Spectroscopic parameters and transition probabilities of several doublet and quartet states of sulfoxide cation

Journal of Quantitative Spectroscopy & Radiative Transfer 244 (2020) 106845

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Spectroscopic parameters and transition probabilities of several doublet and quartet states of sulfoxide cation Kaige Guo, Zunlue Zhu∗ College of Physics, Henan Normal University, Xinxiang 453007, China

a r t i c l e

i n f o

Article history: Received 26 September 2019 Revised 11 January 2020 Accepted 13 January 2020 Available online 23 January 2020 Keywords: Sulfoxide cation Radiative lifetime Transition probability Potential energy curves Transition dipole moment

a b s t r a c t In this work, the potential energy curves of 11 doublet, 11 quartet, and 2 sextet states of a sulfoxide cation are reported and the transition dipole moments between them were calculated. The radiative lifetimes were on the order of 10–6 s for the A2 , C2 , b4  – , and 12  + states, approximately 10–6 – 10–7 s for the 12  and B2  – states, and on the order of 10–7 s for the 22  + state. The radiative lifetimes of these states were short, suggesting that the spontaneous emissions generated from them occurred readily. Many transitions are strong, including A2  – X2 , C2  – X2 , C2  – A2 , B2  – – X2 , B2  – – A2 , 12  – X2 , 12  – A2 , and b4  – – a4 . In addition, the transitions from the the first well of the 12  + state to the X2  state, and from the first well of the 22  + state to the X2 , A2 , and C2  states, are also strong. These strong transitions were easy to measure through spectroscopy. The properties of the transitions between certain  states generating from the X2 , A2 , a4 , and b4  – states were investigated, including several electric dipole–forbidden transitions. The A2 1/2 – X2 1/2 , A2 3/2 – X2 1/2 , b4  – 3/2 – a4 5/2 , b4  – 1/2 – a4 1/2 , and b4  – 1/2 – a4 3/2 transitions are strong. The radiative litimes of the A2 1/2 , A2 3/2 , a4 –1/2 , a4 1/2 , a4 3/2 , a4 5/2 , b4  – 1/2 , and b4  – 3/2 states were calculated and the distributions of their radiative lifetime varying as rotational angular momentum quantum number J were studied. The radiative lifetimes of the a4 1/2 and a4 3/2 states are on the order of several ms; those of the a4 5/2 state are approximately several–ten ms; while those of the a4 –1/2 state are on the order of several s. The results calculated herein can provide some useful guidelines for further experimental and astronomical observations and theoretical studies. © 2020 Published by Elsevier Ltd.

1. Introduction Sulfoxide cation (SO+ ) is a major component of plasma containing sulfur and oxygen. This cation has been extensively identified in outer space by rovibrational spectroscopy, as reviewed by Andreazza and Marinho [1]. For example, it has been detected in giant molecular clouds and a cold dark cloud [2], in comet Halley [3], in shocked interstellar object IC 443G [4,5], in interstellar clouds of varied morphological types [6,7], in the plasma torus of Io [8–10], and in the O–rich evolved star OH231.8 + 4.2 [11]. In addition, this cation is a species of considerable chemical and physical interest. Its role in the ion chemistry of the Earth’s atmosphere has also been a subject of investigation [12]. It has extensive industrial applications, such as for the surface treatment for biomedical devices [13]. A number of research groups [14–28] have measured the spectroscopic parameters and transition properties of the SO+ cation.

∗

Corresponding author. E-mail address: [email protected] (Z. Zhu).

https://doi.org/10.1016/j.jqsrt.2020.106845 0022-4073/© 2020 Published by Elsevier Ltd.

The first spectroscopic measurements were made by Dyke et al. [14], who determined several spectroscopic parameters of the X2 , B2  – , a4 , and b4  – states by vacuum ultraviolet (UV) photoelectron spectroscopy of transient species. Tsuji et al. [15] observed ˚ assigned the emissions within the spectral region of 250 0–540 0 A, them to the A2  – X2  transition, and determined the spectroscopic parameters of the X2  and A2  states. Dujardin and Leach [16] measured the photoion–fluorescence photon coincidence during the radiative and dissociative relaxation processes in vacuum UV photoexcited SO2 and detected the fluorescence of the A2  – X2  and b4  – – a4  systems. Murakami et al. [17] recorded the emissions of the A2  – X2  system of S16 O+ and S18 O+ cations within a spectral range of 180 0–70 0 0 A˚ and measured the origins of many bands. Cossart et al. [18] detected the emissions of the A2  – X2  and b4  – – a4  systems and carried out the first rotational analysis. Hardwick et al. [19] recorded the emissions from the A2  – X2  system photographically at high resolution, measured the line positions of the 0–5, 0–6, 1–5, and 1– 6 vibronic bands, and fitted the molecular constants of the two states. Coxon and Foster [20] recorded six vibronic bands in the

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K. Guo and Z. Zhu / Journal of Quantitative Spectroscopy & Radiative Transfer 244 (2020) 106845

0 – υ   progression and three bands in the 1 – υ   progression of the A2  – X2  system, and conducted a rotational assignment. Combining previous experimental observations, Reddy et al. [21] determined the ground–state dissociation energy De of SO+ to be 43795.89 ± 1532.45 cm–1 , and evaluated the ground–state vibrational levels using the Rydberg–Klein– Rees (RKR) method. Milkman et al. [22,23] recorded the rotationally cold emissions of the A2  – X2  system at high resolution and reported a considerable number of band origins. Norwood and Ng [24] measured the spin–orbit splitting energy of the ground state as 371 ± 20 cm–1 employing photoion–photoelectron coincidence spectroscopy. Amano et al. [25] measured and analyzed the microwave spectrum of the X2  state. Dyke et al. [26] recorded the constant ionic state and photoelectron spectra of the SO molecule by photo– electron spectroscopy using vacuum UV radiation from a synchrotron, and studied several doublet and quartet states. In addition, Barr et al. [27] recorded the constant ionic state and photoelectron spectra of several doublet and quartet states of this cation. Li et al. [28] recorded the absorption spectrum of the fundamental band of the X2  state using a mid–infrared tunable diode laser spectrometer with a velocity modulation technique. Chen et al. [29] measured the b4  – – a4 5/2 transition at 12600 – 12800 cm–1 using optical heterodyne velocity modulation spectroscopy. Using these experimental data [14–20,22–29], some spectroscopic parameters and molecular constants of several states have been evaluated. A number of research groups [12,13,18,30–33] have calculated the spectroscopic parameters and transition properties of SO+ . The first calculations were conducted by Dyke et al. [14], who calculated the potential energy curves (PECs) of five doublet and quartet states; however, they evaluated only a few spectroscopic parameters. Klotz et al. [30] studied the dependence of the spin–orbit matrix elements on the atomic orbital (AO) basis set composition for inner and valence shells of the X2  state in SO+ . Using the self–consistent field (SCF) method followed by the configuration interaction (CI) approach, Cossart et al. [18] calculated the PECs of a number of states and explained the measured spectra. Using the optimally conditioned gradient method of Davidon, Balaban et al. [31] calculated the bond lengths of a considerable number of molecules including SO+ . Using the hybrid density functional HF/DF B3LYP approach, Midda and Das [32] calculated the ground– state molecular properties of this cation, such as the bond length, electric dipole moment, harmonic frequency, atomization energy, infrared intensity, electron affinity, and ionization potential. Using the complete active space SCF (CASSCF) method followed by the internally contracted multireference CI (icMRCI) approach, Houria et al. [13] conducted the spectroscopic and spin–orbit calculations of many states of SO+ . In addition, using the CASSCF method followed by the icMRCI approach, Xing et al. [33] calculated the PECs of the X2  and A2  states, investigated the spin–orbit coupling effect on the two states, and evaluated their spectroscopic properties and molecular constants. In summary, experimental studies [14–27,29] have mainly focused on the transition properties of the X2 , A2 , B2  – , a4 , and b4  – states. However, none of the radiative lifetimes of any states has been experimentally measured until today. In addition, no distributions of the radiative lifetime varying as rotational angular momentum quantum number J are currently available for any  states. Although the transition properties of many vibronic emissions originating from the A2  state have been measured [15– 20,22,23], those of only a few emissions generated from the B2  – and b4  – states [14,16,18,29] have been reported. At the same time, no electric dipole–forbidden transitions originating from the a4  state have been studied. Summarizing the theoretical results above, only Cossart et al. [18] calculated the transition properties of the A2  – X2  system, although the theoretical spectroscopic properties of many doublet and quartet states have been reported

[13,18,31–33]. No theoretical radiative lifetimes of any doublet or quartet states are currently available and no vibronic bands have been calculated. On the basis of these reasons, we calculated the transition properties of emissions arising from several doublet and quartet states belonging to the first two dissociation asymptotes of this cation. 2. Methodology The first ionization energies are 83559.1 and 109837.02 cm–1 for the S and O atoms, respectively [34]. Therefore, the first dissociation asymptote of SO+ is S+ (4 Su ) + O(3 Pg ). Furthermore, the first excited states of a S+ ion and an O atom are 2 Du and 1 Dg , the energy levels of which are 14868.84 and 15867.86 cm–1 , respectively [34]. As a result, the second dissociation asymptote of SO+ is S+ (2 Du ) + O (3 Pg ). According to group theory, the first dissociation limit generates six states, namely X2 , 12  + , 14  + , 14 , 16  + , and 16 , and the second dissociation limit generates 18 states, namely 22  + , 32  + , 12  – , 22 , 32 , 42 , 12 , 22 , 12 , 14  – , 24  + , 34  + , 24 , 34 , 44 , 14 , 24 , and 14 . Of these 24 states originating from the first two dissociation limits, 11 are doublet, 11 are quartet, and two are sextet states. To reduce the complexity of this study, we calculated the spectroscopic properties of only the 11 doublet and 2 quartet states, as well as the transition properties of only 10 of these states. The PECs were calculated using the CASSCF method [35], followed by the icMRCI approach [36,37] with the Davidson correction [38]. All calculations of the potential energies, including the core–valence correlation and scalar relativistic corrections as well as the spin–orbit coupling (SOC) effect, were conducted employing the MOLPRO 2010.1 program in the C2 υ group symmetry [39]. Thus, the CASSCF was used as the reference wavefunction for the icMRCI calculations. The basis sets used to calculate the PECs were aug-cc-pV(5 + d)Z [AV(5 + d)Z] and aug-cc-pV(6 + d)Z [AV(6 + d)Z] basis sets for S atoms [40], as well as aug-cc-pV5Z (AV5Z) and aug-cc-pV6Z (AV6Z) basis sets for O atoms [41–43]. The space point interval of 0.1 A˚ was used to scan the PECs and calculate the core–valence correlation correction, the scalar relativistic correction, and the SOC effect. The core–valence correlation and scalar relativistic corrections were calculated using the cc-pCV5Z [44,45] and cc-pV5Z-DK [46] basis sets, respectively. The SOC effect was calculated employing the all–electron cc-pCV5Z basis set [45]. For convenience, the contributions of core–valence correlation correction, scalar relativistic correction, and the SOC effect to the total potential energies are denoted herein as “CV”, “DK”, and “SOC”, respectively. The approach used to calculate the contributions of these corrections were introduced in previous studies [47,48]. The transition dipole moments (TDMs) used to compute the transition properties were calculated using the icMRCI/AV6Z approach. We used the following scheme to extrapolate the reference and correlation energies to the complete basis set (CBS) limit [49], respectively: re f EXre f = E∞ + Are f X α ,

(1)

corr EXcorr = E∞ + Acorr X β .

