ExoMol line lists - XXXII. The rovibronic spectrum of MgO

TL;DR

This paper presents comprehensive line lists for magnesium oxide (MgO), covering multiple isotopologues and electronic states, with high accuracy, to aid astrophysical research on exoplanets, stars, and brown dwarfs.

Contribution

The study provides extensive, empirically fitted rovibronic line lists for MgO, including transition data for five isotopologues and all relevant electronic states, with improved accuracy over previous data.

Findings

Line list contains 186,842 rovibronic states and 72,833,173 transitions.

Potential energy and coupling curves fitted to empirical energy levels with 0.01 cm$^{-1}$ accuracy.

Line lists are suitable for temperatures up to 5000 K and cover wavelengths >0.3 μm.

Abstract

Line lists for magnesium oxide are computed and extensive comparisons are made with existing experimental spectra. The LiTY line lists cover all ro-vibration transitions within the five lowest-lying electronic states ($X\,{}^1\Sigma^+$, $a\,{}^3\Pi$, $A\,{}^1\Pi$, $B\,{}^1\Sigma^+$ and $b\,{}^3\Sigma^{+}$) and five isotopologues: $^{24}$Mg$^{16}$O, $^{25}$Mg$^{16}$O, $^{26}$Mg$^{16}$O, $^{24}$Mg$^{17}$O, $^{24}$Mg$^{18}$O, $^{24}$Mg$^{17}$O and $^{24}$Mg$^{18}$O. The calculation use potential energy cures, spin-orbit and electronic angular momentum couplings curves determined by fitting to empirical energy levels; these levels are reproduced to within 0.01 \cm\ in most cases. Computed nuclear-motion wavefunctions are combined with {\it ab initio} dipole moment curves to give transition intensities and excited state radiative lifetimes which are compared with laboraroty measurements. The $^{24}$Mg$^{16}$O line list comprises 186 842 ($J\le 320$) ro-vibronic states and 72 833 173 transitions with angular momenta, $J$, up to 300 and covering wavenumbers up to 33 000 cm$^{-1}$ ($\lambda > 0.3$ $\mu$m). The line lists are suitable for temperatures up to about 5000 K. They are relevant to astrophysical studies of exoplanet atmospheres, cool stars and brown dwarfs, and are made available in electronic form at the CDS and ExoMol databases.

Tables

Table 1. Table 1: Experimental sources and maximal values of J 𝐽 J , v 𝑣 v and energy term values (cm -1 ) of high resolution spectroscopic data of MgO used in this work.

Source	Method	State′	State′′	$v^{\prime }$	$v^{\prime \prime }$	$J^{\prime }$	$J^{\prime \prime }$	${\tilde{E}}_{\max }^{\prime }$	${\tilde{E}}^{\prime \prime }\max$
Kagi & Kawaguchi (2006)	Microwave	$a{}_{}{}^3\Pi$	$a{}_{}{}^3\Pi$	0	0	12	7	7	13
		$X{}_{}{}^1\Sigma _{}^{+}$	$X{}_{}{}^1\Sigma _{}^{+}$	1	1	12	7	7	14
				2	2	12	7
				3	3	12	7
Brewer et al. (1962)	Grating	$C{}_{}{}^1\Sigma _{}^{-}$	$A{}_{}{}^1\Pi$	0	0	80	4	26414	26544
				1	1	65	21
Trajmar & Ewing (1965)	Grating	$C{}_{}{}^1\Sigma _{}^{-}$	$A{}_{}{}^1\Pi$	1	1	60	9	26387	26769
		$D{}_{}{}^1\Delta$	$A{}_{}{}^1\Pi$	0	0	61	1
				1	1	43	4
Singh (1973)	Ebert	$G{}_{}{}^1\Pi$	$A{}_{}{}^1\Pi$	0	1	40	14	36388	39870
		$G{}_{}{}^1\Pi$	$X{}_{}{}^1\Sigma _{}^{+}$	0	0	38	16
				0	1	40	16
Singh (1973)	Ebert	$F{}_{}{}^1\Pi$	$X{}_{}{}^1\Sigma _{}^{+}$	0	0	83	14	37686	37896
Antic-Jovanovic et al. (1976)	Grating	$E{}_{}{}^1\Sigma _{}^{+}$	$A{}_{}{}^1\Pi$	0	0	56	36	34304	34173
Azuma et al. (1984)	LIF	$B{}_{}{}^1\Sigma _{}^{+}$	$X{}_{}{}^1\Sigma _{}^{+}$	0	0	40	0	19971	20062
				1	1	56	0
Civis et al. (1991)	DLFR	$X{}_{}{}^1\Sigma _{}^{+}$	$X{}_{}{}^1\Sigma _{}^{+}$	1	0	32	6	766	792
				2	1	28	7
Ip et al. (1991)	LIF	$B{}_{}{}^1\Sigma _{}^{+}$	$a{}_{}{}^3\Pi$	0	0	48	4	16735	27256
				1	1	48	6
		$D{}_{}{}^1\Delta$	$a{}_{}{}^3\Pi$	0	0	33	1
				1	1	17	1
Kagi et al. (1994)	FTS	$A{}_{}{}^1\Pi$	$X{}_{}{}^1\Sigma _{}^{+}$	1	0	56	0	2644	5452
				2	0	67	3
				3	0	58	3
				2	1	31	11
				0	1	37	8

