ExoMol molecular line lists XIX: high accuracy computed hot line lists for H_2^{18}O and H_2^{17}O
Oleg L. Polyansky, Aleksandra A. Kyuberis, Lorenzo Lodi, Jonathan Tennyson, Sergei N. Yurchenko, Roman I. Ovsyannikov, Nikolai F. Zobov
MMon. Not. R. Astron. Soc. , 1– ?? (2014) Printed 13 November 2018 (MN L A TEX style file v2.2)
ExoMol molecular line lists XIX: high accuracy computedhot line lists for H
O and H O Oleg L. Polyansky , , Aleksandra A. Kyuberis , Lorenzo Lodi ,Jonathan Tennyson (cid:63) , Sergei N. Yurchenko , Roman I. Ovsyannikov ,and Nikolai F. Zobov Department of Physics and Astronomy, University College London, London WC1E 6BT, UK Institute of Applied Physics, Russian Academy of Sciences, Ulyanov Street 46, Nizhny Novgorod, Russia 603950.
Accepted XXXX. Received XXXX; in original form XXXX
ABSTRACT
Hot line lists for two isotopologues of water, H
O and H
O, are presented. Thecalculations employ newly constructed potential energy surfaces (PES) which takeadvantage of a novel method for using the large set of experimental energy levels forH
O to give high quality predictions for H
O and H
O. This procedure greatlyextends the energy range for which a PES can be accurately determined, allowingaccurate prediction of higher-lying energy levels than are currently known from directlaboratory measurements. This PES is combined with a high-accuracy, ab initio dipolemoment surface of water in the computation of all energy levels, transition frequenciesand associated Einstein A coefficients for states with rotational excitation up to J = 50 and energies up to 30 000 cm − . The resulting HotWat78 line lists complement thewell-used BT2 H O line list (Barber et.al, 2006, MNRAS, , 1087). Full line listsare made available in the electronic form as supplementary data to this article and at . Key words: molecular data; opacity; astronomical data bases: miscellaneous; planetsand satellites: atmospheres; stars: low-mass; stars: brown dwarfs.
Water spectra can be observed from many different regimes in the Universe, several of which are discussed further below. Thespectrum of water, particularly at elevated temperatures, is rich and complex. A few years ago Barber et al. (2006) presenteda comprehensive line list, known as BT2, which used well-established theoretical procedures to compute all the transitions ofH
O of importance in objects with temperatures up to 3000 K. BT2 contains about 500 million lines. A similar line list forHD O, known as VTT, was subsequently computed by Voronin et al. (2010).The BT2 line list has been extensively used. It forms the basis of the most recent release of the HITEMP high-temperaturespectroscopic database (Rothman et al. 2010) and for the BT-Settl model (Allard 2014) for stellar and substellar atmospherescovering the range from solar-mass stars to the latest-type T and Y dwarfs. BT2 has been used to detect and analyse waterspectra in objects as diverse as the Nova-like object V838 Mon (Banerjee et al. 2005), atmospheres of brown dwarfs (Riceet al. 2010) and M subdwarfs (Rajpurohit et al. 2014), and extensively for exoplanets (Tinetti et al. 2007; Birkby et al.2013). Within the solar system BT2 has been used to show an imbalance between nuclear spin and rotational temperaturesin cometary comae (Dello Russo et al. 2004, 2005) and assign a new set of, as yet unexplained, high energy water emissionsin comets (Barber et al. 2009), as well as to model water spectra in the deep atmosphere of Venus (Bailey 2009).Although BT2 was developed for astrophysical use, it has been applied to a variety of other problems including thecalculation of the refractive index of humid air in the infrared (Mathar 2007), high speed thermometry and tomographicimaging in gas engines and burners (Kranendonk et al. 2007; Rein & Sanders 2010), as the basis for an improved theory (cid:63)
Email: [email protected]© 2014 RAS a r X i v : . [ a s t r o - ph . E P ] F e b Polyansky et al of line-broadening (Bykov et al. 2008), and to validate the data used in models of the earths atmosphere and in particularsimulating the contribution of weak water transitions to the so-called water continuum (Chesnokova et al. 2009).There are several water line lists published in the literature (Viti et al. 1997; Partridge & Schwenke 1997; Barber et al.2006; Mikhailenko et al. 2005). Two linelists have also been computed specifically for the isotopologues: Shirin et al. (2008)created the 3mol room-temperature line lists for H
O, H
O and H
O based on the PES of Shirin et al. (2006); Tashkuncreated a number of line lists based on the work of Partridge & Schwenke (1997), see Mikhailenko et al. (2005). These areconsidered further below.At present hot line lists are only published for H
O and HD O. However isotopically-substituted water containing Oor O provides important markers for a variety of astronomical problems (Nittler & Gaidos 2012). For example Matsuuraet al. (2014) recently detected H