(2)

re f EX and

Here, EXcorr are the reference and correlation energies calculated using the aug-cc-pVXZ basis set (X = 5 and 6), respecre f corr are the reference and correlation energies extively. E∞ and E∞ trapolated to the CBS limit, respectively. In addition, α and β are 3.4 and 2.4 for the reference and correlation energies, respectively. Aref and Acorr are calculated by solving Eqs. (1) and (2), respectively. We used the PECs calculated by the icMRCI + Q/AV(5 + d)Z approach for S atoms and icMRCI + Q/AV5Z method for O atoms,

K. Guo and Z. Zhu / Journal of Quantitative Spectroscopy & Radiative Transfer 244 (2020) 106845

as well as the PECs calculated by the icMRCI + Q/AV(6 + d)Z approach for S atoms and icMRCI + Q/AV6Z method for O atoms for the CBS extrapolation scheme. The PEC obtained by the extrapolation scheme is denoted as “56–d” in this work. In addition, we also extrapolated the PECs of these states to the CBS limit using the potential energies obtained by the icMRCI + Q/AV5Z and icMRCI + Q/AV6Z calculations. Accordingly, the extrapolated PEC is denoted as “56” herein. For spontaneous emissions between two electronic states, for which the upper and lower vibrational levels are υ  and υ   , respectively, the total transition probability for a certain υ  is calculated by Roberto-Neto and Ornellas [50]

Aυ  =

 υ 

Aυ  υ 

(3)

and

Fig. 1. Curves of TDM vs. internuclear separation for 9 pairs of doublet states.

1 τυ  = . Aυ 

(4)

Here, Aυ  υ  is the Einstein A coefficient of the transition from level υ  to level υ   , Aυ  is the total transition probability of υ  , and τυ  is the radiative lifetime of this level υ  . When several spontaneous systems (i = 1, 2,…) are generated from a certain upper state, the total Einstein A coefficient of an upper-level υ  is

Aυ  =



Ai,υ  .

(5)

i

Here, Ai,υ  is the total Einstein A coefficient of emissions from the υ  for the ith system. 3. Results and discussion Houria et al. [13] investigated the PECs of the 24 -S states in detail. To avoid the repetition with [13] and shorten the length of this paper, we depict these PECs in Figs. S1–S3 only as a supplementary material. Note that these PECs were calculated using the icMRCI + Q/56–d + CV + DK approach. To display more information, Figs. S1–S3 show the PECs over only a small range of internuclear separations. As can be seen in these figures, both the 12  + and 22  + states have a double well, which cannot be clearly seen due to a shallow depth. The 12  state also has a shallow well and a potential barrier, the latter of which is also so low that it cannot be clearly seen. Table S1 lists the potential energies of these states at selected internuclear separations as the supplementary material. For brevity, we calculate the transition properties of only eight doublet and two quartet states, namely X2 , A2 , B2  – , C2 , 12  + , 22  + , 12 , 12 , a4 , and b4  – . Based on the transition selection rule, there are 18 pairs of electric dipole–allowed transitions between these states. Figs. 1 and 2 show the curves of the TDM versus the internuclear separation for the 18 pairs of states. To show more information, Figs. 1 and 2 display the TDM curves over only a small range of internuclear separations. Table S2 lists their TDMs at selected internuclear separations as a supplementary material. Including the SOC effect, we calculated the TDMs of the transitions between certain  states generated from the X2 , A2 , a4 , and b4  – states. Figs. 3 and 4 depict these TDMs. Table S3 lists the TDMs at selected internuclear separations as the supplementary material. To study the electric dipole–forbidden transitions, we also calculated the TDMs of several transitions between  states generated from the a4  and X2  states. The TDM curves of these transitions are displayed in Fig. 5. The TDM values at selected internuclear separations are also listed in Table S3 as the supplementary material.

Fig. 2. Curves of TDM vs. internuclear separation for 9 pairs of doublet states.

Fig. 3. Curves of TDM vs. internuclear separation for five pairs of  states

Fig. 4. Curves of TDM vs. internuclear separation for five pairs of  states.

3

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K. Guo and Z. Zhu / Journal of Quantitative Spectroscopy & Radiative Transfer 244 (2020) 106845

Fig. 5. Curves of TDM vs. internuclear separation for six pairs of  states.

3.1. Spectroscopic parameters Using the PECs obtained by the icMRCI + Q/56 + CV + DK and icMRCI + Q/56–d + CV + DK calculations, we fitted the spectroscopic parameters of these states, respectively, supported by the LEVEL program [51]. As these spectroscopic parameters were studied in detail by Houria et al. [13], we list the spectroscopic parameters obtained in this study in Table S4 only as supplementary material. Note that Table S4 lists the spectroscopic parameters of only 11 doublet and 2 quartet states. A considerable number of experimental [14–23,26] and theoretical [13,18,30–32] investigations have reported the spectroscopic parameters of these states. Table S4 lists the spectroscopic parameters together with only the selected experimental [14,15,17,19–21,23] and theoretical values [13,31]. Note that in Table S4, only the theoretical results with an accuracy equivalent or superior to ours when compared with the available experimental results are listed. We also calculated the vibrational levels of these states using the LEVEL program [51]. The results calculated by the icMRCI + Q/56–d + CV + DK approach are presented in Table S5 as supplementary material. As highly vibrational states are difficult to populate experimentally, Table S5 lists only the first 15 levels of each electronic state whenever available. Reddy et al. [21] calculated the vibrational levels employing the RKRV potential. General agreement exists between the vibrational levels calculated in this work and their values [21]. By detail comparison of the spectroscopic parameters listed in Table S4, we confirm that the quality of the results obtained by the icMRCI + Q/56–d + CV + DK calculations is slightly superior to that calculated by the icMRCI + Q/56–d + CV + DK method. For clarity, in this study, we make a comparison between only the results obtained by the icMRCI + Q/56–d + CV + DK calculations with the experimental and other theoretical values. Many experimental [14,15,19–23,26] and theoretical [13,18,30– 32] studies have investigated the ground–state spectroscopic properties of this cation and evaluated the its spectroscopic parameters. Based on Table S4, the De value calculated in this study deviates from the experimentally determined value by 858.98 cm–1 (1.96%) [21], which is a large deviation. However, it should be noteworthy that the RKR value of De [21] has a large uncertainty of 1532.45 cm–1 . Our De result is within the level of uncertainty. Dyke et al. [14] determined the ground–state D0 value as 43553.92 cm–1 . The recent ωe and ωe xe values measured by Milkman et al. [23] are 1306.78 and 7.698 cm–1 , respectively. Using De ≈ D0 + ωe /2 - ωe xe /4 and these results [23], we can estimate the ground– state “experimentally determined” De value obtained in [14] as 44205.39 cm–1 . The difference between the present De and the estimated value is only 449.48 cm–1 , showing a good agreement. The theoretical De results reported previously deviate from this

deduced experimental De value by 1135.40 [13] and 643.40 cm–1 [32], respectively. The deviations are large when compared with the De value calculated herein. Note that the previous calculations [32] did not extrapolate the potential energies to the CBS limit, whereas the present study does. The experimentally determined Re values reported previously are 1.4238 to 1.4250 A˚ [14,18–20,23]. The Re value calculated herein deviates from this range by 0.0 0 03 (0.02%) to 0.0 015 A˚ (0.11%). No other theoretical Re values are superior to the present result when compared with the measurements [14,18– 20,23]. Many experimental studies have reported the ground– state ωe [14,15,17–20,23,26]. We can clearly see that large differences exist between these experimentally determined values of ωe [14,15,17–20,23,26]. As shown in Table S4, the ωe value calculated here deviates from the measurements by only 1.95 (0.14%) [19], 2.44 (0.19%) [20], and 2.81 cm–1 (0.22%) [23]. Clearly, the differences between them are small. The second dissociation limit of SO+ is S+ (2 Du ) + O (3 Pg ). The energy level of S+ (2 Du ) can be obtained by averaging the levels of S+ (2 D3/2 ) and S+ (2 D5/2 ). Based on the energy levels [34], the averaged value is 14868.84 cm–1 . That is, the energy gap between the S+ (2 Du ) + O (3 Pg ) limit and the S+ (4 Su ) + O (3 Pg ) channel is 14868.84 cm–1 . Combining the experimentally determined De value of the ground state [14] and the recent experimental Te result of the A2  state [23], we deduce the “experimentally determined” De result of the A2  state as 27652.74 cm–1 . The De value calculated here compares well with this deduced value. The difference between them is only 128.21 cm–1 (0.46%). In addition, the De value calculated here is slightly closer to the deduced results than the previous theoretical result [32]. The experimentally determined Re values of the A2  state are 1.657 to 1.658 A˚ [19,20,23]. The Re result of this state calculated herein deviates from the measurements [19,20,23] by 0.0040 (0.24%) to 0.0050 A˚ (0.30%). Only the theoretical Re value calculated in [32] is comparative with the present result. The experimentally determined ωe results are 804 to 805.59 cm–1 [15,17– 20,23]. The ωe value calculated in this study deviates from these measurements by 4.15 (0.52%) to 2.56 cm–1 (0.30%). These deviations are slightly large compared with those of the ground state. Only one group of experimentally determined spectroscopic results are currently available [14]. The Re value calculated here agrees well with only the measurements with the difference between them being 0.0051 A˚ (0.33%). Considering that the D0 and ωe results in [14] are 8952.75 and 10 0 0 cm–1 , respectively, we estimate the experimentally determined “semi–empirical” De result as 9452 cm–1 . Clearly, the De value calculated herein agrees well with this experimentally determined result. When we compare the specrtoscopic parameters of the a4  and b4  – states calculated here with the experimental results [12,14,18,52], we can also find a good agreement. In summary, the spectroscopic parameters calculated in this work compare well with the available experimental values and achieve a high level of accuracy. 3.2. Properties of the transition between several –S states For convenience of the discussion regarding the transition properties, we divide the discussion into four sections. First, we discuss the transition properties of the A2  – X2  system, for which many experimentally determined transition properties are currently available [14,16–20,22,23]. Next, we evaluate the transition properties of spontaneous emissions generated from the B2  – , C2 , and 12  states with a single well. Third, we study the transition properties of the emissions originated from the 12  + and 22  + states with a double well. Finally, we investigate the properties of the b4  – – a4  transition.

