Fine-structure transitions of Si and S induced by collisions with atomic hydrogen

Using a quantum-mechanical close-coupling method, we calculate cross sections for fine structure excitation and relaxation of Si and S atoms in collisions with atomic hydrogen. Rate coefficients are calculated over a range of temperatures for astrophysical applications. We determine the temperature-dependent critical densities for the relaxation of Si and S in collisions with H and compare these to the critical densities for collisions with electrons. The present calculation should be useful in modeling environments exhibiting the [S i] 25 {\mu}m and [S i] 57 {\mu}m far-infrared emission lines or where cooling of S and Si by collisions with H are of interest.


INTRODUCTION
The abundances of elements are particularly important parameters for understanding the composition of the gas and dust in the interstellar medium (ISM) and the physics and chemistry behind it. Silicon and sulfur are abundant elements that are arguably less-studied than, respectively, isovalent carbon and oxygen. Collisional rates for cooling of Si and S in collisions with H, such as we provide here, are useful (Hollenbach & McKee 1989) and may be efficiently calculated and presented together, similarly to complementary calculations that were given for C and O in collisions with H, e.g. Launay & Roueff (1977b); Abrahamsson et al. (2007). In the case of sulfur, observed concentrations in dense regions of the interstellar medium (ISM) appear significantly less than those in diffuse clouds (Savage & Sembach 1996;Joseph et al. 1986). Observations of sulfur fine-structure lines can assist in trying to fill this "missing sulfur" gap. For example, the [S ] 25 µm emission line was observed in shocked environments by Neufeld et al. (2009); Goicoechea et al. (2012); Anderson et al. (2013) using the Spitzer Infrared Spectrograph (IRS) and by Rosenthal et al. (2000) with the Short-Wavelength-Spectrometer on the Infrared Space Observatory (ISO-SWS). In another application, recently, Betelgeuse was observed using the Echelon Cross Echelle Spectrograph (EXES) on SOFIA, where the [S ] 25 µm line was used to investigate circumstellar flow (Harper et al. 2020). Since emission from the sulfur atom can arise in these certain circumstellar (wind of Betelgeuse) and neutral shocked (C-shock) environments of the ISM, it is useful to have predictions of the critical densities (the densities where collisional deexcitation rates equal the radiative decay rates) for collisions with neutral hydrogen to reliably establish the diagnostic potential of sulfur fine-structure lines at various densities and temperatures in such environments where warm atomic hydrogen is present.
In the present paper, we present our calculations on the finestructure transitions of Si and S induced by collisions with atomic hydrogen, where A can be Si or S and and can be 0, 1 or 2, respectively. In collisions with H, there is little theoretical work on the Si and S systems, compared to C and O where fully quantum calculations are available. Semiclassically calculated rate coefficients were given by Bahcall & Wolf (1968, Eq. (42)), Tielens & Hollenbach (1985 , Table  4), and by Hollenbach & McKee (1989, Table 8). The present rate coefficients obtained from fully-quantum calculated cross sections could be used for modeling or simulation and other astrophysical applications.
The cross sections for fine-structure transitions are given by where → ( ) are the partial cross sections, is the wave number defined by 2 = 2 ( − ), is the reduced mass of systems and , is the total collision energy, and is the finestructure state splitting energy of the Si or S atom. The matrix is defined by where is the open channel reaction matrix defined in Johnson (1973). To solve for the radial wave functions, which determine the scattering matrices, we used the quantum close-coupling formalism with the wave functions expanded in the space-fixed basis (Yan & Babb 2022; Yan & Babb 2023b).

INTERATOMIC POTENTIALS
For both Si( 3 ) and S( 3 ) the electronic states involved in the fine-structure transitions with H( 2 ) are the b 4 Π, B 2 Σ − , a 4 Σ − , and X 2 Π states. The potentials for SiH and SH were obtained by using the multireference configuration interaction Douglas-Kroll-Hess (MRCI-DKH) method with the augmented-correlation-consistent Table 1. Rate coefficients (in units of 10 −9 cm 3 s −1 ) for the fine-structure excitation and relaxation of Si( 3 ) by H.   Table 2. Rate coefficients (in units of 10 −9 cm 3 s −1 ) for the fine-structure excitation and relaxation of S( 3 ) by H.   polarized valence 5-tuple zeta (aug-cc-pV5Z-dk) basis within M -2010.1 (Werner et al. 2010) in the 2 Abelian symmetry point group. For both SiH and SH these potentials were calculated for internuclear distances from 1.13 to 9.07 0 at an interval of 0.095 0 and from 9.07 to 10.4 0 at an interval of 0.19 0 . Note that we have given the values of in units of 0 , because the scattering calculations were carried out in atomic units (a.u.) (1 0 ≈ 0.529 Å). For the SiH system, 10 molecular orbitals (MOs) are used for the active space: 6 1 , 2 1 and 2 2 symmetry MOs. The remaining five electrons are put in the closed-shell orbitals. The potentials calculated from M are shown in Fig. 1. Our calculated potentials are in good agreement with Zhang et al. (2018); for example, for the X 2 Π state we find = 3.210 eV at = 1.524 Å compared to their value of = 3.193 eV at = 1.517 Å. For the SH system, 12 molecular orbitals (MOs) are used for the active space: 6 1 , 3 1 and 3 2 symmetry MOs. The remaining five electrons are put in the closed-shell orbitals. The potentials calculated from M are shown in Fig. 2. Our calculated potentials are in accordance with those of Hirst & Guest (1982). For the X 2 Π state, our value of = 3.77 eV agrees with the value 3.791 eV adopted by Gorman et al. (2019); Császár et al. (2003).
In order to have the potentials correlate asymptotically to the sep- arated atom limits, which are taken to be the reference energies for the scattering calculations, the potentials for SiH and SH as shown in Figs. 1 and 2 were shifted in energy and joined at = 9.83 0 to the long-range form − 6 / 6 . Smooth fits were obtained using estimates of 6 from (Gould & Bučko 2016); the values used were

