Marvel Analysis of the Measured High-resolution Rovibronic Spectra of ⁹⁰Zr¹⁶O

The Marvel approach (Császár et al. 2007; Furtenbacher et al. 2007; Furtenbacher & Császár 2012a) is an algorithm that enables a set of assigned experimental transition frequencies to be converted into empirical energy levels with associated uncertainties propagated from the input transition data to the output energy levels. This conversion relies on the construction of experimental spectroscopic networks (SNs; Császár & Furtenbacher 2011; Furtenbacher & Császár 2012b; Furtenbacher et al. 2014; Árendás et al. 2016) which contains all interconnected transitions. For a detailed description of the approach, algorithm, and program, we refer readers to Furtenbacher & Császár (2012a).

The Marvel approach has been used to compile empirical energy levels for the very important and electronically similar species⁴⁸Ti¹⁶O (McKemmish et al. 2017). Other Marvel studies on astronomically important molecules include those for (Furtenbacher et al. 2016), acetylene (Chubb et al. 2018a), ammonia (Al Derzi et al. 2015; Furtenbacher et al. 2018), SO₂ (Tóbiás et al. 2018), H₂S (Chubb et al. 2018b), and isotopologues of (Furtenbacher et al. 2013a, 2013b). These are in addition to energies for the isotopologues of water (Tennyson et al. 2009, 2010, 2013, 2014b) for which the MARVEL procedure was originally developed (Tennyson et al. 2014a).

This paper utilized the Marvel algorithm through a specially designed web interface, available at http://kkrk.chem.elte.hu/marvelonline (Furtenbacher & Császár 2018), making it highly accessible across computer systems without installation of a specialized code. Numerous updates to the online interface were also made during this project and related projects in order to optimize the speed, ease, and quality of data processing; for example, options were made available to automatically update uncertainties within thresholds when processing initial data to find a self-consistent SN.

ZrO and TiO share similar features in their electronic structure, as Zr is directly below Ti on the periodic table. Specifically, both have the same qualitative ordering of many low-lying electronic states (in terms of symmetry and spin), with slight differences in T_e so that, e.g., unlike in TiO, the ground electronic state of ZrO is a spin singlet, . Those states with well-characterized experimental electronic states below 25,000 cm⁻¹ are shown in Figure 1, which also gives the observed bands linking these states. Note that we did not find any rotationally resolved spectral data involving the , , , or states.

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The singlet ground state has allowed excitations to the and states. Significant absorption also occurs from thermal population of the states to the higher singlet states , , , and . In the high temperature gaseous environments where is present astrophysically, transitions from the state to the and states may also be relevant.

The most obvious information to include in the label of a rovibronic state of ZrO is the electronic state, state, the total angular momentum, J, and the vibrational quantum number, v. We find these to be relatively unambiguous to define.

For the triplet states, we also need to provide information about the electronic spin state; in this case, we choose to include this as part of the label for the electronic state. The parity of energy levels usually only influences the energy in a measurable manner for Π states; we absorb the e and f parity labels (Brown et al. 1975) into the electronic state label to reduce the overall number of labels.

In the first half of the twentieth century, there was considerable interest in studying the visible and ultraviolet spectrum of , with many bandheads measured by Lowater (1932), Herbig (1949), and Afaf (1949, 1950a, 1950b). These studies include many involving transitions to electronic states that have yet to be investigated using rotationally resolved spectra.

More recently, there was an extensive experimental effort over the 1970s to early 1980s by several groups to obtain rotationally resolved assigned experimental spectra for various important bands; these studies as well as more recent rotationally resolved studies are summarized in Table 1.