K. Guo and Z. Zhu / Journal of Quantitative Spectroscopy & Radiative Transfer 244 (2020) 106845

5

Table 1 FC factors and Einstein A coefficients (s–1 , 2nd line) of vibronic emissions from the A2  – X2  system for υ  and υ   = 0–10.

υ’

υ" = 0

υ" = 1

υ" = 2

υ" = 3

υ" = 4

υ" = 5

υ" = 6

υ" = 7

υ" = 8

υ" = 9

υ " = 10

0

1.76E-4 82 1.14E-3 547 3.93E-3 19,355 9.59E-3 4829 1.86E-2 9538 3.03E-2 15,872 4.35E-2 23,159 5.65E-2 30,447 6.74E-2 36,789 7.52E-2 41,441 7.92E-2 44,025

1.91E-3 845 9.74E-3 4458 2.61E-2 12,299 4.86E-2 23,474 7.00E-2 34,634 8.27E-2 41,798 8.25E-2 42,498 7.04E-2 36,855 5.10E-2 27,110 3.04E-2 16,363 1.36E-2 7374

9.83E-3 4112 3.77E-2 16,339 7.31E-2 32,741 9.35E-2 43,132 8.56E-2 40,562 5.61E-2 27,172 2.32E-2 11,470 3.21E-3 1592 1.23E-3 660 1.24E-2 6588 2.75E-2 14,719

3.18E-2 12,486 8.51E-2 34,762 1.06E-1 44,831 7.39E-2 32,327 2.46E-2 11,049 4.07E-4 179 1.02E-2 4913 3.32E-2 16,322 4.63E-2 23,155 4.11E-2 20,917 2.45E-2 12,620

7.27E-2 26,578 1.21E-1 46,341 7.45E-2 29,617 1.07E-2 4389 5.52E-3 2390 3.76E-2 16,713 5.25E-2 23,957 3.54E-2 16,527 9.82E-3 4670 2.04E-5 13 9.09E-3 4619

1.25E-1 42,307 1.07E-1 37,879 1.25E-2 4626 1.35E-2 5300 5.56E-2 22,497 4.77E-2 19,942 1.14E-2 4875 1.04E-3 480 2.00E-2 9262 3.68E-2 17,403 3.27E-2 15,752

1.68E-1 52,361 4.72E-2 15,436 9.02E-3 3164 6.52E-2 23,665 4.19E-2 15,831 1.28E-3 489 1.61E-2 6633 4.17E-2 17,674 3.30E-2 14,382 8.16E-3 3629 4.99E-4 244

1.82E-1 51,728 2.05E-3 596 6.42E-2 20,591 5.12E-2 17,173 8.40E-5 25 3.14E-2 11,605 4.66E-2 17,906 1.48E-2 5881 5.66E-4 246 1.97E-2 8459 3.40E-2 14,958

1.60E-1 41,500 1.97E-2 5506 8.41E-2 24,652 2.56E-3 765 3.70E-2 12,178 4.79E-2 16,410 4.69E-3 1649 1.16E-2 4401 3.77E-2 14,698 2.67E-2 10,708 3.14E-3 1280

1.17E-1 27,355 8.34E-2 21,015 3.68E-2 9768 2.51E-2 7228 5.84E-2 17,556 2.99E-3 916 2.33E-2 7834 4.27E-2 14,876 1.12E-2 4049 2.38E-3 927 2.48E-2 9799

7.13E-2 14,925 1.36E-1 30,742 5.74E-5 9.61 7.48E-2 19,432 1.09E-2 2947 2.61E-2 7705 4.55E-2 13,998 3.62E-3 1138 1.47E-2 5028 3.67E-2 13,005 1.70E-2 6219

1 2 3 4 5 6 7 8 9 10

3.2.1. Transition properties of the A2  – X2  system For the A2  – X2  transition, neither experimentally nor theoretically determined TDMs have been reported to date. As displayed in Fig. 1, the TDMs of the A2  – X2  transition are large near the internuclear equilibrium separations of the two states. Using the LEVEL program [51], we calculated the Einstein A coefficients, band origins, and Franck–Condon (FC) factors of all spontaneous emissions from the A2  – X2  system. Based on the results calculated herein, the A2  – X2  system has many strong vibronic emissions. This conclusion is consistent with the previous experimental observations. The spectral range of this system extends from the infrared region to the UV region; the strong emissions are in the visible and UV regions. Note that the A2  – X2  transition has the P, R, and Q branches. On the whole, the strength of the vibronic emissions of each branch is similar. Table 1 lists the FC factors and Einstein A coefficients of vibronic emissions for the υ  , υ   = 0 – 10 levels, which are those of the P(2.5) line of each vibronic band. We do not list the origins of these vibronic bands in Table 1 since they are determined easily according to the results presented in Tables S4 and S5. Several experimental groups [15,17,19,20,22,23] have measured a considerable number of band origins of the A2  – X2  transition. To avoid a comparison of tedious data, we take only some of the vibronic bands reported in [20] as examples for a brief comparison with the present results. For the 0–4, 0–5, 0–6, 1– 4, 1–5, and 1–6 bands, Coxon and Foster [20] measured the origins as 26098.39, 24868.62, 23654.40, 26891.07, 25661.33, and 24446.92 cm–1 , respectively; the corresponding band origins calculated in this work are 26459.49, 25222.63, 240 0 0.48, 27256.73, 26019.87, and 24797.72 cm–1 . The differences between theory and experiment are 361.10, 354.01, 346.08, 365.66, 358.54, and 350.80 cm−1 . The differences are somewhat large. If we employ the recent experimentally determined Te of the A2  state [23] to improve the quality of the ab initio PEC, the differences between theory and experiment can be reduced to 23.45, 30.54, 38.47, 18.89, 26.01, and 33.75 cm–1 , respectively, indicating a good agreement. By comparison of the results calculated in this work with other measurements [15,17,19,20,22,23], we can obtain the same conclusion. The A2  state is the first excited doublet state of the SO+ . Therefore, the contribution to the radiative lifetimes of the A2  state comes from only the A2  – X2  transition. Using the Ein-

stein A coefficients obtained herein, we calculated the radiative lifetimes of all vibrational levels of the A2  state. Note that the radiative lifetimes of all vibrational levels were calculated at J = 1.5. Table 2 lists those of the first 16 vibrational levels. As can be seen in Table 2, the radiative lifetimes of these vibrational levels are on the order of 1μs. However, there are no radiative lifetimes of the A2  state available in the literature, whether experimentally or theoretically determined. 3.2.2. Transition properties of emissions generated from C2 , B2  – , and 12  states According to Fig. S1 and Table S4, the C2  state can spontaneously decay into the X2 , A2 , and 12  + states. However, neither experimentally nor theoretically determined TDM values have been reported in the literature. As shown in Figs. 1 and 2, the TDM values of the C2  – X2  and C2  – A2  transitions are clearly larger than those of the C2  – 12  + system. As a result, the FC principle predicts that the emissions of the C2  – X2  and C2  – A2  systems should be stronger than those from the C2  – 12  + transition. Using the LEVEL program [51], we calculated the FC factors, Einstein A coefficients, and band origins, of all spontaneous emissions from the C2  – X2 , C2  – A2 , and C2  – 12  + systems. Based on the results calculated in this study, the C2  – X2  and C2  – A2  transitions are strong. The 12  + state has a double well. The spontaneous emissions from the C2  state to the double well are weak, and the largest Einstein A coefficient of all vibronic emissions is on the order of only 10 s–1 . This conclusion is in accordance with the expectation from the FC principle. It should be noted that all of the C2  – X2 , C2  – A2 , and C2  – 12  + transitions have three branches; and the strengths of the spontaneous emissions of each branch are generally similar. For brevity, Table 3 lists the FC factors and Einstein A coefficients of the vibronic bands of emissions from only the C2  – X2  system for υ  , υ   = 0–10, which are those of the P(2.5) rovibrational transitions of each band. Table S6 lists those of the C2  – A2  transition for υ  , υ   = 0–10 as supplementary material. It should be noted that the data listed in Table S6 are also those of the P(2.5) line of each band. According to the band origins calculated in this study, we determine that the specral range of vibronic emissions from the C2  – X2  system extends from the infrared region to the UV region.

6

K. Guo and Z. Zhu / Journal of Quantitative Spectroscopy & Radiative Transfer 244 (2020) 106845 Table 2 Radiative lifetimes (μs) of the first 16 vibrational levels of A2 , C2 , 12 , B2  – , 12  + , 22  + , and b4  – states.

υ

A2 

C2 

12 

B2  –

b4  –

1st well of the 22  +

1st well of the 12  +

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

2.34 2.41 2.49 2.57 2.65 2.73 2.81 2.90 2.99 3.08 3.17 3.27 3.36 3.45 3.55 3.64

1.17 1.19 1.20 1.21 1.23 1.24 1.24 1.26 1.28 1.31 1.36 1.45 1.56 1.70 1.86 2.05

0.89 0.90 0.92 0.93 0.95 0.98 1.01 1.05 1.10 1.15 1.23 1.32 1.40 1.54 1.80 2.24

0.77 0.78 0.78 0.78 0.79 0.81 0.83 0.86 0.89 0.95 1.04 6.63 6.89 7.18 7.54 7.99

7.85 7.21 6.79 6.49 6.29 6.18 6.12 6.12 6.17 6.28 6.43

0.67 0.70 0.72 0.76 0.81 0.88

1.02 1.24 1.70

Table 3 FC factors and Einstein A coefficients (s–1 , 2nd line) of vibronic emissions of the C2  – X2  system for υ  and υ   = 0–10.

