Spectroscopic properties and transition probabilities of SiC+ cation

Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 204 (2018) 164–173

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Spectroscopic properties and transition probabilities of SiC+ cation Dan Zhou, Deheng Shi ⁎, Jinfeng Sun, Zunlue Zhu College of Physics and Material Science, Henan Normal University, Xinxiang 453007, China

a r t i c l e

i n f o

Article history: Received 24 April 2018 Received in revised form 8 June 2018 Accepted 11 June 2018 Available online 15 June 2018 Keywords: Potential energy curve Electric dipole moment Radiative lifetime Franck–Condon factor Transition probability

a b s t r a c t This study calculates the potential energy curves of 12 Λ-S and 27 Ω states, which belong to the first dissociation channel of SiC+ cation. The potential energy curves are computed with the complete active space self–consistent field method, which is followed by the valence internally multireference configuration interaction approach with the Davidson correction. The transition dipole moments are determined. Core-valence correlation and scalar relativistic correction, as well as extrapolation of the potential energies to the complete basis set limit are included. The spin-orbit coupling effect on the spectroscopic parameters and vibrational properties is evaluated. The vibrational band origins, Franck–Condon factors, and Einstein coefficients of spontaneous emissions are calculated. The rotationless radiative lifetimes of the vibrational levels are approximately 10−5 s long for the e2Π state. The partial radiative lifetimes of vibrational levels are approximately 10−7 s long for the 24Π and 24Σ− states, 10−5 to 10−6 s long for the 22Σ− state and the first well of the 14Π state, and very short for the second well of the 14Π state. Overall, the emissions are strong for the 22Σ−–c2Σ−, 24Σ−–X4Σ−, 24Π–X4Σ− transitions, and for the second well of the 14Π–14Σ+ transition. The spectral range of emissions is determined. In terms of the radiative lifetimes and transition probabilities obtained in this paper, some guidelines for detecting these states are proposed via spectroscopy. These results can be used to measure the emissions from the SiC+ cation, in particular, in interstellar clouds. © 2018 Elsevier B.V. All rights reserved.

1. Introduction The SiC radical is astrophysically important [1–3]. Early in 1979, Suzuki [4] theoretically studied the evolutional feature of all molecules containing Si and C atoms in interstellar clouds. Cernicharo et al. [5] in 1989 first detected the rotational spectra of this radical in the ground state in the circumstellar shell of IRC + 10216. Naturally, it is expected that the SiC+ cation would be observed in these places. Since the observations require accurate spectroscopic information and transition knowledge, the SiC radical has been widely studied for several decades both experimentally and theoretically, as reviewed in the previous works [6, 7]. Unfortunately, no spectroscopic experimental investigations have been performed for the SiC+ cation, although three groups of spectroscopic calculations have been reported in the literature [8–10]. Theoretically, Bruna et al. [8] in 1981 first reported the ab initio calculations of potential energy curves (PECs) for the 12 low–lying quartet and doublet states of SiC+ cation using the multireference configuration interaction (MRCI) approach. With the PECs obtained in their work, they evaluated the spectroscopic parameters of six states. Boldyrev et al. [9] in 1994 predicted the Re and ωe values of the X4Σ−, b2Π, and d2Σ+ states of this cation with the second-order Møller–Plesset ⁎ Corresponding author. E-mail address: [email protected] (D. Shi).

https://doi.org/10.1016/j.saa.2018.06.041 1386-1425/© 2018 Elsevier B.V. All rights reserved.

perturbation theory. Pramanik et al. [10] in 2008 calculated the PECs of 14 Λ-S states and 14 Ω states of this cation with the multireference singles and doubles configuration interaction approach. Using their PECs, they evaluated the Te, Re, and ωe values of these states. In addition, they also calculated some transition dipole moments (TDMs), estimated the partial radiative lifetimes of the vibrational levels, and briefly discussed few transition probabilities of this cation. Summarizing the above results, we confirm that few transition probabilities are currently available, although the transition properties are necessary for observing the SiC+ cation in the interstellar clouds. For this reason, this work will investigate the vibrational band origins, Franck–Condon (FC) factors, and Einstein coefficients of all the spontaneous emissions between several low–lying states, so as to accurately understand the transition probabilities of this cation. This paper is organized as follows. The methodology employed will be briefly introduced in the next section. The PECs and TDMs are reported in Section 3. The spectroscopic parameters, vibrational levels Gυ and rotational constants Bυ are predicted. The vibrational band origins, Franck–Condon factors, and Einstein coefficients of all spontaneous emissions are calculated. The rotationless radiative lifetimes of vibrational levels are estimated for the e2Π, 24Σ−, 22Σ−, 24Π, and 14Π states. The spectral range of the spontaneous emissions is briefly evaluated. The transition probabilities are discussed. The spin–orbit coupling (SOC) effect on the spectroscopic parameters and vibrational levels of each state is briefly studied. A summary is presented in Section 4.

D. Zhou et al. / Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 204 (2018) 164–173

2. Theory and Method

corr −β ΔEcorr ¼ Ecorr X : X ∞ þA

The ionization energies of Si and C atoms are 65,747.76 and 90,820.35 cm−1 [11], respectively. In terms of these ionization energies, the ionization of a C atom is easier than that of a Si atom. Therefore, the first dissociation limit of SiC+ cation is Si+(2Pu) + C(3Pg). According to the Wigner–Witmer rules [12], the C(3Pg) atom and the Si+(2Pu) ion generate a total of 12 Λ-S states. These states are X4Σ−, 12Δ (a2Δ), 12Π (b2Π), 12Σ− (c2Σ−), 12Σ+ (d2Σ+), 22Π (e2Π), 14Δ (A4Δ), 14Σ+, 22Σ−, 14Π, 24Σ−, and 24Π. All the PECs are calculated employing the complete active space selfconsistent field (CASSCF) method, which is followed by the valence internally contracted MRCI (icMRCI) approach with the Davidson correction (icMRCI + Q) [13, 14]. Thus, the CASSCF is used as the reference wavefunctions for the icMRCI calculations. The basis sets used here are aug-cc-pV5Z (AV5Z) and aug-cc-pV6Z (AV6Z). The point spacing interval is 0.02 nm for each state. As with the previous work [15], to determine the detailed information of each PEC, the point spacing is 0.005 nm near the internuclear equilibrium separations of these states. It should be noticed that the point spacing intervals stated here are employed to calculate all the PECs, including the calculations of corevalence correlation and scalar relativistic corrections as well as the SOC effect. All the PECs and TDMs are calculated within the MOLPRO 2010.1 program package [16] in the C2v point group. The state–averaged technique is used in the CASSCF calculations. As with the previous work [17], the molecular orbitals (MOs) used for the icMRCI calculations are generated by the CASSCF calculations. Eight outermost MOs (4a1, 2b1, and 2b2) are put into the active space, corresponding to the 5–8σ, 2π and 3π MOs in the SiC+ cation. The seven valence electrons are distributed into the eight valence MOs. Hence, this active space is referred to as CAS [7, 8]. The remaining 12 inner electrons are put into the six lowest MOs (4a1, 1b1, and 1b2), corresponding to the 1–4σ and 1π MOs in the SiC+ cation. No additional MOs are added into the active spaces when we calculate the core–valence correlation and scalar relativistic corrections as well as the SOC effect. For the icMRCI calculations with the AV6Z basis set, the total of external orbitals is 368, which are 126a1, 90b1, 90b2, and 62a2, respectively. In the CASSCF calculations with the AV6Z basis set, the A1, A2, B1, and B2 symmetries corresponding to the doublet states have 616, 560, 588, and 588 configuration–state functions (CSFs), and those corresponding to the quartet states have 320, 352, 336, and 336 CSFs, respectively. In the icMRCI calculations with the AV6Z basis set, the totals of contracted configurations of A1, A2, B1, and B2 symmetries corresponding to the doublet states are 2451432, 3550576, 2450380, and 2450380, and those corresponding to the quartet states are 2379112, 3483208, 2380616, and 2380616, respectively. To improve the quality of PECs, core–valence correlation and scalar relativistic corrections are included into the PECs with the cc-pCV5Z and cc-pV5Z-DK basis sets, whose approaches have been introduced in our previous paper [18]. For purposes of clarity, the PECs including the core–valence correlation correction is denoted as “+ CV”; and the PECs including the scalar relativistic correction is denoted as “+ DK”. For example, when the PECs are obtained by the icMRCI + Q/AV5Z calculations and the PECs have included the core–valence correlation and scalar relativistic corrections, the method calculated is denoted as “icMRCI + Q/AV5Z + CV + DK”. To obtain more reliable PECs, we extrapolate the potential energies to the complete basis set (CBS) limit with the AV5Z and AV6Z basis sets. It has been proved that the convergence speed of the reference energy is faster than that of the correlation energy [19]. Therefore, in order to improve the accuracy of PECs, we separately extrapolate the reference and correlation energies in this study. The extrapolation scheme is as follows [19],