Table 1.  Data Sources and Their Characteristics for

Tag	References	Band		Range (cm−1)	J Range	Trans. (A/V)	Uncertainties (cm−1)
							Min	Av	Max
54LaUhBa	Lagerqvist et al. (1954)	2–1	0–0	15282–15442	11–89	159/159	0.1	0.1	0.28
		3–2	0–0	15612–15755	11–93	149/149	0.1	0.1	0.32
		4–3	0–0	15898–16048	11–95	165/165	0.1	0.11	0.34
		1–1	0–0	21351–21542	20–76	105/105	0.1	0.1	0.31
		2–2	0–0	21351–21555	20–80	111/111	0.1	0.1	0.1
		3–3	0–0	21457–21640	20–81	106/106	0.1	0.1	0.1
54Uhler	Uhler (1954b)	0e–1	0–0	17884–18002	15–60	106/106	0.1	0.1	0.14
		0f–1	0–0	17889–18006	27–59	96/91	0.1	0.1	0.3
		1e–2	0–0	17619–17757	20–74	103/91	0.1	0.12	0.31
		1f–2	0–0	17653–17758	19–59	99/63	0.1	0.19	0.62
		2–3	0–0	17326–17483	13–85	143/139	0.1	0.15	0.43
57Akerlind	Akerlind (1957)	–	0–0	18994–19280	17–102	156/156	0.1	0.1	0.11
		–	1–0	19843–20106	35–94	110/110	0.1	0.1	0.15
73BaTa	Balfour & Tatum (1973)	e–	0–0	15136–15391	18–107	149/145	0.01	0.031	0.16
		f–	0–0	15185–15382	8–96	85/83	0.01	0.024	0.13
73Lindgren	Lindgren (1973)	1e–1	0–0	17995–18050	30–61	53/53	0.07	0.086	0.19
		1f–1	0–0	17991–18048	30–60	51/50	0.07	0.09	0.23
		2–2	0–0	17761–17820	47–65	36/29	0.07	0.12	0.35
76PhDa.CX	Phillips & Davis (1976a)	–	0–0	16732–17060	2–121	232/203	0.02	0.044	0.19
76PhDa.BX	Phillips & Davis (1976b)	e–	0–0	15102–15391	5–132	201/188	0.02	0.039	0.14
		e–	0–1	14292–14423	1–102	144/135	0.02	0.046	0.18
		e–	0–2	13244–13431	17–116	101/100	0.02	0.042	0.1
		e–	1–0	16023–16244	1–107	149/148	0.02	0.046	0.16
		e–	1–2	14038–14313	4–116	177/175	0.02	0.045	0.31
		e–	1–3	13246–13359	1–102	135/134	0.02	0.042	0.15
		e–	2–0	16817–17091	4–117	159/142	0.02	0.056	0.2
		e–	2–1	15936–16122	1–104	146/136	0.02	0.046	0.27
		e–	2–3	14046–14205	1–106	136/135	0.02	0.048	0.17
		e–	2–4	13122–13257	2–108	150/147	0.02	0.034	0.14
		e–	3–1	16690–16963	1–90	114/111	0.02	0.038	0.14
		e–	3–5	13051–13157	1–100	135/132	0.02	0.036	0.24
		e–	3–6	11990–12223	1–136	188/185	0.02	0.036	0.15
		e–	4–2	16547–16836	2–116	139/139	0.02	0.04	0.3
		e–	4–5	13746–13991	1–104	102/100	0.02	0.043	0.3
		e–	4–6	12824–13057	2–108	132/132	0.02	0.042	0.22
		e–	5–3	16659–16710	2–61	46/46	0.02	0.029	0.064
		e–	5–7	12822–12959	2–108	130/130	0.02	0.026	0.13
		f–	0–0	15108–15383	1–113	114/109	0.02	0.036	0.17
		f–	0–1	14289–14414	2–80	77/73	0.02	0.047	0.44
		f–	0–2	13250–13431	34–107	70/70	0.02	0.034	0.14
		f–	1–0	16021–16237	1–96	93/93	0.02	0.037	0.1
		f–	1–3	13248–13349	3–76	72/72	0.02	0.036	0.16
		f–	2–0	16759–17084	2–113	96/94	0.02	0.046	0.2
		f–	2–1	15938–16115	3–87	81/76	0.02	0.04	0.19
		f–	2–3	14039–14195	9–90	72/72	0.02	0.04	0.15
		f–	2–4	13122–13248	1–85	80/77	0.02	0.033	0.097
		f–	3–1	16703–16957	1–100	90/87	0.02	0.04	0.19
		f–	3–5	13050–13148	2–75	69/68	0.02	0.035	0.2
		f–	3–6	11986–12213	3–121	99/97	0.02	0.033	0.11
		f–	4–2	16555–16830	1–104	81/79	0.02	0.037	0.19
		f–	4–5	13748–13983	1–110	74/73	0.02	0.033	0.19
		f–	4–6	12834–13047	8–111	96/96	0.02	0.035	0.19
		f–	5–3	16661–16704	3–41	30/30	0.02	0.025	0.06
		f–	5–7	12822–12950	1–86	77/77	0.02	0.022	0.06
79GaDe	Gallaher & Devore (1979)	–	1–0	952–986	1–20	40/33	0.02	0.075	0.42
79PhDa	Phillips & Davis (1979a)	2–1	0–0	15132–15442	2–150	372/371	0.02	0.037	0.14
		2–1	0–1	14172–14515	2–150	351/350	0.02	0.042	0.13
		2–1	1–0	15862–16289	2–151	393/393	0.02	0.04	0.14
		2–1	1–1	15089–15361	2–150	327/318	0.02	0.046	0.2
		2–1	1–2	14093–14440	2–151	358/357	0.02	0.041	0.16
		2–1	2–1	15814–16191	3–151	356/355	0.02	0.047	0.17
		2–1	2–2	14928–15277	1–149	327/314	0.02	0.046	0.2
		2–1	2–3	14015–14364	2–151	363/362	0.02	0.045	0.17
		2–1	3–2	15679–16113	2–151	411/407	0.02	0.049	0.2
		2–1	3–3	14929–15194	2–137	332/329	0.02	0.039	0.12
		2–1	3–4	13944–14289	3–151	371/370	0.02	0.038	0.17
		2–1	3–5	13214–13384	2–101	206/206	0.02	0.051	0.14
		2–1	4–3	15802–16024	2–101	261/261	0.02	0.041	0.13
		2–1	4–5	13877–14214	2–151	348/348	0.02	0.039	0.18
		2–1	5–6	13947–14136	2–101	209/209	0.02	0.023	0.13
		3–2	0–0	15417–15754	2–144	375/361	0.02	0.033	0.2
		3–2	0–1	14499–14675	83–150	83/77	0.02	0.052	0.18
		3–2	1–0	16163–16602	2–150	397/395	0.02	0.036	0.53
		3–2	1–1	15423–15665	1–133	307/291	0.02	0.037	0.17
		3–2	1–2	14490–14749	2–151	363/358	0.02	0.037	0.2
		3–2	2–1	16065–16514	2–151	375/361	0.02	0.037	0.29
		3–2	2–2	15198–15591	1–147	374/353	0.02	0.049	0.2
		3–2	2–3	14309–14673	2–151	365/336	0.02	0.041	0.17
		3–2	3–2	15972–16422	2–151	387/353	0.02	0.047	0.2
		3–2	3–3	15102–15508	2–151	358/333	0.02	0.053	0.24
		3–2	3–4	14230–14594	3–150	308/303	0.02	0.031	0.19
		3–2	3–5	13516–13696	2–101	178/177	0.02	0.043	0.15
		3–2	4–3	16103–16335	2–101	244/244	0.02	0.037	0.13
		3–2	4–5	14150–14518	3–151	336/336	0.02	0.038	0.2
		3–2	5–6	14250–14445	2–101	222/222	0.02	0.024	0.081
		4–3	0–0	15704–16040	3–151	360/330	0.02	0.053	0.23
		4–3	0–1	14833–15117	3–151	370/350	0.02	0.04	0.19
		4–3	1–0	16488–16898	3–147	363/348	0.02	0.045	0.24
		4–3	1–1	15666–15967	3–150	336/321	0.02	0.053	0.2
		4–3	1–2	14686–15042	4–151	395/382	0.02	0.042	0.17
		4–3	2–1	16358–16809	3–151	403/397	0.02	0.044	0.2
		4–3	2–2	15604–15885	3–147	311/298	0.02	0.059	0.2
		4–3	2–3	14667–14970	3–136	372/365	0.02	0.038	0.18
		4–3	3–2	16270–16724	3–151	393/386	0.02	0.045	0.2
		4–3	3–3	15566–15796	3–108	239/237	0.02	0.049	0.17
		4–3	3–4	14685–14897	3–105	262/262	0.02	0.034	0.18
		4–3	3–5	13813–13991	3–101	187/186	0.02	0.045	0.15
		4–3	4–3	16410–16636	3–101	251/250	0.02	0.045	0.16
		4–3	4–5	14469–14824	3–151	385/385	0.02	0.031	0.16
		4–3	5–6	14558–14752	3–101	224/224	0.02	0.023	0.11
80HaDa	Hammer & Davis (1980)	1e–	0–0	19047–19121	29–41	24/23	0.01	0.025	0.085
		1e–	0–1	18094–18152	32–39	8/6	0.01	0.032	0.071
		1e–2	0–0	17693–17735	30–41	22/21	0.01	0.024	0.097
		1f–	0–0	19085–19095	29–37	9/9	0.01	0.022	0.042
		1f–	0–1	18120–18128	29–35	7/6	0.01	0.012	0.015
		1f–2	0–0	17695–17736	26–41	28/26	0.01	0.022	0.081
81HaDa	Hammer & Davis (1981)	e–	0–0	9102–9507	2–147	394/354	0.01	0.017	0.14
		e–	1–0	10057–10359	3–109	176/176	0.01	0.024	0.13
		e–	2–1	10022–10271	3–115	151/149	0.01	0.023	0.17
		f–	0–0	9152–9507	2–149	373/336	0.01	0.016	0.09
		f–	1–0	10110–10359	3–118	179/178	0.01	0.021	0.12
		f–	2–1	10033–10271	3–115	159/155	0.01	0.022	0.13
81HaDaZo	Hammer et al. (1981)	e–	1–0	10324–10359	14–22	21/21	0.01	0.011	0.02
		e–	1–1	13915–13948	15–21	11/11	0.01	0.01	0.01
88SiMiHuHa	Simard et al. (1988b)	–	0–0	17011–17060	0–30	59/57	0.006	0.025	0.1
90SuLoFrMa	Suenram et al. (1990)	–	0–0	0–1	0–1	1/1	3 × 10−7	3 × 10−7	3 × 10−7
94Jonsson	Jonsson (1994)	0e–1	0–0	10535–10714	7–107	180/180	0.006	0.012	0.069
		0f–1	0–0	10579–10702	14–92	113/113	0.006	0.0078	0.026
		1e–2	0–0	10611–10728	11–100	137/137	0.006	0.0088	0.045
		1f–2	0–0	10625–10731	20–90	104/104	0.006	0.0097	0.067
		2e–3	0–0	10614–10750	11–111	171/171	0.006	0.0075	0.049
		2f–3	0–0	10614–10750	11–111	168/168	0.006	0.0079	0.049
95KaMcHe	Kaledin et al. (1995)	1e–1	0–0	17993–18050	2–63	67/66	0.007	0.013	0.057
		1f–1	0–0	17984–18048	2–67	79/79	0.007	0.016	0.084
		2–2	0–0	17748–17820	2–64	95/92	0.007	0.021	0.31
99BeGe	Beaton & Gerry (1999)	–	0–0	1–1	0–1	1/1	1 × 10−7	1 × 10−7	1 × 10−7
		–	1–1	0–1	0–1	1/1	1 × 10−7	1 × 10−7	1 × 10−7
		–	2–2	0–1	0–1	1/1	1 × 10−7	1 × 10−7	1 × 10−7
		–	3–3	0–1	0–1	1/1	1 × 10−7	1 × 10−7	1 × 10−7