υ

υ = 0

υ = 1

υ = 2

υ = 3

υ = 4

υ = 5

υ = 6

υ = 7

υ = 8

υ = 9

υ   = 10

0

1.05E-4 66 6.99E-4 443 2.48E-3 1567 6.17E-3 3890 1.22E-2 7630 2.02E-2 12,587 2.93E-2 18,145 3.82E-2 23,485 4.56E-2 27,790 5.06E-2 30,560 5.30E-2 31,764

1.20E-3 760 6.45E-3 4083 1.81E-2 11,478 3.54E-2 22,383 5.39E-2 33,975 6.78E-2 42,613 7.32E-2 45,751 6.92E-2 43,008 5.81E-2 35,886 4.38E-2 26,841 2.98E-2 18,117

6.58E-3 4129 2.71E-2 17,095 5.71E-2 36,053 8.05E-2 50,904 8.42E-2 53,182 6.75E-2 42,500 4.07E-2 25,537 1.67E-2 10,392 2.99E-3 1819 1.51E-4 106 4.38E-3 2764

2.20E-2 14,075 6.77E-2 42,246 9.68E-2 60,744 8.37E-2 52,678 4.34E-2 27,350 9.40E-3 5888 2.27E-4 153 1.11E-2 7075 2.65E-2 16,775 3.48E-2 21,926 3.36E-2 21,040

5.55E-2 33,847 1.09E-1 67,217 8.82E-2 54,703 2.83E-2 17,599 6.84E-5 40 1.55E-2 9791 3.93E-2 24,813 4.35E-2 27,499 2.92E-2 18,369 1.14E-2 7141 1.38E-3 845

1.03E-1 61,297 1.15E-1 69,226 3.14E-2 19,116 1.21E-3 762 3.47E-2 21,447 5.25E-2 32,755 3.19E-2 19,928 6.05E-3 3770 7.69E-4 501 1.16E-2 7399 2.30E-2 14,575

1.49E-1 87,007 7.03E-2 41,504 1.43E-4 96 4.42E-2 26,904 5.61E-2 34,346 1.70E-2 10,441 3.49E-4 227 1.81E-2 11,368 3.37E-2 21,150 2.88E-2 18,074 1.36E-2 8518

1.75E-1 99,483 1.47E-2 8404 3.70E-2 21,897 6.52E-2 38,811 1.19E-2 7117 6.64E-3 4095 3.68E-2 22,630 3.58E-2 22,072 1.17E-2 7191 1.09E-5 5.05 6.93E-3 4381

1.68E-1 93,335 3.12E-3 1832 8.12E-2 46,639 2.09E-2 12,036 9.93E-3 5955 4.76E-2 28,496 2.81E-2 16,904 8.40E-4 493 9.96E-3 6164 2.70E-2 16,741 2.60E-2 16,118

1.35E-1 72,790 4.81E-2 26,645 6.32E-2 35,126 2.89E-3 1718 5.47E-2 31,616 2.61E-2 15,129 6.55E-4 413 2.65E 15,938 3.37E-2 20,314 1.28E-2 7757 1.31E-4 73.61

9.13E-2 47,622 1.09E-1 58,514 1.22E-2 6462 4.96E-2 27,665 3.90E-2 21,753 1.30E-3 798 3.77E-2 21,946 3.04E-2 17,734 1.98E-3 1133 7.40E-3 4494 2.37E-2 14,335

1 2 3 4 5 6 7 8 9 10

Most of the strong emissions are within the UV region. Only a few are of visible light. As discussed above, the vibronic emissions of this system are strong and therefore easy to measure through spectroscopy, although they have not been detected so far in a spectroscopy experiment. The spectral range of emissions from the C2  – A2  system covers the infrared and visible regions. The strong emissions are within the near–infrared and visible regions. Similar to those of the C2  – X2  system, the emissions of the C2  – A2  system are also strong and therefore easy to measure through spectroscopy. The spectral range of the C2  – 12  + system is within the infrared region. The contribution to the radiative lifetimes of the C2  state comes from the C2  – X2 , C2  – A2 , and C2  – 12  + transitions. According to the Einstein A coefficients calculated in this study, the contribution from the C2  – X2  transition to the radiative lifetime is predominant, whereas that from the C2  – 12  + system is insignificant. Table 2 lists the radiative lifetimes of the first 16 levels of the C2  state. Note that the radiative lifetimes of all levels were calculated at J = 1.5. As can be seen in Table 2, the radiative lifetimes of the C2  state are on the order of 1 μs, which are slightly shorter than those of the A2  state. These radiative lifetimes suggest that the spontaneous emissions originated from the C2  state occur readily. However, neither experimentally

nor theoretically determined radiative lifetimes of the C2  state are currently available in the literature. According to the electric dipole–allowed transition selection rule, the B2  – state can spontaneously decay into three lower– lying states, X2 , A2 , and C2 . As can be seen in Fig. 2, the TDM values of the B2  – – X2  and B2  – – A2  systems are clearly larger than those of the B2  – – C2  transition around the internuclear separations of the X2 , A2  and B2  – states. Based on the FC princlple, the vibronic emissions of the B2  – – X2  and B2  – – A2  systems should be stronger than those of the B2  – – C2  transition. To our knowledge, neither experimentally nor theoretically determined TDMs are currently available for any transitions originating from the B2  – state. Using the PECs and TDMs obtained herein, we calculated the band origins, Einstein A coefficients, and FC factors of spontaneous vibronic emissions from the B2  – – X2 , B2  – – A2 , and B2  – – C2  systems, supported by the LEVEL program [51]. According to the results calculated in this work, the B2  – – X2  transition is strong; then the B2  – – A2  system. For the B2  – – C2  transition, the largest Einstein A coefficient of the spontaneous emissions is of the order of only 10 s–1 . These results confirm that the emissions of the B2  – – X2  and B2  – – A2  systems are obviously stronger than those of the B2  – – C2  system as a whole,

K. Guo and Z. Zhu / Journal of Quantitative Spectroscopy & Radiative Transfer 244 (2020) 106845

7

Table 4 FC factors and Einstein A coefficients (s–1 , 2nd line) of vibronic emissions of the B2  – – X2  system for υ  and υ   = 0–10.

υ

υ = 0

υ = 1

υ = 2

υ = 3

υ = 4

υ = 5

υ = 6

υ = 7

υ = 8

υ = 9

υ   = 10

0

7.33E-2 78,308 1.61E-1 193,276 2.05E-1 275,069 1.90E-1 285,289 1.44E-1 240,624 9.54E-2 176,486 5.77E-2 117,816 3.29E-2 73,833 1.81E-2 44,274 9.71E-3 25,721 5.19E-3 14,739

2.15E-1 195,977 1.84E-1 187,647 4.92E-2 55,707 1.80E-4 299.53 3.71E-2 53,821 8.53E-2 135,765 1.05E-1 183,253 9.64E-2 185,307 7.51E-2 157,850 5.24E-2 119,751 3.32E-2 84,281

2.88E-1 224,707 2.76E-2 23,928 3.94E-2 39,319 1.11E-1 122,866 7.15E-2 87,129 1.18E-2 15,637 2.75E-3 4334 2.93E-2 48,947 5.65E-2 102,727 6.74E-2 132,910 6.31E-2 134,262

2.32E-1 154,643 3.35E-2 25,282 1.32E-1 110,780 2.07E-2 19,111 1.72E-2 18,576 7.37E-2 86,715 6.82E-2 87,912 2.59E-2 36,437 1.33E-3 1933 4.93E-3 8655 2.00E-2 37,165

1.25E-1 71,204 1.71E-1 109,879 2.98E-2 21,397 4.92E-2 40,196 8.85E-2 79,824 1.48E-2 14,516 8.81E-3 10,144 4.94E-2 60,950 5.83E-2 78,342 3.47E-2 50,319 1.05E-2 16,244

4.85E-2 23,501 2.07E-1 114,486 2.46E-2 15,278 1.07E-1 74,243 9.73E-4 692.99 5.38E-2 46,910 6.53E-2 62,421 1.29E-2 13,326 3.19E-3 3894 2.86E-2 36,571 4.25E-2 58,175

1.44E-2 5917 1.35E-1 63,442 1.48E-1 78,874 1.16E-2 6964 7.62E-2 50,929 5.06E-2 37,395 1.64E-3 1458 4.88E-2 44,781 5.17E-2 51,500 1.48E-2 15,809 1.74E-5 36.88

3.38E-3 1184 5.78E-2 23,026 1.84E-1 83,436 4.23E-2 21,490 7.71E-2 44,247 9.28E-3 5964 7.51E-2 53,369 2.26E-2 17,486 4.05E-3 3662 3.75E-2 35,660 4.18E-2 42,631

6.23E-4 181.99 1.80E-2 6070 1.19E-1 45,593 1.55E-1 67,557 1.46E-4 82.82 9.46E-2 51,993 1.16E-2 7040 3.14E-2 21,377 5.75E-2 42,784 1.33E-2 10,605 1.86E-3 1754

8.70E-5 20.01 4.27E-3 1203 5.01E-2 16,224 1.64E-1 60,701 7.40E-2 30,890 3.50E-2 18,430 4.30E-2 22,503 3.80E-2 33,802 5.07E-6 1.77 2.48E-2 24,657 4.31E-2 32,878

1.02E-5 1.94 7.71E-4 176.11 1.53E-2 4135 9.65E-2 30,043 1.62E-1 57,620 9.00E-3 3540 8.63E-2 39,042 8.49E-4 405.39 6.15E-2 34,017 2.78E-2 16,857 5.82E-4 407.26

