Rotational relaxation of AlO⁺(¹Σ⁺) in collision with He

Abstract

The rate coefficients for the rotational de-excitation of AlO⁺ by collisions with He are determined. The possible production mechanisms of the AlO⁺ ion in both diffuse and dense molecular clouds are first discussed. A set of ab initio interaction energies is computed at the CCSD(T)-F12 level of theory, and a three-dimensional analytical model of the potential energy surface is obtained using a linear combination of reproducing kernel Hilbert space polynomials together with an analytical long range potential. The nuclear spin free close-coupling equations are solved and the de-excitation rotational rate coefficients for the lower 15 rotational states of AlO⁺ are reported. A propensity rule to favour Δj = −1 transitions is obtained while the hyperfine resolved state-to-state rate coefficients are also discussed.

astrochemistry, molecular data, molecular processes, scattering, ISM: molecules

1 INTRODUCTION

Aluminum is the 12th most abundant element in space but it is mainly depleted into grains. In diffuse clouds, for example, it is assumed to be 90 per cent to 99 per cent depleted into grains (Turner 1991). AlO is supposed to be a crucial gas-phase Al-bearing species for alumina dust formation, because the dimerisation of AlO followed by the oxygenation of the AlO dimers produces Al₂O₃. This molecule in turns forms dimers and small alumina clusters (Sarangi & Cherchneff 2015). Therefore, the chemical production of AlO is expected to measure the efficiency of alumina-dust formation and to be linked to the gas-to-dust mass ratio. In contrast to C-rich circumstellar envelopes, where more volatile molecules such as AlF, AlCl, are detected, AlO can be found around O-rich circumstellar envelopes if small amount of Al is in the gas phase. It is, for example, the case of the envelope of the oxygen rich supergiant star VY Canis Majoris, where six rotational transitions in the 2 and 1 mm regions of the aluminum monoxide, AlO radical, have been detected (Tenenbaum & Ziurys 2009, 2010). More recently, AlO was also observed in the circumstellar shell of Mira (Kamiński et al. 2016). AlO⁺ is not yet detected but its production from AlO is highly possible following the schemes discussed below. First, in diffuse interstellar clouds, photoionization by the ambient interstellar radiation field dominates the ionization process. The ionization energy of AlO being 9.70 eV (Sghaier et al. 2016) is lower than the ionization energy of H (13.60 eV), which is the maximum photon energy in diffuse interstellar clouds. It is also lower than the ionization energy of the most abundant detected neutral diatomic molecules up to now such as OH (13.02 eV), CO (14.01 eV), HCl (12.74 eV), SH (10.42 eV), CH (10.64 eV), and SO (10.29 eV) (Linstrom & Mallard 2001). The photoionization of AlO in diffuse interstellar clouds where it would exist would then be quite probable. In dense molecular clouds ionization is produced by cosmic ray sufficiently energetic to penetrate the interior. Since H₂ and He are the dominant species, the ions produced are mainly H⁺, He⁺ and H|$_{2}^{+}$|⁠. The possible production of AlO⁺ in these regions could follow the scheme proposed by Herbst & Klemperer (1973) long time ago after collisions with He⁺ or H⁺.

\begin{equation} {\rm AlO} + {\rm He} ^{+} \longrightarrow {\rm Al}^{+} + {\rm O} + {\rm He}, \end{equation}

(1)

or

\begin{equation} {\rm AlO} + {\rm H}^{+} \longrightarrow {\rm Al}^{+} + {\rm O} + {\rm H}, \end{equation}

(2)

followed by:

\begin{equation} {\rm Al}^{+} + {\rm O}_{2} \longrightarrow {\rm AlO}^{+} + {\rm O}. \end{equation}

(3)

This last reaction is however endothermic whereas the charge exchange channels (4) and (5) of reactions (1) and (2) remain possible:

\begin{equation} {\rm AlO} + {\rm He}^{+} \longrightarrow {\rm AlO}^{+} + {\rm He} \end{equation}

(4)

or

\begin{equation} {\rm AlO} + {\rm H}^{+} \longrightarrow {\rm AlO}^{+} + {\rm H}. \end{equation}

(5)

In this paper, we study the collisions of AlO⁺ with He, which will be required to evaluate the abundance of AlO⁺ in interstellar clouds whenever it will be detected. We present both the determination of the potential energy surface (PES) of the colliding system and close-coupling calculations. The methodology used is briefly reminded in Section 2, while the results of our study are presented in Section 3. The present data together with the recent spectroscopic data given by Sghaier et al. (2016) should help for the identification of this cation in diverse astrophysical media.

