Near-Infrared Transitions from the Singlet Excited States to the Ground Triplet State of the S 2 Molecule

: Intensity of transitions from the b 1 ∑ + g and a 1 ∆ g states to the ground state X 3 ∑ − g in the near IR emission spectrum of the S 2 molecule has been calculated by the multireference conﬁguration interaction method taking into account spin-orbit coupling (SOC). The intensity of the b 1 ∑ + g − X 3 ∑ − g, Ms = ± 1 transition is largely determined by the spin interaction with the electromag-netic wave, which comes from the zero-ﬁeld splitting of the ground X multiplet and the SOC-induced mixing between b and X 3 ∑ − g,0 states. The Einstein coefﬁcients for the experimentally detected 0 − 0, 0 − 1, 1 − 1 bands of the b 1 ∑ + g − X 3 ∑ − g, Ms = ± 1 emission system are calculated in good agreement with observations. The Einstein coefﬁcient of the a 1 ∆ g − X 3 ∑ − g, Ms = ± 1 magnetic dipole transition is very low, being equal to 0.0014 s − 1 . Nonetheless, the weakest of all experimentally observed bands (the 0 − 0 band of the a − X Ms= ± 1 transition) qualitatively corresponds to this calculation. Most importantly, we provide many other IR bands for magnetic dipole b 1 ∑ + g − X 3 ∑ − g, Ms = ± 1 and a 1 ∆ g − X 3 ∑ − g, Ms = ± 1 transitions, which could be experimentally observable in the S 2 transparency windows from a theoretical point of view. We hope that these results will contribute to the further experimental exploration of the magnetic infrared bands in the S 2 dimer.