1 2 3 4 5 6 7 8 9 10

as predicted above by the FC principle. Based on these results calculated herein, the emissions of the B2  – – X2  and B2  – – X2  systems should be easy to measure through spectroscopy, whereas significant effort will be made to detect the emissions of the B2  – – C2  system in a spectroscopy experiment. Table 4 lists the Einstein A coefficients and FC factors of vibronic emissions from the B2  – – X2  system for υ  , υ   = 0–10 levels. We do not list the band origins in Table 4 because they are easily calculated according to the spectroscopic parameters listed in Table S4 and the vibrational levels tabulated in Table S5 in the supplementary material. Note that the minimum value of J is 0.5 for the B2  – state; in addition, the Einstein A coefficients and FC factors listed in Table 4 are those of the P(2.5) line of each band. We list those of the B2  – – A2  and B2  – – C2  transitions in Table S6 as the supplementary material. According to the band origins calculated in this study, we determine the spectral range of vibronic emissions from the B2  – – X2 , B2  – – A2 , and B2  – – C2  systems. The spectral range of emissions from the B2  – – X2  system extends from the near– infrared range to the UV region. The strong emissions of this system are of UV light. The emissions falling within the infrared region are so weak that they are difficult to measure through spectroscopy. The spectral range of emissions from the B2  – – A2  system extends from the visible to the near–UV region. The strong emissions of this system are of near–UV light. The spectral range of vibronic emissions from the B2  – – C2  system falls into the infrared region. According to the above discussion, we can clearly see that the emissions of the B2  – – X2  system are strong, whereas those from the B2  – – C2  system are weak. Accordingly, the contribution from the B2  – – X2  transition to the radiative lifetimes of the B2  – state is predominant, whereas that from the B2  – – C2  system is insignificant. As listed in Table S5, the B2  – state contains the 11 levels. We calculated the radiative lifetimes for each of these levels of the B2  – state, which are listed in Table 2. Note that the radiative lifetimes of all vibrational levels were calculated at J = 1.5. As can be seen in Table 2, the radiative lifetimes are on the order of 10−6 – 10−7 s for all levels of the B2  – state. To a certain extent, the radiative lifetimes of all vibrational levels are so short that the emissions originating from the B2  – state readily occur.

Based on the electric dipole–allowed transition selection rule, the 12  state can spontaneously decay into at least four doublet states, X2 , A2 , C2 , and 12 . On the basis of the results herein, the 12 – X2 , 12 – A2 , and 12 – 12  transitions are strong, whereas the spontaneous emissions of the 12 – C2  system are very weak. The largest Einstein A coefficient of emissions from the 12 – C2  system is on the order of only 10 s–1 . To a certain extent, the emissions of the 12 – C2  system are difficult to measure, whereas those of the 12 – X2 , 12 – A2 , and 12 – 12  transitions are relatively easy to detect via spectroscopy. Neither experimentally nor theoretically determined transition properties of the 12 – X2 , 12 – A2 , 12 – C2 , and 12 – 12  systems have been reported to date. In addition, only one group of theoretical spectroscopic properties is currently available in the literature. We hope that the transition properties reported herein can provide some useful help for further experimental and theoretical investigations. We present the FC factors and Einstein A coefficients of the P(3.5) rovibrational transitions of each band only from the 12 – X2  system in Table 5, and list those from the 12 – A2  and 12 – 12  systems only in Table S7 as the supplementary material. We evaluated the spectral range of vibronic emissions from the 12 – X2 , 12 – A2 , and 12 – 12  systems according to the band origins calculated in this study. For the 12 – X2  transition, the spectral range of emissions extends from the infrared region to the UV region. Most of the strong emissions are within the UV region; only a few are of visible light. For the 12 – A2  system, the spectral range of emissions covers the infrared and visible regions; the strong emissions of this system are also in these regions. For the 12 – 12  system, the spectral range of vibronic emissions is only within the infrared region. As per the dipole–allowed transition selection rule, at least four transitions, namely, 12 – X2 , 12 – A2 , 12 – C2 , and 12 – 12 , contribute to the radiative lifetimes of the 12  state. According to the discussion above, the emissions of the 12 – X2  system are the strongest, whereas those of the 12 – C2  transition are the weakest. Accordingly, the contribution of the 12 – X2  system to the radiative lifetime is predominant, while that of the 12 – C2  transition can be negligible. Employing the Einstein A coefficients obtained in this study, we calculated the radiative lifetimes of all vibrational levels of the 12  state, the first 16 levels

8

K. Guo and Z. Zhu / Journal of Quantitative Spectroscopy & Radiative Transfer 244 (2020) 106845 Table 5 FC factors and Einstein A coefficients (s–1 , 2nd line) of vibronic emissions of the 12  – X2  system for υ  and υ   = 0–10.

υ

υ = 0

υ = 1

υ = 2

υ = 3

υ = 4

υ = 5

υ = 6

υ = 7

υ = 8

υ = 9

υ   = 10

0

6.58E-2 54,862 1.52E-1 140,240 1.94E-1 198,703 1.84E-1 208,511 1.46E-1 182,651 1.02E-1 141,326 6.51E-2 99,271 3.90E-2 65,517 2.23E-2 41,912 1.28E-2 26,077 7.13E-3 15,866

1.94E-1 138,093 1.87E-1 147,403 6.09E-2 52,323 6.03E-4 450.59 2.37E-2 26,405 6.99E-2 84,203 9.62E-2 126,633 9.60E-2 138,164 8.10E-2 127,376 6.13E-2 104,960 4.29E-2 79,735

2.69E-1 164,636 4.25E-2 28,471 2.31E-2 17,723 1.00E-1 83364.7 8.46E-2 76,356 2.53E-2 24,429 1.98E-5 0.19 1.48E-2 19,120 4.17E-2 57,019 6.02E-2 88,796 6.47E-2 102,785

2.32E-1 123,003 1.39E-2 8326.1 1.26E-1 81,494 4.10E-2 28,736 3.40E-3 2897 5.58E-2 47,912 7.39E-2 69,776 4.32E-2 43,843 1.01E-2 10,676 8.12E-5 198.53 9.34E-3 13,336

1.42E-1 64,928 1.27E-1 64,933 5.55E-2 30,967 2.03E-2 12,737 8.86E-2 59,960 3.85E-2 27,934 2.17E-5 52.45 2.70E-2 24,825 5.45E-2 53,255 4.88E-2 51,018 2.52E-2 27,939

6.52E-2 25,968 1.92E-1 84,594 2.72E-3 1382 1.06E-1 56,793 1.88E-2 10,899 2.14E-2 14,026 6.83E-2 47,944 3.73E-2 27,856 1.93E-3 1343 9.16E-3 8900 3.21E-2 32,107

2.37E-2 8196 1.54E-1 58,933 9.06E-2 38,439 4.51E-2 20,980 3.19E-2 16,491 7.57E-2 42,395 7.51E-3 4423 1.75E-2 11,891 5.27E-2 37,970 3.86E-2 29,422 8.46E-3 6584

6.99E-3 2088 8.36E-2 27,802 1.66E-1 60,780 3.35E-3 1371.1 9.57E-2 42,772 3.29E-3 1579 4.58E-2 24,520 5.41E-2 31,113 5.48E-3 3247.6 1.01E-2 7154 3.74E-2 27,468

1.71E-3 436.48 3.39E-2 9724.2 1.46E-1 46,490 8.52E-2 29,853 2.93E-2 11,338 4.99E-2 21,301 5.26E-2 24,475 5.17E-4 269.98 4.31E-2 23,629 4.35E-2 25,445 7.68E-3 4631

3.51E-4 75.88 1.08E-2 2646.5 8.43E-2 23,122 1.53E-1 46,587 8.33E-3 2772.4 8.35E-2 30,862 7.51E-4 311.9 6.11E-2 27,036 2.71E-2 12,967 1.98E-3 1034 3.36E-2 18,648

6.17E-5 11.03 2.81E-3 582.21 3.57E-2 8356.8 1.35E-1 35,482 9.14E-2 26,510 1.36E-2 4402 6.61E-2 23,361 2.71E-2 10,415 1.30E-2 5484.8 5.22E-2 23,564 1.86E-2 8987

1 2 3 4 5 6 7 8 9 10

of which are listed in Table 2. Note that the radiative lifetimes of all of these levels were calculated at J = 2.5. As can be seen in Table 2, the radiative lifetimes of the 12  state are on the order of 10–6 –10–7 s, indicating that the spontaneous emissions generated from the 12  state easily occur. 3.2.3. Transition properties of emissions generated from the 12  + and 22  + states The 12  + state has both a double well and a barrier. The potential energy at the top of the barrier is higher than that at the dissociation asymptote. Because the double well is shallow, we cannot see it clearly in Fig. S1. The first well is at 1.5058 A˚ with a well depth of 1737.72 cm –1 ; the second well lies at 2.8864 A˚ with a ˚ well depth of 1851.85 cm –1 . The barrier is at approximately 1.95 A. The first well has only three vibrational levels, whereas the second well contains a few more. All vibrational levels of the double well are listed in Table S5 as the supplementary material. Cossart et al. [18] clearly demonstrated the barrier of this state. However, Houria et al. [13] showed that the 12  + state is repulsive. The second well cannot be seen as truly bound when compared with other bound states having short internuclear equilibrium distances. The second well the 12  + state should be originated from the charge–induced dipole interaction. Therefore, we do not investigate the transitions generated from the second well. The 12  + state is the third excited double state of this cation. Based on the electric dipole–allowed transition selection rule, this state can spontaneously decay into the X2  and A2  states. According to the results calculated herein, the transition from the first well to the X2  state is strong. However, the transition from the first well to the A2  is obviously weaker than the transition from the first well to the X2  state. Note that both the 12  + – X2  and 12  + – A2  transitions have P, Q, and R branches. Table S7 lists the Einstein A coefficients and FC factors of emissions from the first well to the X2  and A2  states for υ  = 0, 1, and 2 as well as υ   = 0–10. Note that the results listed in Table S7 are those of the P(2.5) line of each vibronic band. According to the band origins calculated in this work, the spectral range of vibronic emissions from the first well to the X2  state extends from the infrared region to the UV region. The strong emissions fall into the UV region. The spectral range of vibronic emissions from the first well to the A2  state is within the in-