Note. A/V means available/validated.

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Two further studies in the 1980s, Hammer et al. (1981) and Stepanov et al. (1988), investigated higher vibrational levels of some of the most important electronic states but without rotational resolution.

Beyond the position of the lines, many experimental studies have focused on the intensity of transitions (e.g., Herbig 1949; Murthy & Prahllad 1980; Littleton & Davis 1985; Littleton et al. 1993), radiative lifetimes (e.g., Hammer 1978; Hammer & Davis 1979; Simard et al. 1988b), and permanent dipole moments (e.g., Suenram et al. 1990; Pettersson et al. 2000).

The partition function and dissociation constants for zirconium oxide have been considered by various authors, including Shankar & Littleton (1983).

There are a number of other studies of ZrO spectra that we have not used in this study for various reasons. These data sources are collated in Table 2 with brief comments.

Table 2.  Experimental ZrO Papers Not Used in the Rotationally Resolved Marvel or Bandhead Analysis

References	Comment
Lowater (1935)	Good analysis of vibronic bands (including triplet splitting), but the rotational analysis of the – is incorrect.
Tanaka & Horie (1941)	Incorrect rotational analysis of the – bands and more recent data are available
Kiess (1948)	Data not available online.
Herbig (1949)	Unassigned.
Uhler (1954a)	Rotational analysis with band constants, but assigned line positions are given in the associated papers (Lagerqvist et al. 1954; Uhler 1954b).
Uhler & Åkerlind (1955)	Rotational analysis with band constants for singlet A system, but assigned line positions are given in the associated paper (Uhler & Akerlind 1956).
Åkerlind (1956)	Band constants from analysis of a system assigned as the singlet B system at 8192 Å, but this is not consistent with Akerlind (1957).
Åkerlind (1957)	Rotationally resolved data from a system assigned as the singlet B system, but with frequencies around 19,000 cm−1, not 12,000 cm−1 as indicated by the 8192 Å labeling. Due to this confusion, and some later papers (Phillips & Davis 1979b; Balfour & Lindgren 1980) that provide good evidence that the 8192 Å band (around 12,000 cm−1) is a ZrO+ band, these data are not included in our compilation.
Tatum & Balfour (1973)	Data very poorly reproduced digitally, and higher resolution spectra for the – bands studied are available.
Weltner & McLeod (1965)	Ground state determination in Ne matrix.
Schoonveld & Sundaram (1974)	Complete and systematic analysis of available data for triplet systems, but provides no assigned rotationally resolved data.
Bijc et al. (1974)	Determination of Singlet–Triplet separation, but no assigned rotational data.
Lauchlan et al. (1976)	ZrO in Ne inert matrix at 4 K.
Phillips et al. (1979)	The rotational analysis here was shown to be incorrect by the subsequent reanalysis by Jonsson (1994); see the text.
Gallaher & Devore (1979)	Rotationally unresolved infrared study; used for comparison against bandheads but not as part of the Marvel data set.
Murty (1980b)	Contains molecular constants for and , but provides no assigned rotationally resolved data.
Hammer et al. (1981)	Identification of bands in astronomical versus laboratory spectra, no rotational analysis.
Afaf (1987)	Reanalysis of data and recommended molecular constants; also proposes a singlet C band from to a singlet at 7870 cm−1 above; this was later discounted (e.g., Afaf 1995).
Davis & Hammer (1988)	Consolidation of data and proposed electronic structure.
Simard et al. (1988a)	High-resolution study of the – 0–0 band; assigned line positions were not published with the original data and could not be located.
Afaf (1995)	Discusses the δ(3Π–) and ϕ(3Δ–) bands, but provides no assigned rotationally resolved data.
Balfour & Chowdhury (2010)	Low-resolution data with bandhead information on the – state.

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The data in Tatum & Balfour (1973) was of very poor readability, which meant the accuracy of digitization even manually could not be guaranteed. As there are substantial more modern data available for the same transitions, we did not use these data.

A key omission to our Marvel collation is the Phillips et al. (1979) paper. The subsequent study by Jonsson (1994) performed a complete reanalysis of the ZrO spectra in the region around 10,750 cm⁻¹, that assigned all bands, whereas the Phillips et al. (1979) analysis omitted many bands. A key feature of the Jonsson (1994) analysis was a large Λ-doubling splitting between the _0e and the _0f state. This is attributed to spin–orbit coupling with the nearby state whose T_e was only predicted semi-quantitatively with computational techniques in the 1990s.