Introduction
Diatomic sulfur as an important component of sulfur vapor has been widely studied by optical and microwave spectroscopy in laboratories at high temperatures near the sulfur melting point [1][2][3][4]. The dimer S 2 radical was also found in combustion processes, volcanic activity, and electric discharges through the sulfur vapor and sulfur-containing compounds [1][2][3]5]. At the same time, the S 2 molecule is one of the most important species in astrophysics. Compared with the research of O 2 , the importance of the experimental spectral observation of S 2 cannot be ignored. For example, S 2 was monitored regularly, and it could track the star's apparent path in the sky through the images. It possesses a short lifetime due to ultraviolet (UV) photo-dissociation in space. Thus, the diatomic sulfur (S 2 ) radical has been detected in comet spectra [6,7]. Particularly, its emission is seen in the impact site of the comet Shoemaker-Levy 9 on Jupiter. For the same reason, the S 2 radical is an ideal monitor for behavior of a cometary nucleus. Moreover, the S gas was also found in the absorption spectrum of the lo's Pele plume against Jupiter by the HubSpace Telescope in 1999 [8].
The quantum mechanical calculations of the spin-allowed electronic transitions in S 2 molecular spectra [9] and photo-dissociation dynamics [10][11][12][13] have been quite successful. In the well-known analogy of the dioxygen Schumann-Rung bands [14], the B 3 ∑ − u − X 3 ∑ − g system of S 2 was studied by many authors [9][10][11][12][13]. Wieland et al. [15] have observed strong absorption bands of the S 2 molecule in the far UV region 160-187 nm, which were later reinvestigated [3] and classified as C 3 II u − X 3 ∑ − g and D 3 II u − X 3 ∑ − g transitions. In 1962, Tanaka [2] detected four systems of emission bands, all of which are of the Σ-Σ type in the region 1760-2130 Å. They differ from the C 3 II u − X 3 ∑ − g and D 3 II u − X 3 ∑ − g strong bands [3] in the absorption of the S 2 molecule. It is known that the high vibrational energy levels of the B 3 ∑ − u state are strongly predissociated by the crossing with the B 3 II u state [4,9]. The presence of the B 3 ∑ − u ← X 3 ∑ − g Schumann−Rung analogous bands system [10][11][12][13] and other optically allowed B 3 II u − X 3 ∑ − g transitions [11] in the UV and visible regions of S2 absorption spectra was investigated and interpreted [9]. In addition, there is one transient absorption in the far ultraviolet region that was assigned as the f 1 ∆ u −a 1 ∆ g transition from a metastable state a 1 ∆ g [13,16].
Four progressions of S 2 in the visible and near-infrared regions were observed using the laser-induced emission technique [17]; they were assigned as the c 1 ∑ − u −X 3 ∑ − g and c 1 ∑ − u − a 1 ∆ g transitions. According to the above, we know that many transition bands of S 2 have been observed; however, they are basically the transitions allowed by the electric dipole moment, and some weak transitions have not been mentioned. On the other hand, we find that the study of the transitions of S 2 molecules is still insufficient.
Both O 2 and S 2 are important components of astrophysics [6,7]. The spectral assignments that are based mainly on the vibrational spacings of the lower states and the electronic energy separations sometimes become difficult. Therefore, it is necessary to calculate some low-energy transitions. Thus, it is important to know as much as possible about their magnetic and electro-quadrupole properties. The intensity of the transitions was predicted by quantum mechanics, which would be complicated from a theoretical point of view since such evaluation of the method needs to take into account spin-orbit coupling, which leads to the admixture of wave functions for the different multiplet states.
The group VI A diatomic species O 2 and S 2 have the ground X 3 ∑ − g state, the nearinfrared a 1 ∆ g , and the red b 1 ∑ + g counterpart of the ( . . . σ g 2 π u 2 ) configuration [18][19][20][21]. Transitions between these states are strictly forbidden by electric dipole selection rules. However, they acquire weak magnetic dipole and electro-quadrupole transition moments due to spin-orbit coupling induced mixing of various electronic states [18,21]. Thus, the transitions between these states are easily obtained in the absorption and emission spectra of O 2 and S 2 molecules [17,[22][23][24][25]. The b 1 ∑ + g −X 3 ∑ − g system is the analog of the atmospheric O 2 bands; it has been observed in emission in the near infrared using a discharge flow system in which metastable O 2 ( 1 ∆ g ) molecules sensitized the emission of the S 2 molecules [22]. The study of transitions is helpful to compare the electronic structures of the two molecules.
This procedure involves the spin-orbit matrix elements between b 1 ∑ + g singlet and X 3 ∑ − g triplet states of the S 2 molecule, which were obtained in a series of AO basis sets [20,21]. Transitions require accurate energy levels and transition moments to better reduce errors. The b 1 ∑ + g state of S 2 could be efficiently quenched by collisions; it still produces the emission spectra of magnetic dipole b 1 ∑ + g − X 3 ∑ − g,Ms=±1 (0−0,0−1,1−1) transitions of S 2 which were measured by the Fourier-transform spectrometer at high resolution [22]. In addition, dating back to 1979, the weak emission spectra for the (v = 0, v = 0) band of the magnetic dipole a 1 ∆ g − X 3 ∑ − g,Ms=±1 transition was detected in the near-infrared region [23,24]. The intensity of the a 1 ∆ g − X 3 ∑ − g,Ms=±1 band for a chemically unstable S 2 molecule is very low; its short radiative lifetime was calculated to be about 350 s (the Einstein coefficient A = 0.00286 s −1 ) [21]. At present, few studies have given a complete and accurate calculation of the electronic-state energy level and transition moment of the S 2 molecule. The current work makes up for the lack of predecessors and provides theoretical support for future generations.
The primary purpose of the present work is to study the intensity of parity-forbidden singlet-triplet (S-T) transitions between X 3 ∑ − g and a 1 ∆ g , b 1 ∑ + g states in the S 2 dimer using the MRCI+Q method, which could explain how to overcome the spin-forbidden transition of the electric-dipole approximation taking into account spin-orbit coupling. Previous studies of the S 2 dimer have mainly focused on the UV and visible regions. In addition, the near-infrared b 1 ∑ + g − X 3 ∑ − g,Ms=±1 (0−0, 0−1, 1−1) and a 1 ∆ g − X 3 ∑ − g,Ms=±1 (0−0) bands [22,25] have also been observed experimentally. The current work calculates the magnetic dipole moment of the b 1 ∑ + g − X 3 ∑ − g,Ms=±1 and a 1 ∆ g − X 3 ∑ − g,Ms=±1 transitions of S 2 for other vibronic bands in the microwave-infrared regions, which could be observed from a theoretical point of view. The transition properties of the b 1 ∑ + g and a 1 ∆ g states with respect to the ground state X 3 ∑ − g have been evaluated by SOC perturbations from the 3 Π g and 1 Π g states of the S 2 systems. We also note that the magnetic dipole transition moment is decided by the spin part due to the zero-field splitting of the X 3 ∑ − g state [18][19][20][21]. The calculated magnetic dipole a 1 ∆ g − X 3 ∑ − g,Ms=1 transition only contains the orbital angular-momentum contribution [18]. In addition, the error source for the Einstein coefficient is discussed. It is most important that we successfully reproduce the experimentally observed magnetic dipole vibronic bands in our efforts to obtain full predictions. The weak intensities of electroquadrupole a 1 ∆ g − b 1 ∑ + g and a 1 ∆ g − X 3 ∑ − g,Ms=0 bands are also investigated since they should exhibit a particular rovibronic structure. However, the results show that there is little hope that these quadrupole bands can be measured experimentally. Nevertheless, the present results may help further search for the magneto-infrared band in the S 2 systems.

Method of Calculations
The electronic structure calculation of S 2 has been carried out with the Molpro 2012 suite of the quantum chemical package [26]. The "aug−cc−pwCVTZ" basis set with diffuse s-type (0.018684493) and p-type (0.011220492) Gaussian functions is used for the S atom. There are two electrons (excluding the 1s 2 electron of S) participating in the MRCI calculation of S 2 , and the corresponding active space is determined by eight molecular orbitals (MOs), including two σ g , one π u , one π u , two σ u , one π g , and one π g symmetry (3s 2 3p 4 ). The SOC is considered with the aid of the state-interacting approach utilizing the Breit-Pauli operator. After taking the eigenfunction of the unperturbedĤ el Hamiltonian as the basis of diagonalization, the diagonalizedĤ el +Ĥ so matrix can obtain the eigenfunction and eigenvalue of the electronic state (Ω presentation).
The magnetic dipole operator M consists of two terms, which are spin (S) and orbital angular momentum (L); it is determined by where the electronic g-factor g e is equal to 2.0023. Then, after taking the SOC into account, the singlet and the triplet states are mixed, and the perturbed wave function Φ of the electronic state is expressed as Φ a = a 1 ∆ g > + ∑ n C a,n n 3 ∑ g,Ms=±1 > where Physchem 2023, 3