O in the emission-line spectrum of the luminous M-supergiant VY CMa. Astronomicalspectra of water isotopologues (Neufeld et al. 2013) and their direct analysis in cometary dust particles (Floss et al. 2010) andcarbonaceous chrondrites (Clayton & Mayeda 1984; Vollmer et al. 2008) have been used to determine formation mechanismsand constrain formation models. Water isotope ratios are also used to monitor stellar evolution (Abia et al. 2012) and toprobe the atmosphere of Mars (Villanueva et al. 2015). The seemingly minor isotopologues of water can be important speciesin their own right with, for example, H
O being the fifth largest absorber of sunlight the earth’s atmosphere.There is therefore a need for line lists equivalent to BT2 for H
O and H
O to aid spectroscopic studies, and it is thesethat are presented here. These lists form part of the ExoMol project (Tennyson & Yurchenko 2012) which aims to provide acomprehensive set of molecular line lists for studies of molecular line lists for exoplanet and other hot atmospheres.Although our new line lists in some way mimic BT2, they also take advantage of a number of recent theoretical devel-opments. In particular a IUPAC task group (Tennyson et al. 2014) used a systematic procedure (Furtenbacher et al. 2007)to derive empirical energy levels for all the main isotopologues of water (Tennyson et al. 2009, 2010, 2013, 2014). Theselevels are combined with a newly-developed procedure for enhancing the accuracy of calculations on isotopically substitutedspecies, which is used for the first time here. This ensures that most of the key frequencies in our new line lists are determinedwith an accuracy close to experimental, even though many of them are yet to be observed. Furthermore, theoretical work onimproving the accuracy and representation of the water dipole moment (Lodi et al. 2008, 2011) has improved the accuracywith which water transition intensities are predicted (Grechko et al. 2009). Some of these advances have already been usedto create improved room temperature line lists for H
O and H
O (Lodi & Tennyson 2012) which were included in theirentirety in the 2012 release of HITRAN (Rothman L. S. et al. 2013).The paper is structured as follows: section 2 outlines our overall methodology and presents the derivation of potentialenergy surfaces (PES). The details of the calculation of the new line lists, along with comparison with previous line lists, aregiven in section 3. Section 4 discusses further improvement of the line list by the substitution of calculated energy levels withempirical ones, together with the procedure used to label energy levels with approximate vibrational and rotational quantumnumbers. Our results are discussed in section 5.
The fitting of water (H
O) PESs to experimental spectroscopic data has a long history. The first fitted PES giving nearto experimental accuracy was PJT1 (Polyansky et al. 1994). Partridge & Schwenke (1997) constructed a fitted PES startingfrom a highly accurate ab initio calculation; all subsequent water potentials followed this procedure and have been based on ab initio studies of increasing sophistication. As a result there are several very good water PESs available (Shirin et al. 2003,2008; Bubukina et al. 2011).Here we need a PES which satisfies two criteria. First, it should be at least as accurate as the PES used for the BT2 linelist with the calculated energies ranging up to 30 000 cm − . Second, the PES should be adapted to the calculation of energylevels of the two water isotopologues H O and H
O. This second requirement is harder to fulfill, as the characterisation ofthe experimental energy levels of both H
O and H
O is significantly less extensive than for H
O (Tennyson et al. 2014).To take advantage of the accumulated knowledge on the spectrum H
O in constructing a PES for H
O and H
O andfollowing previous work (Zobov et al. 1996; Voronin et al. 2010; Bubukina et al. 2011), we decided to fit a Born-Oppenheimer(BO) mass-independent PES to the available data for H
O and fix the adiabatic BO diagonal correction (BODC), mass-dependent surface to the ab initio value of Polyansky et al. (2003). Obviously this procedure requires the accuracy ofpredictions for H
O and H
O to be verified. This is done by comparing the calculated H
O and H
O energy levels tothe available experimentally-determined ones (Tennyson et al. 2009, 2010).We used the same fitting procedure as Bubukina et al. (2011). Nuclear motion calculations were performed with
DVR3D (Tennyson et al. 2004). As elsewhere, in the fit the experimentally derived energies of H
O for the J = 0 , and 5 rotationalstates by Tennyson et al. (2013) were used.In the following our new empirical PES obtained using the fitting procedure described above will be referenced to asPES1, while the PES by Bubukina et al. (2011) will be referenced to as PES2. Tables 1 and 2 present a comparison between © 2014 RAS, MNRAS , 1– ?? xoMol XIX: H O and H O Table 1.
