# Ro-vibrational excitation of SiO by collision with helium at high temperature

Christian Balança, Fabrice Dayou

► To cite this version:

Christian Balança, Fabrice Dayou. Ro-vibrational excitation of SiO by collision with helium at high temperature. Monthly Notices of the Royal Astronomical Society, 2017, 469 (2), pp.1673-1681. 10.1093/mnras/stx925. hal-02173772

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# Ro-vibrational excitation of SiO by collision with helium at high temperature

Christian Balança\* and Fabrice Dayou

LERMA, Observatoire de Paris, PSL Research University, CNRS-UMR 8112, UPMC Univ. Paris 06, Sorbonne Universités, F-92195 Meudon, France

Accepted 2017 April 12. Received 2017 April 7; in original form 2017 February 23

## ABSTRACT

We report theoretical rate coefficients for the ro-vibrational excitation of SiO by collision with He. A new, three-dimensional, *ab initio* potential energy surface, which includes the dependence on the vibrational coordinate of SiO, has been developed for the X¹A' ground electronic state of the SiO–He system. Quantum dynamics calculations were performed using the vibrational close-coupling rotational infinite order sudden (VCC-IOS) method. Collisional rate coefficients have been determined for the de-excitation of the 41 first rotational states (j = 0–40) of the first six vibrational levels (v = 0–5) of SiO, in the temperature range between 300 and 6000 K. The rotational de-excitation rate coefficients corresponding to the vibrationally elastic process SiO(v, j) → SiO(v, j') are found to be almost insensitive to the selected vibrational level. The rate coefficients for the vibrationally inelastic process SiO(v, j) → SiO(v', j') are significantly enhanced by the kinetic temperature and, to a lesser extent, by the vibrational excitation of SiO, but they are smaller than the vibrationally elastic ones by several orders of magnitude for the whole temperature range. The agreement between the present vibrational close-coupling rotational infinite order sudden (VCC-IOS) results and the available literature data for the SiO–He colliding system is discussed.

**Key words:** astrochemistry – molecular data – molecular processes – scattering – ISM: molecules.

## 1 INTRODUCTION

Rotational emission lines of SiO are commonly observed in warm shocked layers of molecular outflows surrounding young stellar objects (Nisini et al. 2007; Gusdorf et al. 2008; Flower & Pineau des Forêts 2012; Codella et al. 2013) and circumstellar envelopes of asymptotic giant branch (AGB) stars (Pardo et al. 2004; Soria-Ruiz et al. 2004; Cho, Lee & Park 2007; Desmurs et al. 2014; Wong et al. 2016). Rotational lines that exhibit maser amplification, involving the low-lying vibrational levels of SiO, are generally associated with evolved ABG stars, although in a few cases SiO masers have been observed from star-forming regions (Tercero et al. 2011; Niederhofer, Humphreys & Goddi 2012; Cho et al. 2016). Recently, an SiO mega-maser emission has been also detected from the nuclear disc of the NGC 1068 galaxy (Wang et al. 2014). The radiative transfer modelling of the SiO emission lines can provide valuable information on the SiO abundance and the physical properties of the emitting regions (Humphreys et al. 1996; Nisini et al. 2007; Gusdorf et al. 2008; Flower & Pineau des Forêts 2012; Codella et al. 2013; Wong et al. 2016). When the local thermodynamical equilibrium is not fulfilled, rate coefficients for the excitation of SiO by collision

with the most abundant species (H₂, He and H in interstellar and circumstellar environments) are required to describe the collisional excitation effects on the SiO level populations.

The pure rotational excitation of SiO(v = 0) by para-H₂(j = 0) has been studied by Turner et al. (1992) for the low temperature range T ≤ 300 K. The rate coefficients were calculated within the coupled-states approximation, using a two-dimensional (2D) rigid-rotor SiO–He potential energy surface (PES) derived from the three-dimensional (3D) electron-gas PES of Bieniek & Green (1981). In such a case, it was assumed that the interaction of SiO with para-H₂(j = 0) can be approximated by that with He, and the dynamics calculations were conducted using the reduced mass of SiO–H₂ (Green & Chapman 1978). More recently, rate coefficients for the rotational excitation of SiO(v = 0) by He were obtained by means of close-coupling (CC) calculations for T ≤ 300 K, using an accurate 2D rigid-rotor *ab initio* PES (Dayou & Balança 2006). Additional CC calculations were performed using the latter SiO–He PES and the SiO–H₂ reduced mass to provide estimated values of SiO/para-H₂(j = 0) rate coefficients. Important discrepancies with the results of Turner et al. (1992) were found, which are mainly attributed to inaccuracies of the SiO–He electron-gas PES.