RESULTS
Rate coefficients as functions of temperature are obtained by averaging the cross sections over a Maxwellian energy distribution, where is the temperature, is the Boltzmann constant, and is the kinetic energy. The scattering equations were integrated from = 1.2 0 to = 30 0 . Over this range, accordingly the potentials were utilized as described in Sec. 3 with the potential values for < 9.83 0 interpolated using cubic splines. Cross sections were calculated using Eq. (4) from threshold, Eqs. (2) and (3), to sufficiently high collisional energies (10 5 K or about 8.62 eV) to ensure that the integration in Eq. (5) could be accurately evaluated up to temperatures of 10,000 K. [Note that the Si( 1 )-H and S( 1 )-H channels, respectively, open at about 0.8 eV and 1.2 eV. Our cross sections do not include atomic Si or S electronic excitations higher than the ground 3 terms.] In Fig. 3 we present our calculated excitation and relaxation cross sections of the fine structure transitions for the Si-H system. The corresponding rate coefficients are given in Fig. 4 and in Table 1. In Fig. 4, we also show the fine-structure excitation and relaxation rate coefficients for the C-H system (Yan & Babb 2023b) for the purpose of comparison. Because Si and C are isovalent and share the same fine-structure level ordering with the relaxation cross sections for Si and H collisions, Eq. (2), are similar to those for C and H (dotted lines) (Yan & Babb 2023b), though we find the Si-H values are larger than the C-H values for relatively high energy > 1000 K. Also we find that the 0 → 1 excitation rate coefficients for Si and H collisions are much larger than those for C and H collisions.
For the S atom in collision with H, the excitation and relaxation cross sections for the fine structure transitions as functions of energy are shown in Fig. 5, where we find that the cross sections for relaxation for S and H show some resonances for collisional energies around 10 K. The corresponding rate coefficients are given in Fig. 6 and Table 2. In Fig. 6, we also show our calculations on fine-structure excitation and relaxation rate coefficients for the O-H system (Yan & Babb 2022) for the purpose of comparison. We find that the excitation rate coefficients for S and H are similar to those of O and H. However, the 0 → 1 relaxation rate coefficient for S and H is also much larger than that for O and H.
From the Figs. 3 to 6, we can find that the cross sections and rate coefficients of Si in collision with H are relatively larger than those of S in collision with H, which are caused by the more "attractive" (deeper) potentials of SiH compared to SH (the a 4 Σ − electronic state of SiH has numerous bound ro-vibrational levels, while that of SH is repulsive). The present calculations for Si in collision with H and for S in collision with H can replace rate coefficients estimated by scaling values for Si in collision with He and for S in collision with He values (Lique et al. 2018).
The critical density is defined as  where ( → ) is the transition probability (Mendoza 1983) and → ( ; ) is the relaxation rate coefficient for collisions with species , which maybe be hydrogen atoms (H) or electrons (e). For the collisions of Si with H, the present calculations of critical densities are shown in Fig. 7 along with the values from Draine (2011 ,  Table 17.1) and from Tielens & Hollenbach (1985). For the collisions of S with H, the present calculations are shown in Fig. 8 along with the semiclassical calculations of Hollenbach & McKee (1989). In Fig. 8, we also present the critical densities for the relaxation of S in collisions with electrons which we evaluated using the quantum R-matrix effective collision strengths from Tayal (2004); for comparison, we also include the the semiclassical calculations of Hollenbach & McKee (1989).

CONCLUSION
Fine-structure excitation and relaxation cross-sections and rate coefficients for the Si and S atoms in collision with atomic hydrogen are obtained by using quantum-mechanical close-coupling methods. The electronic potential curves of SiH and SH are obtained by using the multireference configuration interaction Douglas-Kroll-Hess (MRCI-DKH) method. The critical densities for the Si and S atoms in H collisions and their comparisons with other calculations or with electron collisions are also given. The present calculations may be useful in diagnostics of astrophysical environments such as shocked and circumstellar environments.

DATA AVAILABILITY
The rates for collisional relaxation by H presented in Table 1 for Si and in Table 2 for S are given in LAMDA format (Schöier et al. 2005) at Figshare Yan & Babb (2023a).  Table 8). For relaxation in S-electron collisions, small dotted lines are the critical densities from Hollenbach & McKee (1989) and large dotted lines are the critical densities calculated using the quantum R-matrix calculations of (Tayal 2004). The blue denotes the (1 → 2) transition and red denotes the (0 → 1) transition. Note that the (2 → 0) transition is forbidden (Hollenbach & McKee 1989).