Table 2. Table 2: Selections of the Obs. − - Calc. residuals, in cm -1 , for the microwave transition frequencies ( X 𝑋 X , v ′ = v ​ " superscript 𝑣 ′ 𝑣 " v^{\prime}=v" , R ​ ( J ) 𝑅 𝐽 R(J) ) of the three isotopologues of Mg 16 O. The Obs. term values are from Törring & Hoeft ( 1986 ) .

Iso	$J$	$v$	Obs	Calc	Obs.-Calc.
26MgO	1	0	2.217914	2.217857	0.000057
26MgO	2	0	3.326801	3.326716	0.000085
26MgO	2	1	3.296283	3.296159	0.000124
26MgO	7	0	8.869430	8.869204	0.000226
25MgO	2	0	3.377244	3.377161	0.000083
24MgO	1	0	2.288127	2.288066	0.000061
24MgO	1	1	2.266805	2.266722	0.000083
24MgO	2	0	3.432117	3.432026	0.000091
24MgO	2	1	3.400133	3.400008	0.000125
24MgO	2	2	3.368112	3.368431	-0.000319
24MgO	7	0	9.150138	9.149896	0.000242
24MgO	7	1	9.064847	9.064513	0.000334
24MgO	7	2	8.979447	8.980303	-0.000856

Table 3. Table 3: Selections of the Obs. − - Calc. residuals, in cm -1 , for the X 𝑋 X state and a 𝑎 a states ( v = 0 𝑣 0 v=0 ) for the refined model of 24 Mg 16 O. The Obs. term values are represented by our MARVEL values.

$X{}_{}{}^1\Sigma _{}^{+}$, $v=0$					$a{}_{}{}^3\Pi$, $v=0$, $\Omega =1$
$J$	Obs	Calc	Obs.-Calc.		$J$	Obs	Calc	Obs.-Calc.
1	1.1441	1.144	0.0001		1	2551.297	2551.290	0.007
3	6.8671	6.8641	0.0030		2	2553.290	2553.300	-0.010
5	17.1596	17.1597	-0.0001		3	2556.289	2556.297	-0.008
7	32.0276	32.0296	-0.0020		4	2560.319	2560.326	-0.006
9	51.4667	51.4724	-0.0057		5	2565.293	2565.309	-0.015
11	75.4784	75.486	-0.0076		6	2571.361	2571.365	-0.004
13	104.0584	104.0679	-0.0095		7	2578.322	2578.324	-0.002
15	137.2076	137.2153	-0.0077		8	2586.413	2586.416	-0.003
17	174.9176	174.9246	-0.0070		9	2595.338	2595.342	-0.005
19	217.1868	217.1919	-0.0051		10	2605.470	2605.477	-0.007
21	264.0096	264.0129	-0.0033		11	2616.352	2616.360	-0.008
23	315.3829	315.3826	0.0003		12	2628.539	2628.545	-0.007
25	371.2966	371.2959	0.0007		13	2641.369	2641.376	-0.006
27	431.7481	431.7467	0.0014		14	2655.606	2655.617	-0.011
29	496.733	496.729	0.0040		15	2670.374	2670.386	-0.012
31	566.2387	566.2358	0.0029		16	2686.657	2686.689	-0.032
33	640.2592	640.2601	-0.0009		17	2703.360	2703.387	-0.027
35	718.7909	718.794	-0.0031		18	2721.727	2721.757	-0.030
37	801.828	801.8294	-0.0014		19	2740.350	2740.375	-0.026
39	889.3578	889.3576	0.0002		20	2760.792	2760.815	-0.023
41	981.3724	981.3695	0.0029
43	1077.8619	1077.856	0.0064
45	1178.8162	1178.805	0.0108
47	1284.2229	1284.209	0.0142
49	1394.0739	1394.054	0.0195
51	1508.3514	1508.331	0.0205
53	1627.0503	1627.026	0.0240
55	1750.1546	1750.128	0.0266
57	1877.6509	1877.623	0.0279
59	2009.5313	2009.498	0.0333
61	2145.7752	2145.739	0.0362
61	2145.7752	2145.739	0.0362

Table 4. Table 4: Selections of the Obs. − - Calc. residuals, in cm -1 , for the A 𝐴 A and B 𝐵 B term values ( v = 0 𝑣 0 v=0 , + + ) for the refined model of 24 Mg 16 O. The Obs. term values are represented by our MARVEL values.