Unfortunately, the data of Simard et al. (1988a) could not be located; however, the spectra and analysis by Kaledin et al. (1995) covers the same spectral transitions.

Finally, we want to briefly discuss Balfour & Chowdhury (2010) in more detail, particularly their claim to observe the a¹Π– system near 19,480 cm⁻¹. We strongly question this assignment because there is no predicted ¹Π state in this energy range from either ab initio calculations or analogy to the TiO electronic states. Based on vibrational frequencies, the spectrum Balfour & Chowdhury (2010) observed does not appear to come from overtones of a – band. The only experimental reference to a ¹Π state in this energy range is from Simard et al. (1988a) who explain perturbation in the triplet splittings using a ¹Π state originally predicted theoretically by Green (1969). The energies of electronic states in this energy range for transition metal diatomics are notoriously challenging to predict accurately even with today’s methods (Tennyson et al. 2016) and thus this early theoretical investigation cannot be trusted even qualitatively for higher electronic states, especially since more recent theoretical papers (Langhoff & Bauschlicher 1990) make no such predictions for a ¹Π state in this energy range.

Attempts to assign the clearly visible bands seen by Balfour & Chowdhury (2010) to a transition were unconvincing. Given the low resolution of these data and its inconsistencies with current knowledge of the electronic structure of ZrO from both a theoretical and experimental perspective, we suggest these unassigned peaks are due to ZrO⁺. The method used by Balfour & Chowdhury (2010) does involve the creation of ions, and there is precedence for the ionization occurring. Phillips & Davis (1979b) conducted a study on bands in what was understood to be the ZrO spectrum with heads at 7811 and 8192 Å that had previously been observed by Afaf (1950a) and analyzed by Uhler & Åkerlind (1955) and Uhler & Akerlind (1956) as belonging to a new system. They found that these bands belonged to a ²Π–²Σ system of ZrO⁺. While further work has been done on ZrO⁺, none of these studies have examined the same wavelengths as Balfour & Chowdhury (2010), and thus no definitive assignment can be made at this stage.

We do not extensively review the literature of quantum computations on ZrO, but notable calculations include those of Langhoff & Bauschlicher (1988, 1990) and Shanmugavel & Sriramachandran (2011).

Our analysis started by digitizing available assigned rovibronic transitions data, then converting them to Marvel format. The full list of compiled data converted to Marvel format is given in the supplementary information; an extract is given in Table 3. The full list of data sources used in the rotationally resolved Marvel analysis are summarized in Table 1; we provide information on the vibronic bands measured, the wavenumber range and J range, as well as the number of transitions measured. In total, we use 12 data sources, involving 9 electronic states with 23,317 transitions and 72 total unique spin-vibronic bands (ignoring Λ splitting).

Table 3.  Extract from the 90Zr-16O.marvel.inp Input File for

1	2	3	4	5	6	7	8	9
		State′	v′	J′	State″	v″	J″	ID
17059.5189	0.006000	C1Sigma+	0	18	X1Sigma+	0	17	88SiMiHuHa.46
17059.9101	0.006000	C1Sigma+	0	21	X1Sigma+	0	20	88SiMiHuHa.49
17059.9295	0.006000	C1Sigma+	0	25	X1Sigma+	0	24	88SiMiHuHa.53
17059.9792	0.006000	C1Sigma+	0	24	X1Sigma+	0	23	88SiMiHuHa.52
10710.4902	0.006622	b3Pi_2e	0	46	a3Delta_3	0	46	94Jonsson.540
10710.4902	0.006622	b3Pi_2f	0	46	a3Delta_3	0	46	94Jonsson.730
10617.7781	0.006660	b3Pi_2e	0	79	a3Delta_3	0	80	94Jonsson.690
Column	Notation
1		Transition frequency (in cm−1)
2		Estimated uncertainty in transition wavenumber (in cm−1)
3	State′	Electronic state of upper energy level; also includes parity for Π states and Ω for triplet states
4	v′	Vibrational quantum number of upper level
5	J′	Total angular momentum quantum number of upper level
6	State″	Electronic state of lower energy level; also includes parity for Π states and Ω for triplet states
7	v″	Vibrational quantum number of lower level
8	J″	Total angular momentum quantum number of lower level
9	ID	Unique ID for transition, with reference key for source (see Table 1) and counting number

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Comments related to Table 1, particularly regarding the initial uncertainty chosen for the data, are as follows:

•  An uncertainty of 0.1 cm⁻¹ was chosen, enabling a high number of validated transitions within these data sets and those by one of the same authors in the same year.
•  Uncertainty as for 54LaUhBa, though this data set could be compared against later data more directly and thus had a bigger influence on setting the maximum uncertainty used.
•  Uncertainty as for 54LaUhBa; these data were the only source of state information, so uncertainty reflects only requirements for self-consistency within this data set.
•  The data table has poor readability and it is likely that minor errors in digitization may exist, though major errors were corrected by hand using the systematic nature of the transition frequencies. We adopted 0.01 cm⁻¹ as the minimum uncertainty for the data (with higher values adopted as necessary up to 0.16 cm⁻¹), as this yielded a reasonable number of self-consistent results.
•  No uncertainty is stated in the paper; however, 0.07 cm⁻¹ gave reasonable self-consistent calculations for most bands. Note that these are satellite bands and hence had lower intensities and higher positional uncertainties than for the main bands.
•  The original paper did not use the label for the upper state; this has been named in subsequent discussions of and adopted here. There are no uncertainties given; however, we found that at least 0.02 cm⁻¹ was required to enable a significant number of these data to self-validate. Some data were substantially more inaccurate than this; we have removed all data that required uncertainties of more than 0.2 cm⁻¹ to be consistent with the rest of the data.
• The data were of low resolution for an infrared spectra and at high temperature; thus a relatively high uncertainty of 0.02 cm⁻¹ minimum up to 0.42 cm⁻¹ was used. Seven lines did not validate.
•  Data obtained from Kurucz and given uncertainties of 0.02 cm⁻¹, as for other Phillips and Davis data of this era. The ₃–₂(0–1) band appears to be largely incorrectly assigned; we have used only those transitions that agree well with assignments from other bands.
•  0.01 cm⁻¹ was stated as the measurement accuracy for at least some bands; this was adopted for the whole data set by multiple papers by the same authors in a similar time period. Note that this is a factor of two more accurate than earlier data from Davis and coworkers.
•  The stated uncertainty was 200 MHz, with reproducibility to 50 MHz; we therefore adopted 0.006 cm⁻¹ as an initial uncertainty for our data.
•  The stated uncertainty was 1 kHz (3 × 10⁻⁸ cm⁻¹); however, consistency with 99BeGe required an uncertainty estimate of 3 × 10⁻⁷ cm⁻¹.
•  The stated uncertainty was 0.016 cm⁻¹; however, we found a smaller uncertainty of 0.006 cm⁻¹, as an initial estimate was warranted due to the consistency of the data both internally and with other results.
•  The stated uncertainty was 0.03 cm⁻¹; however, we found a smaller uncertainty of 0.007 cm⁻¹, as an initial estimate was warranted due to the consistency of the data both internally and with other results.
•  The stated uncertainty was 1 kHz (10⁻⁸ cm⁻¹); however, consistency with 90SuLoFrMa required uncertainty of 10⁻⁷ cm⁻¹, so this was adopted for all values.