Comparison of calculated J = 0 term values for H O using three potentials with experimental data. Experimental (obs) datais taken from Tennyson et al. (2009). v v v Observed PES1 Obs.-Calc. PES2 Obs.-Calc. PES3 Obs.-Calc.0 0 1 3748.318 3748.334 -0.02 3748.326 -0.01 3748.463 -0.150 0 2 7431.076 7431.103 -0.03 7431.059 0.02 7431.467 -0.390 0 3 11011.883 11011.936 -0.05 11011.860 0.02 11012.268 -0.380 1 0 1591.326 1591.297 0.03 1591.342 -0.02 1591.413 -0.090 1 1 5320.251 5320.241 0.01 5320.251 0.00 5320.378 -0.130 1 2 8982.869 8982.868 0.00 8982.844 0.03 8983.118 -0.250 1 3 12541.227 12541.267 -0.04 12541.207 0.02 12541.614 -0.390 2 0 3144.980 3144.934 0.05 3144.993 -0.01 3145.085 -0.100 2 1 6857.273 6857.260 0.01 6857.266 0.01 6857.476 -0.200 7 1 13808.273 13808.224 0.05 13808.371 -0.10 13809.171 -0.901 0 0 3653.142 3653.147 0.00 3653.121 0.02 3653.193 -0.051 0 1 7238.714 7238.773 -0.06 7238.726 -0.01 7238.932 -0.221 0 2 10853.505 10853.545 -0.04 10853.504 0.00 - -1 0 3 14296.280 14296.340 -0.06 14296.265 0.01 14296.584 -0.301 1 0 5227.706 5227.691 0.01 5227.704 0.00 5227.881 -0.181 1 1 8792.544 8792.578 -0.03 8792.546 0.00 8792.816 -0.271 2 0 6764.726 6764.747 -0.02 6764.722 0.00 6764.905 -0.181 2 1 10311.202 10311.247 -0.05 10311.199 0.00 10311.421 -0.221 3 1 11792.822 11792.861 -0.04 11792.834 -0.01 11793.172 -0.352 0 0 7193.246 7193.265 -0.02 7193.257 -0.01 7193.394 -0.152 0 1 10598.476 10598.550 -0.07 10598.483 -0.01 10598.763 -0.292 1 1 12132.993 12133.056 -0.06 12132.984 0.01 12132.365 0.632 2 1 13631.500 13631.542 -0.04 13631.489 0.01 13631.650 -0.153 0 1 13812.158 13812.215 -0.06 13812.170 -0.01 13812.394 -0.243 2 1 16797.168 16797.182 -0.01 16797.177 -0.01 16797.011 0.164 0 1 16875.621 16875.662 -0.04 16875.643 -0.02 16875.474 0.15 the J = 0 energy levels calculated using PES1, PES2 for H O and H
O respectively. For comparison as a third columnwe present the J = 0 levels and corresponding discrepancies using the PES (called PES3 in the tables) due to Partridge &Schwenke (1997) taken from the linelist calculated by Dr. S.A. Tashkun and summarised by Mikhailenko et al. (2005). Theline list based on PES3 was calculated for three temperatures: T=296 K, 1000 K and 3000 K. For all versions the highestvalue of the rotational quantum number J considered is 28 and the spectral range is 0-28500 cm − . The number of lines forH O is 108 784 and for H
O 109 083.Indeed, one can see that the agreement with the experiment is very good. Although the results obtained using PES2are somewhat better than those for PES1. However employing PES1 gives us the opportunity to use the information onH
O experimental energy levels to predict very accurately energy levels of H
O and H
O. We call these predicted levelspseudo-experimental energies for the reasons explained below. Table 3 illustrates the unprecedented accuracy of the predictionof the H
O energy levels for those states whose energies are known experimentally. The slightly less good, but still veryaccurate, energy levels predicted for H
O are shown in the column 2 of Table 3. We might expect a similar level of accuracyfor predictions of the H
O and H
O energy levels for states yet to be measured for these isotopologues, but known forH
O. We note that the standard deviations given in Table 3 are rather systematic suggesting that further improvement inthe predictions may be possible. This and details of our final pseudo-experimental energy levels are discussed in section 4.Recently, highly lying energy levels of H
O have been measured using multiphoton spectroscopy (Makarov et al. 2015).These levels lie at about 27 000 cm − and therefore provide a stringent test of our procedure. The highest upper energy levelconsidered in this work, as for BT2, is 30 000 cm − ; Table 4 illustrates the high quality of our calculations over the wholerange considered. In fact recent studies confirm that BT2 is not so accurate for these high energy states (Lampel et al. 2016).Thus, the line lists, details of whose calculations are given in the following section, are computed using a higher qualityPES than that used to compute BT2. Three sets of energy levels are provided as part of this line list. The first set is thevariationally calculated energy levels obtained using PES2. The second set comprises these energy levels substituted by theexperimental values (Tennyson et al. 2009) where available. The third set is further with pseudo-experimental energy levelssubstituted whenever H O experimental energy levels (Tennyson et al. 2013) are available (see below). This third set is theone we recommend for creating spectra with HotWat78 because of its increased accuracy.