$A{}_{}{}^1\Pi$, $v=0$, $\Omega =1$, $+$					$B{}_{}{}^1\Sigma _{}^{+}$, $v=0$, $+$
$J$	Obs	Calc	Obs.-Calc.		$J$	Obs	Calc	Obs.-Calc.
9	3548.585	3548.563	0.022		0	20003.59	20003.593	0.002
11	3569.710	3569.685	0.025		2	20007.07	20007.073	0.003
13	3594.843	3594.825	0.018		4	20015.19	20015.194	0.004
14	3608.936	3608.919	0.017		6	20027.95	20027.954	0.002
15	3623.999	3623.980	0.020		8	20045.35	20045.353	0.001
16	3640.197	3640.085	0.111		10	20067.38	20067.388	-0.007
17	3657.260	3657.146	0.113		12	20094.05	20094.057	-0.006
18	3675.277	3675.263	0.014		14	20125.34	20125.359	-0.013
19	3694.343	3694.321	0.022		16	20161.27	20161.290	-0.011
20	3714.460	3714.447	0.013		18	20201.84	20201.845	-0.006
21	3735.522	3735.500	0.021		20	20247.01	20247.023	-0.005
22	3757.646	3757.634	0.011		22	20296.81	20296.818	-0.001
23	3780.699	3780.678	0.021		24	20351.22	20351.224	0.004
24	3804.825	3804.819	0.007		26	20410.24	20410.238	0.002
25	3829.869	3829.851	0.018		28	20473.85	20473.854	0.001
26	3856.002	3855.996	0.006		30	20542.08	20542.065	0.019
27	3883.034	3883.012	0.022		32	20614.86	20614.864	0.000
28	3911.167	3911.159	0.007		34	20692.23	20692.245	-0.009
29	3940.174	3940.157	0.018		36	20774.18	20774.200	-0.011
31	4001.292	4001.277	0.014		38	20860.71	20860.721	-0.007
32	4033.413	4033.422	-0.009		40	20951.81	20951.800	0.009
					42	21047.41	21047.428	-0.019

Table 5. Table 5: Selections of Obs. - Calc. residuals, in cm -1 , showing the vibrational accuracy of the refined model. The Obs. term values are represeted by our MARVEL values.

$J$	parity	State	$v$	$\Omega$	Obs	Calc	Obs.-Calc.
1	$-$	$X$	0	0	1.1441	1.144	0.0001
1	$-$	$X$	1	0	775.8723	775.8711	0.0012
6	+	$X$	2	0	1562.7462	1562.715	0.0314
1	+	$a$	0	0	2551.2967	2551.29	0.0068
3	+	$a$	1	0	3193.0857	3193.105	-0.0190
7	$-$	$A$	0	-1	3539.537	3539.509	0.0280
1	$-$	$A$	1	-1	4160.8965	4160.902	-0.0058
3	+	$A$	2	1	4814.6699	4814.617	0.0527
3	+	$A$	3	1	5455.8189	5455.807	0.0123
0	+	$B$	0	0	20003.5941	20003.59	0.0016
0	+	$B$	1	0	20818.1573	20818.14	0.0126

Table 6. Table 6 : Extract from the states file for 24 Mg 16 O.

$i$	$\tilde{E}$	$g_i$	$J$	$\tau$	$g$	$+/-$	$e/f$	State	$v$	$\Lambda$	$\Sigma$	$\Omega$
636	1.144048	3	1	1.77983E+05	0.000000	-	e	X1Sigma+	0	0	0	0
637	775.871080	3	1	1.35976E+00	0.000000	-	e	X1Sigma+	1	0	0	0
638	1540.259721	3	1	6.33385E-01	0.000002	-	e	X1Sigma+	2	0	0	0
639	2294.147145	3	1	3.98566E-01	0.000034	-	e	X1Sigma+	3	0	0	0
640	2551.292291	3	1	8.23030E-02	0.527645	-	e	a3Pi	0	-1	0	-1
641	2622.222505	3	1	2.58488E-02	-0.027585	-	e	a3Pi	0	-1	1	0
642	3038.170256	3	1	2.22267E-01	0.000236	-	e	X1Sigma+	4	0	0	0
643	3193.107760	3	1	3.27360E-02	0.526013	-	e	a3Pi	1	-1	0	-1
644	3265.983987	3	1	2.08686E-02	-0.026155	-	e	a3Pi	1	-1	1	0
645	3504.293682	3	1	1.05523E-04	0.499992	-	e	A1Pi	0	-1	0	-1
646	3771.346664	3	1	9.69221E-02	0.002761	-	e	X1Sigma+	5	0	0	0
647	3824.775932	3	1	2.06126E-02	0.520372	-	e	a3Pi	2	-1	0	-1
648	3901.850493	3	1	1.82913E-02	-0.023039	-	e	a3Pi	2	-1	1	0
649	4160.902293	3	1	7.47291E-05	0.499992	-	e	A1Pi	1	-1	0	-1
650	4446.228208	3	1	1.51760E-02	0.538037	-	e	a3Pi	3	-1	0	-1
651	4479.741149	3	1	2.31664E-02	-0.028181	-	e	a3Pi	3	-1	1	0
652	4544.158030	3	1	2.77669E-02	-0.009762	-	e	X1Sigma+	6	0	0	0
653	4809.732862	3	1	5.84090E-05	0.499993	-	e	A1Pi	2	-1	0	-1
654	5057.431336	3	1	1.21026E-02	0.530620	-	e	a3Pi	4	-1	0	-1
655	5117.344124	3	1	1.20824E-02	-0.030073	-	e	a3Pi	4	-1	1	0
656	5239.189020	3	1	7.80934E-02	-0.000452	-	e	X1Sigma+	7	0	0	0
657	5450.913256	3	1	4.83072E-05	0.499994	-	e	A1Pi	3	-1	0	-1
658	5658.311999	3	1	1.01359E-02	0.528937	-	e	a3Pi	5	-1	0	-1