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The mixing coefficient of the infrared band (b 1 ∑ + g − X 3 ∑ − g,Ms=±1 transition) is calculated as C b,X = 0.05504i, which is derived from the SOC matrix element (426.53 cm −1 ) and energy difference ∆E b−X = 7749 cm −1 ; the former is determined by the multiplet splitting for the S( 3 P) state.
The spin contribution of magnetic operator Equation (1), taking into account Equation (4), becomes Besides the spin contribution, the contribution of the orbital angular momentum L to the magnetic dipole transition moment is also essential. Taking the orbital section of the b 1 ∑ + g −X 3 ∑ − g,Ms=1 transition as an example, we obtain The role of the intermediate 3 Π g and 1 Π g states is to establish a connection between the singlet (a 1 ∆ g and b 1 ∑ + g ) state and the triplet ground (X 3 ∑ − g ) state, allowing the S 2 molecule to arise from these magnetic transitions in the near-IR region. In addition, the intensity of the quadrupole transition moment is determined by Here, Q is the quadrupole part of the multipole expansion of radiation. For different transitions, the formulas of the Einstein coefficient A have different expressions, Equations (10) and (11), corresponding to the magnetic dipole moment and the electroquadrupole moment, respectively.
The unit of energy E is cm −1 ; the magnetic dipole moment M, electric quadrupole moment Q, and Einstein coefficient A are in units of the Bohr magneton, ea 0 2 , and s −1 , respectively.
It can be seen that the calculated Re values of the X  ground state and low-lying a 1 Δg excited state are in relatively good agreement with the corresponding previous experimental [37] and theoretical [9,38] data. Compared with the experimental ωe values for the X  , a 1 Δg, and  PECs, our results are a bit lower, and the differences are 15.99 cm −1 (2.25%), 19.66 cm −1 (2.88%), and 18.84 cm −1 (2.89%), respectively [37]. In addition, the complete basis set (CBS) limit for vertical excitation energy was calculated with the following extrapolation schemes, which use two-or three-point extrapolation formulas. Three-point extrapolation generally underestimates the limit,

3,4,5
Two-point extrapolation typically overestimates it,   It can be seen that the calculated R e values of the X 3 ∑ − g ground state and lowlying a 1 ∆ g excited state are in relatively good agreement with the corresponding previous experimental [37] and theoretical [9,38] data. Compared with the experimental ω e values for the X 3 ∑ − g , a 1 ∆ g , and b 1 ∑ + g PECs, our results are a bit lower, and the differences are 15.99 cm −1 (2.25%), 19.66 cm −1 (2.88%), and 18.84 cm −1 (2.89%), respectively [37].
In addition, the complete basis set (CBS) limit for vertical excitation energy was calculated with the following extrapolation schemes, which use two-or three-point extrapolation formulas. Three-point extrapolation generally underestimates the limit, Two-point extrapolation typically overestimates it, The reliability of vertical excitation energy in quantum chemistry is evidently dependent on the basis set. The optimized results are obtained from complete basis set calculations. We expect them to be more accurate than other results [21,38]. In Table S1, the influence of the AO basis is negligible for the vertical excitation energy, as can be seen from the results obtained by employing two-or three-point extrapolation formulas in different basises. The maximum difference of the vertical excitation energy from X 3 ∑ − g state to the a 1 ∆ g , b 1 Σ + g ,1 3 Π g , 1 1 Π g states is only a few percent (1.10%, 2.96%, 0.51%, 0.16%). The relative positions of the a and b levels are of crucial importance in our study. Experimental energy data for the gas phase T 0 and T e values are 4394.2 and 4745.4 cm −1 for the a 1 ∆ g state [23]. The T 0 value is close to other experimental data for Ne matrix (4395 cm −1 ), where the light Nobel gas environment provides negligible perturbaqtion for S 2 molecule [5]. Our T 0 value is a bit lower (4118 cm −1 ) but the ratio of the T 0 energies of the a 1 ∆ g and b 1 ∑ + g states (0.562) is quite close to experimental data (0.551) [23] being also in accordance with the corresponding ratios 0.60, 0.56, and 0.54 observed for O2, SO, and Se2, respectively [39].