O AND H O The line list calculations were performed with the
DVR3D program suite (Tennyson et al. 2004) using the PES1 and PES2discussed above, and the ab initio dipole moment surfaces LTP2011S of Lodi et al. (2011). As for BT2, the highest rotational © 2014 RAS, MNRAS , 1– ?? Polyansky et al
Table 2.
Comparison of calculated J = 0 term values for H O using three potentials with experimental data. Experimental (obs) datais taken from Tennyson et al. (2009). v v v Observed PES1 Obs.-Calc. PES2 Obs.-Calc. PES3 Obs.-Calc.0 0 1 3741.57 3741.581 -0.01 3741.567 0.00 3741.575 -0.010 0 2 7418.72 7418.741 -0.02 7418.693 0.03 7418.759 -0.030 0 3 10993.68 10993.734 -0.05 10993.659 0.02 10993.689 -0.010 1 0 1588.28 1588.240 0.04 1588.271 0.00 1588.299 -0.020 1 1 5310.46 5310.443 0.02 5310.438 0.02 5310.388 0.070 1 2 8967.57 8967.552 0.01 8967.519 0.05 8967.491 0.070 1 3 12520.12 12520.153 -0.03 12520.089 0.03 12520.068 0.060 2 0 3139.05 3138.999 0.05 3139.038 0.01 3139.031 0.020 2 1 6844.60 6844.580 0.02 6844.566 0.03 6844.539 0.060 2 2 10483.22 10483.264 -0.04 10483.202 0.02 10483.212 0.010 3 0 4648.48 4648.435 0.04 4648.469 0.01 4648.452 0.030 3 1 8341.11 8341.109 0.00 8341.086 0.02 8341.114 -0.010 3 2 11963.54 11963.580 -0.04 11963.507 0.03 11963.615 -0.080 4 0 6110.42 6110.408 0.02 6110.433 -0.01 6110.410 0.010 4 1 9795.33 9795.354 -0.02 9795.324 0.01 9795.329 0.001 0 0 3649.69 3649.688 0.00 3649.649 0.04 3649.667 0.021 0 1 7228.88 7228.934 -0.05 7228.883 0.00 7228.888 0.001 0 2 10839.96 10839.986 -0.03 10839.942 0.01 - -1 0 3 14276.34 14276.389 -0.05 14276.318 0.02 14276.229 0.111 1 0 5221.24 5221.233 0.01 5221.227 0.02 5221.298 -0.051 1 1 8779.72 8779.747 -0.03 8779.707 0.01 8779.722 0.001 1 2 12372.71 12372.723 -0.02 12372.679 0.03 - -1 2 0 6755.51 6755.528 -0.02 6755.483 0.03 6755.501 0.011 2 1 10295.63 10295.673 -0.04 10295.616 0.02 10295.524 0.111 3 0 8249.04 8249.063 -0.03 8249.023 0.01 8249.073 -0.041 3 1 11774.71 11774.742 -0.03 11774.701 0.01 11774.670 0.042 0 0 7185.88 7185.894 -0.02 7185.879 0.00 7185.880 0.002 0 1 10585.29 10585.357 -0.07 10585.292 -0.01 10585.300 -0.012 0 2 14187.98 14188.069 -0.09 14187.985 0.00 - -2 1 0 8739.53 8739.530 0.00 8739.520 0.01 8739.589 -0.062 1 1 12116.80 12116.851 -0.05 12116.778 0.02 12116.833 -0.042 2 0 10256.58 10256.604 -0.02 10256.569 0.02 10256.537 0.052 2 1 13612.71 13612.745 -0.04 13612.688 0.02 13612.468 0.242 3 0 11734.53 11734.543 -0.02 11734.517 0.01 11734.625 -0.103 0 0 10573.92 10573.955 -0.04 10573.927 -0.01 10573.898 0.023 0 1 13795.40 13795.455 -0.06 13795.410 -0.01 13795.280 0.123 1 0 12106.98 12107.025 -0.05 12106.974 0.00 12107.006 -0.033 2 1 16775.38 16775.396 -0.01 16775.385 0.00 16774.779 0.604 0 1 16854.99 16855.126 -0.14 16855.099 -0.11 16854.534 0.46 state, J , in the calculation was taken as J = 50 and the limiting energy as 30 000 cm − . Analysis using the H O partitionfunction (Vidler & Tennyson 2000) performed in BT2 suggests that these parameters are sufficient to cover all transitionslongwards of 0.5 µ m for temperatures