2 METHODOLOGY

2.1 Potential energy surface

2.1.1 Ab initio calculations

Since the vibrational frequency of AlO⁺ is relatively small [ω_e(AlO⁺) = 1000 cm⁻¹, Sghaier et al. 2016], a surface including the vibrational motion of AlO⁺ is required for computing the collisional rate coefficients up to 1000 K. We then developed a three-dimensional PES in Jacobi coordinates (r, R, θ), where r is the diatomic distance, R is the distance between the He atom and the centre of mass of AlO⁺ and θ is the Jacobi angle between the diatom and R, with the linear configuration He-O-Al being associated with θ = 0°, see Fig. 1. A grid of geometries of the atom-diatom system was made of a direct product of nine diatomic distances of r around the equilbrium distance of AlO⁺ (r_e = 3.0 a₀) ranging from 2.6 a₀ to 3.4 a₀, 34 values of R varying from 3.6 a₀ to 50 a₀ and of 19 values of θ chosen between 0° and 180° by step of 10°.

Figure 1.

Countor plot of the He-AlO⁺ interaction PES in Jacobi coordinates at r = 3.0 a₀. Negative energies are shown in blue contours lines in increments of 10 cm⁻¹. The zero energy is when AlO⁺ and He are separated.

We used the ab initio approach proposed by Hochlaf and coworkers for calculating multi dimensional PESs of weakly bound systems, which offers a high level of accuracy (Lique, Kłos & Hochlaf 2010; Ajili et al. 2013; Al Mogren et al. 2014). Briefly, the grid of three body interaction energies was computed at the CCSD(T)-F12 level using an aug-cc-pVTZ basis set and the corresponding auxiliary basis sets and density fitting functions (Klopper 2001; Weigend, Köhn & Hättig 2002), which are the default basis sets of Peterson and co-workers implemented in molpro (Werner et al. 2012). The basis set superposition error was corrected using the counterpoise procedure (Boys & Bernardi 1970), while the size-consistency default of the CCSD(T)-F12 (Knizia, Adler & Werner 2009) method was compensated by shifting up the whole ab initio interaction energy grid to make it vanishing at R = 100 a₀. The maximum energy shift was found to be lower than 1 per cent of the well depth.

2.1.2 Analytical representation of the PES

An analytical representation of the three dimensional PES was obtained using the reproducing kernel Hilbert space (RKHS) method of Ho and Rabitz (Ho & Rabitz 1996):

\begin{equation} V^{\rm RKHS}(r,R,\theta )=\sum \limits _{k=1}^N q^{2,4}(r_k,r)q^{2,4}(R_k,R)q^2(z_k,z), \end{equation}

(6)

where q is the usual one dimensional kernel function as defined by Ho & Rabitz (1996). N is the total number of calculated ab initio energies while r_k, R_k, and z_k = (1 − cos θ_k)/2 are defining the ab initio geometry grid of the triatomic system. The long range part of the PES was computed analytically from the summation of the induction and dispersion terms as

\begin{eqnarray} &&{ V^{\rm LR}(r,R,\theta ) = -\frac{\alpha _{\rm He}}{2R^4} +\frac{2\alpha _{\rm He}\mu \cos \theta }{R^5} }\nonumber \\ &&-\frac{\alpha _{\rm He}\mu ^2}{2R^6}(3\cos ^2\theta -1) -\frac{IE_{\rm He}IE_{\rm AlO}\alpha _{\rm He}}{2(IE_{\rm He}+IE_{\rm AlO})}\frac{1}{R^6} \nonumber\\ && \times \left[ \alpha _{\parallel }(r)+ \nonumber 2\alpha _{\perp }(r)+0.5\left[ \alpha _{\parallel }(r)-2\alpha _{\perp }(r)\right]\right. \nonumber\\ &&\left. \times (3\cos ^2\theta -1)\right], \end{eqnarray}