During the Marvel process, many of our initial estimated uncertainties were updated to establish a self-consistent network, while some transitions were removed from consideration (designated through a minus sign at the start of the Marvel input line for that transition). To assess the data, Table 1 provides data on the minimum, average, and maximum uncertainty of each transition; in most cases, we were able to keep the minimum and average uncertainty to within a factor of two. We validated 22,549 of our 23,317 input transitions, i.e., showed that these validated transitions were consistent with other measurements. The Marvel input file can readily be updated in the future with new spectroscopic information to enable an updated set of Marvel energies to be created.

Figure 2 represents the data from Table 1 showing the experimentally measured transitions connecting different vibronic states. From this diagram, it is clear that there are three bands connecting the singlet and triplet manifold, and some satellite transitions for the triplet sub-manifolds, allowing most energy levels to be connected.

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There are three minor free-floating networks connecting the (ν = 6) and (ν = 5) levels. These could be connected through observing additional vibrational transitions, but this is not essential for producing a good model of the electronic states.

Table 4 shows an extract of the final empirical energy levels produced by Marvel for in this work. This list of energy levels includes an estimate for the uncertainty in the provided energy of the quantum state, as well as identifying the number of transitions used to determine the energy level; on average, 5.3 transitions were used to find each energy level.

Table 4.  Extract from the 90Zr-16O.main.energies Output File for

State	v	J		Unc.	No.
X1Sigma+	5	92	8286.730593	0.016290	3
a3Delta_1	4	93	8289.132156	0.013142	3
a3Delta_2	4	89	8295.024932	0.018098	3
X1Sigma+	6	79	8296.993918	0.009580	6
a3Delta_1	1	124	8312.622601	0.022336	7
A1Delta	0	76	8313.129649	0.004368	11
a3Delta_2	5	76	8320.754165	0.013995	5
a3Delta_2	3	101	8322.052377	0.013804	6
a3Delta_2	1	121	8322.707720	0.012197	7
X1Sigma+	4	104	8327.541550	0.020000	1
A1Delta	1	60	8336.081220	0.008485	2
a3Delta_2	0	130	8336.647124	0.013307	6

Note. Energies and their uncertainties are given in cm⁻¹. No indicates the number of transitions that contributed to the stated energy and uncertainty.

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Table 5 summarizes the 8088 empirical energy levels found in the main SN from the Marvel analysis for . We see the minimum, average, and maximum uncertatinty provided for the empirical energies from the Marvel analysis. The minimum is usually very small, often less than 0.01 cm⁻¹, while the maximum can exceed 0.1 cm⁻¹; this is probably for higher J states. There is generally coverage to high rotational number J if the spin-vibronic level is known.

Table 5.  Summary of the 8088 Empirical Energy Levels for Determined by Our Marvel analysis

	v	J Range	Uncertainties (cm−1)
			Min	Av	Max
	0	0–131	1 × 10−7	0.0096	0.055
	1	2–107	0.006	0.016	0.08
	2	2–115	0.0079	0.016	0.11
	3	2–105	0.008	0.014	0.052
	4	2–106	0.012	0.02	0.068
	5	2–107	0.0089	0.015	0.051
	6	2–107	0.0082	0.014	0.067
	7	3–66	0.012	0.016	0.042
	0	2–133	0.0031	0.0058	0.12
	1	3–115	0.0045	0.012	0.14
e	0	1–133	0.0043	0.0075	0.065
	1	1–116	0.0038	0.011	0.057
	2	1–117	0.0048	0.011	0.053
	3	1–108	0.0084	0.015	0.062
	4	1–116	0.0097	0.02	0.21
	5	2–67	0.01	0.018	0.045
f	0	1–133	0.0046	0.0089	0.15
	1	1–118	0.0053	0.012	0.052
	2	2–115	0.0055	0.013	0.076
	3	2–106	0.012	0.02	0.08
	4	2–107	0.012	0.023	0.14
	5	3–66	0.014	0.019	0.042
	0	0–121	0.0041	0.02	0.21
	0	17–102	0.057	0.063	0.14
	1	35–93	0.057	0.064	0.1
1	0	2–150	0.0026	0.0072	0.063
	1	2–150	0.007	0.014	0.12
	2	1–150	0.0074	0.013	0.084
	3	2–150	0.0072	0.013	0.059
	4	3–150	0.012	0.021	0.12
	5	2–106	0.0089	0.017	0.06
2	0	2–149	0.0023	0.0086	0.057
	1	1–148	0.0052	0.014	0.08
	2	1–150	0.0072	0.017	0.2
	3	2–147	0.0075	0.018	0.17
	4	3–150	0.012	0.021	0.15
	5	2–136	0.0089	0.016	0.054
3	0	3–144	0.0024	0.013	0.2
	1	3–144	0.0069	0.015	0.15
	2	3–145	0.0071	0.012	0.04
	3	3–135	0.0077	0.012	0.062
	4	3–105	0.012	0.017	0.043
	5	3–130	0.0082	0.014	0.055
0e	0	7–106	0.0035	0.0065	0.023
0f	0	14–91	0.0042	0.0064	0.037
1e	0	11–100	0.0035	0.0056	0.013
1f	0	20–90	0.0036	0.0058	0.012
2e	0	12–111	0.0035	0.0051	0.015
2f	0	12–111	0.0035	0.0053	0.031
2	0	2–150	0.0081	0.014	0.14
	1	2–151	0.0071	0.013	0.22
	2	2–151	0.0075	0.013	0.056
	3	3–151	0.0069	0.011	0.035
	4	3–106	0.0086	0.014	0.041
3	0	3–148	0.01	0.018	0.12
	1	2–151	0.0067	0.011	0.046
	2	2–147	0.0071	0.015	0.13
	3	2–150	0.0063	0.012	0.09
	4	3–136	0.0088	0.014	0.029
4	0	4–144	0.0081	0.016	0.11
	1	4–145	0.0072	0.013	0.053
	2	4–145	0.0072	0.014	0.16
	3	4–145	0.0068	0.013	0.12
	4	4–131	0.0082	0.012	0.037
0e	0	15–59	0.058	0.07	0.1
0f	0	27–59	0.058	0.064	0.15
1e	0	3–73	0.0043	0.02	0.1
1f	0	3–67	0.0044	0.016	0.13
2	0	2–85	0.0043	0.034	0.1
1	0	20–76	0.071	0.074	0.1
2	0	20–79	0.071	0.075	0.1
3	0	21–81	0.071	0.077	0.1