up to 3000 K.Wavefunctions were obtained by solving the nuclear Schrödinger equation using two-step procedure of calculation of rovi-brational energies (Tennyson & Sutcliffe 1986). The calculations benefitted from recent algorithmic improvements (Tennyson& Yurchenko 2016), in particular in the method used to construct the final Hamiltonian matrices for J > due to Azzamet al. (2016). Transition intensities were computed for ∆ J = 0 and 1 for all four symmetries and every J (cid:54) . The matrixelements of the DMS were calculated using the program Dipole of the suite
DVR3D and the actual spectrum for bothisotopologues was generated with the program Spectra. About 500 million transitions were calculated for each isotopologue.Figure 1 shows the distribution of the H
O lines in HotWat78.Using our calculations we provide the values of partition function for both isotopologues for wide range of temperatures,which are presented in the Table 5 as well as in the supplementary data on a grid of 1 K. We use the HITRAN convention(Fischer et al. 2003) and include the nuclear statistical weights g ns in to the partition function explicitly (Tennyson et al.2016). The nuclear statistical weights for H O are the same as for the main isotopologue, 1 and 3 for the para- and ortho-states, respectively. In case of H O, g ns are 6 (para) and 18 (ortho). For calculation of partition functions for H O andH
O we used all available energy levels with applying the cut-off at 30000 cm − . © 2014 RAS, MNRAS , 1– ?? xoMol XIX: H O and H O Table 3.
Standard deviation in cm − with which our pseudo-experimental energy levels the of H O and H
O predicted the observedones compiled by Tennyson et al. (2010) as a function of rotational state, J , N is number of levels used for calculation of the standarddeviation. J N H O N H O0 27 0.0058 39 0.00921 93 0.0056 124 0.00932 161 0.0071 212 0.01093 199 0.0074 254 0.00904 236 0.0118 316 0.01475 232 0.0103 335 0.01416 263 0.0100 401 0.01167 222 0.0138 385 0.01408 182 0.0146 381 0.01309 138 0.0123 335 0.017410 116 0.0130 288 0.017611 72 0.0080 232 0.016812 47 0.0111 188 0.020113 26 0.0083 135 0.017914 9 0.0096 106 0.019815 3 0.0150 73 0.017616 1 0.0066 46 0.018417 1 0.0015 19 0.015618 11 0.0187
Table 4.
Prediction of experimental energy levels of H
O. Experimental (obs) data is taken from Makarov et al. (2015). J Observed Calculated Obs.-Calc.0 27476.33 27476.24 0.091 27497.03 27496.92 0.111 27510.64 27510.31 0.331 27517.09 27517.44 -0.352 27537.12 27536.96 0.162 27546.82 27546.45 0.371 27509.55 27509.19 0.362 27545.66 27545.28 0.38
Figure 1.
The distribution of the H
O transitions per J in the line HotWat78 list.© 2014 RAS, MNRAS , 1–, 1–
O transitions per J in the line HotWat78 list.© 2014 RAS, MNRAS , 1–, 1– ?? Polyansky et al
Figure 2.
Comparison between BT2 and HotWat78 for H
O at the temperature T =2000 K, and comparison of HotWat78 with 3mol(Shirin et al. 2008) and HITRAN at T =296 K for H O and H
O respectively. © 2014 RAS, MNRAS , 1– ?? xoMol XIX: H O and H O Figure 3.