Table 7. Table 7 : Extract from the transitions file for 24 Mg 16 O.

$f$	$i$	$A_{fi}$(s-1)	$\tilde{\nu }$ (cm-1)
36757	37183	7.7028E-12	0.098041
32237	31809	2.1383E-14	0.098067
32909	32497	1.0981E-11	0.098167
90382	90758	2.8824E-13	0.098189
98009	98376	3.5386E-14	0.098192
49202	49606	1.5586E-12	0.098197
123918	124241	1.7154E-15	0.098208
91640	91262	4.8358E-13	0.098261
83152	82769	9.9825E-15	0.098285
7957	8392	4.4448E-11	0.098298
101679	102046	5.2911E-14	0.098322
82876	83259	2.4564E-18	0.098324

Equations

$$V(R)=V_{\mathrm{e}}+(A_{\mathrm{e}}-V_{\mathrm{e}}){\left[1-\exp \left(-\sum _{k=0}^N{B}_k{\xi }^k(r-r_{\mathrm{e}})\right)\right]}^2,$$

$$\xi =\frac{r^p-r_{\mathrm{ref}}^p}{r^p+r_{\mathrm{ref}}^p},$$

$$F(z)=\sum _{k=0}^N{B}_k{z}^k(1-z)+z{B}_{\infty },$$

$$z=(r-r_{\mathrm{ref}})e^{-{\beta }_2(r-r_{\mathrm{ref}}{)}^2-{\beta }_4(r-r_{\mathrm{ref}}{)}^4},$$

Full text

††pagerange: ExoMol line lists XXXII: The rovibronic spectrum of MgO–References††pubyear: 2018

1 Introduction

Magnesium oxide (MgO) is a metal oxide formed from the cosmically abundant elements of magnesium and oxygen. MgO is a major constituent of chondritic meteorites (Maatouk et al., 2010). It was observed in the exosphere of Mercury by the first and second flyby of the MESSENGER spacecraft (Killen et al., 2010; Sarantos et al., 2011). The observed distribution suggests temperatures of tens of thousands Kelvin. The quantity of MgO in the lunar surface was estimated using X-ray fluorescence observations made by the Chandrayaan-1 X-ray Spectrometer (Weider et al., (2012)). The strongest singlet band ($B\,{}^{1}\Sigma^{+}$ – $X\,{}^{1}\Sigma^{+}$) present in the lunar exosphere were analysed during the 2009 Perseid meteor shower (Berezhnoy et al., 2014).

MgO is also thought to be a major component of interstellar dust (Yoneda & Grossman, 1995; Nozawa et al., 2003; Rietmeijer, 2009). Attempts to detect MgO in the gas phase in the interstellar medium (ISM) have so far proved inconclusive (Turner & Steimle, 1985) or negative (Sakamoto et al., 1998), suggesting that Mg is heavily depleted onto grains (Sakamoto et al., 1998). The discovery of hot, rocky exoplanets has led to suggestions that the many compounds usually in the condensed phase, and in particular MgO (Coppari et al., 2013), should be present in significant concentrations in the gas phases (Schaefer et al., 2012; Tennyson & Yurchenko, 2018). The detection of us such species would require lists of key spectroscopic transitions and here we provide line lists for MgO isotopologues which should be valid over an extended range of temperatures

MgO has a very characteristic electronic spectrum which should be amenable to observations. In particular, the two singlet systems ($B\,{}^{1}\Sigma^{+}$ – $A\,{}^{1}\Pi$  and $B\,{}^{1}\Sigma^{+}$ – $X\,{}^{1}\Sigma^{+}$), called the red and green band systems respectively, have long been studied in the laboratory (Mahanti, 1932; Lagerqvist & Uhler, 1949). Experimental studies of MgO spectra initially used grating spectrographs and more recently laser spectroscopy which allowed experiments to be performed on the low lying states as well as some higher states such as the $E\,{}^{1}\Sigma^{+}$ and $F\,{}^{1}\Pi$ states (Bellert et al., 2003; Wang et al., 2004). A full survey of high resolution spectroscopic studies of MgO is given in the next section.

