[41] of Ref. [7]. Values are given in cm1, except for te which is unitless. For varied parameters, uncertainties are given in parentheses in the same units as the last digit. c Constrained to the value of Ref. [7]. b

table gives the number of transitions excluded from the analysis, n, disregarding those removed from a given reference because available in another one with a more accurate wavenumber. Comparing this table with Table 6 of Coudert et al. [14] shows that similar results were obtained for data sets fitted in both investigations. In the present one, four data sets, not considered in Ref. [14] were added. They consist of the IR transitions measured by Zobov et al. [36,37] and Coheur et al. [38] and of the FIR transitions reported by Horneman et al. [30]. The IR transitions reported by Zobov et al. [37], involving high J-value levels, provide a test for the theoretical model. Only the transitions listed by these authors as not appearing in HITRAN were fitted. Table 4 shows that for these 2481 transitions the RMS is 0.0096 cm1, which is larger than the experimental uncertainty of 0.001 cm1 quoted by the authors [37]. As the residuals did not display any systematic trend, this RMS value was deemed a reliable estimate of the experimental uncertainty

An analysis of previously published line strengths [14,23– 25,29,41–44] and C 2 Stark coefficients [45] was carried out computing the intensity as indicated in Section 4.1 of Coudert et al. [14]. The eigenvectors needed to evaluate the matrix elements of the transition moment, the transition wavenumbers, and the lower level energy were retrieved from the results of Section 4. In the fitting procedure, the spectroscopic constants involved in the transition moment expansion in Eq. (10) of Coudert et al. [14] were determined. In this second analysis, 7054 line strengths and C 2 Stark coefficients were considered. Intensities were given a weight equal to the inverse of their experimental uncertainty squared. In the case of the C 2 Stark coefficients [45], an experimental uncertainty of 0.2% was taken as in Ref. [14]. The strength of transitions corresponding to unresolved K-type doublets was calculated as in Section 4.2 of the same reference. The unitless standard deviation of this second analysis is 1.2 and 233 spectroscopic parameters were varied. A linelist spanning the spectral range 10–7000 cm1 was built using the results of the line strength analysis. An intensity cutoff of 1027 cm1/(molecule cm2) was taken for a temperature of 1500 K; the maximum J-values were the same as in the line position analysis (Section 4). This database is thus well suited for modeling the spectrum of hot water. The database was formatted as a HITRAN linelist and is available in the Supplementary Material. No effort was made to estimate the value of the line shape parameters which were set to zero. The database contains 217 387 lines of which 34 140 can also be found in the 2012 version of HITRAN [46]. For the common lines, there is a satisfactory agreement for line positions and line strengths when the latter is larger than 1025 cm1/(molecule cm2) at the reference temperature of 296 K. The new database was used to compute an emission spectrum which was compared to the experimental one, described in Section 2. The intensity of a spectral line in emission Inm em connecting the upper level n and the lower level m was calculated [47] with:

Nn hcmnm Anm

ð1Þ

where N n is the number of molecules in the upper level n, h is Planck’s constant, c the speed of light, mnm the wavenumber of the line, and Anm is the Einstein transition probability of spontaneous emission. The signal recorded with the experimental setup of Ref. [12] was modeled assuming Boltzmann equilibrium for population distribution:

Nn ¼ Ng n expðhcEn =kTÞ=Q ðTÞ

ð2Þ

40

L.H. Coudert et al. / Journal of Molecular Spectroscopy 303 (2014) 36–41

Fig. 1. Comparison between observed (OBS) and calculated (CAL) emission spectra. The former IOBS ðrÞ is described in Section 2; the latter ICAL ðrÞ is calculated as indicated in Section 5; log½ICAL ðrÞ=IMax  in the range 3 to 0 is plotted.

Table 7 Excluded levels in Coudert et al. [14] and in the present work. Level

Energya

O–Cb

Energyc

O–Cd

ð001Þ208;13 ð030Þ113;8 ð030Þ114;7 ð030Þ135;9 ð030Þ137;7 ð030Þ154;11 ð030Þ158;8 ð030Þ1511;5 ð030Þ1612;5 ð030Þ1714;4 ð030Þ1814;5 ð030Þ1815;4 ð110Þ132;12 ð110Þ155;10 ð110Þ167;9 ð110Þ1710;7 ð011Þ155;10 ð011Þ1616;1 ð011Þ178;9 ð011Þ1716;2 ð011Þ187;12 ð011Þ188;11 ð011Þ1812;7

9588.319 6592.187 6727.938 7523.003 8037.797 8079.373 9015.569 10028.267 10750.440 11849.274 12284.705 12638.388 7253.618 8597.896 9296.947 10364.474 8694.210 11715.475 9982.270 12138.292 10185.347 10399.530 11425.995