Magnetic Dipole Mechanisms for the X
Transitions in the S 2 Spectrum Transitions between the low-lying "gerade" states are forbidden by dipole selection rules (∆Λ = 0, ±1, ∆S = 0, g−u), but SOC-induced mixing between electronic multiplets provides additional source of weak magnetic dipole transitions (which satisfy the selection ∆Λ = 0, ∓1, ∆S = 0, g−g, u−u) in the near-infrared region of the S 2 spectrum.
The calculated {1}-{6} transitions between a 1 ∆ g , b 1 ∑ + g , and X 3 ∑ − g states are collected in Figure 2; these transitions can be detected as weak magnetic and quadrupole bands. What we want to stress is that the weak intensity of the vibrational near-IR band in the S 2 spectra can be measured at the liquid-nitrogen-cooled germanium detector facilities [22]. Among them, the {3} transition can be expressed as the difference in the quadrupole moments between a and X states multiplied by the corresponding C a,x coefficient. However, the actual calculation leads to neglected electro-quadrupole transition {3} intensity. The magnetic S-T transition moment number {2} "borrows" intensity from transition number ; this is a typical EPR {4} transition moment equal to two Bohr magnetons (2 µ B ) in zero magnetic fields, which can be considered as the contribution of the spin magnetic dipole from nature. In addition, the orbital angular-momentum operator also contributes to the S-T transition {2} intensity. A similar orbital analysis is also applied to the transition number {1} intensity, which is dominated by the orbital magnetic interaction. According to perturbation theory, the SOC mixing of the electronic states is responsible for "intensity borrowing" of transition numbers {1} and {2}. Transition {5} is dominated by the contribution of the orbital magnetic dipole, and the intensity of magnetic dipole moment transition {5} is very weak, almost negligible.
In Table S2, three different sets, basis A (aug−cc−pwCVTZ), basis B (aug-cc-pwCVQZ), and basis C (aug-cc-pwCV5Z), have been employed in order to obtain more accurate spinorbit matrix elements, which are responsible for the discussed intensity borrowing schemes. The difference between basis X (X = A, B, C) and basis X 1 (X 1 = A 1 , B 1 , C 1 ) is that X 1 correlates the 2s 2 2p 6 3s 2 3p 4 electrons of the S atom to account for the core-valence (CV) effect. Its result shows that the impact of the CV effect cannot be ignored in the calculation process. Basis A 2 contains additional diffuse s-type (0.018684493) and p-type (0.011220492) Gaussian functions relative to basis A 1 in order to better understand the influence of the CV effect and diffusion function on spin-orbit coupling matrix elements, which indicates that the effect of the diffusion function is tiny or even negligible. In Table S2, three different sets, basis A (aug-cc-pwCVTZ), basis B (aug-cc-pwCVQZ), and basis C (aug-cc-pwCV5Z), have been employed in order to obtain more accurate spin-orbit matrix elements, which are responsible for the discussed intensity borrowing schemes. The difference between basis X (X = A, B, C) and basis X1 (X1 = A1, B1, C1) is that X1 correlates the 2s 2 2p 6 3s 2 3p 4 electrons of the S atom to account for the core-valence (CV) effect. Its result shows that the impact of the CV effect cannot be ignored in the calculation process. Basis A2 contains additional diffuse s-type (0.018684493) and p-type (0.011220492) Gaussian functions relative to basis A1 in order to better understand the influence of the CV effect and diffusion function on spin-orbit coupling matrix elements, which indicates that the effect of the diffusion function is tiny or even negligible.
In Table 2, it is obvious that the best CBS limit produces spin-orbit matrix elements with the X  and  , 1 Π , 1 Π states which are slightly higher than the previous theoretically calculated values [20,40]. However, our values are within a reasonable error range. The best CBS limit results for spin-orbit matrix elements between the 1 Π and Δ ,  states are predicted to be close to the result of basis G; the differences are 5.90% and 6.39%, respectively.  [20] 423.0 202.7 − 204.0 basis E [20] 410.3 200.9 − 200.8 basis F [20] 416.6 203.9 − 204.9 basis G (core2) [21] − 200.  In Table 2, it is obvious that the best CBS limit produces spin-orbit matrix elements with the X 3 ∑ − g and b 1 ∑ + g , 1 1 Π g , 1 3 Π g states which are slightly higher than the previous theoretically calculated values [20,40]. However, our values are within a reasonable error range. The best CBS limit results for spin-orbit matrix elements between the 1 3 Π g and a 1 ∆ g , b 1 ∑ + g states are predicted to be close to the result of basis G; the differences are 5.90% and 6.39%, respectively.  [20] 423.0 202.7 −204.0 basis E [20] 410.3 200.9 −200.8 basis F [20] 416.6 203.9 −204.9 basis G (core2) [21] −200. The calculated spin-orbit matrix elements between the Ω-Ω states of the S 2 system (Figure 3) show very interesting R dependence behavior. For example, the SOC integral <b 1 ∑ + g L y S y 1 3 Π x g,Ms=1 > is a rising function with R while the important matrix element <b 1 ∑ + g |L z S z |X 3 ∑ − g,Ms=0 > declines with R > R 0 . The SOC integrals <b 1 ∑ + g L y S y 1 3 Π x g,Ms=1 > and <b 1 The spin-orbit interaction operator subscript (L y S y ) and state superscript (1 3 Π x g,Ms=1 ) are defined as y components of the SOC operator and x components of the degenerate irreducible representation Π g , respectively. At the same time, it is clearly visible that the value of the SOC integral <b 1 ∑ + g |L z S z |X 3 ∑ − g,Ms=0 > is the largest, which is one of the reasons why the magnetic dipole moment transition intensity is much stronger than other transitions. , and state superscript (1 Π , ) are defined as y components of the SOC operator and x components of the degenerate irreducible representation Π , respectively. At the same time, it is clearly visible that the value of the SOC integral <  | S |X  , > is the largest, which is one of the reasons why the magnetic dipole moment  X  , transition intensity is much stronger than other transitions. The calculated orbital angular-momentum parts are necessary for the magnetic dipole transition intensity estimations, which are collected in Table 3, while the corresponding theoretically calculated values of the reference are also listed. They show that the difference in orbital transitions between states range from 0.26% to 3.72% at the equilibrium internuclear distance (Rs-s = 1.899 Å). Among them, the Δ 1 Π transition has the largest difference between our results and the calculated values of Ref. [21], which is about 3.72%. Table 3. Calculated values of the L matrix elements (in atomic unit (a.u.)) for the quantities that are necessary for calculation of the magnetic transition probabilities in S2.  The calculated orbital angular-momentum parts are necessary for the magnetic dipole transition intensity estimations, which are collected in Table 3, while the corresponding theoretically calculated values of the reference are also listed. They show that the difference in orbital transitions between states range from 0.26% to 3.72% at the equilibrium internuclear distance (R s−s = 1.899 Å). Among them, the a 1 ∆ g L x 1 1 Π y g transition has the largest difference between our results and the calculated values of Ref. [21], which is about 3.72%. Table 3. Calculated values of the L matrix elements (in atomic unit (a.u.)) for the quantities that are necessary for calculation of the magnetic transition probabilities in S 2 .