Rate coefficients for the ro-vibrational excitation of SiO(v) by para-H₂(j = 0) have been reported by Bieniek & Green (1983) for the vibrational levels v ≤ 2 in the temperature range

<footnote>\* E-mail: Christian.Balanca@obspm.fr</footnote>
1000 ≤ T ≤ 3000 K. They were obtained within the infinite order sudden (IOS) approximation, using the SiO–He electron-gas PES (Bieniek & Green 1981) and the SiO–H₂ reduced mass. Owning to the known deficiencies of the electron-gas PES (Dayou & Balança 2006), we decided to investigate the ro-vibrational excitation of SiO by He employing a more accurate *ab initio* methodology to describe the interaction potential. A new 3D *ab initio* PES has thus been developed, which extends the previous 2D *ab initio* PES by including the dependence on the SiO bond length. In order to provide collisional data suitable to the modelling of SiO lines in high-temperature environments, the rate coefficients have been determined for transitions among the states of the vibrational levels v ≤ 5 in a wide temperature range 300 ≤ T ≤ 6000 K. Due to the small rotational constant of SiO, the number of ro-vibrational states to be accounted for is quite large in this case. Nowadays, exact CC dynamics calculations are feasible for systems involving a huge number of quantum states (Stoecklin et al. 2016), but they are still computationally demanding. For this study, given the high collision energy domain considered and the small rotational energy spacings of SiO, we chose to employ the more approximate vibrational close-coupling rotational infinite order sudden (VCC-IOS) approach (Goldflam, Kouri & Green 1977a,b; Parker & Pack 1978), which is expected to be well-suited to this case. The same approach was used to treat the ro-vibrational excitation of analogous colliding systems, such as CO (Cecchi-Pestellini et al. 2002), SO (Lique et al. 2006), CS (Lique & Spielfiedel 2007) and SiS (Tobola et al. 2008) with He as the collider.

This paper is organized as follows. The calculation of the SiO–He PES and its main features are presented in Section 2. The details of dynamics calculations are given in Section 3. In Section 4, we present the collisional cross-sections and rate coefficients, and discuss our results through comparisons with the available literature data.

## 2 SIO–HE POTENTIAL ENERGY SURFACE

### 2.1 Ab initio calculations

The 3D PES for the X¹A' ground electronic state of the SiO–He system has been computed by means of *ab initio* quantum chemistry calculations, using the partially spin-restricted coupled-cluster method at the RCCSD(T) level of theory (Hampel, Peterson & Werner 1992; Watts, Gauss & Bartlett 1993). The standard aug-cc-pVQZ basis sets (Dunning 1989; Kendall, Dunning & Harrison 1992; Woon & Dunning 1993, 1994) were used for the three atoms, together with the [3s3p2d2f1g] bond functions of Cybulski & Toczylowski (1999) placed at the mid-point between the He atom and the SiO centre-of-mass. The calculations were conducted on a 3D grid in the Jacobi coordinate system (R, r, θ), R being the distance of the helium atom to the centre-of-mass of SiO, r the SiO bond length and θ the angle between R and r (θ = 0° corresponding to linear He–OSi). At each geometry, the calculated energy $E_{SiO–He}$ was corrected for the basis set superposition error (BSSE) using the Boys & Bernardi (1970) counterpoise procedure, yielding the interaction energy:

$$V_{\text{int}}(R, r, \theta) = E_{\text{SiO-He}}(R, r, \theta) - E_{\text{SiO}}(R, r, \theta) - E_{\text{He}}(R, r, \theta), \quad (1)$$

where the SiO and He subsystems are computed with the SiO–He basis set. All electronic structure calculations have been carried out with the MOLPRO (Werner et al. 2003) suite of programs.

<figure><img src="image_3.png" /><figcaption>**Figure 1.** RKR diatomic potential for $^{28}$Si$^{16}$O. The vibrational wavefunctions are displayed for selected energy levels.</figcaption></figure>

A total of 5794 geometries were computed for the interaction potential $V_{\text{int}}(R, r, \theta)$. Owing to the requirements of the fitting procedure (see Section 2.2), we employed a rectangular grid for the (r, θ) coordinates. A set of $N_r = 7$ values of the SiO bond length r were chosen in the range $2.45 \le r \le 3.35$ bohr, so that the PES maps the turning points of the SiO vibrational levels up to v = 6. We considered $N_\theta = 13$ values of the angle θ in the range 0 ≤ θ ≤ 180°, with uniform steps of 15°. For each (r, θ) value, the radial scattering coordinate R was varied from 4 to 25 bohr, by steps of 0.25 bohr up to 16 bohr, and then steps of 1–2 bohr up to 25 bohr. Additional geometries were computed along the short-range repulsive wall, down to R = 1.5 bohr for some (r, θ) values, in such a way that the potential energy $V(R, r, \theta)$ (see equation 2) be higher than 42 000 cm$^{-1}$ at the shortest R for any (r, θ). Accordingly, the *ab initio* PES $V(R, r, \theta)$ corresponding to the calculated interaction energies $V_{\text{int}}(R, r, \theta)$ spans an energy range of at least 35 000 cm$^{-1}$ above the vibrational energy level v = 5 of SiO.

### 2.2 Analytic representation of the PES

The PES suitable for use in collision dynamics calculations can be written as follows:

$$V(R, r, \theta) = V_{\text{int}}(R, r, \theta) + V(r), \quad (2)$$

where $V_{\text{int}}(R, r, \theta)$ is the interaction potential between the collision partners SiO and He, and $V(r)$ is the interaction potential associated with the vibrating SiO diatom. We chose to describe the diatomic potential $V(r)$ by a Rydberg–Klein–Rees (RKR) potential, using the procedure of Le Roy (2004) and the spectroscopic constants derived by Lovas, Maki & Olson (1981) for $^{28}$Si$^{16}$O from infrared and microwave spectra. The SiO vibrational energy levels $\epsilon_v$ and vibrational wavefunctions $\chi_v(r)$ were determined from the RKR potential by the Fourier Grid Hamiltonian (FGH) method (Marston & Balint-Kurti 1989). Selected vibrational states are shown in Fig. 1 together with the RKR diatomic potential.