frared and visible regions. Most strong emissions are within the infrared region. Only a few are of visible light. Table 2 lists the radiative lifetimes of all vibrational levels of the first well. These lifetimes were calculated at J = 1.5 for all levels. According to Table 2, the radiative lifetimes are on the order of 10–6 s, suggesting that the spontaneous emissions generated from this well occur easily. Since only a few spontaneous emissions originate from this well, a significant effort must be made to detect them via spectroscopy. The 22  + state contains a double well and a potential bar˚ The potential energy at rier. The barrier lies at about 1.91 A. the highest position of this barrier is slightly lower than that at the dissociation asymptote. Therefore, the depths of the double well are relative to the barrier; however, the De values of the double well are relative to the dissociation limit. The first ˚ the depth and dissociation energy of which well is at 1.5778 A, are 4237.51 and 5917.69 cm–1 , respectively. The second well is at ˚ the depth and dissociation energy of which are 4857.62 2.3009 A, and 6554.06 cm–1 , respectively. The first and second wells have 6 and 18 vibrational levels, respectively. The first 16 levels of this double well are listed in Table S5 as a supplementary material whenever available. Houria et al. [13] and Cossart et al. [18] clearly showed the barrier of this state. Similar to the 12  + state, the second well of the 22  + state cannot be seen as truly bound, which should be originated from the charge–induced dipole interaction. Therefore, we do not either investigate the transitions generated from it. Using the LEVEL program [51], we calculated the FC factors, Einstein A coefficients, band origins and of the spontaneous emissions from the 22  + – X2 , 22  + – A2 , 22  + – C2 , and 22  + – 12  + systems. Based on the results calculated herein, the transitions from the first well to the X2  and A2  states are strong, whereas the transitions from the first well to the C2  and 12  + states are obviously weak compared with those from the first well to the X2  and A2  states. For brevity, Table 6 shows the FC factors and Einstein A coefficients of spontaneous vibronic emissions from only the first well of the 22  + state to the X2  state for υ  and υ   = 0–10 whenever available. It is noteworthy that the 22  + – X2 , 22  + – A2 , 22  + – C2  transitions have P, Q, and R branches, whereas the 22  + – 12  + system has only the P and R branches. The FC factors and Einstein A coefficients listed in Table 6 are those of the P(2.5) line of each vibronic band. Table

K. Guo and Z. Zhu / Journal of Quantitative Spectroscopy & Radiative Transfer 244 (2020) 106845

9

Table 6 FC factors and Einstein A coefficients (s–1 , 2nd line) of emissions from the first well of the 22  + state to the X2  state for υ  = 0–5 and υ   = 0–10.

υ

υ = 0

υ = 1

υ = 2

υ = 3

υ = 4

υ = 5

υ = 6

υ = 7

υ = 8

υ = 9

υ   = 10

0

1.80E- 2 22,216 5.68E-2 75,749 9.85E-2 141,848 1.26E-1 195,710 1.33E-1 220,592 1.21E-1 214,202

7.78E-2 83,658 1.46E-1 169,294 1.34E-1 167,653 7.45E-2 99,497 2.15E-2 30,316 6.52E-4 841.23

1.62E-1 152,485 1.40E-1 142,039 3.10E-2 33,481 1.72E-3 2158 3.68E-2 46,951 6.33E-2 85,456

2.15E-1 178,272 4.47E-2 39,611 1.24E-2 12,189 7.47E-2 77524.5 6.52E-2 71,929 1.89E-2 21,881

2.07E-1 150,758 6.71E-4 567.43 8.88E-2 75,689 5.55E-2 50,391 6.49E-4 575.26 2.20E-2 23,192

1.53E-1 98,753 5.70E-2 39,824 8.29E-2 62,063 1.96E-6 0.090 4.74E-2 41,145 5.38E-2 50,780

9.18E-2 52,236 1.33E-1 82,056 1.41E-2 9323 5.02E-2 35,921 5.84E-2 44,507 2.65E-3 2081.73

4.58E-2 22,936 1.56E-1 84,775 1.04E-2 6039.7 8.40E-2 52,832 2.95E-3 1961 3.26E-2 23,628

1.94E-2 8520 1.25E-1 59,811 7.70E-2 39,641 3.25E-2 18,148 2.95E-2 17,570 5.82E-2 35,490

7.11E-3 2719 7.70E-2 32,199 1.32E-1 59,759 4.13E-4 191.73 7.56E-2 39,794 7.66E-3 4315

2.28E-3 756.19 3.84E-2 14,012 1.32E-1 52,319 4.43E-2 19,055 4.37E-2 20,423 1.63E-2 8068.8

1 2 3 4 5

S7 in supplementary material lists the spontaneous emissions from the first well of the 22  + state to the A2  and C2  states for υ  = 0–5 and υ   = 0–10. The spectral range of spontaneous emissions from the first well to the X2  state is within the visible and UV regions, and the strong emissions are of UV light. The spectral range of emissions from the first well to the A2  state covers the infrared region to the UV region; and the strong emissions are of visible light. The spectral range of emissions from the first well to the C2  state covers the infrared region. As listed in Table 3, we can clearly see that the radiative lifetimes are on the order of 10–7 s for all vibrational levels of the first well of the 22  + state. Note that the radiative lifetimes were calculated at J = 1.5 for each vibrational state of the double well. 3.2.4. Transition properties of the b4  – – a4  system No experimental TDMs of the b4  – – a4  transition are currently available, and only Ornellas and Borin [12] calculated the transition properties of this system. The TDMs obtained herein agrees well with the theoretical values reported in [12]. Using the PECs calculated by the icMRCI + Q/56-d + CV + DK approach and the TDMs determined by the icMRCI/AV6Z method, we calculated the band origins, Einstein A coefficients, and FC factors of vibronic emissions from the b4  – – a4  system, supported by the LEVEL program [51]. Based on the results obtained herein, there is a considerable number of emissions with this intensity, although the largest Einstein A coefficient of all vibronic transitions is only 104 s–1 . Therefore, the b4  – – a4  transition is easy to measure in a spectroscopy experiment. In fact, at least five experimental groups [16,18,27,29,52] have measured the b4  – – a4  transition thus far. Both spectral range of emissions and the strong emissions from this system fall in the infrared and visible regions. Table 7 lists the FC factors and Einstein A coefficients of certain vibronic emissions from the b4  – – a4  system for υ  and υ   = 0– 10. Note that the b4  – – a4  transition has P, Q, and R branches. The results listed in Table 7 are those of the P(2.5) lines of vibronic bands. The radiative lifetime of the b4  – state is only from the contribution of the b4  – – a4  system. The lifetimes of the first 15 vibrational levels are listed in Table 2. According to Table 2, we can conclude that the radiative lifetimes of the b4  – state is approximately 10–6 s, predicting that the emissions originating from this state occur easily. To the best of our knowledge, no experimental or other theoretical lifetimes have been found in the literature. Therefore, we cannot make any comparison. 3.3. Properties of the transition between several  states To avoid a lengthy paper, we calculated the properties of transitions between the  states generated only from the X2 , A2 ,

a4 , and b4  – states. With the SOC effect taken into account, the X2 , A2 , a4 , and b4  – states split into ten  states. Using the PECs obtained by the icMRCI + Q/56–d + CV + DK + SOC calculations, we evaluated the De , Te , Re , and ωe values of these  states, supported by the LEVEL profram [51]. The results are listed in Table S8 as a supplementary material together with some experimental [14,15,17,18] and other theoretical [14,18,30,33] values for comparison. In addition, we list the vibrational levels and rotational constants of the first 15 vibrational states of these  states in Table S9 so as to calculate the origins of each vibronic band conveniently. We only briefly study the SOC effect on the spectroscopic properties since detailed discussion has been performed previously [13,33] about these states. The splitting energy in the X2  state calculated herein is 339.53 cm−1 , which is in excellent agreement with the experimental value of 340 ± 25 cm–1 [14]. The splitting energy obtained in this study is somewhat smaller than that calculated previously [33]. As can be seen in Table S8, the SOC effct on the Re and ωe values of the X2 1/2 and X2 3/2 states is insignificant. For the A2  state, the splitting energy is only 47.41 cm−1 , which is obviously smaller than that of the ground state. Similar to the X2  state, the SOC effct on the Re and ωe values of the A2 1/2 and A2 3/2 states is also insignificant. The energy separations between two neighboring  states from the a4 –1/2 to a4 5/2 are 58.16, 66.94, and 74.62 cm−1 , respectively, while the splitting energy in the b4  – state is also only 6.51 cm−1 , suggesting that the splitting energies of the a4  and b4  – states is obviously smaller than that of the X2  state. As with the X2  and A2  states, the SOC effect on the Re and ωe of the a4  and b4  – states is small. 3.3.1. Properties of transitions from the A2 1/2 , A2 3/2 to the X2 1/2 , X2 3/2 states Four transitions between these  states can occur, namely, A2 1/2 – X2 1/2 , A2 3/2 – X2 1/2 , A2 1/2 – X2 3/2 , and A2 3/2 – X2 3/2 . Using the PECs obtained by the icMRCI + Q/56– d + CV + DK + SOC calculations and the TDMs by the icMRCI/AV6Z approach, we calculated the FC factors, band origins, and Einstein A coefficients of vibronic emissions of these systems, supported by the LEVEL program [51]. Based on the results calculated herein, on the whole, (1) the A2 1/2 – X2 1/2 and A2 3/2 – X2 1/2 transitions are strong and thus, easy to measure in a spectroscopy experiment. The spectral intensity of emissions from these two systems is very similar to that of the A2 – X2  transition. In contrast, the A2 1/2 – X2 3/2 and A2 3/2 – X2 3/2 transitions are very weak and therefore, difficult to detect via spectroscopy; (2) The spectral range of the A2 1/2 – X2 1/2 and A2 3/2 – X2 1/2 systems is the same as that of the A2 – X2  transition. The distribution range of strong emissions from the A2 1/2 – X2 1/2 and A2 3/2 – X2 1/2 systems is also the same as that of the A2 – X2  transi-

10

K. Guo and Z. Zhu / Journal of Quantitative Spectroscopy & Radiative Transfer 244 (2020) 106845 Table 7 FC factors and Einstein A coefficients (s–1 , 2nd line) of vibronic emissions from the b4  – – a4  system for υ  and υ   = 0 – 10.

