Single- and Double-electron Capture Cross Sections for O⁶⁺ Ion in Collisions with H₂ Molecules

In atomic physics, charge exchange (or electron capture) is a fundamental process in which an ion takes one or more electrons from the target atom or molecule during collisions. The charge-exchange process was initially studied as a diagnostic for laboratory plasmas (Isler 1977; Hoekstra et al. 1998). Since the first discovery of X-ray emission from comet C/Hyakutake (Lisse et al. 1996) and its interpretation as the result of charge-exchange collisions between highly charged solar-wind oxygen ions and the cometary neutral atmosphere (Cravens 1997), the investigations on charge-exchange collisions have expanded considerably. Over the past decades, the presence and importance of charge-exchange X-ray emission have been confirmed in a broad range of astrophysical environments, particularly in the interfaces where solar-wind ions interact with neutrals in comets and planetary atmospheres (Hoekstra et al. 1998; Cravens 2000, 2002; Holmström et al. 2001; Beiersdorfer et al. 2003; Lallement 2004; Branduardi-Raymont et al. 2007; Robertson et al. 2009; Wargelin et al. 2014). In addition, charge exchange is also considered as a potential mechanism responsible for X-ray emission from stellar winds of supergiants (Pollock 2007), shock rims of supernova remnants (Katsuda et al. 2011; Cumbee et al. 2014), star-forming galaxies (Liu et al. 2012; Zhang et al. 2014), active galactic nuclei (Gu et al. 2017), etc.

Elaborate models (Cravens 1997; Gunell et al. 2005; Smith et al. 2012; Cumbee et al. 2014, 2017; Gu et al. 2016; Liang et al. 2021) have been developed to simulate the astrophysical X-ray emissions induced by charge exchange. In practice, detailed knowledge of charge-exchange cross sections for the relevant ion–neutral collisions are needed as input data. In the charge-exchange model of Smith et al. (2012), which does not include realistic electron-capture cross sections, the hydrogenic method was adopted to obtain the total cross sections, and the approximation of Janev & Winter (1985) was used to obtain the n ℓ-resolved cross sections. Cumbee et al. (2014, 2017) constructed a comprehensive model using the KRONOS ⁵ charge-exchange database, where the cross sections are primarily obtained via multichannel Landau–Zener calculations, supplemented by the results from more accurate quantum-mechanical molecular-orbital close-coupling, atomic-orbital close-coupling, and classical trajectory Monte Carlo methods when available. However, only single-electron capture processes are taken into account in the model. More recently, Liang et al. (2021) set up a charge-exchange model with the inclusion of double-electron capture processes, showing a nonnegligible contribution from the latter processes. The development of these sophisticated charge-exchange models presents an urgent requirement of accurate electron-capture cross sections for relevant ion–neutral collisions.

In the present work, we are interested in the most abundant solar wind-heavy ion O⁶⁺ (Schwadron & Cravens 2000) in collisions with the H₂ molecule. The electronic processes can be broadly categorized into three classes; they are: (i) single-electron capture (SEC) processes,

$$\begin{eqnarray}&&{{\rm{O}}}^{6+}(1{s}^{2})+{{\rm{H}}}_{2}({X}^{1}{{\rm{\Sigma }}}_{g}^{+})\to {{\rm{O}}}^{5+}(1{s}^{2}n\ell )+{{\rm{H}}}_{2}^{+}({X}^{2}{{\rm{\Sigma }}}_{g}^{+})\end{eqnarray} \tag{ 1a }$$

$$\begin{eqnarray}&&\to {{\rm{O}}}^{5+}(1{s}^{2}n{\ell })+2{{\rm{H}}}^{+}+{e}^{-}\end{eqnarray} \tag{ 1b }$$

$$\begin{eqnarray}&&\to {{\rm{O}}}^{5+}(1{s}^{2}n{\ell })+{{\rm{H}}}^{+}+{\rm{H}}(n^{\prime} {\ell }^{\prime} ),\end{eqnarray} \tag{ 1c }$$