ref −α ref ; ΔEref X ¼ E∞ þ A X

ð1Þ

165

ð2Þ

corr Here, ΔEref are the reference and correlation energies, reX and ΔEX spectively, calculated by the aug-cc-pVXZ basis set (here X = 5 and 6). ΔE ∞ ref and ΔE ∞ corr are the reference and correlation energies, respectively, obtained by the CBS extrapolation. The extrapolation parameters α and β are taken as 3.4 and 2.4 for the reference and correlation energies [19], respectively. Here, the PECs obtained by the CBS extrapolation with the AV5Z and AV6Z basis sets is denoted as “56”. For example, when the PECs are obtained by the icMRCI + Q calculations and the PECs have included the extrapolation noted above, the approach calculated is denoted as “icMRCI + Q/56” for convenience of description in this paper. The SOC effect is calculated by the state interaction approach [20] at the level of icMRCI theory with the all–electron cc-pCV5Z basis set. The calculations are performed within the MOLPRO 2010.1 program package in the C2v point group. The all–electron cc-pCV5Z basis set with and without the SOC operator is used to determine the potential energies, respectively. The difference between the two energies is the contribution to the total energy by the SOC effect, which is denoted as SOC in this paper. For example, when the PECs are obtained by the “icMRCI + Q/56 + CV + DK” calculations and the PECs have included the SOC effect, the approach calculated is denoted as “icMRCI + Q/56 + CV + DK + SOC”. Employing the PECs obtained by the icMRCI + Q/56 + CV + DK and icMRCI + Q/56 + CV + DK + SOC calculations, we evaluate the spectroscopic parameters, Te, De, Re, ωe, ωexe, ωeye, αe, and Be with MOLCAS program [21]. The TDMs are calculated by the valence icMRCI approach along with the AV6Z basis set. In this paper, we use these TDMs and PECs to calculate the transition probabilities and FC factors of all the spontaneous emissions, with the LEVEL program [22]. Supposing that the upper and lower vibrational levels are υ′ and υ″, respectively, the total transition probability of a certain upper level is obtained by summing the Einstein coefficients of emissions from this level to all the levels of lower states. The rotationless radiative lifetime of a certain upper level is determined as the reciprocal of total transition probability [23–25],

Aυ0 ¼

X

Aυ0 υ00

ð3Þ

υ00

τυ0 ¼

1 Aυ0

ð4Þ

Here, Aυ'υ'' is the rotationless Einstein coefficient of spontaneous emissions from an upper–level υ′ to the lower–level υ″; Aυ' is the total transition probability of an upper–level υ′; and τυ' is the radiative lifetime of level υ′. Sometimes, the total Einstein coefficient of an upper level is determined by several contributions. To explain how this value is calculated, we take the 24Π state as an example. According to transition selection rules, the spontaneous emissions from the 24Π state to the X4Σ−, A4Δ, 14Σ+, 14Π, and 24Σ− states can occur. That is, the total Einstein coefficient of a certain vibrational–level υ’ of the 24Π state comes from the contributions of five groups of transitions, 24Π–X4Σ−, 24Π–A4Δ, 24Π– 14Σ+, 24Π–14Π, and 24Π–24Σ−. When several spontaneous emission systems (i = 1, 2, …) are generated from a certain upper state, the total Einstein coefficient of emissions from an upper–level υ’ is, Aυ0 ¼

X

Ai;υ0

ð5Þ

i

Here, Ai, υ' is the total Einstein coefficient of emissions from an upper–level υ′ for the ith emission system.

166

D. Zhou et al. / Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 204 (2018) 164–173

50000

To better study the transition probabilities, we list the leading valence electronic configurations around their respective internuclear equilibrium positions in Table 1. These valence configurations are obtained by the icMRCI approach along with the AV6Z basis set. In Table 1, we only tabulate those valence configurations with the CSF coefficients squared larger than 0.08. According to this table, we can clearly describe how the electronic transitions from one state to another occur. The curves of TDM versus internuclear separation, which are used to compute the electric dipole transitions in this paper, are depicted in Figs. 3 and 4. As with the PECs, to clearly show the dominant features of each TDM curve, we demonstrate them in Figs. 3 and 4 only over a small range around the internuclear equilibrium separations. For convenience of discussion, we divide the 12 Λ-S states into two categories according to the spin multiplicity. One group is double states. The other group is quartet states. 3.1.1. Transition Probabilities of Emissions Generated From the Quartet States Using the PECs obtained by the icMRCI + Q/56 + CV + DK calculations, we fit the spectroscopic parameters of six quartet states with the MOLCAS program [21]. The spectroscopic results are listed in Table 2. No experimental spectroscopic parameters are currently available and only three groups of theoretical spectroscopic results have been reported in the literature [8–10] thus far. For purposes of comparison, we also tabulate these theoretical spectroscopic values in Table 2.

50000 Potential energy /cm

-1

6 40000

20000

Si+(2Pu)+C(3Pg)

5 3

4

2

10000 1 0

0.2

0.3

0.4

40000

Si+(2Pu) + C(3Pg)

30000

7

20000 10000

65 4 2

3 1

0

0.2

0.3

0.4

0.5

Internuclear separation /nm

3.1. Spectroscopic Parameters and Transition Probabilities

30000

Potential energy /cm

Using the icMRCI + Q/56 + CV + DK approach, we calculate the PECs of the X4Σ−, a2Δ, b2Π, c2Σ−, d2Σ+, e2Π, A4Δ, 14Σ+, 22Σ−, 14Π, 24Σ−, and 24Π states, which are generated from the first dissociation asymptote of SiC+ cation. Figs. 1 and 2 depict the PECs of these states. To display more information of each PEC, we show them only over a small range of internuclear separations. For convenience of discussion on the transition properties, the PECs with the same spin multiplicity are demonstrated in the same figure. From Figs. 1 and 2, we can summarize some leading features. All the 12 Λ-S states are bound. The b2Π, 14Π, 24Π, and 24Σ− states have a single barrier. Further investigations show that each barrier is generated by the avoided crossing of this state with another. Furthermore, both the b2Π and 14Π states have a double well. Only the barrier of 24Π state can be clearly seen in Fig. 1.

-1

3. Results and Discussion

0.5

Internuclear separation /nm Fig. 1. PECs of the six quartet states of SiC+ radical. 1–X4Σ−; 2–A4Δ; 3–14Σ+; 4–14Π; 5–24Σ−; 6–24Π.

Fig. 2. PECs of the six doublet states of SiC+ radical. 1 X4Σ−; 2–a2Δ; 3–b2Π; 1–c2Σ−; 5– d2Σ+; 6–e2Π; 7–22Σ−.