Following the procedure proposed by Werner, Follmeg & Alexander (1988), we chose to represent both the short-range and long-range parts of the interaction potential by the analytical form:

$$V_{\text{int}}(R, r, \theta) = \sum_{n=0}^{N_r-1} \sum_{l=0}^{N_\theta-1} P_l(\cos \theta) A_{ln}(R) (r - r_{eq})^n, \quad (3)$$
where $N_r = 7$ and $N_\theta = 13$ are the number of bond lengths $r$ and angles $\theta$ considered for the calculation of ab initio interaction energies, $P_l(\cos\theta)$ are Legendre polynomials and $r_{eq}$ is the SiO equilibrium bond length. At the AVQZ/RCCSD(T) level of theory, we obtain $r_{eq} = 2.874$ bohr. Since the RKR diatomic potential $V(r)$ was built using the experimental value $r_{eq} = 2.853$ bohr (Huber & Herzberg 1979), a shift of -0.021 bohr was applied to each $r$ value of $V_{\text{int}}(R, r, \theta)$. For each bond length $r_m$ and angle $\theta_k$, the one-dimensional (1D) ab initio potentials $V_{\text{int}}(R, r_m, \theta_k)$ were fitted to the following form:

$$B_{km}(R) = \exp(-a_{km} R) \left[ \sum_{i=0}^{l_{\max}} b_{km}^{(i)} R^i \right] - \frac{1}{2} [1 + \tanh(R)] \left[ \sum_{j=j_{\min}}^{j_{\max}} c_{km}^{(j)} R^{-j} \right], \quad (4)$$

where the first and second parts hold for the short-range and long-range parts of the potential, respectively. In the present case, to ensure a proper feature of the PES at short range, without damaging the accuracy of the fit in the potential well region, it was found necessary to employ two distinct sets of functional forms $B_{km}(R)$ for each $V_{\text{int}}(R, r_m, \theta_k)$. The first set, $B_{km}^{(a)}(R)$, was built accounting for all the calculated interaction energies (5794 geometries). We used $i_{\max} = 5$ for the short-range part, and $j = 6, 8, 10$ so as to represent the main dispersion and induction contributions to long-range interactions (involving dipole/quadrupole moments of SiO and dipole/quadrupole-induced moments of both fragments). The second set, $B_{km}^{(b)}(R)$, was built only from ab initio data points located along the short-range repulsive wall, for which interaction energies lie above 2000 cm$^{-1}$ (about 1000 geometries). In that case, we used $i_{\max} = 2$ and $j = 0$. Then, a switching function $f(R)$ was used to form the composite 1D potentials:

$$B_{km}(R) = B_{km}^{(a)}(R)[1 - f(R)] + B_{km}^{(b)}(R)f(R), \quad (5)$$

where $0 \le f(R) \le 1$. The switching function $f(R)$ is defined by equations (9)–(12) of Dayou & Spiefiedel (2003) and depends on two parameters, $\alpha$ and $R_0$. The dimensionless parameter $\alpha$ controls the width of the matching region, and was set to $\alpha = 50$ in the present case. The distance $R_0$ controls the location of the matching region. We chose a specific value of $R_0$ for each $B_{km}(R)$, such that the interaction energy at $R_0$ be at least equal to 24 000 cm$^{-1}$ for any $r$ and $\theta$ values. Once the 1D potentials $B_{km}(R)$ are known, the radial coefficients $A_{ln}(R)$ of equation (3) are determined from the matrix product $\mathbf{A}(R) = \mathbf{P}^{-1}\mathbf{B}(R)\mathbf{S}^{-1}$, with matrix elements of $\mathbf{P}$ and $\mathbf{S}$ given by $P_{kl} = P_l(\cos\theta_k)$ and $S_{nm} = (r_m - r_{eq})^n$, respectively. The ab initio interaction energies are well reproduced by the analytic representation of $V_{\text{int}}(R, r, \theta)$. For interaction energies below 42 000 cm$^{-1}$ (5624 geometries), the root-mean-square (rms) error between the analytical potential and the ab initio energies is 47 cm$^{-1}$, corresponding to a mean relative error of 0.1 per cent, with a maximal deviation of 4 per cent. For the PES $V(R, r, \theta)$, it leads to an rms error of 24 cm$^{-1}$ for potential energies below 42 000 cm$^{-1}$ (5585 geometries), which reduces to 5 cm$^{-1}$ below 10 000 cm$^{-1}$ (4320 geometries) and 0.6 cm$^{-1}$ below 1000 cm$^{-1}$ (657 geometries).