Comparison of H
O and H
O between 3mol (Shirin et al. 2008) and HotWat78 at the temperature T =3000 K. The series of IUPAC papers on the various isotopologues of water (Tennyson et al. 2009, 2010, 2013, 2014) used measuredtransition frequencies to derive ro-vibrational energy levels using the so-called MARVEL (measured active rotation-vibrationenergy levels) procedure (Furtenbacher et al. 2007; Furtenbacher & Császár 2012). These energy levels can be used to generatepseudo-experimental values of the line frequencies in our line lists when the calculated energy level is substituted by thecorresponding (pseudo-)experimental one. The comparison of these generated line frequencies with actual experimental onesdemonstrate near-perfect coincidence. The number of generated pseudo-experimental lines is significantly higher than thenumber of the directly observed lines because line frequencies between pseudo-experimental levels can be predicted to highaccuracy even when the lines have not been measured, as demonstrated by Tennyson et al. (2013). Less than 200 000experimentally observed H
O lines give rise to about 5 000 000 lines with pseudo-experimental frequencies generated inthis way. Use of such a procedure provides significantly more accurate line lists than just the calculated ones. We thereforesubstituted our computed energy with those of Tennyson et al. (2009) where possible.However as described in section 2, the procedure for fitting PES using H
O data opens the way for us to further improvethe accuracy of the calculated line lists. Looking at Table 6, we can see that the obs − calc residuals for a particular H Ovibrational state are very similar to the residuals for the same states of H
O and H
O. The following procedure can beused to exploit this. First let us consider the idealised situation when all the residuals for energy levels of H O, R v,J (16) ,are exactly equal to those of H O, R v,J (18) , where ( v, J ) represent the vibrational and rotational quantum numbers. Inthis case we can predict the precise “estimated” value of an H O level, E est v,J (18) , from the empirically-determined levels ofH O, E obs v,J (18) E est v,j (18) = E calc v,J (18) + R v,J (18) = E calc v,J (18) + R v,J (16) (1)where E calc v,J (18) is the corresponding calculated H O energy level. So even if the level of the H
O isotopologue has yet tobe observed, its pseudo-experimental value can be retrieved from the calculated level of H
O using our calculations plus theresidual for H
O provided the experimental level of H
O is known.Table 6 shows that residuals for H
O and H
O are slightly different, we can therefore improve this procedure. Wenotice from the Table 6, that the H
O and H
O residuals differ by similar amounts. If we average this value: ∆ R (18) = 1 N N (cid:88) v =1 R v, (18) − R v, (16) . (2) © 2014 RAS, MNRAS , 1–, 1–
O residuals differ by similar amounts. If we average this value: ∆ R (18) = 1 N N (cid:88) v =1 R v, (18) − R v, (16) . (2) © 2014 RAS, MNRAS , 1–, 1– ?? Polyansky et al
Table 5.
Partition Function of H
O and H O. T ( K ) H O H
O10 7.97970859 1.3313500720 20.1629004 3.3707446540 56.7292812 9.4886067460 101.331587 16.950963980 153.237432 25.6357152100 211.822453 35.4382143200 587.053283 98.2237727296 1052.12202 176.043783300 1073.45356 179.613285400 1654.78625 276.895547500 2328.51505 389.655412600 3099.26294 518.674912800 4966.65892 831.3523021000 7346.85187 1230.028251200 10357.5304 1734.467241400 14140.2160 2368.432921500 16371.1820 2742.404041600 18857.9004 3159.293451800 24694.5428 4137.938952000 31855.8230 5338.909082200 40570.4778 6800.617462400 51091.7815 8565.599492500 57116.1119 9576.292002600 63698.8388 10680.72742800 78697.3411 13197.33443000 96419.4218 16171.18733200 117222.299 19662.25433400 141485.523 23734.24093500 155038.487 26008.84113600 169606.832 28453.89043800 201996.792 33890.08294000 239072.534 40112.78344200 281250.969 47191.90284400 328941.890 55196.14174500 354979.000 59566.04294600 382541.321 64191.87534800 442425.403 74242.12995000 508945.054 85405.68855200 582421.516 97736.34705400 663142.877 111282.3335500 706300.716 118524.5155600 751361.549 126085.8835800 847292.676 148990.8616000 951113.377 159603.233 where N runs over the number of vibrational states for which J = 0 levels are known, which corresponds to 40 for H O and24 for H
O. Then we can use this average difference to further correct our estimated H
O energy levels using the revisedformula: E est v,j (18) = E calc v,J (18) + R v,J (16) + ∆ R (18) . (3)Calculating the observed values of energies of H O using Eq. (1) gives a standard deviation for E est v,j (18) levels from theknown experimental values, E obs v,j (18) , of 0.009 cm − . However, ∆ R (18) is 0.006 cm − . If instead we use Eq. (3), then thestandard deviation reduces to 0.003 cm − . Although ∆ R (18) is evaluated for J = 0 only, this procedure still works for higher J values. For example it also results in a standard deviation of 0.003 cm − when applied to the J = 10 levels of the (010)state.This procedure, which can clearly also be applied to H O, leads to the generation of about 5 million transitions whichinvolve the pseudo-experimental levels of H
O and H
O. It therefore provides a line list with much more accurate valuesof the frequencies of these transitions: in general better by about 0.005 cm − for H O and somewhat worse for H
O, butstill much more accurate than possible with variational calculations.The reason this procedure can be applied to the construction of the pseudo-experimental values of the energy levels ofminor isotopologues is that for the major water isotopologue H
O the number of energy levels known experimentally is © 2014 RAS, MNRAS , 1– ?? xoMol XIX: H O and H O Table 6.