(ii) double-electron capture (DEC) into bound states,

$$\begin{eqnarray}&&{{\rm{O}}}^{6+}(1{s}^{2})+{{\rm{H}}}_{2}({X}^{1}{{\rm{\Sigma }}}_{g}^{+})\to {{\rm{O}}}^{4+}(1{s}^{2}2\ell n^{\prime} \ell ^{\prime} )+2{{\rm{H}}}^{+},\end{eqnarray} \tag{ 2 }$$

and (iii) autoionization double capture (ADC),

$$\begin{eqnarray}&&{{\rm{O}}}^{6+}(1{s}^{2})+{{\rm{H}}}_{2}({X}^{1}{{\rm{\Sigma }}}_{g}^{+})\to {{\rm{O}}}^{4+}(1{s}^{2}n\ell n^{\prime} \ell ^{\prime} )+2{{\rm{H}}}^{+}.\end{eqnarray} \tag{ 3 }$$

Experimentally, these electronic processes have been extensively studied in the past. Crandall et al. (1977, 1979) have measured total SEC cross sections in the energy range of 1–6 keV u⁻¹ by using the Oak Ridge National Laboratory Penning ion gauge (ORNL-PIG) ion source, and the charge-state analysis from the electrostatic analyzer was used to determine the charge-exchange cross sections. Phaneuf et al. (1982) extended the SEC measurements to a wider energy region from 0.01 to 10 keV u⁻¹ using the ORNL-PIG ion source for energies above 0.4 keV u⁻¹ and the pulsed-laser ion source for lower energies; the time-of-flight technique permitted cross sections for different charge states to be measured simultaneously. Later on, by means of photon emission spectroscopy, Dijkkamp et al. (1985) and Lubinski et al. (2000) measured absolute state-resolved SEC cross sections for energies 0.94–7.5 keV u⁻¹ and 0.007–1.3 keV u⁻¹, respectively. In the former experiment, total SEC cross sections were also measured by means of charge-state analysis at 0.5–5 keV u⁻¹. More recently, both total SEC and DEC cross sections were measured by Machacek et al. (2014) at 1.17 and 2.33 keV u⁻¹ and Han et al. (2021) for energies ranging from 2.63 to 19.50 keV u⁻¹. Though with different apparatuses, electron cyclotron resonance ion source was used in both experiments, and cross sections for ions in different charge states due to the electron-capture process were measured by applying an appropriate potential field to deflect the ions (see also Greenwood et al. 2001 and Zhang et al. 2018 for more details about the experiments). Despite extensive experimental studies on O⁶⁺ and H₂ collisions, only very few data are available for state-resolved SEC processes limited in the energy range of 0.007–10 keV u⁻¹ (Dijkkamp et al. 1985; Lubinski et al. 2000), and the DEC ones for energies within 1–20 keV u⁻¹ (Machacek et al. 2014; Han et al. 2021). From the theoretical point of view, investigating this collision system remains challenging due to the strong coupling between various channels, the multicenter features of the H₂ molecular target, and the role of the electronic correlation, notably in the important two-electron processes invoked just above. To our knowledge, no detailed theoretical investigation for O⁶⁺ and H₂ collisions has been reported so far.

In this work, we study theoretically single- and double-electron capture processes in a wide-energy domain ranging from 0.1 to 100 keV u⁻¹. We use a two-active-electron, three-center, semiclassical, asymptotic-state close-coupling (SCASCC) method with large basis sets, ensuring a controlled convergence of the cross sections and providing physical insight on this collision system. Total and state-resolved SEC, DEC, and ADC cross sections are presented and compared with available experimental results. In particular, the contributions of ADC processes to the SEC and DEC cross sections are analyzed in detail.

The organization of the paper is as follows. In Section 2, we briefly outline the SCASCC method used in the present calculations. In Section 3, we present a detailed analysis of the total and partial cross sections for SEC, DEC, and ADC processes, including direct comparisons with available experimental data. A brief conclusion follows in Section 4. Atomic units are used throughout, unless explicitly indicated otherwise.