R s-s (Å)
<b 1   In Table 4, the spin and orbit contributions to the total magnetic dipole transition moments between the X 3 ∑ − g and a 1 ∆ g , b 1 ∑ + g states in the S 2 spectrum are given; see also Figure 4 for details. The spin operator contribution to the magnetic b 1 ∑ + g − X 3 ∑ − g,Ms=±1 transition comes from the microwave band (X 3 ∑ − g,Ms=0 −X 3 ∑ − g,Ms=1 ) and the C b,X coefficient. The intensity of the magnetic transition between spin sublevels (Ms = ±1−Ms = 0) of the ground triplet state X 3 ∑ − g is equal to 2µ B . The orbital part of the magnetic b 1 ∑ + g − X 3 ∑ − g,Ms=±1 transition remains of the order of magnitude less than the spincurrent part for R distances from 1.5 Å to 2.2 Å; in fact, it accounts for about one-seventh of the total b 1 ∑ + g − X 3 ∑ − g,Ms=±1 transition moment value. Note that the spin contribution in Table 4 must be multiplied by the g-factor value of the free electron g e = 2.0023. In addition, the magnetic dipole a 1 ∆ g −X 3 ∑ − g,Ms=±1 transition is mainly determined by the contribution of the orbital angular momentum L; it is equal to 0.026369 µ B at R e distance. Table 4. Spin (S), orbit (L), and the total magnetic dipole transition moments as functions of the internuclear distance.