Using the equation D0(SiC+) = D0(SiC) + I.P. (Si) − I.P. (SiC) and corresponding experimental data, Bruna et al. [8] estimated that experimental D0 of the ground–state SiC+ cation was 3.4–3.8 ± 0.6 eV. The De value obtained in this study equals 3.6381 eV, which falls into the range of the deduced D0 result. Theoretically, Bruna et al. [8] determined the theoretical D0 of the SiC+ cation in the ground state to be 3.71 eV and Pramanik et al. [10] obtained the theoretical De of the ground state as 3.22 eV. The present De agrees favorably with the theoretical result determined by Bruna et al. [8], whereas it is in poor agreement with the value reported by Pramanik et al. [10]. As for Re, our result agrees well with the theoretical one reported by Boldyrev et al. [9], but it obviously deviates from those computed by Bruna et al. [8] and Pramanik et al. [10]. As seen in Table 2, the ωe value calculated in this study agrees poorly with all the theoretical results reported previously [8–10]. The ground state has a deep well and possesses many vibrational levels. To conveniently determining the vibrational band origins of spontaneous emissions, some Gυ and Bυ values (υ ≤ 39) of this state are collected in Table S1 of the Supplementary material. The A4Δ state is the first excited quintet state. This state has a well depth of only 6418.56 cm−1. It has 43 vibrational levels, some of which are collected in Table S1. For the A4Δ, the Te, Re, and ωe results obtained in this study more or less deviate from those reported by Pramanik et al. [10]. The spectroscopic parameters obtained here are more reliable than those [10] since the present results have included various corrections, such as core–valence correlation correction, scalar relativistic correction, Davidson correction, and extrapolation of potential energies to the CBS limit. Similar to the X4Σ− state, the A4Δ state also has the single reference character around the internuclear equilibrium separation. No spontaneous emissions are generated from A4Δ state according to the transition selection rules. The Te, Re, and ωe results of 14Σ+ state is very close to those of the A4Δ state for the present study. For example, the difference of Te between the two states is only 445.78 cm−1. From the spectroscopic standpoint, both the A4Δ and 14Σ+ states are insignificant, since the spontaneous emissions generated from the two states are forbidden. The 14Π state has a double well and a single barrier. The barrier lies at the internuclear separation of approximately 0.185 nm, which is generated by the avoided crossing of this state with the 24Π state. It is the avoided crossing that generates the potential well of 24Π state. The potential energy at the top of this barrier is higher than that at the dissociation asymptote only by 418.35 cm−1. Therefore, both the well depth and dissociation energy of the first well are relative to the barrier and those of the second well are relative to the dissociation asymptote. Because the height of this barrier is very small, it is not clearly seen in

D. Zhou et al. / Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 204 (2018) 164–173

167

Table 1 Leading valence configurations of 12 states of the SiC+ cation around the internuclear equilibrium positions. State

Leading valence configuration

State

X4 Σ− a2Δ b2Π 1st well 2nd well

5σ26σ27σ12π03π28σ0 (0.8081)a. 5σ26σ27σ12π03π28σ0 (0.8284).

14Π 1st well 2nd well 22Σ− 24Σ−

5σ26σ27σ02π03π38σ0 (0.7207). 5σ26σ27σ22π03π18σ0 (0.6908); 5σ26σ27σ02π03π38σ0 (0.1483). 5σ26σ27σ12π03π28σ0 (0.8404). 5σ26σ27σ12π03π28σ0 (0.7779). 5σ26σ27σ12π03π18σ1 (0.8634). 5σ26σ27σ22π03π18σ0 (0.4711); 5σ26σ27σ02π03π38σ0 (0.2958).

c2Σ− d2Σ+ 14Σ+ e2Π a

Leading valence configuration 5σ26σ17σ12π03π38σ0 (0.7503). 5σ26σ27σ12π13π18σ0 (0.8801). 5σ26σ27σ12π03π18σ1 (0.7766). 5σ26σ17σ22π03π28σ0 (0.5558); 5σ26σ27σ12π03π18σ1 (0.2463). 5σ26σ27σ02π03π28σ1 (0.4967); 5σ26σ17σ12π03π38σ0 (0.3186). 5σ26σ27σ12π03π18σ1 (0.5976); 5σ26σ27σ02π13π28σ0 (0.0885); 5σ26σ27σ12π03π08σ2 (0.0824).

24Π A4Δ

Values in parentheses are the coefficients squared of CSF associated with the electronic configuration.

0.6 0.5

4

0.4 7

0.3 0.2

5

6

2 0.1 3 0.0 0.2

Table 2 and the vibrational levels listed in Table S1. The same explanations are applicable to the vibrational band origins of all the other transitions discussed in this paper. For transitions from the first well to the X4Σ− state, there are 43 and 89 emissions whose orders of magnitude of the Einstein coefficients are 4 and 3, respectively. Employing the Te and Gυ values of 14Π state as well as the Gυ result of X2Π state listed in Tables 2 and S1, we estimate that the spectral range of these emissions covers the near-UV to the farinfrared regions. Further studies confirm that the strong emissions are near-UV and visible light. For transitions from the second well to the X4Σ− state, almost all the spontaneous emissions are very weak. For transitions from the first well to the A4Δ state, there are 14 and 38 emissions whose Einstein coefficients are approximately 102 and 101, respectively. The spectra of these emissions extend from the nearinfrared to the far-infrared light. The strong emissions are only near–infrared light. For transitions from the second well to the A4Δ state, all the spontaneous emissions are very weak and therefore, very difficult to detect via spectroscopy. Of all the emissions, the largest Einstein coefficients are only approximately 10−2. For the transitions from the first well to the 14Σ+ state, there are 10 and 36 emissions, with magnitude orders of the Einstein coefficients being only 2 and 1, respectively. Similar to the transitions from the second well to the A4Δ state, the emissions from the first well to the 14Σ+ state are very weak according to the Einstein coefficients. In terms of the vibtational band origins calculated in this study, the spectra of this system extend from the nearinfrared to the far–infrared light. It should be noted that the emissions from the second well to the 14Σ+ state are very intense. Calculations confirm that there are 31, 61, and 51 emissions, whose orders of magnitude of the Einstein coefficients are 18, 17, and 16, respectively. All the emissions of this system fall into the infrared region.

Transition dipole moment /a.u.

Transition dipole moment /a.u.

Fig. 1. The first well of this state lies at 0.16646 nm. Its depth relative to the barrier is 5620.36 cm−1. It has 12 vibrational levels, as collected in Table S1. The second well lies at 0.41419 nm, whose depth relative to the dissociation limit is 415.38 cm−1. It has 18 vibrational levels, as listed in Table S1, although the second well is much shallower than the first one. The avoided crossing was observed by Bruna et al. [8] and Pramanik et al. [10]. They [8, 10] did not evaluate the spectroscopic properties of the second well, possibly because the second well is too shallow. The well depth of the first well obtained in this study is 0.6968 eV, which is much deeper than that of 0.29 eV reported by Pramanik et al. [10]. According to the transition selection rules, the 14Π state can spontaneously decay to the X4Σ−, A4Δ, and 14Σ+ states. As listed in Table 1, the leading valence configurations of the X4Σ−, A4Δ, 14Σ+, and 14Π states have the single reference character around their respective internuclear equilibrium separations. The dominant electronic transitions from the first well to the X4Σ−, A4Δ, and 14Σ+ states are the 6σ13π3 –6σ23π2, 6σ13π38σ0–6σ23π18σ1, and 6σ13π38σ0–6σ23π18σ1 promotions; and those from the second well to the X4Σ−, A4Δ, and 14Σ+ states are the 2π13π1–2π03π2, 2π18σ0–2π08σ1, and 2π13π1–2π03π2 promotions, respectively. Using the TDMs and PECs obtained in this paper, we evaluate the FC factors, vibrational band origins, and Einstein coefficients of all the spontaneous emissions from the 14Π state to the X4Σ−, A4Δ, and 14Σ+ states, with the LEVEL program [22]. For purposes of clarity, we collect some of the relatively large Einstein coefficients of spontaneous emissions along with the FC factors in Table 3 for the 14Π–X4Σ− transition only, and list those in Table S2 of the Supplementary Material for the 14Π–A4Δ and 14Π–14Σ+ transitions. For purposes of brevity, we omit the vibrational band origins from Table 3 and Table S2 because they can be approximately calculated using the Te values presented in

6 7 0.2 4 0.1 3 5 0.0

1

8

0.3

1

2 0.2

0.3

0.4

0.3

0.4

Internuclear separation /nm

Internuclear separation /nm Fig. 3. Curves of TDM versus internuclear separation for the 7 pairs of states. 1: 22Σ−–e2Π; 2: 24Σ− −14Π; 3: e2Π–d2Σ+; 4: 24Σ− −X4Σ−; 5: 22Π–a2Δ; 6: e2Π–c2Σ−; 7: 22Σ− −c2Σ−.

Fig. 4. Curves of TDM versus internuclear separation for the 8 pairs of states. 1: 14Π–14Σ+; 2: 14Π–X4Σ−; 3: 14Π–A4Δ; 4: 24Π–A4Δ; 5: 24Π–14Σ+; 6: 24Π–14Π; 7: 24Π–24Σ−; 8: 24Π–X4Σ−.