In the present work, the cross sections of the electronic processes occurring in O⁶⁺ and H₂ collisions are calculated using a two-active-electron SCASCC method that has been previously described (e.g., in Sisourat et al. 2011; Gao et al. 2017, 2019, 2021; Zhang et al. 2020). Here we just outline the main features of the approach as they relate to the present calculations. The two-electron time-dependent Schrödinger equation (TDSE) is written as

$$\begin{eqnarray}&&\left[H-i\displaystyle \frac{\partial }{\partial t}\right]{\rm{\Psi }}({{\boldsymbol{r}}}_{1},{{\boldsymbol{r}}}_{2},{\boldsymbol{R}}(t))=0,\end{eqnarray} \tag{ 4 }$$

with H the electronic Hamiltonian defined as

$$\begin{eqnarray}\begin{array}{rcl}H & = & \displaystyle \sum _{i=1,2}\left[-\displaystyle \frac{1}{2}{{\rm{\nabla }}}_{i}^{2}+{V}_{T}({r}_{i})+{V}_{P}({r}_{i}^{p})\right]+\displaystyle \frac{1}{\left|{{\boldsymbol{r}}}_{1}-{{\boldsymbol{r}}}_{2}\right|},\end{array}\end{eqnarray} \tag{ 5 }$$

where r _i and ${{\boldsymbol{r}}}_{i}^{p}={{\boldsymbol{r}}}_{i}-{\boldsymbol{R}}(t)$ are the position vector of the electrons with respect to the center of mass of the target and to the projectile nuclei, respectively. The relative projectile-target position R (t) defines the trajectory in the usual straight-line, constant velocity approximation R (t) = b + v t, where b and v are the impact parameter and velocity, respectively (see Figure 1). Note that our approach is based on the rovibrational sudden approximation: the nuclei of the molecular target are frozen (i.e., fixed R _AB ). This assumption is valid in the energy range considered in this work since the electronic processes take place in a timescale much shorter (sub-femtosecond) than the vibrational motion of the molecular target (>10 fs).

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The term V_T (V_P ) in Equation (5) defines the electron–target (electron–projectile) nuclei interaction. For the present collision system, these terms are defined as

$$\begin{eqnarray}\begin{array}{rcl}{V}_{T}({r}_{i}) & = & \,-\displaystyle \frac{1}{\left|{{\boldsymbol{r}}}_{i}-\tfrac{1}{2}{{\boldsymbol{R}}}_{{AB}}\right|}-\displaystyle \frac{1}{\left|{{\boldsymbol{r}}}_{i}+\tfrac{1}{2}{{\boldsymbol{R}}}_{{AB}}\right|}\\ {V}_{P}({r}_{i}^{p}) & = & \,-\displaystyle \frac{6}{{r}_{i}^{p}}-\displaystyle \frac{2}{{r}_{i}^{p}}(1+\alpha \ {r}_{i}^{p}){e}^{-\beta {r}_{i}^{p}}.\end{array}\end{eqnarray} \tag{ 6 }$$

V_P describes the interaction between an active electron and the O⁶⁺ ion in the frozen-core electron approximation in which the inner-shell electrons are assumed to be inactive. The variational parameters α = 5.074 and β = 10.148 were obtained through an optimization procedure in order to reproduce the O⁵⁺ doublet and O⁴⁺ singlet spectrum (see Table 1).

The TDSE, Equation (4), is solved by expanding the wave function onto a basis set composed of states of the isolated collision partners (i.e., asymptotic states):

$$\begin{eqnarray}\begin{array}{rcl}{\rm{\Psi }}({{\boldsymbol{r}}}_{1},{{\boldsymbol{r}}}_{2},t) & = & \displaystyle \sum _{i=1}^{{N}_{{TT}}}{c}_{i}^{{TT}}(t){{\rm{\Phi }}}_{i}^{{TT}}({{\boldsymbol{r}}}_{1},{{\boldsymbol{r}}}_{2}){e}^{-{{iE}}_{i}^{{TT}}t}\\ & & +\displaystyle \sum _{j=1}^{{N}_{{PP}}}{c}_{j}^{{PP}}(t){{\rm{\Phi }}}_{j}^{{PP}}({{\boldsymbol{r}}}_{1},{{\boldsymbol{r}}}_{2},t){e}^{-{{iE}}_{j}^{{PP}}t}\\ & & +\displaystyle \sum _{k=1}^{{N}_{T}}\displaystyle \sum _{l=1}^{{N}_{P}}{c}_{{kl}}^{{TP}}(t)[{\phi }_{k}^{T}({{\boldsymbol{r}}}_{1}){\phi }_{l}^{P}({{\boldsymbol{r}}}_{2},t)\\ & & +{\phi }_{k}^{T}({{\boldsymbol{r}}}_{2}){\phi }_{l}^{P}({{\boldsymbol{r}}}_{1},t)]{e}^{-i({E}_{k}^{T}+{E}_{l}^{P})t},\end{array}\end{eqnarray} \tag{ 7 }$$