(7)

where the experimental values of the polarizability and ionization energy of the Helium atom are, respectively, α_He = 1.41 a|$_0^3$| (Olney et al. 1997) and IE_He = 24.587 eV (Linstrom & Mallard 2001). The ionization energy of AlO⁺ is fixed to its experimental value |$IE_{\rm AlO^+}= 9.46$| eV (Linstrom & Mallard 2001), while the ab initio parallel and perpendicular components of the polarizability of AlO⁺ were interpolated to

\begin{eqnarray} \mu (r) &=& \sum _{k=0}^{4} a_k(r-r_e)^k,\nonumber \\ \alpha _{\perp }(r) &=& \sum _{k=0}^{7} b_k(r-r_e)^k, \nonumber\\ \alpha _{\parallel }(r) &=& \sum _{k=0}^{5} c_k(r-r_e)^k. \end{eqnarray}

(8)

The final PES is obtained from the following combination of the RKHS and long range contributions:

\begin{eqnarray} V(r,R,\theta )\!=\! V^{\rm RKHS}(r,R,\theta )S(R)\!+\!V^{\rm LR}(r,R,\theta )(1-S(R)), \end{eqnarray}

(9)

where the switching function S(R) is defined by

\begin{equation} S(R)=\frac{1}{2}[1-\tanh {(A_{0}(R-R_{0}))}], \end{equation}

(10)

with A₀ = 0.4 a₀, and R₀ = 24.0 a|$_0^{-1}$|⁠.

2.2 Scattering calculations

The collision dynamics was computed using the Newmat code (Stoecklin, Voronin & Rayez 2002). This code which solves the rovibrational inelastic close-coupling equations in the space-fixed frame was employed previously in the study of several other atom-ionic systems of astrochemical interest like He-CH⁺ (Turpin, Halvick & Stoecklin 2010), He-NO⁺ (Denis-Alpizar & Stoecklin 2015), or He-C₃N⁻ (Lara-Moreno, Stoecklin & Halvick 2017). The close-coupling equations (Arthurs & Dalgarno 1960) were solved using the log-derivative propagator (Manolopoulos 1988) starting from R = 4.0 a₀ (e.g. E(r = 3.0 a₀, R = 4.0 a₀, θ = 180⁰) = 27167 cm⁻¹), while the minimum value of the largest propagation distance was set to 60 a₀ and extended up to 90 a₀ for the lowest collision energy (10⁻³ cm⁻¹) considered in the calculations. The vibration of the diatomic molecule is treated by first solving the exact diatomic equations using the ab initio diatomic potential and a finite basis representation of imaginary exponential, as described by Colbert & Miller (1992). The diatomic rovibrational wave functions are then evaluated along a Gauss-Hermite grid and used to calculate the vibrational part of the intermolecular potential matrix elements. The inclusion of 30 rotational states for each of the three vibrational levels included in the diatomic basis set was sufficient to converge the cross-sections in the [10⁻³, 5000] cm⁻¹ collision energy interval of our calculations. The relative convergence criterion of the inelastic cross-sections as a function of the total angular momentum was taken to be 1  per cent for the whole range of energy and the maximum value of the total angular momentum needed to reach convergence was equal to J = 146 for E = 5000 cm⁻¹. The state-to-state rotational transition rate coefficients were then computed by Boltzman averaging the corresponding cross-sections |$\sigma _{j\rightarrow j^{\prime }}$| at a given temperature T as

\begin{equation} k_{j\rightarrow j^{\prime }}(T)=\sqrt{\frac{8}{\pi \mu k_{\rm B}^{3}T^{3}}}\int \limits _{0}^{\infty }\sigma _{j\rightarrow j^{\prime }}(E_{\rm c})e^{-\frac{E_{\rm c}}{k_{\rm B}T}}{\rm d}E_{\rm c}, \end{equation}

(11)

where k_B is the Boltzmann constant, E_c the collision energy, and the initial and final rotational quantum numbers are, respectively, denoted j and j^΄.