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Table 2 Spectroscopic parameters of the SiC+ cation obtained by the icMRCI + Q/56 + CV + DK calculations and comparison with other theoretical results for the six quartet states.

X4 Σ − Cal. [8] Cal. [9] Cal. [10] A4 Δ Cal. [10] 14Σ+ Cal. [10] 14Π 1st well Cal. [10] 2nd well 24Σ− Cal. [10] 24Π Cal. [10] a

Te/cm−1

Re/nm

ωe/cm−1

ωexe/cm−1

102ωeуe/cm−1

Be/cm−1

102αe/cm−1

De/eV

0.00 0.00 0.00 0.00 22,937.95 21,173 23,383.71 21,473

0.18072 0.18415 0.1804 0.183 0.23856 0.246 0.24074 0.248

852.14 797 890 817 307.35 285 307.42 281

5.735 –

5.676 –

0.61461 –

0.520 –

3.6381 3.71a

– 5.579

– 134.4

– 0.35468

– 0.505

3.32 0.7958

2.558

28.97

0.34719

0.384

0.7402

23,957.85 24,464 28,957.04 28,085.51 27,447 35,864.35 35,254

0.16646 0.170 0.41419 0.18247 0.189 0.18442 0.185

1013.51 875 59.31 719.23 537 1044.86 965

11.68

4.241

0.72389

0.772

0.6968

2.615 14.01

311.1 237.1

0.11696 0.60231

0.500 0.933

0.0515 0.3400

3.249

106.1

0.59012

0.139

1.4931

D0 result obtained from the estimated full CI with basis set B from Ref. [8].

The rotationless radiative lifetime of a certain vibrational level of the 14Π state comes from the three systems, 14Π–X4Σ−, 14Π–A4Δ, and 14Π–14Σ+. Since we cannot entirely determine the sequence of quintet states, we are not certain that Te of only three states (X4Σ−, A4Δ, and 14Σ+) is lower than that of the 14Π state. In other words, we are not sure the rotationless radiative lifetimes of vibrational levels calculated in this study are total ones. Using Eqs. (3)–(5) in combination with the Einstein coefficients obtained in this work, we calculate the partial radiative lifetimes for some vibrational levels of the double well of 14Π state. The results are collected in Table 4. In terms of Table 4, for the first well, the partial radiative lifetimes are approximately several to several–dozen microseconds long; for the second well, the partial radiative lifetimes are very short, in particular for highly–vibrational levels. It should be noted that the actual radiative lifetimes are shorter

than those tabulated in Table 4. In terms of the radiative lifetimes collected in Table 4, we confirm that the spontaneous emissions generated from the 14Π state easily occur. Overall, the emissions from the first well to the X4Σ− state are very intense, which are the most important contributor to the rotationless radiative lifetimes of the first well, whereas the emissions from the second well to the 14Σ+ state are the predominant contributor to the rotationless radiative lifetimes of the second well. The 24Σ− state has a single barrier. The barrier lies at the internuclear separation of approximately 0.221 nm. Further calculations confirm that the barrier is formed by the avoided crossing of 24Σ− state with the 34Σ− state. The potential energy at the top of the barrier is higher than that at the dissociation limit by approximately 1249.50 cm−1. Therefore, both the well depth and dissociation energy

Table 3 Some of the relatively large Einstein coefficients (s−1, 2nd line) of spontaneous emissions with the FC factors (1st line) for the 14Π–X4Σ−, 24Σ−–X4Σ−, and 24Π–X4Σ− transitions. υ′–υ″

FC

1st well of the 14Π–X4Σ− 0–0 0.1071 1.89 × 104 1–0 0.3178 4.07 × 104 1–8 0.0575 1.12 × 104 2–9 0.0699 1.35 × 104 4–6 0.0423 1.21 × 104

υ′–υ″

FC

υ′–υ″

FC

υ′–υ″

FC

υ′– υ″

FC

υ′–υ″

FC

0–1

0.1974 4.27 × 104 0.1510 3.21 × 104 0.3721 2.37 × 104 0.3183 1.59 × 104 0.2194 1.79 × 104

0–2

0.2160 5.09 × 104 0.0711 1.62 × 104 0.1119 1.61 × 104 0.0834 1.68 × 104 0.4162 5.82 × 104

0–3

0.1802 4.29 × 104 0.1057 2.40 × 104 0.0944 2.40 × 104 0.0525 1.58 × 104 0.0253 1.16 × 104

0–4

0.1267 2.95 × 104 0.1048 2.29 × 104 0.0663 1.43 × 104 0.0579 1.10 × 104 0.0379 1.55 × 104

0–5

0.0791 1.76 × 104 0.0833 1.73 × 104 0.0775 1.59 × 104 0.3091 2.28 × 104 0.1016 1.86 × 104

0.0441 5.08 × 104 0.0260 1.63 × 104 0.0767 6.63 × 103 0.1557 5.81 × 103

0–2

0.0055 5.41 × 103 0.0007 3.75 × 103 0.0055 2.60 × 103 0.0311 1.67 × 103

0–3

0.0005 1.75 × 103 0.0002 6.21 × 104 0.0058 3.72 × 105 0.0000 2.73 × 104

1–0

0.1158 9.04 × 105 0.6642 6.03 × 106 0.1809 1.25 × 106 0.0575 6.62 × 105 0.0889 8.92 × 105

1–0

0.0919 1.12 × 106 0.1900 1.38 × 106 0.0312 3.73 × 105 0.0094 1.35 × 105 0.0117 1.75 × 105

1–1

0.7282 6.83 × 106 0.0132 1.81 × 106 0.0674 7.39 × 105 0.0858 9.04 × 105 0.1030 1.03 × 106

1–2

1–1 2–0 3–1 5–2

2nd well of the 14Π–X4Σ− Emissions of this system are very weak. 24Σ−–X4Σ− 0–0 0.9499 0–1 2.81 × 106 1–2 0.1187 1–3 1.03 × 105 2–3 0.2046 2–4 6.42 × 104 3–4 0.2239 3–5 1.27 × 104 4 4 − 2 Π–X Σ 0–0 0.8823 0–1 8.49 × 106 2–1 0.1020 2–2 1.34 × 106 3–3 0.6439 3–4 5.63 × 106 4–5 0.1531 4–6 9.95 × 105 5–6 0.1153 5–7 6.92 × 105

1–4 2–2 3–4 5–3

1–5 2–6 3–7

2–3 3–5 5–2 6–3

1–5 2–3 3–5 5–4

2–0 3–1 4–1

2–4 4–2 5–3 6–4

1–6 2–7 3–10 5–5

2–1 3–2 4–2

3–1 4–3 5–4 6–5

1–7 2–8 4–2 5–6

0.0498 8.58 × 105 0.1494 1.54 × 106 0.2855 1.42 × 106 0.0407 7.12 × 105

1–1

0.1718 1.30 × 106 0.0466 5.27 × 105 0.0597 1.02 × 106 0.0365 7.95 × 105 0.0185 5.92 × 105

2–0

2–2 3–3 4–3

3–2 4–4 5–5 6–6

0.8005 1.74 × 106 0.5408 9.93 × 105 0.2042 5.89 × 105 0.2846 7.85 × 105 0.0228 2.64 × 105 0.0844 1.23 × 106 0.6438 5.43 × 106 0.6501 5.29 × 106 0.6537 5.14 × 106

D. Zhou et al. / Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 204 (2018) 164–173

169

Table 4 Partial radiative lifetimes (μs) for the vibrational levels of the 14Π, 24Σ−, 24Π, e2Π, and 22Σ− states of the SiC+ cation. State

υ′

1st well of the 14Π 0 6 2nd well of the 14Π 0 6 12 24Σ− 0 0 24Π 6 12 2 0 e Π 6 12 0 22Σ− 6

τ

υ′

τ

υ′

τ

υ′

τ

υ′

τ

υ′

τ

4.54 5.90

1 7

5.42 7.97

2 8

6.92 11.82

3 9

9.05 17.50

4 10

9.54 26.48

5

6.31

0.0577 5.75 × 10−10 3.84 × 10−14 0.348 0.104 0.114 0.148 39.41 24.48 23.71 19.77 7.58