where the superscripts T and TT (P and PP) denote states, including pseudostates, and corresponding energies for which respectively one and two electrons are on the target (projectile). For both electrons, the projectile states contain plane-wave electron translation factors (ETFs), ${e}^{i{\boldsymbol{v}}\cdot {\boldsymbol{r}}-i\tfrac{1}{2}{v}^{2}t}$, ensuring Galilean invariance of the results. Note that for the collision system under consideration, the initial state is singlet so that only singlet one- and two-center states are considered in the expansion Equation (7). The insertion of Equation (7) into Equation (4) results in a system of first-order coupled differential equations, which can be written in matrix form as

$$\begin{eqnarray}&&i\displaystyle \frac{\partial }{\partial t}c(t)={{\rm{S}}}^{-1}({\boldsymbol{b}},{\boldsymbol{v}},t){\rm{M}}({\boldsymbol{b}},{\boldsymbol{v}},t)c(t),\end{eqnarray} \tag{ 8 }$$

where c(t) is the column vector of the time-dependent expansion coefficients, i.e., c^TT , c^PP , and c^TP in Equation (7) and S and M are the overlap and coupling matrices, respectively. These equations are solved using the predictor-corrector variable time-step Adams–Bashford–Moulton method for a set of initial conditions: initial electronic state i, velocity v, impact parameter b, and molecular target orientation defined by (Θ_m , Φ_m ) (see Figure 1).

The probability of a transition i → f is given by the coefficient c_f (≡c^TT , c^PP or c^TP ) as

$$\begin{eqnarray}&&{P}_{{fi}}(v,{\boldsymbol{b}},{R}_{{AB}},{{\rm{\Phi }}}_{m},{{\rm{\Theta }}}_{m})=\mathop{\mathrm{lim}}\limits_{t\to \infty }| {c}_{f}(t){| }^{2}.\end{eqnarray} \tag{ 9 }$$

The corresponding cross sections for a given molecular geometry can be calculated from these probabilities as follows:

$$\begin{eqnarray}&&{\sigma }_{{fi}}(v,{R}_{{AB}},{{\rm{\Theta }}}_{m})=\int {d}^{2}{\boldsymbol{b}}{P}_{{fi}}(v,{\boldsymbol{b}},{R}_{{AB}},{{\rm{\Phi }}}_{m},{{\rm{\Theta }}}_{m}),\end{eqnarray} \tag{ 10 }$$

where the dependence upon Φ_m disappears after integration over b . The total cross sections (averaged over the degrees of freedom of the molecule) are then given by

$$\begin{eqnarray}\begin{array}{rcl}{\sigma }_{{fi}}^{\mathrm{tot}}(v) & = & {\displaystyle \int }_{0}^{\pi }{\displaystyle \int }_{0}^{\infty }\rho ({{\rm{\Theta }}}_{m})\rho ({R}_{{AB}})\times {\sigma }_{{fi}}(v,{R}_{{AB}},{{\rm{\Theta }}}_{m}){{dR}}_{{AB}}d{{\rm{\Theta }}}_{m},\end{array}\end{eqnarray} \tag{ 11 }$$

where ρ(Θ_m ) and ρ(R_AB ) are respectively the distribution of the molecular geometry over Θ_m and R_AB . The former is assumed to be isotropic, i.e.,

$$\begin{eqnarray}&&\rho ({{\rm{\Theta }}}_{m})=\displaystyle \frac{1}{2}\sin ({{\rm{\Theta }}}_{m}).\end{eqnarray} \tag{ 12 }$$