Vibrational band origins, in cm − , for H O, H
O and H
O. Observed (obs) data is taken from Tennyson et al. (2013)and Tennyson et al. (2009); calculated results are given as observed minus calculated (o–c). ( v v v ) H O H
O H
Oobs o–c obs o–c obs o–c(010) 1594.75 0.019 1591.33 0.028 1588.28 0.036(020) 3151.63 0.040 3144.98 0.046 3139.05 0.051(100) 3657.05 -0.007 3653.14 -0.005 3649.69 -0.002(110) 5234.97 0.005 5227.71 0.014 5221.24 0.010(120) 6775.09 -0.028 6764.73 -0.022 6755.51 -0.018(200) 7201.54 -0.024 7193.25 -0.019 7185.88 -0.016(012) 9000.14 -0.009 8982.87 0.001 8967.57 0.013(102) 10868.88 -0.049 10853.51 -0.040 10839.96 -0.030(001) 3755.93 -0.017 3748.32 -0.015 3741.57 -0.014(011) 5331.27 -0.002 5320.25 0.010 5310.46 0.019(021) 6871.52 0.004 6857.27 0.012 6844.60 0.019(101) 7249.82 -0.063 7238.71 -0.059 7228.88 -0.051(111) 8807.00 -0.044 8792.54 -0.034 8779.72 -0.027(121) 10328.73 -0.055 10311.20 -0.045 10295.63 -0.039(201) 10613.36 -0.074 10598.48 -0.075 10585.29 -0.072(003) 11032.40 -0.061 11011.88 -0.053 10993.68 -0.053(131) 11813.20 -0.041 11792.82 -0.039 11774.71 -0.034(211) 12151.25 -0.072 12132.99 -0.064 12116.80 -0.054(113) 12565.01 -0.050 12541.23 -0.041 12520.12 -0.030(221) 13652.66 -0.045 13631.50 -0.042 13612.71 -0.035(301) 13830.94 -0.062 13812.16 -0.057 13795.40 -0.057(103) 14318.81 -0.069 14296.28 -0.061 14276.34 -0.053 significantly higher, then that for H
O and H
O. For example the assignment of weak H
O lines in various regions isavailable (Tolchenov et al. 2005; Polyansky et al. 1998; Schermaul et al. 2002), where isotopologues data are not known. As aresult very highly-excited bending (Polyansky et al. 1997; Zobov et al. 2005) and stretching energy levels (Maksyutenko et al.2007; Grechko et al. 2009; Császár et al. 2010) are known, which form the basis upon which our pseudo-experimental energylevels are constructed.