υ

υ = 0

υ = 1

υ = 2

υ = 3

υ = 4

υ = 5

υ = 6

υ = 7

υ = 8

υ = 9

υ   = 10

0

1.28E-1 15,912 3.19E-1 46,312 3.23E-1 52,570 1.72E-1 30,123 5.08E-2 9049 7.79E-3 1292 4.77E-4 55.99 3.87E-6 0.0099 2.4E-11 0.0209 1.53E-7 0.031 8.73E-9 0.0030

2.22E-1 24,010 1.39E-1 18,209 4.89E-3 614.8 2.01E-1 32,265 2.72E-1 46,858 1.31E-1 22,777 2.72E-2 4362 2.16E-3 241.15 3.46E-5 0.0615 6.14E-9 0.12 4.3E-10 0.0082

2.20E-1 19,968 4.34E-3 535.87 1.30E-1 16,948 6.11E-2 9558 4.45E-2 6715 2.59E-1 43,620 2.13E-1 36,259 6.17E-2 9517 6.19E-3 659.38 1.05E-4 0.051 2.29E-6 1.83

1.71E-1 12,754 2.79E-2 2543 9.77E-2 11,292 1.64E-2 2042 1.25E-1 19,224 4.28E-4 168.30 1.76E-1 28,588 2.69E-1 44,348 1.05E-1 15,560 1.22E-2 1196 1.51E-4 1.35

1.13E-1 6831 8.70E-2 6717 1.51E-2 1537 9.19E-2 10,665 9.13E-3 1396 9.92E-2 14,984 3.62E-2 6720 8.82E-2 13,438 2.89E-1 46,172 1.51E-1 21,359 1.94E-2 1721

6.80E-2 3215 1.12E-1 7104 4.74E-3 358.83 7.11E-2 7218 2.78E-2 3189 5.53E-2 7945 4.03E-2 5857 8.18E-2 14,447 3.00E-2 3896 2.85E-1 43,587 1.96E-1 26,418

3.79E-2 1375 1.03E-1 5206 4.09E-2 2686 1.56E-2 1368 7.36E-2 7572 2.33E-5 8.89E-5 8.04E-2 11,411 4.50E-3 534.48 1.03E-1 17,690 3.87E-3 216.89 2.66E-1 38,484

2.01E-2 542.34 7.85E-2 3079.7 7.26E-2 3873 9.42E-4 57.08 5.31E-2 4708 3.56E-2 3687 1.64E-2 2194 6.76E-2 9508 2.75E-3 601.34 9.65E-2 16,290 7.91E-4 575.06

1.02E-2 199.24 5.29E-2 1573 8.14E-2 3428 2.28E-2 1268 1.30E-2 982.24 6.08E-2 5508 5.16E-3 505.58 4.38E-2 5750 3.74E-2 5164 2.06E-2 3673 7.47E-2 12,320

5.05E-3 68.46 3.29E-2 718.08 7.22E-2 2339 4.94E-2 2218 2.30E-4 10.59 3.99E-2 3091 3.91E-2 3609 1.74E-3 235.41 5.67E-2 7430 1.20E-2 1552 4.12E-2 7030

2.45E-3 21.98 1.93E-2 298.49 5.55E-2 1345 6.29E-2 2209 1.44E-2 679.70 1.00E-2 654.47 5.01E-2 3982 1.31E-2 1203 1.69E-2 2039.8 5.08E-2 6630 6.46E-4 48.00

1 2 3 4 5 6 7 8 9 10

Table 8 FC factors and Einstein A coefficients (s–1 , 2nd line) of vibronic emissions from the A2 1/2 – X2 1/2 system for υ  and υ   = 0–10.

υ

υ = 0

υ = 1

υ = 2

υ = 3

υ = 4

υ = 5

υ = 6

υ = 7

υ = 8

υ = 9

υ   = 10

0

1.75E-4 72.23 1.13E-3 483.48 3.91E-3 1718 9.53E-3 4300 1.84E-2 8515 3.02E-2 14,208 4.33E-2 20,773 5.62E-2 27,357 6.71E-2 33,102 7.49E-2 37,332 7.90E-2 39,714

1.89E-3 740.25 9.69E-3 3926 2.60E-2 10,883 4.83E-2 20,865 6.97E-2 30,916 8.25E-2 37,465 8.25E-2 38,248 7.05E-2 33,316 5.12E-2 24,636 3.07E-2 14,981 1.38E-2 6842

9.78E-3 3580 3.75E-2 14,316 7.28E-2 28,868 9.33E-2 38,274 8.57E-2 36,242 5.63E-2 24,481 2.35E-2 10,469 3.34E-3 1516 1.14E-3 538.25 1.21E-2 5775 2.71E-2 13,103

3.17E-2 10,795 8.49E-2 30,287 1.06E-1 39,390 7.41E-2 28,698 2.48E-2 9982 4.58E-4 191.17 9.92E-3 4234 3.29E-2 14,416 4.61E-2 20,704 4.12E-2 18,912 2.48E-2 11,578

7.24E-2 22,803 1.21E-1 40,134 7.47E-2 25,972 1.09E-2 3972 5.33E-3 1998 3.73E-2 14,579 5.24E-2 21,231 3.56E-2 14,887 1.01E-2 4343 7.42E-6 3.42 8.77E-3 3966

1.26E-1 36,005 1.07E-1 32,620 1.28E-2 4106 1.33E-2 4473 5.54E-2 19,523 4.79E-2 17,622 1.17E-2 4462 9.28E-4 362.79 1.96E-2 7999.68 3.66E-2 15,388 3.29E-2 14,198

1.68E-1 44,199 4.76E-2 13,256 8.77E-3 2621 6.49E-2 20,287 4.23E-2 13,861 1.40E-3 482.55 1.57E-2 5612 4.15E-2 15,413 3.33E-2 12,833 8.50E-3 3391 4.04E-4 164.90

1.82E-1 43,324 2.17E-3 534.54 6.38E-2 17,342 5.16E-2 14,778 1.19E-4 34.13 3.09E-2 9836 4.67E-2 15,575 1.53E-2 5311 4.65E-4 166.97 1.92E-2 7205.9 3.38E-2 13,142

1.61E-1 34,512 1.93E-2 4507 8.42E-2 20747 2.74E-3 697.66 3.65E-2 10,200 4.81E-2 14,124 4.97E-3 1525 1.12E-2 3630 3.75E-2 12,664 2.70E-2 9522 3.42E-3 1252

1.17E-1 22,610 8.27E-2 17,278 3.73E-2 8247.9 2.46E-2 5924.7 5.86E-2 14,844 3.23E-3 847.28 2.27E-2 6499 4.27E-2 12,781 1.17E-2 3663 2.12E-3 698.78 2.43E-2 8314

7.16E-2 12,273 1.35E-1 25,209 9.10E-5 13.54 7.45E-2 16,090 1.14E-2 2561 2.55E-2 6318 4.58E-2 11,890 3.94E-3 1061 1.41E-2 4105 3.66E-2 11125 1.75E-2 5573

1 2 3 4 5 6 7 8 9 10

tion. Table 8 lists the FC factors and Einstein A coefficients of certain bands from the A2 1/2 – X2 1/2 system. We do not include the band origins in Table 8 since they can be calculated easily according to the results listed in Tables S8 and S9. To avoid a lenthy table, we present those of the A2 3/2 – X2 1/2 , A2 1/2 – X2 3/2 and A2 3/2 – X2 3/2 transitions in Table S10 as the supplementary meterial. Note the the results listed in Tables 8 and S10 are those of P(2.5) rovibrational transitions of vibronic bands. The radiative lifetime of the A2 1/2 state comes from the contribution of the A2 1/2 – X2 1/2 and A2 1/2 – X2 3/2 systems; those of the A2 3/2 state are from the contribution of the A2 3/2 – X2 1/2 and A2 3/2 – X2 3/2 transitions. Based on the Einstein A coefficients calculated herein, the contributions of the A2 1/2 – X2 1/2 and A2 3/2 – X2 1/2 systems to the lifetimes are significant, whereas those from the A2 1/2 – X2 3/2 and A2 3/2 – X2 3/2 transitions can be negligible. Using Eqs. (3)–(5), we calculated the radiative lifetimes of all levels of the A2 1/2 and A2 3/2 states and

list those of the first 16 levels in Table 9. As can be seen in Table 9, the lifetimes of these two  states are on the order of several μs and are slightly larger than the corresponding values of the A2  state. As with the A2  state, the radiative lifetime increases gradually as increasing level. To our knowledge, no experimental and theoretical lifetimes have not been found for present comparison. For the A2 1/2 and A2 3/2 states, neither theoretically nor experimentally determined distributions of the radiative lifetime with a variation of the rotational angular momentum quantum number J are currently available for a certain vibrational level. For this reason, in this study, we briefly calculated these missing distributions, which are shown in Figs. 6 and 7 at J ≤ 70.5. Note that the minimum J of both A2 1/2 and A2 3/2 states is 1.5. As can be seen in Figs. 6 and 7, for different vibrational levels, the radiative lifetime slowly increases as J increases. Furthermore, the largest variation of radiative lifetime as J is less than 0.5 μs at J ≤ 70.5. We hope that the results calculated herein can provide some useful guidelines for further experimental and astronomical

K. Guo and Z. Zhu / Journal of Quantitative Spectroscopy & Radiative Transfer 244 (2020) 106845

11

Table 9 Radiative lifetimes (μs) of the first 16 levels of the A2 1/2 , A2 3/2 , b4  – 1/2 , and b4  – 3/2 states.