The global minimum of the PES $V(R, r, \theta)$ is at the linear He–OSi geometry for $R = 7.999$ bohr and $r = 2.853$ bohr, with a well depth $\epsilon = -26.764$ cm$^{-1}$. A similar result was obtained for the rigid-rotor SiO–He PES of Dayou & Balança (2006), computed at fixed $r = r_{eq}$ by the same ab initio methodology (global minimum at $R = 7.993$ bohr with a well depth $\epsilon = -26.596$ cm$^{-1}$). For $r = r_{eq}$, the two PESs differ mainly by the description of the short-range repulsive

<figure><img src="image_4.png" /><figcaption>**Figure 2.** First angular expansion coefficients $V_l(R)$ for the 2D rigid-rotor SiO–He PES $V(R, \theta)$ obtained from the present 3D PES computed at $r = r_{eq}$ (solid lines) and the rigid-rotor PES of Dayou & Balança (2006, dashed lines).</figcaption></figure>

wall. This is due to the range of ab initio interaction energies included in the fitting procedure, much higher in the present case ($\ge 42\,000$ cm$^{-1}$) than for the rigid-rotor PES ($\ge 3500$ cm$^{-1}$). Further details are provided by the comparison of the radial terms $V_l(R)$ associated with the two PESs, corresponding to the angular expansion coefficients of each PES in a basis of Legendre polynomials. In the present case, they are obtained by selecting the relevant column vector from the matrix product $\mathbf{V}(R) = \mathbf{P}^{-1}\mathbf{B}(R)$. As can be seen in Fig. 2, the isotropic ($l = 0$) and anisotropic ($l \ne 0$) radial terms are almost identical for the two PESs, with only slight deviations for the odd anisotropy components at short $R$ distances and high interaction energies. In the case of pure rotational energy transfers, the even/odd anisotropy components shown in Fig. 2 are the coupling terms that provide the main contribution to the rotationally inelastic scattering process SiO($j \to j'$) for even/odd values of $\Delta j = |j - j'|$, with the requirement $|j - j'| \le l \le j + j'$ (Arthurs & Dalgarno 1960; Flower 2007).

## 2.3 Vibrationally averaged matrix elements

The dynamics calculations involve coupling matrix elements corresponding to the interaction potential averaged over the vibrational states of SiO:

$$V_{v,v'}(R, \theta) = \langle \chi_v(r) | V_{\text{int}}(R, r, \theta) | \chi_{v'}(r) \rangle. \quad (6)$$

The vibrationally averaged matrix elements $V_{v,v'}(R, \theta)$ can be readily obtained from the expansion of equation (3) by computation of the matrix elements $\langle \chi_v(r) | (r - r_{eq})^n | \chi_{v'}(r) \rangle$. The latter quantities were determined by numerical quadrature using the vibrational wavefunctions $\chi_v(r)$ obtained from the RKR diatomic potential by the FGH method. Contour plots for the diagonal $V_{0,0}(R, \theta)$ and off-diagonal $V_{0,1}(R, \theta)$ matrix elements are shown in Figs 3 and 4, respectively. The diagonal matrix elements $V_{v,v'}(R, \theta)$ drive the vibrationally elastic scattering processes SiO($v, j \to j'$) to SiO($v, j'$). Whatever the vibrational state $v$ considered in the range $v = 0-6$, at the scale of the figure all diagonal matrix elements are found almost identical, and, as expected from the results of Fig. 2, quite similar to the rigid-rotor SiO–He PES previously reported (see fig. 1 of
<figure><img src="image_5.png" /><figcaption>**Figure 3.** Contour plot of the vibrationally averaged matrix elements $V_{0,0}(R, \theta)$ for the SiO–He system as a function of Jacobi coordinates $R$ and $\theta$. The contours are drawn by steps of $1 \text{ cm}^{-1}$ for negative energies (black lines) relative to the SiO+He dissociation limit, and by steps of $10 \text{ cm}^{-1}$ for positive energies (red lines).</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>**Figure 4.** Contour plot of the vibrationally averaged matrix elements $V_{0,1}(R, \theta)$ for the SiO–He system as a function of Jacobi coordinates $R$ and $\theta$. The contours are drawn by steps of $0.1 \text{ cm}^{-1}$ for values higher than $-1 \text{ cm}^{-1}$ (black lines), and by steps of $1 \text{ cm}^{-1}$ below (red lines).</figcaption></figure>

Dayou & Balança 2006). For the diagonal matrix elements $V_{0,0}(R, \theta)$ shown in Fig. 3, the global minimum is located at $R = 8.005$ bohr with a well depth $\epsilon = -26.779 \text{ cm}^{-1}$. The off-diagonal matrix elements $V_{v,v \pm 1}(R, \theta)$ are the main contributors to the vibrationally inelastic scattering processes SiO($v, j$) → SiO($v \pm 1, j'$). As can be seen in Fig. 4, the couplings between adjacent vibrational levels become significant only for short SiO–He distances and geometries close to linearity. Since the interaction potential is highly repulsive in this case (see Fig. 3), the vibrationally inelastic processes are expected to be rather inefficient unless high kinetic energies are probed. All others $V_{v,v \pm 1}(R, \theta)$ matrix elements are found closely similar to the $V_{0,1}(R, \theta)$ one, although a slight shift in $R$ is observed with increasing the vibrational state $v$. The $V_{v,v \pm 2}(R, \theta)$ matrix elements also behave as in Fig. 4, but are much smaller in magnitude.