The most recent and comprehensive ab initio study of MgO is by Bauschlicher & Schwenke (2017), who presented a high level study for a large number of electronic states of this molecule using the SA-CASSCF/IC-MRCI/aug-cc-pV5Z level of theory. They reported high level ab initio potential energy and (transition) dipole moment curves as well as lifetimes for a number of electronic bands. We have adopted these ab initio curves for the present study. Another recent and accurate ab initio study of MgO, also relevant to our work, is by Maatouk et al. (2010), who used the MRCI/cc-pV5Z method to compute potential energy, spin-orbit and transition dipole moment curves for the valence and valence-Rydberg electronic states of MgO. Earlier, but also comprehensive, ab initio studies of MgO were performed by Thommel et al. (1989b) and Thommel et al. (1989a) using the MRDCI method for various states. None of these works reported the electronic angular momenta of MgO.

Limited work have been performed on the construction of MgO line lists. While studies have reduced the observed spectra into spectroscopic constants (Pyykkö et al., 1987; Thommel et al., 1989b; Daily et al., 2002), the only sourcesof astronomical transition data is the JPL list of 44 pure rotational transitions associated with the $v=0$ and $v=1$ vibrational states of the $X\,{}^{1}\Sigma^{+}$ electronic ground state (Pickett et al., 1998). These data are designed for detecting MgO in the interstellar medium but do not provided the information required to study the spectra of hotter sources.

The ExoMol project aims to provide a catalogue of spectroscopic transitions for molecular species which may be present in the atmospheres of the exoplanets, brown dwarfs and cool stars (Tennyson & Yurchenko, 2012). The ExoMol database (Tennyson et al., 2016) currently contains 52 molecules ranging from diatomic to polyatomic molecules and ions (Tennyson & Yurchenko, 2018). However, there are only a handful of metal oxides are available, namely CaO (Yurchenko et al., 2016), VO (McKemmish et al., 2016), AlO (Patrascu et al., 2015) and SiO (Barton et al., 2013); an updated line list for TiO has just been completed (McKemmish et al., 2019). Such line lists are also important for analysing and modelling spectra in laser induced plasmas (Woods et al., 2012; De Giacomo et al., 2014; Parigger et al., 2015).

Here we present new extensive line lists for MgO covering the spectroscopy of its five lowest electronic states, $X\,{}^{1}\Sigma^{+}$, $A\,{}^{1}\Pi$, $B\,{}^{1}\Sigma^{+}$, $a\,{}^{3}\Pi$ and $b\,{}^{3}\Sigma^{+}$. The line list are computed using nuclear-motion program Duo (Yurchenko et al., 2016a) using a combination of empirical and ab initio potential energy curves (PECs), spin-orbit curves (SOCs) and electronic angular momenum curves (EAMCs) in conjunction with high level ab initio (transition) dipole moment curves (T)DMCs. The ab initio curves were taken from the literature, where available, or were computed as part of this work. Given the astronomical interest in MgO, the line lists are produced to cover an extensive energy and spectroscopic range, up to 37 250 cm*-1* and should be applicable for temperatures up to at least 5000 K. Line lists are generated for the three stable isotopes of Mg and three stable isotopes of O.

2 Method

2.1 Experimental data

Experimental transitions frequencies for 24Mg16O involving a total nine different electronic states were collected from various papers which are summarised in Table 1. The MARVEL technique (Measured Active Rotational-Vibrational Energy Levels) (Furtenbacher et al., 2007; Furtenbacher & Császár, 2012; Tóbiás et al., 2019) was used to determine empirical energy levels. A total of 2457 transitions were collected from the listed sources and validated yielding 757 distinct empirical energy levels. The MARVEL transitions and energy files are given as part of the supplementary data. The MARVEL energy levels together with the experimental transitions were used as Duo inputs to refine our spectroscopic model (PECs, SOCs and EAMCs).

2.2 Spectroscopic model

As with other metal oxides studied within the ExoMol projects, the procedure used here to calculate the line list for MgO is to refine the ab initio PECs, SOCs and EAMCs using available experimental data (Tennyson, 2011).

Our model of MgO consists of five PECs, $X\,{}^{1}\Sigma^{+}$, $A\,{}^{1}\Pi$, $B\,{}^{1}\Sigma^{+}$, $a\,{}^{3}\Pi$ and $b\,{}^{3}\Sigma^{+}$ which are shown in Figure 1. They are augmented by five SOCs, $X$–$a$, $a$–$a$, $A$–$a$, $B$–$a$ and $a$–$b$, which are shown in Figure 2, and two EAMCs $X$–$A$ and $a$–$b$, the $x$ components of which are shown in Figure 3. The initial, ab initio, PECs were taken from Bauschlicher & Schwenke (2017). The SOCs and EAMCs were obtained ab initio using the MOLPRO electronic structure package (Werner et al., 2012) at the multi-reference configuration interaction (MRCI) level of theory in conjunction with the aug-cc-pwCVQZ basis sets with relativistic, core-correlation effects and Davidson correction, as part of this study. The active space is given by (8,4,4,0). The states-averaged CASSCF (SA-CASSCF) comprised (3,2,1,1) states of symmetries $\Sigma^{+}$, $\Pi$, $\Delta$ and $\Sigma^{-}$, respectively. All electron were correlated. The relativistic correction was estimated using the Douglas-Kroll method.