The nuclear spin of Aluminium (I) is equal to 5/2 and leads to a large hyperfine splitting of the rotational levels of AlO⁺, which was recently measured (Breier et al. 2014). Therefore, the rates including the hyperfine structure need to be evaluated. In what follows, the hyperfine levels will be designated by the quantum number F, which varies from |I − j| to |I + j|. We computed the hyperfine transitions rate coefficients using a statistical approach combined with the infinite-order-sudden approximation (IOS). Recently, Faure & Lique (2012) showed that this combined statistical-IOS approach gives results comparable to those obtained using the recoupling method. For a molecule in a ¹Σ electronic state, such as AlO⁺, and within the IOS approximation, the inelastic state-to-state rate coefficients for a transition starting from an excited rotational state can be obtained from those calculated for the initial rotational state j = 0 as

\begin{eqnarray} k_{j\rightarrow j^{\prime }}^{\rm ios}(T)=(2j^{\prime }+1)\sum \limits _L\left(\begin{array}{ccc}j^{\prime }& j& L\\ 0& 0& 0 \end{array}\right)^2k_{0\rightarrow L}(T), \end{eqnarray}

(12)

where the (:::) is a Wigner 3-j symbol, and the k_{0 → L} are considered as the fundamental close coupling rates (Faure & Lique 2012). Using a combination of the IOS and the recoupling formula, the IOS hyperfine rate coefficients can be obtained as

\begin{eqnarray} k_{jF\rightarrow j^{\prime }F^{\prime }}^{\rm ios}(T)=(2j+1)(2j^{\prime }+1)(2F^{\prime }+1)\sum \limits _L \nonumber \\ \left(\begin{array}{ccc}j^{\prime }& j& L\\ 0& 0& 0 \end{array}\right)^2 \left\lbrace \begin{array}{ccc}j^{\prime }& j& L\\ F^{\prime }& F& I \end{array} \right\rbrace ^2k_{0\rightarrow L}(T), \end{eqnarray}

(13)

where the {: : :} is the 6-j symbol. The hyperfine rates are then computed as

\begin{equation} k_{jF\rightarrow j^{\prime }F^{\prime }}^{\rm IOS}(T)=\frac{k_{jF\rightarrow j^{\prime }F^{\prime }}^{\rm ios}(T)}{k_{j\rightarrow j^{\prime }}^{\rm ios}(T)}k_{j\rightarrow j^{\prime }}^{\rm CC}(T), \end{equation}

(14)

where |$k_{j\rightarrow j^{\prime }}^{\rm CC}$| is the rotational close coupling rate coefficients computed in equation (11). Eventually, the results of equation (14) were also compared with those obtained when using the statistical method, in which the hyperfine rates are estimated as

\begin{equation} k_{jF\rightarrow j^{\prime }F^{\prime }}^{\rm STA}(T)=\frac{(2F^{\prime }+1)}{(2I+1)(2j^{\prime }+1)}k_{j\rightarrow j^{\prime }}^{\rm CC}(T), \end{equation}

(15)

3 RESULTS

The contour plot of the PES for the interaction of AlO⁺ (X¹Σ⁺) with He is represented in Fig. 1 in Jacobi coordinates for the equilibrium distance of AlO⁺ (at r = 3.0 a₀). The minimum interaction energy (−320.03 cm⁻¹) represented on this figure is found for R = 5.90 a₀ and θ = 26.0°. A secondary minimum (−55.13 cm⁻¹) can also be seen on the same figure for R = 7.30 a₀ and θ = 180|$_{.}^{\circ}$|0.

This PES was then employed for performing nuclear spin free close-coupling calculations of the rotational de-excitation cross-section of AlO⁺ by collision with He and for collision energies ranging from 10⁻³ up to 5000 cm⁻¹. As this large range of collision energy allows the opening of excited vibrational levels of AlO⁺, we first compare in Fig. 2 the magnitudes of the vibrational and rotational quenching rate coefficients for the (ν = 1, j = 2) initial state of AlO⁺ where the rotational and vibrational quenching rates are, respectively, defined by

\begin{equation} k^{\rm quen-rot}_{\nu ,j} = \sum \limits _{j^{\prime } = 0,j-1}k_{\nu ,j^{\prime }} \end{equation}

(16)

and

\begin{equation} k^{\rm quen-vib}_{\nu ,j} = \sum \limits _{\nu ^{\prime }<\nu ,j^{\prime }}k_{\nu ^{\prime },j^{\prime }}. \end{equation}

(17)

This figure clearly shows that the vibrationnal quenching rates while increasing with temperature remain always at the least (around 1000 K) two orders of magnitude smaller than the rotational one. They can then be safely neglected in the comments of the rotational transition rate coefficients presented below.