Electro-Quadrupole Mechanisms for Transitions between and
 ,  States in the S2 Spectrum According to the selection rules for the electro-quadrupole transition probability (ΔΛ = 0, ±1, ±2, S=0, g-g, u-u), the transition numbers {3} and {6} are not as effective as magnetic dipole transitions, being extremely weak in the S2 molecule.
In Figure 5, the quadrupole transition moment of the infrared band b  → a  (analogous with the Noxon band of O2) is calculated as 3.8873 ea0 2 at equilibrium internuclear distance Re = 1.889 Å. Moreover, after considering the SOC effect, the wave functions of the singlet b  state and triplet X  state are mixed, resulting in the  X  , transition quadrupole activity. The corresponding electro-quadrupole 3.3. Electro-Quadrupole Mechanisms for Transitions between a 1 ∆ g and X 3 ∑ − g , b 1 ∑ + g States in the S 2 Spectrum According to the selection rules for the electro-quadrupole transition probability (∆Λ = 0, ±1, ±2, ∆S=0, g−g, u−u), the transition numbers {3} and {6} are not as effective as magnetic dipole transitions, being extremely weak in the S 2 molecule.
In Figure 5, the quadrupole transition moment of the infrared band b 1 ∑ + g → a 1 ∆ g (analogous with the Noxon band of O 2 ) is calculated as 3.8873 ea 0 2 at equilibrium in-ternuclear distance R e = 1.889 Å. Moreover, after considering the SOC effect, the wave functions of the singlet b 1 ∑ + g state and triplet X 3 ∑ − g state are mixed, resulting in the a 1 ∆ g − X 3 ∑ − g,Ms=0 transition quadrupole activity. The corresponding electro-quadrupole transition moment to the Ω = 0 component of the ground state X 3 ∑ − g from the a 1 ∆ g state is equal to Q a−X,0 = −C * b,X Q b−a = 0.2140 ea 0 2 at the equilibrium distance R e = 1.889 Å. Comparison with the oxygen near IR analogous b-a and a-X,0 bands indicates that the Noxon band in O2 has comparable intensity with the corresponding S2 quadrupole transition; their rate constants are 0.0014 and 0.0012 s −1 , respectively [41,42]. However, the a-X,0 quadrupole transition which depends on SOC, Equation (9), is much more intense in sulfur dimer than in oxygen. Its rate constant in S2 is equal to 10 −5 s −1 (Table 5), being impossible to measure in discharge through sulfur vapor. However, an even weaker electric quadrupole contribution to the a 1 Δg-X 3 Σg transition rate constant in O2 was identified through its rotational analysis and it was measured this way in order to be equal to (1.02 ± 0.10) × 10 −6 s −1 [43]. Such a weak oxygen band was detected in the solar absorption spectra because of a long light absorption path through the whole atmosphere [42][43][44]. The rotational ladder sublevels of the a 1 Δg and X 3 Σg states can be treated in terms of the Hund's (b) case coupling [42][43][44]. The spin sublevels of the ground state in rotating molecules are mixed and more accurate and can be treated in terms of the intermediate Hund's coupling case [42]. The rotational "e parity" sublevels of the Ω = 0 and Ω = 1 ground X 3 Σg triplet states are mixed as the spin-rotational terms F1(J) and F3(J); at the same time, the "f parity" rotational sublevel Ω = 1 corresponds to the pure F2(J) rotational term. The rotating molecule wave functions in Equations (2) and (3) depend on rotational quantum number J because of Coriolis coupling BeL terms [42]. Gordon et al. [44] made a comparison with the low line intensity for the observed P, Q, R branches of the quadrupole a-X transitions of O2 corresponding to selection rule ΔJ = ±2, ±1, 0, taking into account eigenfunctions of rotational spin sublevels. Good agreement is obtained assuming that the a-X transitions occur only from the SOC-induced mixing between the b and X0 states, as shown in Equation (2), and through intensity borrowing from the Noxon-type band in the oxygen spectrum [42][43][44][45]. Thus, similar predictions could be relevant for the quadrupole lines in sulfur dimer a-X absorption (or emission) since their intensity is predicted to be 10 times more efficient with our calculation. Table 5. Energy (∆E) of the (0,0) vibrational band, its Einstein coefficient (A), wavelengths (λ), the Franck-Condon factor, and radiative lifetime (τ) of the (0,0) vibrational transition.

Ω-Ω Transition
Comparison with the oxygen near IR analogous b−a and a−X,0 bands indicates that the Noxon band in O 2 has comparable intensity with the corresponding S 2 quadrupole transition; their rate constants are 0.0014 and 0.0012 s −1 , respectively [41,42]. However, the a−X,0 quadrupole transition which depends on SOC, Equation (9), is much more intense in sulfur dimer than in oxygen. Its rate constant in S 2 is equal to 10 −5 s −1 (Table 5), being impossible to measure in discharge through sulfur vapor. However, an even weaker electric quadrupole contribution to the a 1 ∆ g − X 3 Σ g transition rate constant in O 2 was identified through its rotational analysis and it was measured this way in order to be equal to (1.02 ± 0.10) × 10 −6 s −1 [43]. Such a weak oxygen band was detected in the solar absorption spectra because of a long light absorption path through the whole atmosphere [42][43][44]. The rotational ladder sublevels of the a 1 ∆ g and X 3 Σ g states can be treated in terms of the Hund's (b) case coupling [42][43][44]. The spin sublevels of the ground state in rotating molecules are mixed and more accurate and can be treated in terms of the intermediate Hund's coupling case [42]. The rotational "e parity" sublevels of the Ω = 0 and Ω = 1 ground X 3 Σ g triplet states are mixed as the spin-rotational terms F1(J) and F3(J); at the same time, the "f parity" rotational sublevel Ω = 1 corresponds to the pure F2(J) rotational term. The rotating molecule wave functions in Equations (2) and (3) depend on rotational quantum number J because of Coriolis coupling B e L terms [42]. Gordon et al. [44] made a comparison with the low line intensity for the observed P, Q, R branches of the quadrupole a−X transitions of O 2 corresponding to selection rule ∆J = ±2, ±1, 0, taking into account eigenfunctions of rotational spin sublevels. Good agreement is obtained assuming that the a−X transitions occur only from the SOC-induced mixing between the b and X0 states, as shown in Equation (2), and through intensity borrowing from the Noxon-type band in the oxygen spectrum [42][43][44][45]. Thus, similar predictions could be relevant for the quadrupole lines in sulfur dimer a−X absorption (or emission) since their intensity is predicted to be 10 times more efficient with our calculation. Table 5. Energy (∆E) of the (0,0) vibrational band, its Einstein coefficient (A), wavelengths (λ), the Franck-Condon factor, and radiative lifetime (τ) of the (0,0) vibrational transition. The emission probabilities for b−X Ms=±1 , a−X Ms=±1 , b−a, a−X Ms=0 bands of the S 2 spectrum are calculated in Table 5. The Einstein coefficient of the (0, 0) vibrational band for the b−X Ms=±1 transition is evaluated as 0.403824 s −1 (τ = 2.4763 s), which is obtained by employing the calculated energy of the (0, 0) vibrational band (∆E = 7641 cm −1 ). The experimental value ∆E 0,0 = 7981 cm −1 [39] and theoretical value ∆E vertical = 8970 cm −1 [21] were employed to obtain A 0,0 by R. Klotz [21], who deduced that A 0,0 is equal to 0.3 s −1 and 0.423 s −1 , respectively. Our transition probabilities are closer to the experimental results [21]. In addition, the errors of all these parameters are determined and listed in Table S3. The maximum errors of ∆E b−X and C b,X are 1.86 cm −1 and 0.000037i, respectively. By using the error propagation formula, the error of the Einstein coefficient (∆A) is calculated to be = 0.0003218 s −1 ; thus, the typical relative error of A is about 0.08%.