The nuclear motion program Duo (Yurchenko et al., 2016b) is used to solve the fully-coupled Schrödinger equation for the five lowest bound electronic states of MgO. Duo calculations used a grid-based sinc DVR basis of 501 points spanning 0.02 to 8 Å to obtain vibrational basis functions by solving five independent $J=0$ problems for the corresponding electronic state. The final vibrational basis set comprised 100, 62, 100, 57, and 70 lowest vibrational eigenfunctions of the $X$, $a$, $A$, $B$ and $b$ states, respectively. These numbers were chosen to include all vibrational states below the corresponding dissociation limits (see below). A detailed description of the methodology is given by Yurchenko et al. (2016b) and Tennyson et al. (2016).

In order to facilitate the refinement of the experimental model, the four lowest PECs ($X$, $a$ $A$, and $B$) were represented using an extended Morse oscillator (EMO) (Lee et al., 1999) potentials with the form:

[TABLE]

where $D_{\rm e}=A_{\rm e}-V_{\rm e}$ is the dissociation energy, $r_{\rm e}$ is an equilibrium distance of the PEC, and $\xi$ is the Šurkus variable (Šurkus et al., 1984). The Šurkus variable is defined as:

[TABLE]

where $p$ is a non-zero real number parameter, $r_{\rm ref}$ is a reference position and $r_{\rm ref}=r_{\rm e}$ in this case. The $b$ state ab initio PEC was not used in the refining procedure and therefore did not need to be represented analytically. The ab initio points values were interpolated on the Duo grid using the cubic splines Yurchenko et al. (2016a).

The lowest dissociation limit from MgO is to the $a\,{}^{3}\Pi$ asymptote (Mg(${}^{1}S$) and O(${}^{3}P$)), which Bauschlicher & Schwenke (2017) estimated as lying at 2.7$\pm 0.1$ eV. Their adiabatic dissociation energy for the $X\,{}^{1}\Sigma^{+}$ state is 4.65 eV, which is similar to the value obtained using the atomic asymptotic separation to O(${}^{1}D$), 4.67 eV. This is also the asymptote as for the $A\,{}^{1}\Pi$ state. Adding the atomic separation of O(${}^{3}P$) and Mg(${}^{3}P$) to this limit gives the asymptote for the $B\,{}^{1}\Sigma^{+}$ state of 5.4 eV. These asymptotes were adopted for the values of $A_{\rm e}$ in Eq. (1) used in this work.

We did not represent the ab initio SOCs and EAMCs analytically directly. Instead, in order to allow for their refinement, the morphing procedure (Meuwly & Hutson, 1999; Skokov et al., 1999; Patrascu et al., 2014) was used. According to this procedure, as implemented in Duo, the ab initio SOCs and EAMCs are multiplied by a morphing function $F(z)$ represented by following expansion:

[TABLE]

where $z$ is either taken as a damped-coordinate (for the $a$–$S$ SOC) given by:

[TABLE]

see also Prajapat et al. (2017) and Yurchenko et al. (2018), or as the Šurkus variable $z=\xi$ (all other SOCs and EAMCs). Here $r_{\rm ref}$ is a reference position equal to $r_{\rm e}$ by default and $\beta_{2}$ and $\beta_{4}$ are damping factors. When used with morphing, the parameter $B_{\infty}$ is usually fixed to 1.

EAMCs give rise to the $\Lambda$ doubling (parity doubling) for states with $\Lambda$ (projection of the electronic angular momentum) greater than 0.

The PECs, SOCs and EAMCs, representing the final, refined spectroscopic model are shown in Figs. 1–3 and given as part of the supplementary material inside the Duo input file.

MgO is an ionic system. This leads to large permanent dipole moments which vary strongly with $r$ and, because the degree of charge separation changes strongly between excited states, large transition dipoles. Similar behaviour was observed in the previous ExoMol study on CaO (Yurchenko et al., 2016). The diagonal DMCs ($X-X$, $a-a$, $A-A$, $b-b$) as well as the transition DMC for $B-A$ are taken from the ab initio work of Bauschlicher & Schwenke (2017) who used an MRCI calculation with an aug-cc-pV5Z. All other TDMCs ($A-X$, $B-X$, $a-c$) were computed ab initio as part of the present work. The corresponding electronic transition dipole moments are shown in Fig. 4. All (T)DMCs except for $B-B$, $B-A$, $b-b$ were represented analytically using the damped-$z$ expansion in Eq. (3). This was done in order to reduce the numerical noise in the calculated intensities for high overtones, see recommendations by Medvedev et al. (2016). The corresponding expansion parameters as well as their grid representations can be found in the Duo input files provided as supplementary data. Büsener et al. (1987) reported experimental electric dipole moments for the $X\,{}^{1}\Sigma^{+}$ and $A\,{}^{1}\Pi$ states ($v=0$) using Stark quantum-beat spectroscopy, 6.2(6) D and 5.94(24) D, respectively. Our values of the vibrationally ($v=0$) averaged dipole moments based on the ab initio dipole moment of Bauschlicher & Schwenke (2017) are 5.99 D and 5.68 D, respectively, which agree well with these experimental value. The dipole moment curves (DMCs) are shown in Figure 4.