The newly constructed H
O and H
O line lists are named HotWat78. The new HotWat78 line lists are calculated for J (cid:54)
50 and for the spectral range 0-30000 cm − . HotWat78 contains 519 461 789 lines for H O is 519 461 789 and 513 112 779lines for H
O. The new linelist is both the most complete and the most accurate one, see Tables 1 and 2. They are storedin the ExoMol format (Tennyson et al. 2013) which uses the compact storage of results originally developed for BT2. Thisinvolves using a states file ( .states ), see Table 7, and a transitions file ( .trans ), see Table 8. The energy levels in the statesfiles are marked as ‘observed’ if the results are taken from the IUPAC compilation, ‘estimated’ if they are generated usingEq. (3) or as ‘calculated’, for which the results of the PES2 calculation are used.The states file lists all the ro-vibrational levels for each J and for four C v symmetries. It is common to further label theevery level with (approximate) vibrational quantum numbers ( v , v , v ) which correspond to the symmetric stretch, bendingand asymmetric stretch modes, respectively and the Rotational levels within each vibrational state by J, K a , K c , where againthe projection quantum numbers K a and K c are approximate. DVR3D does not provide these approximate labels but thereare several methods available for labeling water energy levels (Partridge & Schwenke 1997; Szidarovszky et al. 2012; Shirinet al. 2008). Here we label levels with J (cid:54) and energies below 20 000 cm − . As our energy levels differ by less than 1 cm − from those of Shirin et al. (2008), transferring the labels from this previous study proved to be straightforward. We note thatthe labels we use are based on the normal modes from a harmonic oscillator model. It is well know that the higher stretchingstates of water are better represented with a local-mode model (Child & Halonen 1984). However, there is a one-to-onecorrespodance between the two labelling schemes (Carleer et al. 1999); the use of normal mode labels are used for simplicity.The accuracy of the present line lists can be established by the comparison with the previous line lists calculations. Twotypes of comparison could be made. The overall picture for the high temperature is that the coverage the HotWat78 H Oand H
O line lists should be very similar to BT2, but that both the predicted intensives and the line positions should besignificantly better. Furthermore lines may shift by between a few cm − to a few tens of cm − between isotopologues. Figure1 demonstrate that, as expected, the overall picture is very similar for BT2 (H O) and HotWat78 (H
O and H
O). Herewe provide the comparison only for H
O but for the H
O it looks the same. © 2014 RAS, MNRAS , 1– ?? Polyansky et al
Table 7.
Extract from the final states file for H O. i ˜ E g tot J Ka Kc v1 v2 v3 S i : State counting number. ˜ E : State energy in cm − . g : Total state degeneracy. J : Total angular momentum K a : Asymmetric top quantum number. K c : Asymmetric top quantum number. ν : Symmetric stretch quantum number. ν : Bending quantum number. ν : Asymmetric stretch quantum number. S : State symmetry in C v . Figures 2 and 3 illustrate the similarity of the HotWat78 line lists with the previous high accuracy H
O and H
Oline lists (called 3mol) of Shirin et al. (2008) for these molecules at the room temperature. Figures 4 and 5 also provide acomparison with the HITRAN data for the room temperature for H
O and H
O. These figures only provide an overview,but a detailed line by line comparison confirms that all the calculations we present here are done correctly.The present line lists are significantly more complete, but this is only apparant at higher temperatures, see Fig. 3. Forthe room temperature the previous line lists should look similar, as they indeed do, see Figures 2.
This paper reports hot line lists for H
O and H
O. These line lists represent significant improvement on both coverage andaccuracy of the previous H
O and H
O line lists (Mikhailenko et al. 2005; Shirin et al. 2008). The predicted frequencies inthese line lists have been significantly improved using information obtained from the corresponding H
O empirical energylevels. This procedure can be adapted to give improved predictions of energy levels and transition frequencies for isotopologuesof molecules for whom the empirical energy levels of the parent molecule are well-known.The complete HotWat78 line lists for H
O and H
O can be downloaded from the CDS, via ftp://cdsarc.u-strasbg.fr/pub/cats/J/MNRAS/ , or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/MNRAS/ . The line lists together with auxiliarydata including the potential parameters, dipole moment functions, and theoretical energy levels can be also obtained at , where they form part of the enhanced Exomol database (Tennyson et al. 2016). The BT2 H
O line list(Barber et al. 2006) is already available from these sources.Finally we note that pressure-broadening has been shown to have a significant effect on water spectra in exoplanets(Tinetti et al. 2012). ExoMol, in common with other databases, assumes that pressure-broadening parameters for H
O andH
O are the same as those for H
O. This assumption is built into the recently updated structure of the ExoMol database(Tennyson et al. 2016). Barton et al. (2016) have recently presented a comprehensive set of pressure-broadening parameters forH
O lines which form the basis for the ExoMol pressure-broadening diet for water (Barton et al. 2016). These parameters,which are available on the ExoMol website, are also suitable for use with the HotWat78 line lists. © 2014 RAS, MNRAS , 1– ?? xoMol XIX: H O and H O Table 8.
Extract from the transitions file for H O f i A fi f : Upper state counting number. i : Lower state counting number. A fi : Einstein-A coefficient in s − . ACKNOWLEDGEMENTS
This work is supported by ERC Advanced Investigator Project 267219 and by the Russian Fund for Fundamental Studies.
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