### 3 DYNAMICS CALCULATIONS

The ro-vibrational excitation of SiO($v, j$) by collision with He has been treated by means of the VCC-IOS method (Goldflam

et al. 1977a,b; Parker & Pack 1978). In such a case, the vibrationally inelastic scattering process is treated by a close-coupling (CC) approach at fixed relative orientations $\theta$ of SiO and He, independently of the rotational states of SiO, which are assumed to be degenerate. The CC equations are solved at each total energy $E$ and employ the coupling matrix elements $V_{v,v'}(R, \theta)$ defined in Section 2.3. The angular expansion of the scattering S-matrix elements $S_{vv'}(\theta; E)$ in a basis of Legendre polynomials $P_l(\cos \theta)$ provides the set of inelastic cross-sections $\sigma_{v,0 \to v',l}(E)$. The vibrationally and rotationally de-excitation cross-sections are then obtained from the following relationship:

$$ \sigma_{v, j \to v', j'}(E) = (2j' + 1) \sum_l \begin{pmatrix} j & j' & l \\ 0 & 0 & 0 \end{pmatrix}^2 \sigma_{v, 0 \to v', l}(E), \quad (7) $$

where $\nu$ is a 3-j symbol. Accordingly, the summation restricts to $l$ values that satisfy the triangular inequality $|j - j'| \le l \le j + j'$, with $j + j' + l$ being an even integer. Since the rotational energy spacings are ignored, the collision energy $E_k$ corresponding to each de-excitation cross-section is given by $E_k = E - \epsilon_v$, where $\epsilon_v$ is the vibrational energy level of SiO($v$). The ro-vibrational de-excitation rate coefficients are obtained by averaging the related cross-sections over a Maxwell distribution of the relative velocity at a kinetic temperature $T$:

$$ k_{v, j \to v', j'}(T) = \left( \frac{8k_B T}{\pi \mu} \right)^{1/2} \int_0^\infty \sigma_{v, j \to v', j'}(E_k) u e^{-u} du, \quad (8) $$

where $\mu$ is the reduced mass of SiO–He, $k_B$ is the Boltzmann constant and $u = (E_k/k_B T)$. The ro-vibrational excitation rate coefficients can be derived applying the usual detailed balance:

$$ k_{v', j' \to v, j}(T) = \frac{(2j + 1)}{(2j' + 1)} \exp \left[ - \frac{(\epsilon_{vj} - \epsilon_{v'j'})}{k_B T} \right] k_{v, j' \to v', j'}(T), \quad (9) $$

where $\epsilon_{vj}$ are the ro-vibrational energy levels of SiO($v, j$).

In this work, we have considered the ro-vibrational excitation of $^{28}$Si$^{16}$O($v, j$) by $^4$He ($\mu = 3.669$ a.m.u.) for the 41 first rotational states ($j = 0-40$) of each of the first six vibrational levels ($v = 0-5$). The vibrational energy levels of SiO are described by a Dunham expansion, $\epsilon_v = \sum_k Y_{k0}(v + 1/2)^k$, based on the spectroscopic constants $Y_{k0}$ ($k = 1-3$) of Lovas et al. (1981). They correspond to the energy levels of the RKR diatomic potential since it was built from the same spectroscopic constants. The VCC-IOS inelastic cross-sections $\sigma_{v,0 \to v',l}(E)$ have been computed with the MOLSCAT program (Hutson & Green 1994), using the modified log derivative propagator of Manolopoulos (1986) to solve the CC equations. The vibrational level $v = 6$ was added to the vibrational basis set in the dynamics calculations. The expansion coefficients of S-matrix elements in the basis of Legendre polynomials were computed by Gauss–Legendre quadrature with $l_{max} + 2$ points, where $l_{max} = 80$ is the maximal value of $l$ required to get converged cross-sections for rotational levels $j \le 40$. The calculations were performed for total energies ranging from 1 to 42 000 cm$^{-1}$. This energy range allowed us to get converged rate coefficients up to 6000 K for the whole set of vibrational levels considered ($v = 0-5$).

### 4 RESULTS AND DISCUSSION

We report in Fig. 5 selected rotational de-excitation cross-sections as a function collision energy for the vibrationally elastic process
<figure><img src="image_7.png" /><figcaption>**Figure 5.** Cross-sections for the vibrationally elastic, rotational de-excitation of SiO($v = 0, j$) by He as a function of collision energy. The cross-sections are labelled according to the related transition $j \rightarrow j'$ between the rotational quantum states of SiO. The VCC-IOS results (solid lines) are compared with the CC results (dashed lines) of Dayou & Balança (2006) obtained from a rigid-rotor SiO–He PES computed at the equilibrium bond-length $r_{eq}$ of SiO.</figcaption></figure>