Duo input files which fully specify our final spectroscopic model are given as part of the supplementary data.

3 Results

4 Accuracy of the fits

The ab initio PECs ans SOCs were refined by fitting to the 2457 experimental transition frequencies (see Table 1) augmented with 756 MARVEL term values for 24Mg16O. The data used in the fit are provided as part of the spectrospic model. The root mean square (RMS) errors as observed minus calculated (Obs. $-$ Calc.) residues are 0.029 cm*-1* for all transition wavenumbers, or 0.009 cm*-1*, 0.002 cm*-1*, 0.016 cm*-1* and 0.04 cm*-1* for the $X$, $a$, $A$ and $B$ states, respectively. The MARVEL term values are reproduced with an RMS error of 0.05 cm*-1*. Figure 5 gives an overview of the Obs. $-$ Calc. residues with more detailed results given in Tables 3, 4 and 5.

Line lists for the other isotopologues of MgO were generated using the curves with no allowance for any failure of the Born-Oppenheimer approximation. This means that while we expect these line lists to still be accurate, some loss of accuracy is to be expected. The pure rotational frequencies of 25Mg16O and 26Mg16O do show excellent agreement with experiment, see Table 2.

4.1 Partition function

Partition functions, $Q(T)$, were computed with Duo in steps of 1 K by explicit summation of the calculated energy levels. ExoMol always produces partition functions according to the HITRAN convention (Gamache et al., 2017) which explicitly includes the full atomic nuclear spin degeneracy, $g_{\rm ns}$. Since the nuclear spins of 24Mg and 16O are both zero, therefore the nuclear statistical weight is

1. There is no need to scale the Sauval & Tatum (1984) or Barklem & Collet (2016) partition functions for comparison. The nuclear spins of 26Mg and 18O are also zero, while the 25Mg and 17O have the nuclear spins of $5/2$, which result in the nuclear spin statistical factor of 6 for 25Mg16O and 24Mg17O.

At 300 K we obtain a value of $Q=374.651$ which is in excellent agreement with the value of 374.621 used in the JPL database (Pickett et al., 1998). Figure 6 plots our temperature-dependent partition function and compared with those computed by Sauval & Tatum (1984) and by Barklem & Collet (2016). All three data sets agree very well below 3500 K with the ExoMol results being higher than Barklem & Collet or Sauval & Tatum. We believe our energies to be more complete which should explain why our partition function is higher. However, above about 5000 K electronically excited states of MgO, not considered in this calculation, will become increasingly thermally occupied. For this reason and because the abundance of MgO is likely to small in this temperature region we used a 5000 K upper temperature limit in this study.

The partition functions for all isotopologues are given in the supplementary data.

4.2 Lifetimes

Lifetimes for the states are provided as part of the Duo calculations, detailed methodology is given by Tennyson et al. (2016). Lifetimes of the excited states have only been studied experimentally for the $B\,{}^{1}\Sigma^{+}$ state and in this case there is disagreement between previous studies on the lifetime of the $v=0$ state. The lifetime was measured to be 32.7$\pm$1.7 ns by Diffenderfer et al. (1983) using fluorescence decay, 22.5$\pm$1.5 ns by Büsener et al. (1987) ($J=1$) using Stark quantum-beat spectroscopy, and 21.5$\pm$1.8 ns by Naulin et al. (1991) ($J=70$) also using fluorescence decay. Maatouk et al. (2010) calculated the $J=0$ and $J=70$ lifetimes as 33.3 and 22.0 ns, respectively suggesting a significant decrease in lifetime with rotational excitation.

Our lifetimes ($B\,{}^{1}\Sigma^{+}$, $v=0$) for $J=0$ and $J$ =70 are 21.8 ns and 21.7 ns, respectively, in agreement with the measurements of Büsener et al. (1987) and Naulin et al. (1991). An overview of the lifetimes of MgO for the four lowest electronic states is presented in Figure 7. The shortest-lived states are from $B\,{}^{1}\Sigma^{+}$. The lowest states ($v=0$, small $J$) of $X\,{}^{1}\Sigma^{+}$ and $a\,{}^{3}\Pi$ have very long lifetime.

4.3 Line lists

Line lists were generated by consider all lower states up to 24 000 cm*-1*, upper states up to 37 500 cm*-1* and rotationally excited states up to $J=300$. These parameters are sufficient for completeness up to temperatures of 5000 K and wavenumbers up to 33 000  cm*-1* or wavelengths longer than 0.3 $\mu$m, although at higher temperatures the line lists will be not quite complete for the highest wavenumbers.

The line list for 24Mg16O contains 72 833 173 transitions between 186 842 states. The line list have been formatted into the ExoMol format (transition files and state files) (Tennyson et al., 2013) with lifetimes of each state quoted. A extract of the states file is shown in Table 6 and a extract of the transition file is shown in Table 7. The line lists for isotopologues have comparable sizes.