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Our Marvel analysis shows that there is good rotationally resolved empirical understanding of a reasonable number of vibrational states of the , , , and electronic states (sufficient for a good potential energy curve to be fitted). However, there is much less vibrational information (only one or two levels) for the , , , , , and states. This will cause significant problems when fitting potential energy curves for a full spectroscopic model and eventual line list for and its isotopologues, particularly for the , , , and states in which only one vibrational level is known. Note that line lists of all isotopologues can be easily produced using variational nuclear-motion techniques using data from only a single isotopologue with reasonably high accuracy; however, ab initio predictions of vibrational constants especially for higher lying electronic states of transition-metal-containing diatomics still have errors of up to 50 cm⁻¹ (Tennyson et al. 2016).

Figure 3 shows the empirical energy levels for the main SN from Marvel against J for each spin-vibronic band. These are clearly quadratic and smooth, indicating there are no major problems with the Marvel network established for .

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Figure 4 shows the difference between the singlet Marvel energy levels for and those from the Plez et al. (2003) line list. For the , , and states, the differences average around 0.05–0.15 cm⁻¹, with somewhat higher deviations, up to 1 cm⁻¹, for large J, especially in the state. The scatter here is probably largely a reflection of inaccuracies in the MARVEL energy levels, though perturbations not considered in the Plez line list might also contribute. The state, however, shows much more significant deviations; the v = 0 state is off by about 2 cm⁻¹ up to J = 100, with much more significant deviations for larger J. The v = 1 state also shows substantial differences of up to 4 cm⁻¹ that changes significantly with J.

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Figures 5–7 show the difference between the triplet Marvel energy levels for and those from the Plez et al. (2003) line list. These deviations are much more significant than for the singlet states.

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Clear systematic errors can be seen throughout the and bands—since these are a major cause of opacity of in stellar conditions, improvements to these energy levels are highly desirable. Note, however, that since the errors in the _n and _n+1 parallel each other, the errors in transition frequencies in the Plez line list will be much smaller than the errors in energies that are plotted here.

The state shows significant and systematic deviations in the Plez database compared to the Marvel data of up to 15 cm⁻¹ for many vibronic levels. Our adoption of the Jonsson (1994) assignments in preference to the Phillips et al. (1979) assignments contributes to much larger Λ doubling in the Marvel data than was adopted in the Plez data. There are also clear systematic differences in the energies of the ₁ levels of more than 15 cm⁻¹ in many regions. Smaller differences of up to 5 cm⁻¹ were found in the ₂ levels that parallel the errors in the ₃ and ₄ energies, indicating that the errors associated with the transition frequencies in this band in the Plez line list will be much smaller.

The data show that Plez’s triplet splitting is in error by about 1 cm⁻¹, with some larger errors at high J.

Band constants were obtained by a quadratic fit of the energies of the lines against rotational quantum number J for each band.

Table 6 shows the rotational band constants, T_v, B_v, and D_v for each singlet vibronic bands. The centrifugal term, D_v, is reasonably constant within a given electronic state, while the rotational constant, B_v, decreases as expected as the bond length increases in higher vibrational states.

Table 6.  Spectroscopic Band Constants, in cm⁻¹, for the Singlet Vibronic Bands, Assuming No Perturbations

State	v	Tv	Bv	107 Dv
	0	−0.028	0.4226	3.18
	1	969.509	0.42065	3.19
	2	1932.154	0.41868	3.18
	3	2887.873	0.41674	3.22
	4	3836.760	0.41475	3.22
	5	4778.739	0.41275	3.21
	6	5713.739	0.41077	3.23
	0	5887.160	0.41646	3.26
	1	6823.105	0.41457	3.26
	0	15383.385	0.40151	3.51
	1	16236.949	0.39959	3.50
	2	17084.607	0.39765	3.51
	3	17926.299	0.39570	3.51
	4	18762.010	0.39375	3.52
	5	19591.668	0.39175	3.40
	0	17050.378	0.40480	3.44
	0	25159.631	0.39736	3.57
	1	25994.872	0.39540	3.66

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Table 7 shows the fitted rotational band constants for each spin-vibronic band in the triplet manifold for from this Marvel analysis.

Table 7.  Spectroscopic Constants, in cm⁻¹, for the Triplet Spin-vibronic Bands, Assuming No Perturbations

		Σ = −1			Σ = 0			Σ = 1
v		Tv	Bv	Dv(107)	Tv	Bv	Dv(107)	Tv	Bv	Dv(107) Δ(SO)
		1			2			3
0		1080.363	0.41333	3.18	1367.750	0.41476	3.26	1703.505	0.41565	3.43
1		2011.656	0.41144	3.19	2299.369	0.41285	3.26	2635.505	0.41374	3.44
2		2936.474	0.40955	3.20	3224.476	0.41096	3.29	3561.029	0.41181	3.45
3		3854.766	0.40765	3.21	4143.140	0.40903	3.28	4480.011	0.40989	3.45
4		4766.602	0.40574	3.22	5055.317	0.40712	3.31	5392.650	0.40791	3.43
5		5672.050	0.40378	3.19	5960.981	0.40516	3.28	6298.588	0.40602	3.48
		0			1			2
0	e	11765.173	0.40801	3.36	12069.846	0.40862	3.42	12427.697	0.40934	3.62
	f	11783.845	0.40754	3.29	12069.859	0.40915	3.45	12427.705	0.40933	3.60
		2			3			4
0		16507.187	0.40312	3.57	17109.068	0.40368	3.60	17737.310	0.40430	3.77
1		17357.358	0.40103	3.58	17958.303	0.40160	3.62	18588.627	0.40222	3.77
2		18200.953	0.39894	3.59	18801.000	0.39949	3.63	19433.859	0.40015	3.78
3		19038.014	0.39683	3.60	19637.131	0.39738	3.65	20273.306	0.39807	3.77
4		19868.503	0.39469	3.58	20466.627	0.39524	3.62	21106.945	0.39602	3.77
		0			1			2
0	e	19074.117	0.39551	0.85	19113.069	0.40387	5.73	19169.508	0.40619	5.02
	f	19078.935	0.39449	0.04	19112.826	0.40266	4.79
		1			2			3
0		22616.840	0.38945	3.43	22916.797	0.39207	0.67	23335.153	0.39572	3.01

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A compilation of band constants is given by Kaledin et al. (1995); we find significant differences of 2 cm⁻¹ in the T₀ for ₂, ₃, and ₃. We prefer our value, however, as the — transitions form part of our input data, whereas it is unclear how the Kaledin et al. (1995) was compiled. Otherwise, the T₀ values agree within 0.1 cm⁻¹.