1 7 13 1 1 7

0.0048 2.30 × 10−11 2.22 × 10−14 0.367 0.105 0.117

2 8

0.0008 2.14 × 10−12

3 9

0.0002 4.35 × 10−13

4 10

3.39 × 10−6 1.46 × 10−13

5 11

3.12 × 10−8 6.77 × 10−14

2 2 8

0.374 0.106 0.120

3 3 9

0.415 0.108 0.124

4 10

0.110 0.128

5 11

0.112 0.134

1 7 13 1 7

34.77 23.75 24.31 15.74 7.06

2 8

31.43 23.31

3 9

28.89 23.11

4 10

26.98 23.12

5 11

25.54 23.32

2 8

12.68 6.75

3 9

10.80 6.67

4 10

9.33 6.96

5

8.33

of this state are relative to the barrier. This state has a well depth of 2742.49 cm−1. It has four vibrational levels, as presented in Table S1 of the Supplementary Material. The 24Σ− state has the obvious multireference characters around the internuclear equilibrium separation. Due to the symmetry limitations, the 24Σ− state spontaneously decays to only the X4Σ− and 14Π states. The dominant electronic transitions between the 24Σ− and the X4Σ− state are the 6σ17σ2–6σ27σ1 and 3π18σ1–3π28σ0 promotions; those between the 24Σ− state and the first well of the 14Π state are the 7σ23π2–7σ13π3 and 6σ22π13π1–6σ12π03π3 promotions; and those between the 24Σ− state to the second well of the 14Π state are the 6σ17σ22π03π2–6σ27σ12π13π1 and 7σ12π08σ1–7σ12π18σ0 promotions. As seen in Table 2, the Re value of 24Σ− state is very close to that of the X4Σ− state. As displayed in Fig. 3, the TDMs of the 24Σ−–X4Σ− transitions are large around the internuclear equilibrium separations of the two states. In terms of these results, the FC principle predicts that the emissions are strong for the 24Σ−–X4Σ− transition. Te of the 14Π state is higher than that of the 24Σ− state. Therefore, spontaneous emissions from the 24Σ− state to the second well of the 14Π state do not occur. On one hand, Re of the 24Σ− state is far away from that of the second well of the 14Π state; on the other hand, as shown in Fig. 3, the TDMs of the transitions between the 24Σ− state and the second well of the 14Π state are very small near the internuclear equilibrium separation of the second well. In combination with these results, the FC principle predicts that the spontaneous emissions from the second well of the 14Π state to the 24Σ− state should be very weak. The results calculated in this paper confirm the expectations from the FC principle. For the 24Σ−–X4Σ− transition, because the vibrational levels of 24Σ− state are only a few, the total of emissions arising from this state is few. In terms of the results obtained in this study, there are only 4, 8, 6, and 8 emissions whose Einstein coefficients are approximately 106, 105, 104, and 103, respectively. Therefore, the emissions of 24Σ−–X4Σ− transition are intense. According to the vibrational band origins calculated here, the spectra of the 24Σ−–X4Σ− transition should be covered within the UV to the infrared range and the strong spontaneous emissions are mainly UV light. The emissions falling into the infrared region are so weak that they are very difficult to observe via spectroscopy. For transitions from the 24Σ− state to the first well of the 14Π state, the emissions are only few and all the emissions are infrared light. Further investigations show that there are only 1, 8, and 12 emissions whose orders of magnitude of the Einstein coefficients are 3, 2, and 1, respectively. Because the potential energies of 24Σ− state are lower than those of the second well of the 14Π state, no spontaneous emissions occur from the 24Σ− state to the second well. As collected in Table 4, the present partial radiative lifetimes are approximately several–hundred nanoseconds long for the four levels of the 24Σ− state, which are mainly contributed by the 24Σ−–X4Σ−

transition. Pramanik et al. [10] determined that the 24Σ− state only had two vibrational levels and the radiative lifetimes of the two levels were approximately several microseconds long, which are slightly longer than those evaluated in this study. The 24Π state has a single barrier. As clearly seen in Fig. 1, the potential energy at the top of the barrier is much higher than that at the dissociation limit. Therefore, both the dissociation energy and well depth are relative to the barrier. The barrier lies at the internuclear separation of approximately 0.227 nm, which is generated by the avoided crossing of this state with the 34Π state. As discussed above, there is an avoided crossing between the 14Π and 24Π states, which forms the potential well of 24Π state. The 24Π state has a well depth of 12,042.53 cm−1 and possesses 13 vibrational levels, as listed in Table S1. Only one group of theoretical Te, Re, and ωe values [10] are currently available. By comparison, we confirm that the spectroscopic parameters obtained in this study agree favorably with the previous ones [10]. As noted in Section 2, the 24Π state can spontaneously decay to the 4 − X Σ , A4Δ, 14Σ+, 14Π, and 24Σ− states according to the transition selection rules. Using the PECs and TDMs obtained in this study, we calculate the vibrational band origins, FC factors, and Einstein coefficients of spontaneous emissions originated from the 24Π state, with the LEVEL program [22]. For purposes of clarity, we collect some of the relatively large Einstein coefficients of emissions in Table 3 along with the FC factors for the 24Π–X4Σ− transition only. For reasons of discussion, we list those in Table S2 for the 24Π–A4Δ, 24Π–14Σ+, 24Π–14Π, and 24Π–24Σ− transitions. According to the results obtained in this study, for the 24Π–X4Σ− transition, there are 35, 56, and 58 spontaneous emissions whose Einstein coefficients are approximately 106, 105, and 104, respectively. The spectra of the transitions cover the near–UV to the near–infrared range, and most intense emissions are near–UV light. Furthermore, the emissions in the infrared region are very weak. For the 24Π–A4Δ transition, there are 5 and 39 spontaneous emissions, with the Einstein coefficients approximately 103 and 102, respectively. The spectra of transitions extend from the visible to the infrared light. Further analyses confirm that most intense emissions are visible light, which originate from the highly vibrational levels of 24Π state. For the 24Π–14Σ+ transition, there are 3 and 86 emissions, whose orders of magnitude of the Einstein coefficients are 3 and 2, respectively. The spectral range of transitions falls into the visible and infrared regions. For the 24Π–24Σ− transition, there are only few emissions since both the 24Π and 24Σ− state have few vibrational levels. However, there are 4, 11, and 32 spontaneous emissions whose Einstein coefficients are approximately 104, 103, and 102, respectively. These results show that the emissions of this system are intense. Further analysis shows that the emissions of 24Π–24Σ− transition are mainly infrared and visible light. Most strong emissions are near-infrared light. For transitions from the 24Π state to the first

170

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well of the 14Π state, there are 48 and 78 spontaneous emissions whose orders of magnitude of the Einstein coefficients are 4 and 3, respectively. The spectral range of transitions covers the infrared and visible light. For transitions from the 24Π state to the second well of the 14Π state, the spontaneous emissions are very weak. The largest Einstein coefficients of emissions are approximately 100. Of these spontaneous emissions generated from the 24Π state, the 24Π–X4Σ− transitions are the most intense. In Table 4, we see that the partial radiative lifetimes of vibrational levels are approximately 10−7 s for the 24Π state. According to these results, we confirm that the emissions generated from the 24Π state should easily occur. The 24Π–X4Σ− transitions are the most important contributors to the radiative lifetimes for the vibrational levels of 24Π state. 3.1.2. Transition Probabilities of Emissions Generated From the Doublet States The spectroscopic parameters of six doublet states are collected in Table 5, which are evaluated with the PECs obtained by the icMRCI + Q/56 + CV + DK calculations. Similar to the quintet states, only a few theoretical results [8–10] are currently available. For purposes of comparison, we also list these results in Table 5. For the a2Δ state, only two groups of theoretical studies [8, 10] are currently available. The Te and Re results obtained in this study are close to the previous values, whereas the ωe values determined here are obviously far away from those reported previously [8, 10]. The dominant valence electronic configuration of a2Δ state is the 5σ26σ27σ12π03π28σ0 around the internuclear equilibrium separation, which is the same as that reported in the literature [10]. This state has many vibrational levels, some of which are tabulated in Table S1. The a2Δ state is the lowest excited state of this cation. Because of the spin limitations, no spontaneous emissions occur from this state to any other electronic states. The b2Π state has a double well and a single barrier. The first wells lies at 0.17392 nm, which compares well with the Re value obtained by Boldyrev et al. [9]. The second well lies at 0.19448 nm, which falls into the Re range determined in Ref. [9]. In addition, the Te and Re values of the second well obtained in this study are in excellent agreement with those calculated by Pramanik et al. [10]. The barrier is so low that it cannot be clearly seen in Fig. 2. Therefore, the depths of double well are relative to the barrier, whereas the De values of the double well are relative to the dissociation limit. The first well has a depth of