5 Simulated spectra

Absorption spectra at different temperatures are presented in Figure 8. The contribution of each band is shown in Figure 9. The green ($B-X$) and red ($B-A$) singlet band systems are the strongest bands and show a significant overlap at visible wavelengths. All spectral simulations employed the ExoCross code (Yurchenko et al., 2018).

Comparisons with the previous experimental works have been made to assess the quality of our computed line list. The rotational $X$ band in the form of a stick spectrum is shown in Figure 10, where it is compared to the experimentally determined transitions from the JPL database (Pickett et al., 1998). The JPL spectrum is based on the dipole moment of 6.88 D taken from an old ab initio calculation due to Fowler & Sadlej (1991). Our vibrational averaged ($v=0$) value is 5.99 D, which is the same as given by Bauschlicher & Schwenke (2017). We expect this new value to be represent an improvement which suggests that the current JPL intensities are about 30% too strong and, correspondingly, the observational upper limits for the ISM (Turner & Steimle, 1985; Sakamoto et al., 1998) to be about 30% too low.

5.1 The red band

Dreyer et al. (2001) made excitation scans using the Laser-induced fluorescence excitation spectroscopy for the $B-A$ transitions. For the 588 nm to 632 nm range there is overall agreement as shown in Figure 11. The $(0,0)$ and $(0,1)$ bandheads give a good match, whereas the bandheads for the higher state are not as clear as the lower ones.

Figure 12 shows a spectrum for the 16115 cm*-1* to 16150 cm -1 region from Dreyer et al. (2001). The main peaks for the $(0,1)$ lines generally agree whereas the magnitudes can differ, see Figure 13. The $(0,1)$ bandhead region cannot be resolved between 1584 nm and 1586 nm otherwise the resolved peaks agree with experiment within 0.1 nm. This figure shows the individual contributions from all three major isotopologues assuming the terrestrial abundance (0.79/0.10/0.11 for 24Mg16O/25Mg16O/26Mg16O, respectively).

Red band studies made by Pasternack et al. (1978) also show good agreement with the ExoMol data around the $(1,1)$ band; Figure 14 gives a comparison of the emission spectrum simulated using the ExoMol line list with the laser-induced fluorescence of MgO in acetylene-air flames (Pasternack et al., 1978). In order to match the experiment, the vacuum wavelength was re-scaled to the air-acetylene wavelength using the refractive indexes from (Ciddor, 1996; Loria, 1909)111The refractive index data are taken from M. N. Polyanskiy’s “Refractive index database”, https://refractiveindex.info. assuming a 45:55 mixture. This ratio was adjusted to match the experimental spectrum.

5.1.1 The green band

For the 495 nm to 501 nm region of the $B-X$ transitions, Figure 15 compares our data with the laser-induced fluorescence spectrum of Pasternack et al. (1978). The shapes, which are largely determined by bandheads, of the two spectra are similar. The same wavelength calibration (vacuum to air-acetylene) as above was applied. The ExoMol spectrum agrees very well also with the higher resolution experimental data from the same region, measured between 499.7 nm and 500.1 nm using Stark quantum-beat spectroscopy by Büsener et al. (1987), see Figure 16, where no calibration was required. For these simulations the non-LTE model was assumed based on two temperatures, rotational $T_{\rm rot}$ and vibrational $T_{\rm vib}$, as implemented in Duo. The temperatures selected to match the experimental emission spectra ere 1000 K and 2500 K, respectively.

6 Conclusion

Accurate and complete line lists, known as LiTY, for 24Mg16O, 25Mg16O, 26Mg16O are computed using high level ab initio DMCs as well as a spectroscopically refined Duo model. Our work displays good agreement with the most theoretical and experimental studies made in the past. The lifetimes computed for ($B\,{}^{1}\Sigma^{+}$, $v=0$) are 2.18 ns for $J=0$ and 2.17 ns for $J=70$, partially matching prior studies which disagree with each other. The partition functions agree with other theoretical studies for the temperature up to 5000 K

With a valid temperature range up to 5000 K, the strong red and green bands between 15000 and 20000 cm*-1* (0.66 to 0.5 $\mu$m) should be useful for detecting MgO in the atmospheres of cool stars, brown dwarfs and some exoplanets.

The LiTY line lists can be downloaded from the CDS, via ftp://cdsarc.u-strasbg.fr/pub/cats/J/MNRAS/, or http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/MNRAS/, or from www.exomol.com.

Acknowledgements

This work was supported by the UK Science and Technology Research Council (STFC) No. ST/R000476/1 and the COST action MOLIM No. CM1405. This work made extensive use of UCL’s Legion high performance computing facility along with the STFC DiRAC HPC facility supported by BIS National E-infrastructure capital grant ST/J005673/1 and STFC grants ST/H008586/1 and ST/K00333X/1. We thank Charles Bauschlicher for providing their ab initio PECs and DMCs of MgO.