SiO($v = 0, j \rightarrow SiO(v = 0, j')$). The results of the present VCC-IOS calculations are compared to the results of CC calculations previously obtained (Dayou & Balança 2006) from the rigid-rotor SiO–He PES. As can be seen, whatever the transition considered, the two sets of results differ significantly at low collision energies, whereas a much better agreement is attained in the high energy range. In the present case, the discrepancies observed between the VCC-IOS and CC results are mainly due to the use of the IOS approximation. Indeed, as the vibrational couplings are weak (see Section 2.3), the rigid-rotor approximation employed for the CC calculations is expected to provide a reliable description of pure rotational energy transfers (Krems 2002; Lique et al. 2006; Lique & Spielfiedel 2007; Tobola et al. 2008). Moreover, the PESs involved in the two treatments [the vibrationally averaged PES $V_{0,0}(R, \theta)$ and the rigid-rotor PES] are highly similar for the range of interaction energies spanned by the CC calculations (below 3500 cm$^{-1}$). In the IOS approach, the rotational motion of the molecule during the collision is ignored, and the rotational levels are assumed to be degenerate. This approximation becomes relevant if the collision energy is large compared with the rotational energy spacings, and the collision time is much shorter than the rotational period of the molecule (Goldflam et al. 1977a; Kouri 1979). The collision energy must also be larger than the potential well depth, since the possible resonances associated with closed rotational states are not correctly described. For SiO–He, these requirements can be fulfilled for moderate collision energies, due to the small rotational constant of SiO ($B_e = 0.73$ cm$^{-1}$) and the relatively shallow potential well ($\epsilon = -27$ cm$^{-1}$). The results of Fig. 5 are consistent with the above requirements. For transitions associated with a given value of $\Delta j = |j - j'|$, the agreement between the VCC-IOS and CC results improve with increasing collision energy, and, by increasing $\Delta j$ values, the agreement is attained at higher energies. At the highest collision energy considered in the CC calculations (around 3000 cm$^{-1}$), the deviations between the cross-sections values go from a few per cent to about 20 per cent with increasing initial rotational state $j$ and $\Delta j$. The increasing discrepancies with $j$ can be related to the faster rotational motion of the SiO molecule for higher rotational states. For a given initial $j$ state, slightly larger discrepancies are obtained for transitions involving odd $\Delta j$ values compared to even $\Delta j$ ones. The small differences between the two PESs at short SiO–He distances may be the cause of this effect, since they result in slightly larger deviations for the odd anisotropy coupling terms (see Fig. 2).

Selected rotational de-excitation rate coefficients for the vibrationally elastic process SiO($v = 0, j \rightarrow SiO(v = 0, j')$) are displayed in Figs 6 and 7 as a function of temperature. The VCC-IOS results are compared with the CC results of Dayou & Balança (2006), which restrict to the temperature range $T \le 300$ K. As can be seen in Fig. 6, for rotational transitions involving small $\Delta j$ values ($\Delta j = 1$ and 2), the two sets of results are in fair agreement. The temperature dependence of the VCC-IOS rate coefficients is consistent with the CC results, as we could expect from the quite similar energy dependence obtained for the cross-sections (Fig. 5). The dependence of the rate coefficients on $j$ and $\Delta j$ is properly described, in particular the propensity rule that favours odd $\Delta j$ transitions over even $\Delta j$ ones. The latter propensity rule relates to the marked asymmetry of the SiO–He PES, which gives rise to a dominant contribution of the odd anisotropy (McCurdy & Miller 1977). At 300 K, for $\Delta j = 1$ and 2, the deviations between the rate coefficients values vary between few per cent to about 15 per cent with increasing $j$ over the range $j \le 26$, the highest rotational state considered in the CC calculations. For transitions associated with larger $\Delta j$
<figure><img src="image_8.png" /><figcaption>**Figure 6.** Rate coefficients for the vibrationally elastic, rotational de-excitation of SiO($v = 0, j$) by He as a function of temperature. The rate coefficients are labelled according to the related transition $j \rightarrow j'$ between the rotational quantum states of SiO. The VCC-IOS results (solid lines) are compared with the CC results (dashed lines) of Dayou & Balança (2006).</figcaption></figure>

values (Fig. 7), the discrepancies between the VCC-IOS and CC rate coefficients can grow up to a factor of 2 for $\Delta j \ge 9$. Nonetheless, at 300 K, a non-negligible contribution to the integrand of equation (8) is provided by the cross-section values at low collision energies, where the reliability of the IOS approximation is highly questionable (see Fig. 5). For transitions involving large $\Delta j$, the accuracy of the VCC-IOS rate coefficients should thus improve at higher temperatures. For the high temperature range, it is worth to compare the present VCC-IOS results with the extrapolation of CC rate coefficients performed by Schöier et al. (2005) from 300 up to 2000 K. As can be seen in Fig. 7, the two sets of results differ significantly for transitions with large $\Delta j$. Flower & Pineau des Forêts (2012) have shown that the rapid increase with $T$ of the extrapolated rate coefficients corresponding to large $\Delta j$ led to an improper modelling of the SiO line intensities and line profiles, which was corrected by assuming that the rate coefficients remain constant for $T \ge 300$ K. The features of the VCC-IOS rate coefficients shown in Fig. 7 are consistent with the latter assumption.

<figure><img src="image_9.png" /><figcaption>**Figure 7.** Rate coefficients for the vibrationally elastic, rotational de-excitation of SiO($v = 0, j$) by He as a function of temperature. The rate coefficients are labelled according to the related transition $j \rightarrow j'$ between the rotational quantum states of SiO. The VCC-IOS results (solid lines) are compared with the CC results (dashed lines) of Dayou & Balança (2006) and their extrapolation to high temperatures (long dashed lines) reported in kinetic data bases (Schöier et al. 2005).</figcaption></figure>