243.05 cm−1, which possesses no vibrational levels. The second well has a depth of 83.75 cm−1. As with the first well, the second well either has no levels. Pramanik et al. [10] reported that the first well had no vibrational levels, in accordance with the results obtained in this study. From a spectroscopic standpoint, the b2Π state is meaningless because it has no vibrational levels. The c2Σ− and d2Σ+ states have the same valence electronic configurations around the internuclear equilibrium separations. The Te and Re values of c2Σ− state are close to those of the d2Σ+ state and also compare well with those reported by Pramanik et al. [10]. The two states have many vibrational levels, some of which are listed in Table S1. Similar to the b2Π, A4Δ, and 14Σ+ states, the c2Σ− and d2Σ+ states have no spectroscopic meanings because the spontaneous emissions generated from the two states are forbidden according to the transition selection rules. For the e2Π state, two groups of theoretical spectroscopic parameters have been reported in the literature [8, 10]. Overall, the present results agree well with them [8, 10]. This state has a well depth of 14,870.44 cm−1 with 24 vibrational levels, which are collected in Table S1. As listed in Table 1, the e2Π state has the obvious multireference characters around the internuclear equilibrium separation. According to the dipole-allowed transition selection rules, the e2Π state can spontaneously decay to the a2Δ, c2Σ−, and d2Σ+ states. The dominant electronic transitions from the e2Π to the a2Δ state are the 7σ23π1–7σ13π2 and 7σ03π3–7σ13π2 promotions; those from the e2Π to the c2Σ− state are the 7σ23π1–7σ13π2 and 7σ03π3–7σ13π2 promotions; and those from the e2Π to the d2Σ+ state are the 7σ23π1– 7σ13π2 and 7σ03π3–7σ13π2 promotions, respectively. Using the PECs and TDMs determined in this study, we calculate the vibrational band origins, FC factors, and Einstein coefficients of spontaneous emissions with the LEVEL program [22]. For purposes of clarity, we collect some of the relatively large Einstein coefficients of emissions in Table 6 along with the FC factors for the e2Π–a2Δ transitions only. For convenience of discussion, we collect those in Table S2 for the e2Π–c2Σ− and e2Π–d2Σ+ transitions. For the e2Π–a2Δ transition, according to the results calculated in this study, there are 9 and 65 emissions whose Einstein coefficients are approximately 104 and 103, respectively. The spectral range of transitions covers the visible to the far–infrared regions. Further analyses confirm that the strong emissions are mainly near–infrared light. The emissions in the visible to far–infrared regions are very weak and therefore, very difficult to detect via spectroscopy. For the e2Π–c2Σ− transition, there

Table 5 Spectroscopic parameters of the SiC+ cation obtained by the icMRCI + Q/56 + CV + DK calculations and comparison with other theoretical results for the six doublet states.

2

a Δ Cal. [8] Cal. [10]

Te/cm−1

Re/nm

ωe/cm−1

ωexe/cm−1

102ωeуe/cm−1

Be/cm−1

102αe/cm−1

De/eV

9848.92 10,162.58 10,266

0.18461 0.18945 0.188

761.61 702 723

6.416

14.97

0.58895

0.627

2.4173

–

–

–

–

2.05

10,558.05 – 10,723.53 11,695.86 10,696 11,154.80 11,453.09 11,492 13,661.86 13,066.18 – 13,666 14,498.06 14,195.35 14,311 25,157.06 23,723

0.17392 0.1675 0.19448 0.175–0.201 0.199 0.18316 0.18786 0.186 0.18804 0.19791 0.1504 0.191 0.18499 0.18892 0.187 0.23001 0.234

– 952 – 741 480, 700a 787.15 731 759 688.61 586 1443 651 1139.15 1017 1013 447.00 402

–

–

–

–

2.3292

–

–

–

–

2.3094

7.249

3.726

0.59826

0.603

2.2554

7.020

12.73

0.56769

0.702

1.9450

29.85

167.3

0.58692

0.544

1.8437

16.34

311.2

0.37916

0.379

0.5213

2

b Π 1st well Cal. [9] 2nd well Cal. [8] Cal. [10] c2Σ− Cal. [8] Cal. [10] d2Σ+ Cal. [8] Cal. [9] Cal. [10] e2Π Cal. [8] Cal. [10] 22Σ− Cal. [10] a

Result estimated from the diabatic curve.

D. Zhou et al. / Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 204 (2018) 164–173

171

Table 6 Some of the relatively large Einstein coefficients (s−1, 2nd line) of emissions along with the FC factors (1st line) for the e2Π–a2Δ and 22Σ−–c2Σ− transitions of the SiC+ cation. υ′–υ″ e2Π–a2Δ 0–0 5–3 8–5 9–11

FC

υ′–υ″

FC

υ′–υ″

FC

υ′–υ″

FC

υ′– υ″

FC

υ′–υ″

FC

0.9832 1.22 × 104 0.0694 1.82 × 103 0.0672 1.89 × 103 0.0914 1.03 × 103

1–1

0.9608 1.31 × 104 0.7926 1.26 × 104 0.0879 2.46 × 103 0.0311 1.75 × 103

2–2

0.9214 1.34 × 104 0.0809 2.18 × 103 0.7106 1.07 × 104 10–8 2.14 × 103

3–3

0.8771 1.33 × 104 0.7580 1.20 × 104 0.0248 1.47 × 103 10–10 9.14 × 103

4–2

0.0535 1.35 × 103 0.0873 2.40 × 103 0.0828 2.37 × 103 11–8 1.98 × 103

4–4

0.8330 1.31 × 104 0.7303 1.14 × 104 0.6986 9.93 × 103 11–90.0570 1.82 × 103

22Σ−–c2Σ− 1–6 0.0648 1.06 × 104 3–4 0.0569 1.83 × 104 5–2 0.0267 1.56 × 104 6–4 0.0475 2.09 × 104 8–1 0.0259 2.38 × 104

5–5 8–6 10–7

1–7 3–5 5–3 6–7 8–2

0.1022 1.33 × 104 0.0844 2.21 × 104 0.0603 2.92 × 104 0.0455 1.13 × 104 0.0560 4.29 × 104

6–4 8–8

1–8 3–6 5–4 7–1 8–3

0.1135 1.16 × 104 0.0696 1.47 × 104 0.0706 2.82 × 104 0.0185 1.56 × 104 0.0439 2.80 × 104

are 7 and 142 spontaneous emissions whose orders of magnitude of the Einstein coefficients are 4 and 3, respectively. The spectral range of transitions is in the visible and the infrared regions. The strong emissions are mainly infrared light. The emissions in the visible range are very weak and therefore, very difficult to measure via spectroscopy. For the e2Π– d2Σ+ transition, there are 42 and 136 spontaneous emissions whose Einstein coefficients are approximately 102 and 101, respectively. In terms of the vibrational band origins, the emissions of this system are mainly infrared light. Only a few are visible light. Overall, the emissions of e2Π–a2Δ transition are as intense as those of the e2Π–c2Σ− transition, and all the emissions of e2Π–d2Σ+ transition are weak according to the Einstein coefficients. As listed in Table 4, the partial radiative lifetimes are about several–hundred microseconds long for all the vibrational levels of e2Π state. The actual radiative lifetimes of e2Π state are shorter than these results because we do not include the rotational radiative lifetimes such as those occurring within this state. Based on the short radiative lifetimes, the spontaneous emissions from the e2Π state to the a2Δ, c2Σ−, and d2Σ+ states easily occur. No experimental or other theoretical spontaneous emissions generated from the e2Π state have been observed to date. Therefore, we cannot make direct comparison between experiment and theory. We expect that the Einstein coefficients, vibrational band origins, and rotationless radiative lifetimes reported in this paper will provide some useful guidelines for detecting these emissions, in particular, in interstellar clouds. Only one group of theoretical spectroscopic results is currently available in the literature [10]. Obvious differences of the Te, Re, and ωe are seen between the present values and the previous ones [10]. The 22Σ− state has a well depth of 4204.57 cm−1, which possesses 11 vibrational levels, as listed in Table S1. Similar to the a2Δ, c2Σ−, d2Σ+, X4Σ−, 14Σ+, and 14Π states, the 22Σ− state also has a single reference character around the internuclear equilibrium separation. Because of the spin and symmetry limitations, this state spontaneously decays to only the