Tables 8–12 show bandheads predicted by the Marvel energy levels, compared with available, low-resolution bandhead observations (note that the high-resolution bandhead observations are included in the Marvel input data set). For the singlet states, there is actually only a small number of data points in the – band that allow for direct comparison of Marvel predictions against low-resolution bandhead studies. There are no assigned low-resolution data readily available for the – band, and the low-resolution bandhead data for – does not overlap with our Marvel predictions. For the triplets, there is good low-resolution bandhead data for the –, –, and – bands against which the Marvel results can be compared; in these cases, there is good agreement for all bands, generally better than 0.5 cm⁻¹ (though up to 1.5 cm⁻¹).

Table 8.  R-branch Bandheads in cm⁻¹ From the State for ; J Gives the Approximate J Value Corresponding to the Rotational Transitions at the Bandhead

	v′–v″	J	MARVEL	Low-res obs.
–	0–0	18	15391.40	⋯
	0–1	20	14422.64	⋯
	0–2	22	13460.94	⋯
	0–3a	25	12506.31	⋯
	0–4a	29	11559.08	⋯
	0–5a	34	10619.12	⋯
	0–6a	41	9687.00	⋯
	0–7a	53	8763.33	⋯
	1–0	18	16244.28	⋯
	1–1a	18	15275.39	⋯
	1–2	21	14313.52	⋯
	1–3	22	13358.69	⋯
	1–4a	24	12410.99	⋯
	1–5a	30	11470.63	⋯
	1–6a	36	10537.59	⋯
	1–7a	43	9612.56	⋯
	2–0	15	17091.34	⋯
	2–1	17	16122.32	⋯
	2–2a	18	15160.33	⋯
	2–3	21	14205.29	⋯
	2–4	22	13257.41	⋯
	2–5a	26	12316.62	⋯
	2–6a	30	11383.20	⋯
	2–7a	25	10457.20	⋯
	3–0a	15	17932.50	⋯
	3–1	14	16963.46	⋯
	3–2a	17	16001.36	⋯
	3–3a	17	15046.20	⋯
	3–4a	21	14098.05	⋯
	3–5	22	13157.07	⋯
	3–6	27	12223.20	⋯
	3–7a	31	11296.67	⋯
	4–0a	14	18767.79	⋯
	4–1a	14	17798.67	⋯
	4–2	16	16836.45	⋯
	4–3a	17	15881.27	⋯
	4–4a	17	14933.00	⋯
	4–5	20	13991.83	⋯
	4–6	21	13057.77	⋯
	4–7a	26	12130.78	⋯
	5–0a	11	19597.03	⋯
	5–1a	13	18627.83	⋯
	5–2a	13	17665.63	⋯
	5–3	14	16710.22	⋯
	5–4a	17	15761.90	⋯
	5–5a	20	14820.57	⋯
	5–6a	20	13886.36	⋯
	5–7	21	12959.14	⋯
–	0–0	22	17059.99	⋯
	0–1a	25	16091.58	⋯
	0–2a	28	15130.40	⋯
	0–3a	33	14176.55	⋯
	0–4a	41	13230.35	⋯
	0–5a	49	12292.37	⋯
	0–6a	82	11365.35	⋯
	1–0	⋯	⋯	17933b
	2–0	⋯	⋯	18799b
	3–0	⋯	⋯	19664b

Notes.

^aBands unobserved in rotationally resolved spectra that have been predicted by Marvel. ^bBalfour & Chowdhury (2010; converted from wavelength assuming vacuum).

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Table 9.  Other Singlet R-branch Bandheads in cm⁻¹ for ; J Gives the Approximate J Value Corresponding to the Rotational Transitions at the Bandhead

	v′–v″	J	MARVEL	Low-res obs.
–	0–0	17	25166.23	⋯
	0–1	18	24197.07	⋯
	0–2	17	23235.27	⋯
	0–3	22	22280.33	⋯
	0–4	22	21332.47	⋯
	0–5	25	20391.58	⋯
	0–6	30	19458.10	⋯
	0–7	35	18531.99	⋯
–	2–0	101	1975.73	⋯
	3–0	68	2917.35	⋯
	3–1	99	1961.43	⋯
	4–0	51	3858.92	⋯
	4–1	65	2896.33	⋯
	4–2	99	1948.15	⋯
	5–0	42	4796.37	⋯
	5–1	51	3831.03	⋯
	5–2	67	2875.26	⋯
	5–3	98	1932.88	⋯
	6–0	33	5728.41	⋯
	6–1	41	4761.69	⋯
	6–2	49	3803.14	⋯
	6–3	67	2854.22	⋯
	6–4	98	1918.42	⋯
	7–0	28	6654.41	⋯
	7–1	33	5686.80	⋯
	7–2	40	4726.86	⋯
	7–3	50	3775.20	⋯
–	0–0	26	9507.28	9507.45a
	0–1	30	8572.87	⋯
	1–0	23	10359.55	10359.62a
	1–1b	25	9424.77	9424.93a
	2–0	21	11206.15	⋯
	2–1	22	10271.14	10271.26a
	3–0b	17	12047.00	⋯
	3–1	21	11111.79	⋯
	4–0b	17	12882.08	⋯
	4–1b	17	11946.74	⋯
	5–0b	15	13711.04	⋯
	5–1b	17	12775.64	⋯
–	0–0b	33	11177.59	⋯
	0–1b	41	10244.31	⋯
–	0–0	22	19281.19	⋯
	0–1	22	18346.21	⋯

Notes.

^aBalfour & Chowdhury (2010; converted from wavelength assuming vacuum). ^bBands unobserved in rotationally resolved spectra that have been predicted by Marvel.