The case of transitions involving vibrationally excited states of SiO is considered in Fig. 8, where we display, as a function of temperature, rotational de-excitation rate coefficients for the vibrationally elastic SiO($v, j \rightarrow SiO(v, j')$) and inelastic SiO($v, j \rightarrow SiO(v-1, j')$) processes. As a typical example, we chose the $j = 2 \rightarrow j' = 1$ transition, for which strong maser emission has been reported for the associated rotational line in $v = 1$ (Soria-Ruiz et al. 2004, 2007; Tercero et al. 2011; Desmurs et al. 2014). Fig. 8 shows that the vibrational excitation of SiO barely affects the vibrationally elastic rate coefficients. For $\Delta j = 1$ transitions, the rate coefficients decrease slightly with the vibrational state $v$, whereas a slight increase with $v$ is obtained for $\Delta j = 2$ (not shown here). Nonetheless, in both cases, the vibrationally elastic rate coefficients are found to vary by less than 10 per cent with $v$, whatever the temperature range considered. This feature was expected owing to the strong similarities between the diagonal matrix elements $V_{v,v'}(R, \theta)$ associated with different vibrational states $v$ (Section 2.3). It is worth to note that similar results are obtained for vibrationally elastic transitions involving higher rotational states $j$ and larger $\Delta j$ values. The vibrationally inelastic rate coefficients involving adjacent vibrational levels ($\Delta v = v - v' = 1$) are much smaller than the elastic ones ($\Delta v = 0$), although they rise sharply with the kinetic temperature $T$ and the initial vibrational state $v$. The magnitude of the vibrationally inelastic rate coefficients decreases rapidly with increasing $\Delta v$ but they display trends with $T$ and $v$, which are similar to the $\Delta v = 1$ case, independently of $\Delta j$. The temperature dependence of the vibrationally inelastic rate coefficients is consistent with the features observed for the coupling matrix elements $V_{v,v'}(R, \theta)$, since the vibrational couplings become significant only if the He atom gets close to SiO, which requires large kinetic energies due to the repulsive part of the interaction potential (see Figs 3 and 4). The dependence on $v$ of the rate coefficients relates to a slight shift of the coupling matrix elements towards larger SiO-He distances that is induced by the vibrational motion of SiO.

To our knowledge, the only rate coefficients available to date for the ro-vibrational excitation of SiO by para-H$_2$($j = 0$) are those
<figure><img src="image_10.png" /><figcaption>**Figure 8.** VCC-IOS rate coefficients for the vibrationally elastic (upper panel) and vibrationally inelastic (lower panel), rotational de-excitation of SiO(*v*, *j*) by He as a function of temperature. The *j* = 2 → *j'* = 1 rotational transition is considered, and the rate coefficients are labelled according to the related transition *v* → *v'* between the vibrational quantum states of SiO.</figcaption></figure>

calculated by Bieniek & Green (1983) for the vibrational levels
$v = 0-2$ (with $\Delta v = 0, 1$) in the temperature range $1000 \le
T \le 3000$ K. They were computed within the IOS approxima-
tion using a SiO–He PES based on an electron-gas model (Bie-
niek & Green 1981). A perturbative distorted-wave approach (Bi-
niek 1980) was employed to treat the vibrationally inelastic scat-
tering at fixed orientations of SiO and He. The SiO–H$_2$ reduced
mass $\mu_{SiO-H_2}$ was used in the dynamics calculations to get an
estimate of SiO/para-H$_2$ (*j* = 0) rate coefficients (Green & Chap-
man 1978). For that purpose, another widely used approximation
(Schöier et al. 2005; Lique et al. 2008; Roueff & Lique 2013; Walker
et al. 2014) is to scale the SiO/He rate coefficients by the reduced
masses ratio ($\mu_{SiO-He}/\mu_{SiO-H_2}$)$^{1/2}$. In our previous study on the title
system (Dayou & Balança 2006), we found that these two approx-
imations provide quite similar results for small $\Delta j$ transitions on
increasing the kinetic temperature (the rates agree within less than
10 per cent at 300 K). In order to be compared with the present
VCC-IOS results, the rate coefficients reported by Bieniek & Green
(1983) have thus been divided by the latter scaling factor to recover
values relevant to the SiO/He colliding system.

The two sets of results are compared in Fig. 9. Rotational de-
excitation rate coefficients are shown for the vibrationally elastic

SiO(*v*, *j*) → SiO(*v*, *j'* ) and inelastic SiO(*v*, *j*) → SiO(*v* − 1, *j'* ) processes involving the first excited vibrational state *v* = 1, at a fixed temperature of 2000 K, for Δ*j* = 1 and 2. As can be seen, both sets of rate coefficients display a similar dependence with the initial *j* state. For a given Δ*j*, the rates increase rapidly with *j* for the low *j* states, while they are almost independent of *j* for the higher states. Following equation (7), this feature suggests a negligible contribu- tion of the IOS cross-sections σ$_{*v*,0→v'*,l}$ for high *l* values (the number of *l* components involved in each transition increases with *j* accord- ing to |*j* − *j*'| ≤ *l* ≤ *j* + *j*'). This reflects the limited influence of the high-order anisotropy components of the SiO–He PES for tran- sitions involving both high *j* and small Δ*j* values. Regarding the magnitude of the rate coefficients, significant differences are ob- served between the two sets of results. The vibrationally elastic rate coefficients differ by about a factor of 3 for Δ*j* = 1, and 2 for Δ*j* = 2, with opposite propensity rules regarding even/odd Δ*j* transitions. The same propensity rules are obtained for the vibrationally inelas- tic rate coefficients, which favour slightly transitions with even Δ*j*, but the discrepancies can grow up to one order of magnitude in this case. Such large discrepancies are most likely due to the dis- tinct SiO–He PESs employed in the dynamics calculations, since the distorted-wave approach was found to introduce deviations of only few per cent on the rate coefficients with respect to the VCC- IOS treatment (Bieniek & Green 1983). The opposite propensity rules observed for the vibrationally elastic rate coefficients reflect the near-homonuclear symmetry of the PES of Bieniek & Green (1981), which is in contrast to the marked asymmetry of the present PES (Fig. 2). Similar deviations were obtained in the rigid-rotor case between the rate coefficients of Dayou & Balança (2006) and those of Turner et al. (1992), which were also based on the PES of Bieniek & Green (1981). Given the improved accuracy of the present ab ini- tio PES with respect to the electron-gas PES, the rate coefficient values reported in this work are thought to be the most appropriate for the modelling of SiO line emission in high-temperature astro- physical media. The complete set of de-excitation rate coefficients will be made available through the LAMDA (Schöier et al. 2005) and BASECOL (Dubernet et al. 2013) data bases.