6–6 9–6 0.0721

2–5 4–3 5–5 7–2 8–6

0.0660 1.53 × 104 0.0435 1.90 × 104 0.0308 1.00 × 104 0.0510 3.59 × 104 0.0359 1.32 × 104

7–7 9–7 0.6933

2–6 4–4 6–2 7–3 9–1

0.0952 1.78 × 104 0.0749 2.69 × 104 0.0400 2.57 × 104 0.0606 3.55 × 104 0.0321 3.18 × 104

7–7 9–9 0.0374

2–7 4–5 6–3 7–6 9–2

0.0841 1.36 × 104 0.0685 2.01 × 104 0.0672 3.59 × 104 0.0373 1.26 × 104 0.0536 4.44 × 104

c2Σ− and e2Π states. For purposes of clarity, here we collect some of the relatively large Einstein coefficients of emissions along with the FC factors in Table 6 for the 22Σ−–c2Σ− transition only, and list those in Table S2 for the 22Σ−–e2Π transition. According to the results calculated in this study, for the 22Σ−–c2Σ− transition, there are 42 and 124 emissions whose Einstein coefficients are approximately 104 and 103, respectively. The spectra of this system cover the visible to the far–infrared range. The strong emissions are mainly near–infrared light. Only a few are visible light. For the 22Σ−– e2Π transition, the largest Einstein coefficients of emissions are approximately 101. In terms of these Einstein coefficients, the emissions of 22Σ−–e2Π transition are very weak and therefore, very difficult to detect via spectroscopy. The partial rotationless radiative lifetimes come from the two groups of transitions, 22Σ−–c2Σ− and 22Σ−–e2Π. As listed in Table 4, the partial radiative lifetimes for the vibrational levels of 22Σ− state are several or several–hundred microseconds long, meaning that the spontaneous emissions generated from the 22Σ− state easily occur. 3.2. Spectroscopic Properties of 27 Ω States Generated Form the 12 Λ-S States With the SOC effect taken into account, the ground state 2Pu of Si+ ion splits into two components, 2P1/2 and 2P3/2; the ground state 3Pg of C atom splits into three components, 3P0, 3P1, and 3P2. In terms of these results, the dissociation channel Si+(2Pu) + C(3Pg) splits into six dissociation limits, as listed in Table 7. The 12 Λ-S states split into the 27 Ω states with the SOC effect included. These Ω states are collected in Table 7 along with the energy difference of a certain dissociation asymptote relative to the lowest one. For purposes of comparison, the corresponding experimental energy differences are also tabulated in this table.

Table 7 Dissociation relationships of 27 Ω states generated from the first dissociation asymptote of the SiC+ cation. Relative energy/cm−1 Atomic state Si+(2P1/2) + C(3P0) Si+(2P1/2) + C(3P1) Si+(2P1/2) + C(3P2) Si+(2P3/2) + C(3P0) Si+(2P3/2) + C(3P1) Si+(2P3/2) + C(3P2) a

Ω state X4Σ−1/2 X4Σ−3/2, b2Π1/2, c2Σ−1/2 a2Δ5/2, e2Π3/2, b2Π3/2, d2Σ+1/2, e2Π1/2 a2Δ3/2, A4Δ1/2 14Π5/2, A4Δ3/2, 14Π3/2, 14Π1/2, 14Σ+1/2, 14Π-1/2 A4Δ7/2, 24Π5/2, A4Δ5/2, 14Σ+3/2, 24Σ−3/2, 24Π3/2, 22Σ−1/2, 24Σ−1/2, 24Π1/2, 24Π-1/2

Obtained by the icMRCI + Q/56 + CV + DK + SOC calculations.

This worka 0.00 12.29 38.34 249.37 274.80 289.24

Exp. [11] 0.00 16.42 43.41 287.24 303.66 330.65

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50000 -1

Of the 12 Λ-S state, the a2Δ, e2Π, and 14Π states are inverted with the SOC effect account for. The PECs of 27 Ω states are demonstrated in Figs. 5–8. As with Figs. 1 and 2, to clearly show the detailed information of each PEC, we depict them only over a small range of internuclear separations. The dissociation channel of each Ω state has been tabulated in Table 7. To avoid repetition, we do not show these asymptotes in Figs. 5–8. Using the PECs obtained by the icMRCI + Q/56 + CV + DK + SOC calculations, we evaluate Te, De, Re, and ωe of the 27 Ω states, with the help of MOLCAS program [21]. For purposes of clarity, the spectroscopic parameters of each Ω state are collected in Table S3 of the Supplementary Material. For reasons of discussion, the dominant Λ-S state compositions of each Ω state around their respective internuclear equilibrium positions are also listed in Table S3. The spectroscopic parameters of X4Σ−1/2 and X4Σ−3/2 states agree well with those of the X4Σ− state; and the SOC splitting energy of the X4Σ− state determined here is only 0.22 cm−1. As with the X4Σ− state, each of the 14Σ− and 24Σ− states also splits into two Ω states. The De, Te, Re, and ωe values of each Ω state agree favorably with those of the corresponding Λ-S states; and the SOC splitting energies of the 14Σ− and 24Σ− states are only 1.09 and 0.22 cm−1, respectively. For the X4Σ−, 14Σ−, and 24Σ− states, the vibrational levels of each Ω state are also very close to those of the corresponding Λ-S state. That is, the SOC effect on the spectroscopic parameters is insignificant for these states. The c2Σ−, d2Σ+, and 22Σ− states do not split with the SOC effect included. The largest deviations of the De, Te, Re, and ωe values of these three Ω states from their corresponding Λ-S states are 79.04, 3.73 cm−1, 0.00003 nm, and 0.33 cm−1, respectively, which are very small. Comparing the vibrational levels of these Ω states with those of the corresponding Λ-S states, we confirm that the SOC effect on the vibrational levels is insignificant. As with the b2Π state, the b2Π1/2 and b2Π3/2 states also have a double well. The splitting energies of the first and second wells are 75.06 and 32.48 cm−1, respectively. The De and Re values of the two Ω states are close to those of the b2Π state. The depths of the first and second wells are 266.14 and 87.18 cm−1 for the b2Π1/2 state and those are 220.27 and 79.25 cm−1 for the b2Π3/2 state, respectively. Similar to the b2Π state, no vibrational levels exist for the b2Π1/2 and b2Π3/2 states. In terms of these results, we confirm that the SOC effect on the spectroscopic parameters is very small. As noted above, the a2Δ and e2Π states are inverted with the SOC effect included. Each of the a2Δ and e2Π states splits into two Ω states. For the a2Δ3/2, a2Δ5/2, e2Π1/2, and e2Π3/2 states, the largest deviations of the De, Re, and ωe values of these Ω states from the a2Δ and e2Π states are

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Internuclear separation /nm Fig. 6. PECs of the 9 states with Ω = 3/2. 1–X4Σ−3/2; 2–b2Π3/2; 3–a2Δ3/2; 4–e2Π3/2; 5– A4Δ3/2; 6–14Σ+3/2; 7–14Π3/2; 8–24Σ−3/2; 9–24Π3/2.