Figure 2.

Quenching rate coefficients of AlO⁺ in collision with He from the initial state ν = 1, j = 2.

Fig. 3 shows the de-excitation cross-section for the lower rotational states j = 1, 2, and 3 of AlO⁺. The typical shape and Feshbach resonances can be observed in this figure. The cross-sections monotonously decrease when |Δj| increases, while the Δj = −1 transition is clearly the most favoured as expected for collisions involving diatomic molecules formed by two atoms with large differences of mass.

Figure 3.

Rotational de-excitation cross-sections for the lowest rotational transitions of AlO⁺.

The state-to-state de-excitation rate coefficients for the rotational levels of AlO⁺j = 1–15 in collisions with He were computed from equation (11) and reported in the supplementary materials for several temperatures. While these rates were computed without considering the nuclear spin of Al, the hyperfine state-to-state rate coefficients can be easily obtained from these data by using equation (14) and (15) (Faure & Lique 2012). Fig. 4 shows state-to-state rate coefficients for hyperfine transitions computed using both the IOS and the statistical formula. They show noticeable differences, not only in the magnitudes but also in the law of variation as a function of temperature. In the IOS approximation, the rotational de-excitation from j = 2 to j = 1 conserving F is favoured, while in the statistical approach the rates leading to F = 3.5 are larger as can be seen in Fig. 4. In their study of the collisions of HCN and CN with para-H₂, Faure & Lique (2012) found that the statistical method was inaccurate. They also found that the IOS approach was in good agreement with the exact recoupling calculations and we then follow their recommendation by suggesting to use formula equation (14) in order to obtain the hyperfine resolved state-to-state rate coefficients. This approach was used to calculate some of the lowest hyperfine transition rates, which are reported in the supplementary materials.

Figure 4.

Hyperfine rate coefficients from j = 2, F = 2.5. The final state is indicated as (j, F). The computed IOS rates are represented with blue solid lines, while the statistical rates are represented with red dashed lines.

4 SUMMARY

As AlO was found in several environments of the ISM, the AlO⁺ molecule is suggested to be a good candidate for being also detected in the ISM. We first discussed the possible production mechanisms of this ion in both diffuse and dense molecular clouds. We then presented a three-dimensional model of the PES for the He-AlO⁺ collisions based on a large grid of high-level ab initio calculations. The minimum of this PES (−320.03 cm⁻¹) is obtained for a bent geometry of the He-AlO⁺ complex (r = 3.0 a₀, and θ = 26|$_{.}^{\circ}$|0). This surface was then employed to perform close coupling calculations including the vibration of AlO⁺ in order to obtain nuclear spin free state-to-state collisional rates coefficients for rotational levels of AlO⁺ up to j = 15 and for temperatures up to 1000 K. The rate coefficients at a given T decrease monotonously when |Δj| increases, while a propensity rule to favour Δj = −1 transitions is clearly at work for this system as expected for collisions involving heteronuclear diatomic molecules. Eventually, the formula needed to obtain straightforwardly from these data the rate coefficients for the hyperfine resolved state-to-state transitions were given and some of the lower state to state hyperfine transition rate coefficients reported.

Acknowledgements

OD-A acknowledges the support from the project CONICYT/FONDECYT/INICIACION/N^o11160005. The support of the COST Action CM1405 (MOLIM: MOLecules In Motion) is also acknowledged. Computer time for this study was provided by the Mésocentre de Calcul Intensif Aquitain, which is the computing facility of Université de Bordeaux et Université de Pau et des Pays de l'Adour.

SUPPORTING INFORMATION

Supplementary data are available at MNRAS online.

hyperfine-rate-AlOp-He.txt

rotational-rates-AlOp-He.txt

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