5 CONCLUSION

The ro-vibrational excitation of SiO by collisions with He has been
investigated by means of quantum dynamics calculations performed
within the VCC-IOS approximation, using a new 3D ab initio PES
developed for the X¹A' ground electronic state of the SiO–He sys-
tem. The wide range of geometries spanned by the ab initio PES al-
lowed the determination of rotational de-excitation rate coefficients
for the vibrationally elastic SiO(*v*, *j*) → SiO(*v*, *j'* ) and inelastic
SiO(*v*, *j*) → SiO(*v'*, *j'* ) processes involving the 41 first rotational
states (*j* = 0 – 40) of the first six vibrational levels (*v* = 0–5) of
SiO, for kinetic temperatures up to 6000 K.

Whatever the rotational transition considered, the vibrationally
inelastic rate coefficients are significantly enhanced by the kinetic
temperature and, to a lesser extent, by the initial vibrational mo-
tion of SiO. Nonetheless, even at the highest temperatures, they
are smaller than the vibrationally elastic ones by several orders of
magnitude, pointing out an inefficient transfer of the vibrational en-
ergy to the other modes of motion. Furthermore, the vibrationally
elastic rate coefficients are almost insensitive to the selected vi-
brational level, indicating that the vibrational motion of SiO has a
quite small influence on pure rotational energy transfers. Similar
conclusions were drawn from the study of the ro-vibrational excita-
tion of analogous colliding systems, such as CO (Krems 2002), SO
<figure><img src="image_11.png" /><figcaption>**Figure 9.** Rate coefficients for the vibrationally elastic (upper panel) and vibrationally inelastic (lower panel), rotational de-excitation of SiO($v = 1, j$) by He at 2000 K as a function of the initial rotational state $j$, for $\Delta j = 1$ (left-hand column) and $\Delta j = 2$ (right-hand column). The VCC-IOS results (filled circles) are compared with the results of Bieniek & Green (1983) (empty circles) relevant to SiO–He (see text).</figcaption></figure>

(Lique et al. 2006), CS (Lique & Spielfiedel 2007) and SiS (Tobola et al. 2008) with He as the collider.

The VCC-IOS rate coefficients for the vibrationally elastic process SiO($v = 0, j$) → SiO($v = 0, j'$) have been compared to the results of CC calculations previously performed for SiO–He within the rigid-rotor approximation (Dayou & Balança 2006). For transitions involving small $\Delta j$ values, the two sets of results are in good agreement, with deviations of less than 15 per cent at 300 K, the highest temperature available for the CC results. For transitions with large $\Delta j$, the comparison is not relevant at 300 K, due to the failure of the IOS approximation, but we expect a similar degree of accuracy at higher temperatures. In the high temperature range, the VCC-IOS rate coefficients display a weak temperature dependence that is markedly different from the extrapolation of the CC rate coefficients (Schöier et al. 2005), and consistent with the expectations of Flower & Pineau des Forêts (2012).

The VCC-IOS results have also been compared to the rate coefficients reported by Bieniek & Green (1983) for the temperature range $1000 \le T \le 3000$ K. For the vibrationally elastic processes, the rates differ by about a factor 3 to 2 for $\Delta j = 1$ and 2, respectively, with opposite propensity rules. For the vibrationally inelastic processes, deviations of about one order of magnitude are observed. Since the *ab initio* PES employed in this work is thought

to be more reliable than the electron-gas PES of Bieniek & Green (1981), we recommend the use of the present rate coefficients for the modelling of SiO line emission in high-temperature astrophysical environments. The scaling factor $(\mu_{SiO-He}/\mu_{SiO-H_2})^{1/2}$ can be applied to get a first estimate of SiO/para-H$_2$ ($j = 0$) rate coefficients, but the validity of this approximation is still to be carefully examined by performing explicit calculations for the SiO–H$_2$ colliding system (Lique et al. 2008; Walker et al. 2014). The complete set of de-excitation rate coefficients will be made available through the LAMDA (Schöier et al. 2005) and BASECOL (Dubernet et al. 2013) data bases.

## ACKNOWLEDGEMENTS

This work was supported by the programme Physique et Chimie du Milieu Interstellaire (PCMI) funded by CNRS and CNES. The potential energy surface calculations were performed on the parallel machine Tycho of Paris Observatory, and all the dynamics calculations were performed at the IDRIS-CNRS center (Institut de Développement et des Ressources en Informatique Scientifique du Centre National de la Recherche Scientifique). This work was granted access to the HPC resources of (TGCC/CINES/IDRIS) under the allocation 2015047344 made by GENCI.
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