0.035 0.005 0.002 0.002 0.072 0.005 0.071 0.119 0.120 0.399 0.225 0.260 0.032 0.043 0.080 0.093 0.000 0.182 0.121 0.058 0.271 0.048 0.082

9588.319 6592.186 6727.938 7523.001 8037.732 8079.374 9015.504 10028.383 10750.553 11849.630 12285.147 12638.651 7253.647 8597.916 9297.026 10364.361 8719.945 11715.331 9982.149 12138.208 10185.075 10399.475 11425.922

0.042e 0.002 0.001 0.003 0.007 0.001 0.007 0.008 0.007 0.117e 0.352e 0.032 0.003 0.016 0.010 0.033 0.009 0.148e 0.002 0.005 0.018 0.013 0.026

a Experimental energy (cm1) in Ref. [14] of the levels excluded from the analysis of that reference. b Experimental minus calculated energy (cm1) in Ref. [14]. c Experimental energy (cm1) in this work. d Experimental minus calculated energy (cm1) in this work. e Indicates levels excluded from the analysis.

where N is the total number of molecules in the cell, g n is the degeneracy of the upper level n; En is its energy in cm1 , k is Boltzmann’s constant, Q(T) is the partition function, and T is the temperature. A calculated emission spectrum ICAL ðrÞ was computed using Eqs. (1) and (2), taking a Gaussian line profile with a half width at half height of 0.004 cm1, and setting the temperature to 1576 K as reported in Ref. [12]. Under these conditions, preliminary comparisons between observed and calculated spectra were not satisfactory as the processes by which hot water vapor is produced in the cell by the radio-frequency discharge are complex and difficult to model [48]. Unexpectedly, the best agreement was obtained comparing the experimental signal and the logarithm of the calculated one. This is illustrated in Fig. 1 displaying a comparison between

IOBS ðrÞ and log½ICAL ðrÞ=IMax , where IMax is the maximum value of the signal for the range in this figure. Although this choice is arbitrary and cannot be theoretically substantiated, it led to a visually satisfactory agreement. 6. Discussion The Bending–Rotation approach is applied to the analysis of a large body of high-resolution data pertaining to the water molecule and involving vibrational states up to the Second Triad. In addition to previously published data, the data set includes FIR transitions recorded at a high temperature in this work and in Ref. [12] and the first microwave transitions [15] measured recently for the Second Triad. The present work extends a previous investigation carried out with the Bending–Rotation approach [14] to higher values of the rotational quantum number J for all vibrational states. The analysis results are displayed in Tables 2–5. The FIR transitions recorded in this work and in Ref. [12] provide a stringent test for the Bending–Rotation approach as they involve levels characterized by high J-values. For these transitions, satisfactory results were obtained up to the First Triad; but they tend to degrade slightly for the Second Triad as evidenced by Table 3. For microwave transitions, Table 5 shows that large STD values are obtained for lines within the (0 2 0) and (0 3 0) states. Such an outcome is not unexpected as it is for these states that the effects of the anomalous centrifugal distortion are the strongest. The data set also contains experimental energy levels from Refs. [7,13,18]. As some of them were excluded from the analysis, it is important to understand if this is due to a failure of the Bending–Rotation approach or to the fact that their experimental energy are not correct. The left hand side of Table 7 displays the 23 experimental levels which, in the previous investigation [14], were excluded from the analysis. The right hand size of this table gives the experimental energies and the residuals obtained in this work. We can see that only four levels are still excluded and this is due to the fact that the present calculation is more satisfactory that the previous one [14] and that the experimental energies used in this work are more accurate. For the four excluded levels, an assignment error, although unlikely, cannot be ruled out. It should be stressed that two of them, the ð030Þ1714;4 and ð030Þ1814;5 levels, are within less than 3 cm1 from levels with the same symmetry which are not considered in the Bending–Rotation approach, the ð050Þ174;14 and the ð210Þ182;17 levels, respectively, with which they may interact.

L.H. Coudert et al. / Journal of Molecular Spectroscopy 303 (2014) 36–41

The results of the line position analysis, along with those of the line intensity analysis reported in Section 5, were utilized to build a HITRAN-type database designed for water vapor at a temperature of 1500 K. A comparison between the experimental spectrum recorded at a high temperature in this work and that calculated with the help of this database was performed. The results are shown in Fig. 1 for the region from 310 to 320 cm1. As stressed in Section 5, a qualitative agreement between both spectra could only be achieved computing the theoretical spectrum with an analytical expression incompatible with a simple light emission process. Appendix A. Supplementary material Supplementary data for this article are available on ScienceDirect (http://www.sciencedirect.com) and as part of the Ohio State University Molecular Spectroscopy Archives (http://library. osu.edu/sites/msa/jmsa_hp.htm). Supplementary data associated with this article can be found, in the online version, at http:// dx.doi.org/10.1016/j.jms.2014.07.003. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]

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