υ

A2 1/2

A2 3/2

b4  – 1/2

b4  – 3/2

υ

A2 1/2

A2 3/2

b4  – 1/2

b4  – 3/2

0 1 2 3 4 5 6 7

2.78 2.85 2.93 3.01 3.10 3.19 3.28 3.38

2.77 2.84 2.91 2.99 3.08 3.17 3.26 3.35

8.00 7.33 6.90 6.59 6.39 6.27 6.21 6.21

16.47 15.07 14.16 13.51 13.09 12.83 12.70 12.69

8 9 10 11 12 13 14 15

3.48 3.58 3.69 3.79 3.90 4.01 4.11 4.18

3.45 3.55 3.66 3.77 3.88 3.98 4.10 4.23

6.26 6.36 6.51 6.73 7.01 7.34 7.71 8.15

12.79 12.99 13.30 13.71 14.23 14.85 15.60 16.50

Fig. 6. Distribution of radiative lifetime varying as J for a certain υ of the A 1/2 state. Lines correspond to υ = 0–15 from the bottom to the top, respectively. 2

Fig. 7. Distribution of radiative lifetime variying as J for a certain υ of the A2 3/2 state. Lines correspond to υ = 0–15 from the bottom to the top, respectively.

observations as well as theoretical investigations, although these radiative lifetimes cannot be confirmed through a comparison with other results. 3.3.2. Properties of transitions from the b4  – 3/2 , b4  – 1/2 to the a4 5/2 , a4 3/2 , a4 1/2 , and a4 –1/2 states As can be seen in Figs. 3 and 4, the TDMs of the b4  – 3/2 – a4 5/2 , b4  – 1/2 – a4 3/2 , and b4  – 1/2 – a4 1/2 systems are large and almost the same, while those of the b4  – 3/2 – a4 3/2 , b4  – 3/2 – a4 1/2 , and b4  – 1/2 – a4 –1/2 transitions are very small. In addition, the SOC effect on the spectroscopic parameters of all these states is small, suggesting that the SOC effect on the PECs is not large. Based on these reasons, we can conclude that the b4  – 3/2 – a4 5/2 , b4  – 1/2 – a4 3/2 , and b4  – 1/2 – a4 1/2 transitions should be strong and the emissions from the b4  – 3/2 –

Fig. 8. Distribution of radiative lifetime varying as J for a certain υ of the b4  – 1/2 state. Lines correspond to υ = 0–6 from the top to the bottom, respectively

a4 3/2 , b4  – 3/2 – a4 1/2 , and b4  – 1/2 – a4 –1/2 systems should be very weak. Based on the results calculated herein, the largest Einstein A coefficient of vibronic emission from the b4  – 3/2 – a4 5/2 , b4  – 1/2 – a4 3/2 , and b4  – 1/2 – a4 1/2 systems are in the order of 104 s–1 , whereas that from the b4  – 3/2 – a4 3/2 , b4  – 3/2 – a4 1/2 , and b4  – 1/2 – a4 –1/2 transitions is only several s–1 , verifying the predictions from the FC principle. For brevity, we list the FC factors and Einstein A coefficients of certain vibronic emissions from all these systems in Table S11. Simple comparison confirms that the spectral range of emissions and the spectral distribution of strong transitions from the b4  – 3/2 – a4 5/2 , b4  – 1/2 – a4 3/2 , and b4  – 1/2 – a4 1/2 systems are the same as those of the b4  – – a4  transition. Note that all these results noted herein are for the P(2.5) lines of vibronic bands. According to the Einstein A coefficients calculated in this work, we conclude that, the contribution to the radiative lifetime of the b4  – 3/2 state comes from b4  – 3/2 – a4 5/2 system, whereas that comes from the b4  – 3/2 – a4 3/2 and b4  – 3/2 – a4 1/2 transitions is tiny and can be negligible. The contribution to the radiative lifetime of the b4  – 1/2 state from the b4  – 1/2 – a4 3/2 and b4  – 1/2 – a4 1/2 systems is predominant, while that from the b4  – 1/2 – a4 –1/2 transition is insignificant. The radiative lifetimes of all vibrational levels of the b4  – 1/2 state is approximately several μs, which are comparable with those of the b4  – state; whereas those of the b4  – 3/2 state are more than ten μs, which are almost equal to twice the lifetimes of the b4  – 1/2 state. No radiative lifetimes of the b4  – 1/2 and b4  – 3/2 states are currently available, whether experimentally or theoretically. The radiative lifetime gradually decreases as increasing υ at υ ≤ 7, while it gradually increases υ at υ ≥ 8 for the b4  – 1/2 and b4  – 3/2 states. To distinguish them in the figure, for the b4  – 1/2 state, we show the distributions of radiative lifetime varying as J for the υ ≤ 7 levels in Fig. 8 and depict the remaining distributions

12

K. Guo and Z. Zhu / Journal of Quantitative Spectroscopy & Radiative Transfer 244 (2020) 106845 Table 10 Radiative lifetimes (ms) of the first 16 levels of the a4 –1/2 , a4 1/2 , a4 3/2 , and a4 5/2 states.

υ

a4 –1/2

a4 1/2

a4 3/2

a4 5/2

υ

a4 –1/2

a4 1/2

a4 3/2

a4 5/2

0 1 2 3 4 5 6 7

3.80×103 4.08×103 4.01×103 3.38×103 2.80×103 2.49×103 2.35×103 2.32×103

4.38 4.27 4.26 4.25 4.22 4.21 4.21 4.21

4.23 4.15 4.16 4.15 4.14 4.15 4.15 4.17

90.24 81.93 76.58 70.87 66.06 62.81 60.59 59.10

8 9 10 11 12 13 14 15

2.33×103 2.34×103 2.32×103 2.29×103 2.25×103 2.23×103 2.22×103 2.24×103

4.23 4.25 4.27 4.31 4.35 4.41 4.48 4.55

4.19 4.22 4.25 4.29 4.34 4.40 4.48 4.56

58.04 57.21 56.51 55.92 55.45 55.14 54.98 54.99

studied their transition properties. The main conclusions are summarized as follows.

Fig. 9. Distribution of radiative lifetime varying as J for a certain υ of the b4  – 1/2 state. Lines correspond to υ = 7–15 from the bottom to the top, respectively.

in Fig. 9. For clarity, we show those of the b4  – 3/2 state in Figs. S4 and S5 as the supplementary material. As can be seen in Figs. S4 and S5, the radiative lifetime gradually decreases as J increases for the υ ≤ 7, while increases as J increases for the υ ≥ 8. 3.3.3. Properties of transitions from the a4 5/2 , a4 3/2 , a4 1/2 , and a4 –1/2 states to the X2 3/2 and X2 1/2 states As demonstrated in Fig. 5, the TDM values of all transitions from the a4 5/2 , a4 3/2 , a4 1/2 , and a4 –1/2 states to the X2 3/2 and X2 1/2 states are small, suggesting that these electric dipole– forbidden transitions should be weak. Great effort would be made when we measure these transitions in a spectroscopy experiment. Based on the results calculated herein, the largest Einstein A coefficients of vibronic emissions are on the order of 101 , 10–1 , 100 , 100 , 101 , and 10–2 s–1 for the a4 3/2 – X2 1/2 , a4 3/2 – X2 3/2 , a4 5/2 – X2 3/2 , a4 1/2 – X2 1/2 , a4 1/2 – X2 3/2 , and a4 – 1/2 – X2 1/2 transitions, respectively. These results prove that all electric dipole–forbidden transitions are weak, as predicted from the FC principle. Emploing Eqs. (3)–(5), we calculated the radiative lifetimes of all vibrational levels of the a4 5/2 , a4 3/2 , a4 1/2 , and a4 – 1/2 states and list the results of the first 16 vibrational levels in Table 10. As can be seen in Table 10, the readitive lifetimes of the a4 3/2 and a4 1/2 states are on the order of several ms, while those of the a4 –1/2 state are approximately several s, suggesting that the a4 3/2 and a4 1/2 states can be measured in a spectroscopy experiment, whereas great effort would be made in order to observe the a4 –1/2 state via spectroscopy. 4. Conclusion This work calculated the PECs of all states belonging to the first two dissociation asymptotes, investigated the TDMs between the X2 , A2 , B2  – , 12 , C2 , 12 , 12  + , 22  + , a4 , and b4  – states as well as the TDMs of the transitions between certain  states generated from the X2 , A2 , a4 , and b4  – states, and

(1) The radiative lifetimes are on the order of 10–6 s for the A2 , C2 , b4  – , and 12  + state, approximately 10–6 – 10–7 s for the 12  and B2  – states, and on the order of 10–7 s for the 22  + state. (2) The A2 – X2 , C2 – X2 , C2 – A2 , B2  – – X2 , B2  – – A2 , 12 – X2 , 12 – A2 , and b4  – – a4  transitions are strong. The transitions from the the first well of the 12  + state to the X2  state and from the first well of the 22  + state to the X2 , A2 , and C2  states are also strong, and are easy to measure in a spectroscopy experiment. (3) The A2 1/2 – X2 1/2 , A2 3/2 – X2 1/2 , b4  – 3/2 – a4 5/2 , b4  – 1/2 – a4 1/2 , and b4  – 1/2 – a4 3/2 transitions are strong. The distributions of the radiative lifetime with the variation of J were calculated at J ≤ 70.5 for the A2 1/2 , A2 3/2 , B4  – 1/2 , and B4  – 3/2 states. (4) The radiative lifetimes of the a4 1/2 and a4 3/2 states are on the order of several ms; those of the a4 5/2 state are approximately several–ten ms; while those of the a4 –1/2 state are on the order of several s. Acknowledgment This work is sponsored by the National Natural Science Foundation of China under Grant no. 11274097. Supplementary material Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.jqsrt.2020.106845. References [1] Andreazza CM, Marinho EP. Formation of SO, SO+ , and S2 by radiative association. Astrophys J 2005;624:1121–5. [2] Woods RC. Sulfur–bearing ions in the ionosphere of comet halley. Astrophys J 1991;378:756–62. [3] Turner BE. Microwave spectroscopy of molecular ions in the laboratory and in space. Phil Trans R Soc Lond A 1988;324:L141–6. [4] Turner BE. Detection of interstellar SO+ : a diagnostic of dissociative shock chemistry. Astrophys J 1992;396:L107–10. [5] Turner BE, Chan K-W, Green S, Lubowich DA. Tests of shock chemistry in IC 433G. Astrophys J 1992;399 144–33. [6] Turner BE. Interstellar SO. Astrophys J 1994;430:727–42. [7] Turner BE. The physics and chemistry of small translucent molecular clouds. VI. Organo–sulfur species. Astrophys J 1996;461:246–64. [8] Kivelson MG, Khurana KK, Walker RJ, Warnecke J, Russell CT, Linker JA, Southwood DJ, Polanskey C. Io’s interaction with the plasma torus: Galileo magnetometer report. Science 1996;274:396–8. [9] Russell CT, Kivelson MG. Detection of SO in Io’s exosphere. Science 20 0 0;287:1998–9. [10] Blanco-Cane X, Russell CT, Strangeway RJ, Kivelson MG, Khurana KK. Galileo observations of ion cyclotron waves in the Io torus. Adv Space Res 2001;28:1469–74. [11] Contreras CS, Prieto LV, Agúndez M, Cernicharo J, Quintana–Lacaci G, Bujarrabal V, Alcolea J, Goicoechea JR, Herpin F, Menten KM, Wyrowski F. Molecular ions in the O–rich evolved star OH231.8 + 4.2: HCO+ , H13 CO+ and first detection of SO+ , N2 H+ , and H3 O+ . Astron Astrophys 2015;577:A52.

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