Ω-Ω Transition
Moreover, the radiative lifetime of the a 1 ∆ g − X 3 Σ − g,Ms=±1 transition is 721.3 s, which is significantly larger than the previous theoretical value (350 s) [21]. The main reason is that the energy difference between our result and the theoretical value [21] is about 1354 cm −1 . All the above conclusions imply the fact that the emission probability is sensitive to energy differences; thus, it is important to use precise energy for the spectroscopic predictions.
The probabilities of the electro-quadrupole transition between a 1 ∆ g and b 1 ∑ + g are determined as A 0,0 = 0.001228 s −1 and τ 0,0 = 814 s. The previous radiative lifetime has been calculated by Klotz as 960 s with the MRD CI method or 745 s with the full CI method [21]; and the τ 0,0 has been estimated to be 780 s with the experimental energy (∆E = 3586 cm −1 ) [5,39] in the calculation [21]. In spite of the fact that the Einstein coefficient of the (0, 0) vibrational band of the electric-quadrupole a 1 ∆ g − X 3 Σ − g,Ms=0 transition is very small (A = 0.000010 s −l ), the comparison with the corresponding oxygen analog [42] allows us to predict all quantitative details of its rovibronic structure. It is known that the quenching of the b 1 ∑ + g state in group VI diatomic molecules by foreign gases proceeds mostly by the near-resonant e-to-v energy transfer also involving the spin-allowed quadrupole b → a transition [23,43]. The late mechanism is very important because of intense electric dipole-quadrupole intermolecular interactions [43]. Moreover, the a 1 ∆ g state quenching by intermolecular electronic-to-vibrational (e-to-v) energy transfer depends on the SOC-induced quadrupole a → X,0 transition [43,46,47]; thus, the exact knowledge of quadrupole moments is important.
The 0−0, 0−1, and 1−1 bands of the b−X Ms=±1 emission system were measured by Fink [22,41], who monitored the 0-0 band at 1.25 µm through a Bomem interferometer in the near-infrared spectral region. the molecular constants for the 0−0, 0−1, and 1−1 bands of the b−X Ms=±1 transition are obtained from by Fink [22]. The intensity of the magnetic transition for the 0−0 band of the a−X Ms=±1 system is very low, and the Einstein coefficient is equal to about 0.001386 s −1 ; in fact, it is hard to measure. However, Setzer [23] observed the weak band (0−0 band of the a−X Ms=±1 transition) of this chemically unstable molecule S 2 near 4400 cm −1 in the early low-resolution emission study. In Figure 6, we give the transition intensity represented by the experimentally observed vibrational b−X Ms=±1 and a−X Ms=±1 bands. Each vertical line represents a vibration energy level band; in order to see each vibrational level more clearly, we use different colors to represent the corresponding bands of different vibrational levels; the ordinate and abscissa represent its transition intensity and band energy position, respectively. Physchem 2023, 3, FOR PEER REVIEW 13 the weak band (0-0 band of the a-XMs=±1 transition) of this chemically unstable molecule S2 near 4400 cm −1 in the early low-resolution emission study. In Figure 6, we give the transition intensity represented by the experimentally observed vibrational b-XMs=±1 and a-XMs=±1 bands. Each vertical line represents a vibration energy level band; in order to see each vibrational level more clearly, we use different colors to represent the corresponding bands of different vibrational levels; the ordinate and abscissa represent its transition intensity and band energy position, respectively. In order to provide more observable bands for the experiment, we take the intensity of b-XMs=±1 (0, 0) and a-XMs=±1 (0, 0) bands as the zero point of the abscissa, respectively. This will prove that the above bands can be detected by modern optical facilities and spectral resolution presented in Figure 6; the relevant experimentally observed vibrational energy band information-including the vibrational levels, Einstein coefficients, and oscillator strengths-are listed in Tables S4 and S5 of the Supplementary Material. Comparison with previous theoretical studies [20,21] indicates that the increase of the basis set and active space of the MRCI method is important for a detailed understanding of spectral peculiarities in the forbidden bands of thе sulfur dimer.
The studied forbidden bands in the S2 spectrum finalize a series of magnetic and quadrupole transition calculations in diatomic molecules [42,47,48]. Despite relatively low intensity and experimental difficulties in their detection, the theoretical studies of the magnetic and quadrupole transitions are important for astrophysical studies and for fundamental knowledge of molecular structure in physical chemistry [48].