136.31 cm−1, 0.0002 nm, and 2.91 cm−1; the SOC splitting energies of the a2Δ and e2Π states are 77.26 and 7.02 cm−1, respectively. For the 24Π state, the largest deviations of the De, Re, and ωe results are only 2.42 cm−1, 0.00013 nm, and 2.87 cm−1 between the 24Π state and the 24Π−1/2, 24Π1/2, 24Π3/2, and 24Π5/2 states. The SOC splitting energies are 21.07, 21.29, and 21.29 cm−1 between the 24Π−1/2 and 24Π1/2 states, the 24Π1/2 and 24Π3/2 states, as well as the 24Π3/2 and 24Π5/2 states. In conclusion, the SOC effect on the spectroscopic parameters is very tiny for the a2Δ, e2Π, and 24Π states. The same conclusion can also be obtained for the A4Δ state. Similar to the a2Δ and e2Π states, Te of the 14Π5/2 state is the lowest, whereas Te of the 14Π−1/2 state is the highest. Therefore, the 14Π state is also inverted when the SOC effect is taken into account. The first well is deep, but the second well is shallow. Accordingly, the SOC effect on the spectroscopic parameters should be small for the first well, but should be obvious for the second well. The reason is that the small SOC effect can bring about the great changes of PECs for the shallow well and therefore, the obvious influence on the spectroscopic parameters may occur. By careful comparison, the largest deviations of De, Re, and ωe of all the Ω states from those of the 14Π state are 127.44 cm−1, 0.00005 nm, and 2.79 cm−1 for the first well, whereas those are 78.24 cm−1, 0.00647 nm, and 10.34 cm−1 for the second well. In addition, the SOC splitting energies of neighboring states from the 14Π5/2 to the 14Π−1/2 state are 21.94, 21.73, and 21.51 cm−1 for the first

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Internuclear separation /nm Fig. 5. PECs of the 11 states with Ω = 1/2. 1–X4Σ−1/2; 2–b2Π1/2; 3–c2Σ−1/2; 4–d2Σ+1/2; 5– e2Π1/2; 6–A4Δ1/2; 7–14Σ+1/2; 8–22Σ−1/2; 9–14Π1/2; 10–24Σ−1/2; 11–24Π1/2.

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Internuclear separation /nm Fig. 7. PECs of the 4 states with Ω = 5/2. 1-a2Δ5/2; 2-A4Δ5/2; 3-14Π5/2; 4-24Π5/2.

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Acknowledgments

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This work is sponsored by the National Natural Science Foundation of China under Grant No. 11274097 and the Program for Science and Technology of Henan Province in China under Grant No. 142300410201.

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Appendix A. Supplementary Data

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Supplementary data to this article can be found online at https://doi. org/10.1016/j.saa.2018.06.041.

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Internuclear separation /nm Fig. 8. PECs of the 3 states with Ω = 7/2 and −1/2. 1-A4Δ7/2; 2-14Π−1/2; 3-24Π−1/2.

well, and those are 9.00, 9.89, and 10.75 cm−1 for the second well, respectively. To some extent, these splitting energies are small. 4. Conclusions The PECs of 12 Λ-S and 27 Ω states of SiC+ cation are calculated. Core-valence correlation and scalar relativistic corrections, Davidson correction, as well as extrapolation of the potential energies to the CBS limit are included. The TDMs between two Λ-S states are computed. The vibrational band origins, FC factors, and Einstein coefficients of emissions are evaluated. Summarizing these results, we have the following main conclusions. (1) The rotationless radiative lifetimes of vibrational levels are approximately 10−7 s for the 24Π and 24Σ− states, 10−5 to 10−6 s for the 22Σ− state and the first well of 14Π state, 10−5 s for the e2Π state, and very short for the second well of the 14Π state. Here, only the lifetimes of e2Π state are full rotationless radiative lifetimes. (2) From a spectroscopic standpoint, the a2Δ, b2Π, c2Σ−, d2Σ+, A4Δ, and 14Σ+ states are insignificant, since the spontaneous emissions generated from these states are forbidden. (3) The Einstein coefficients of many emissions are large between the 22Σ− and the c2Σ− state, the e2Π and the a2Δ state, the 24Σ− and the X4Σ− state, the 24Π and the X4Σ− state, the first well of the 14Π state and the X4Σ− state, as well as the second well of the 14Π state and the 14Σ+ state. In combination with the radiative lifetimes, we confirm that these emissions should be not difficult to detect. (4) The SOC effect on the spectroscopic parameters is insignificant for all the states except for the second well of the 14Π state.

[1] R.P. Diez, J.A. Alonso, Chem. Phys. 455 (2015) 41–47. [2] R. Mollaaghababa, C.A. Gottlieb, J.M. Vrtilek, P. Thaddeus, Astrophys. J. 352 (1990) L21–L23. [3] M. Bogey, C. Demuynck, J.L. Destombes, Astron. Astrophys. 232 (1990) L19–L21. [4] H. Suzuki, Prog. Theor. Phys. 62 (1979) 936–956. [5] J. Cernicharo, C.A. Gottlieb, M. Guélin, P. Thaddeus, J.M. Vrtilek, Astrophys. J. 341 (1989) L25–L28. [6] F.L. Sefyani, J. Schamps, Astrophys. J. 434 (1994) 816–823. [7] D.H. Shi, W. Xing, J.F. Sun, Z.L. Zhu, Eur. Phys. J. D 66 (2012) 262. [8] P.J. Bruna, C. Petrongolo, R.J. Buenker, S.D. Peyerimhoff, J. Chem. Phys. 74 (1981) 4611–4620. [9] A.I. Boldyrev, J. Simons, V.G. Zakrzewski, W. von Niessen, J. Phys. Chem. 98 (1994) 1427–1435. [10] A. Pramanik, S. Chakrabarti, K.K. Das, Chem. Phys. Lett. 450 (2008) 221–227. [11] https://physics.nist.gov/PhysRefData/ASD/Html/verhist.shtml. [12] G.H. Herzberg, Molecular Spectra and Molecular Structure: Spectra of Diatomic Molecules, Vol. 1, Van Nostrand Reinhold, New York, 1951. [13] S.R. Langhoff, E.R. Davidson, Int. J. Quantum Chem. 8 (1974) 61–72. [14] A. Richartz, R.J. Buenker, S.D. Peyerimhoff, Chem. Phys. 28 (1978) 305–312. [15] W. Xing, D.H. Shi, J.C. Zhang, J.F. Sun, Z.L. Zhu, J. Quant. Spectrosc. Radiat. Transf. 210 (2018) 62–73. [16] MOLPRO 2010.1 is a package of ab initio programs written by H. –J. Werner, P. J. Knowles, R. Lindh, F. R. Manby, M. Schütz, P. Celani, T. Korona, A. Mitrushenkov, G. Rauhut, T. B. Adler, R. D. Amos, A. Bernhardsson, A. Berning, D. L. Cooper, M. J. O. Deegan, A. J. Dobbyn, F. Eckert, E. Goll, C. Hampel, G. Hetzer, T. Hrenar, G. Knizia, C. Köppl, Y. Liu, A. W. Lloyd, R. A. Mata, A. J. May, S. J. McNicholas, W. Meyer, M. E. Mura, A. Nicklass, P. Palmieri, K. Pflüger, R. Pitzer, M. Reiher, U. Schumann, H. Stoll, A. J. Stone, R. Tarroni, T. Thorsteinsson, M. Wang, A. Wolf. [17] H. Liu, D.H. Shi, J.F. Sun, Z.L. Zhu, J. Quant. Spectrosc. Radiat. Transf. 121 (2013) 9–22. [18] H. Liu, D.H. Shi, J.F. Sun, Z.L. Zhu, Spectrochim, Spectrochim. Acta A Mol. Biomol. Spectrosc. 181 (2017) 226–238. [19] V.B. Oyeyemi, D.B. Krisiloff, J.A. Keith, F. Libisch, M. Pavone, E.A. Carter, J. Chem. Phys. 140 (2014), 044317. . [20] A. Berning, M. Schweizer, H.-J. Werner, P.J. Knowles, P. Palmieri, Mol. Phys. 98 (2000) 1823–1833. [21] G. Karlström, R. Lindh, P.–.Å. Malmqvist, B.O. Roos, U. Ryde, V. Veryazov, P.–.O. Widmark, M. Cossi, B. Schimmelpfennig, P. Neogrady, L. Seijo, Comput. Mater. Sci. 28 (2003) 222–239. [22] R.J. Le Roy, Level 7.5: A Computer Program for Solving the Radial Schrödinger Equation for Bound and Quasi–Bound Levels, University of Waterloo Chemical Physics Research Report No. CP–642R3, University of Waterloo, Waterloo, ON, Canada, 2001. [23] O. Roberto-Neto, F.R. Ornellas, Chem. Phys. Lett. 226 (1994) 463–468. [24] F.R. Ornellas, S. Iwata, J. Chem. Phys. 107 (1997) 6782–6794. [25] Y. Yin, D.H. Shi, J.F. Sun, Z.L. Zhu, Astrophys. J. Suppl. Ser. 235 (2018) 25.