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Table 10.  Other Singlet R-branch Bandheads in cm⁻¹ for ; J Gives the Approximate J Value Corresponding to the Rotational Transitions at the Bandhead

	v′–v″	J	MARVEL
–	0–0	91	9814.07
	1–0	61	10637.34
	1–1	90	9794.47
–	2–0	53	56.81
	3–0	41	893.65
	4–0	34	1726.14
	5–0	28	2553.57
–	0–0	50	8130.59
	1–0	40	8961.46

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Table 11.  Triplet – and – R-branch Bandheads in cm⁻¹ for ; J Gives the Approximate J Value Corresponding to the Rotational Transitions at the Bandhead

	v′–v″	J	MARVEL	Low-res obs.
1–	0–0	⋯	⋯	19124a
	1–0	⋯	⋯	19963a
	2–0	⋯	⋯	20784a
0e–1	0–0	72	10715.52	10715.26b
	0–1c	101	9798.16	⋯
	1–1	⋯	⋯	10634.15b
0f–1	0–0	67	10731.97	10731.92a
1e–2	0–0	63	10729.02	10728.98a
	0–1c	89	9808.11	⋯
1f–2	0–0	67	10731.44	10731.29a
	1–1	⋯	⋯	10649.82b
2–3	0–0	62	10750.52	⋯
	0–1	81	9828.65	⋯
1e–2	0–0	34	17760.35	⋯
	0–1c	40	16831.39	16833.0d
	0–2c	49	15909.87	15909.2d
	0–3c	58	14996.56	14994.3d
1f–2	0–0	32	17758.67	⋯
	0–1c	38	16829.39	16833.0d
	0–2c	44	15907.41	15909.2d
	0–3c	55	14993.55	14994.5d
1–2	1–1	⋯	⋯	17669.2d
	1–2	⋯	⋯	16747.2d
	1–3	⋯	⋯	15556.1d
	1–4	⋯	⋯	14923.9d

Notes.

^aMeasured (Davis & Hammer 1981) reassigned here. ^bBalfour & Chowdhury (2010; converted from wavelength assuming vacuum). ^cBands unobserved in rotationally resolved spectra that have been predicted by Marvel. ^dStepanov et al. (1988).

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Table 12.  d³Φ–a³Δ R-branch Bandheads in cm⁻¹ for from the Main Spectroscopic Networks; J Gives the Location of the Bandhead in Our Data

v′–v″	J	MARVEL	Low-res obs.	J	MARVEL	Low-res obs.	J	MARVEL	Low-res obs.
	2–1			3–2			4–3
0–0	39	15443	15443.0a	34	15756.3	15756.3a	36	16048.46	16048.6a
0–1	48	14515.12		43	14827.63		42	15119.3	15119.2a
0–2b	57	13595.58		52	13906.89		51	14198.01
0–3b	78	12686.13		67	12995.36		68	13285.67
0–4b	108	11790.97		96	12096.25		90	12385.29
1–0	32	16290.36	16290.4a	30	16603.14	16603.4a	30	16897.5	16897.4a
1–1	37	15361.4	15361.4a	34	15673.57	15673.5a	35	15967.47	15967.9a
1–2	44	14439.86		40	14751.22		40	15044.66	15044.0a
1–3b	57	12526.53		51	13836.77		51	14129.7
1–4b	76	12623.07		68	12931.37		67	13223.55
1–5b	104	11733.3		94	12047.9		87	12329
2–0b	27	17132.02	17132.9a	26	17444.03	17444.7a	25	17741.09
2–1			16202.5a	28	16513.89	16514.0a	29	16810.47	16810.6a
2–2	36	15279.76	15279.8a	34	15590.72	15590.8a	33	15886.85	15886.9a
2–3b	45	14364.58		40	14674.79		39	14970.55	14970.7a
2–4b	54	13457.5		51	13766.6		51	14061.86
2–5b	74	12560.01		64	12867.46		64	13161.97
3–0b	23	17967.59		22	18278.87		23	18579.17
3–1b	26	17037.47	17036.2a	24	17348.38	17347.7a	25	17648.31	17646.7a
3–2	29	16114.24	16114.2a	27	16424.59	16424.7a	29	16724.12	16724.3a
3–3	35	15198.07	15298.4a	34	15507.83	15508.2a	33	15807.01	15807.6a
3–4	41	14289.18		39	14597.25		40	14897.02	14897.2a
3–5	54	13388.34		47	13696.37		51	13994.73
4–0b	21	18796.88		19	19107.25		19	19411.79
4–1b	24	17866.54		21	18176.52		22	18480.69
4–2b	25	16942.87	16942.5a	24	17252.48	17253.3a	24	17556.27	17555.6a
4–3	29	16026.05	16025.0a	28	16335.14	16335.6a	27	16638.61	16638.6a
4–4b	36	15116.29	15115.2a	34	15424.86	15425.2a	31	15727.82	15728.3a
4–5	43	14213.74		39	14521.64		39	14824.26
5–3b			16847.4a						17463.2a
5–4b			15936.9a						16554.2a
5–5b			15034.8a			15342.6a			15648.8a

Notes.

^aStepanov et al. (1988). ^bBands unobserved in rotationally resolved spectra that have been predicted by Marvel.

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The Marvel results provide predictions for 48 vibronic bands previously unmeasured in low or high-resolution spectra. In contrast, there are at least 68 additional low-resolution bandheads whose positions cannot be predicted by our Marvel data due to lack of information on, usually, the excited state. The most significant missed opportunity for rotational resolved data is in the 48 nonsatellite, i.e., ΔΣ = 0, – bandheads for v = 0 to v = 6 observed in low-resolution by Stepanov et al. (1988); note that these data are not reported in this paper. Much lower resolution bandheads are identified by Balfour & Chowdhury (2010) for the – band involving excited vibrational levels of the state; this too warrants further investigation to allow characterization of the state.

There have also been some bandheads discussed in previous spectroscopic studies that our data unfortunately cannot help assign. Specifically, we do not find any recommended assignment of the double bandhead at 12,082.65 cm⁻¹ (R head) and 12,069.9 cm⁻¹ (Q head) observed by Davis & Hammer (1981).

Table 13 shows equilibrium term energy, vibrational, and rotational constants for the , , , , ₁, ₂, ₃, ₂, ₃, and ₄ electronic states based entirely on Marvel energy levels. These equilibrium constants are obtained by fitting to the relevant band constants, with obvious outliers removed for averaging of D_v's to obtain D. Note that we have chosen to provide constants for the various subcomponents of the triplet levels individually rather than use additional constants to unify their treatment.

Table 13.  Equilibrium Vibrational Constants, in cm⁻¹, Based Solely on Marvel Energy Levels

State	Te	ωe	ωeχe	Be	α(103)	D(107)
	0	976.44	3.45	0.42361	1.97	3.19
	5906.58	935.95	⋯	0.41741	1.89	3.26
	15441.70	859.59	2.99	0.40246	1.90	3.50
	25229.40	835.24	⋯	0.39834	1.96	3.60
1	1099.70	937.74	3.23	0.41430	1.91	3.19
2	1386.90	938.09	3.24	0.41573	1.92	3.26
3	1722.45	938.51	3.25	0.41663	1.93	3.43
2	16567.04	856.72	3.27	0.40419	2.11	3.57
3	17169.35	855.84	3.29	0.40475	2.11	3.61
4	17796.92	857.09	2.94	0.40533	2.07	3.77

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