Conclusions
Transitions from the  and a 1 Δg states to the ground state X  are forbidden in the framework of electric dipole selection rules for the S2 molecule. The present paper has estimated the intensity of the parity-forbidden singlet-triplet (S-T) transitions of the S2 molecule for the inter-combinations between groups of states a 1 Δg,  and X  in the near-IR region by employing high-level ab initio MRCI calculations with the SOC and CV correlation. The radiative lifetime of the  -X Σ , transition is largely determined by spin angular-momentum term contributions to the magnetic moment, due to the zero-field splitting of the X  state for S2. The intensity of the Δ -X Σ , transition is dominated by the orbital part in the magnetic dipole transition, and it arises In order to provide more observable bands for the experiment, we take the intensity of b−X Ms=±1 (0, 0) and a−X Ms=±1 (0, 0) bands as the zero point of the abscissa, respectively. This will prove that the above bands can be detected by modern optical facilities and spectral resolution presented in Figure 6; the relevant experimentally observed vibrational energy band information-including the vibrational levels, Einstein coefficients, and oscillator strengths-are listed in Tables S4 and S5 of the Supplementary Material. Comparison with previous theoretical studies [20,21] indicates that the increase of the basis set and active space of the MRCI method is important for a detailed understanding of spectral peculiarities in the forbidden bands of the sulfur dimer.
The studied forbidden bands in the S 2 spectrum finalize a series of magnetic and quadrupole transition calculations in diatomic molecules [42,47,48]. Despite relatively low intensity and experimental difficulties in their detection, the theoretical studies of the magnetic and quadrupole transitions are important for astrophysical studies and for fundamental knowledge of molecular structure in physical chemistry [48].

Conclusions
Transitions from the b 1 ∑ + g and a 1 ∆ g states to the ground state X 3 ∑ − g are forbidden in the framework of electric dipole selection rules for the S 2 molecule. The present paper has estimated the intensity of the parity-forbidden singlet-triplet (S−T) transitions of the S 2 molecule for the inter-combinations between groups of states a 1 ∆ g , b 1 ∑ + g and X 3 ∑ − g in the near-IR region by employing high-level ab initio MRCI calculations with the SOC and CV correlation. The radiative lifetime of the b 1 ∑ + g −X 3 Σ − g,Ms=±1 transition is largely determined by spin angular-momentum term contributions to the magnetic moment, due to the zero-field splitting of the X 3 ∑ − g state for S 2 . The intensity of the a 1 ∆ g − X 3 Σ − g,Ms=±1 transition is dominated by the orbital part in the magnetic dipole transition, and it arises from the a 1 ∆ g − 3 Π g and X 3 ∑ − g − 1 Π g transitions. Comparing with the radiative lifetimes of the electro-quadrupole transition for the a 1 ∆ g − b 1 ∑ + g band, our result was found to be within a reasonable error of existing calculations. Though the electro-quadrupole transition is almost negligible for the a 1 ∆ g − X 3 Σ − g,Ms=0 (0,0) band (either in emission or in absorption), comparison with the analogous oxygen near-IR spectrum [44] indicates the detailed prediction for more intense quadrupole branches in rotational structure ratios for the S 2 molecule.
The current results show that the experimentally detected 0−0, 0−1, and 1−1 bands of the b 1 ∑ + g −X 3 Σ − g,Ms=±1 emission system [22,41] are well reproduced by ab initio theory. There is no doubt that we also fully calculate and interpret the weak 0−0 vibrational band of the a 1 ∆ g − X 3 Σ − g,Ms=±1 transition for chemically unstable S 2 molecules experimentally studied by Setzer and Fink [23]. We also successfully provide some other bands for the magnetic IR spectrum of the S 2 dimer. A preliminary attempt to calculate the low transition probabilities of the b−X Ms=±1 , a−X Ms=±1 , b−a, a−X Ms=0 bands with the perturbation theory formulas was satisfactory. Our study of the S 2 molecular spectrum in the near-IR region provides more useful information for the exploration of matter in the field of astrophysics. This study provides the systematic calculations of forbidden magnetic and quadrupole transitions in O 2 , N 2 , Ge 2 , and Sn 2 homonuclear diatomic molecules and illustrates the increasing role of spin-orbit coupling effects in such fields of physical chemistry.

Supplementary Materials:
The following supporting information can be downloaded at: https: //www.mdpi.com/article/10.3390/physchem3010009/s1, Table S1: The vertical excitation energies with the X 3 ∑ − g state for other states (cm −1 ) in R s−s = 1.899Å; Table S2: Calculated spin-orbit matrix elements in S 2 (R s−s = 1.899Å); Table S3: The error analysis for the Einstein coefficient A of the b−X Ms=0 transition; Table S4: The theoretically inferred energy, Einstein coefficient and oscillator strength of the experimentally observable b−X Ms=±1 band; Table S5: The theoretically inferred energy, Einstein coefficient and oscillator strength of the experimentally observable a−X Ms=±1 band.

Data Availability Statement:
The data presented in this study are available in article and supplementary material.

Conflicts of Interest:
The authors declare no conflict of interest.