The distribution ρ(R_AB ) over the internuclear distance of the target is generally that corresponding to a given (e.g., ground) vibrational state or a superposition of vibrational states of the molecular target (Caillat et al. 2004). The cross sections should then be evaluated for different values of R_AB and generally averaged over the initial vibrational state distribution. However, it has been shown that considering only the equilibrium distance value R_eq is sufficient to obtain accurate cross sections (Caillat et al. 2004, 2006; Sisourat et al. 2011; Rabadán et al. 2017; Zhang et al. 2020). In the following we therefore consider only a molecular target frozen at the equilibrium distance, the so-called Franck–Condon approximation (Errea et al. 1997). Thus, the ρ(R_AB ) can be expressed as

$$\begin{eqnarray}\begin{array}{rcl}\rho ({R}_{{AB}}) & = & 1\ {if}\ {R}_{{AB}}={R}_{\mathrm{eq}}\\ \rho ({R}_{{AB}}) & = & 0\ {if}\ {R}_{{AB}}\ne {R}_{\mathrm{eq}},\end{array}\end{eqnarray} \tag{ 13 }$$

with R_eq = 1.4 au for H₂.

The total cross sections (Equation (11)) can be rewritten as

$$\begin{eqnarray}&&{\sigma }_{{fi}}^{\mathrm{tot}}(v)=\displaystyle \frac{1}{2}{\int }_{0}^{\pi }d{{\rm{\Theta }}}_{m}\sin ({{\rm{\Theta }}}_{m}){\sigma }_{{fi}}(v,{R}_{\mathrm{eq}},{{\rm{\Theta }}}_{m}).\end{eqnarray} \tag{ 14 }$$

In order to reduce the computational costs, it is often chosen to evaluate the cross sections through an approximate averaging procedure using only three characteristic molecular orientations, namely (Θ_m , Φ_m ) = (0, 0), (π/2, 0), and (π/2, π/2):

$$\begin{eqnarray}\begin{array}{rcl}{\sigma }_{{fi}}^{\mathrm{tot}}(v,{R}_{\mathrm{eq}}) & = & \displaystyle \frac{2\pi }{3}[{\varrho }_{{fi}}(v,{R}_{\mathrm{eq}},0,0)+{\varrho }_{{fi}}(v,{R}_{\mathrm{eq}},\pi /2,0)\\ & & +{\varrho }_{{fi}}(v,{R}_{\mathrm{eq}},\pi /2,\pi /2)]\end{array}\end{eqnarray} \tag{ 15 }$$

and

$$\begin{eqnarray}&&{\varrho }_{{fi}}(v,{R}_{\mathrm{eq}},{{\rm{\Theta }}}_{m},{{\rm{\Phi }}}_{m})={\displaystyle \int }_{0}^{+\infty }{{bP}}_{{fi}}(v,b,{R}_{\mathrm{eq}},{{\rm{\Theta }}}_{m},{{\rm{\Phi }}}_{m}){db}.\end{eqnarray} \tag{ 16 }$$

The approximation has been widely used and proven to give good estimates of the cross sections (see, e.g., Lühr & Saenz 2010; Sisourat et al. 2011; Zhang et al. 2020).

In the present calculations, the one- and two-electron states and pseudostates included in the expansion Equation (7) are expressed in terms of Gaussian-type orbitals (GTOs) and products of them. For the present collision system, a set of 70 GTOs (12 for l = 0; 8 × 3 for l = 1; 4 × 5 for l = 2; and 2 × 7 for l = 3) are used on the O⁶⁺ center, and 7 GTOs for l = 0 taken from Sisourat et al. (2011) are used for each center of H₂. In the wave-function expansion, we have included 2479 singlet states and pseudostates in total: 38 TT (H₂); 575 TP (H₂ ⁺, O⁵⁺); and 1866 PP (O⁴⁺) ones. In Table 1, we show the energies of the relevant O⁵⁺ and O⁴⁺ states compared with the corresponding available data from NIST (Kramida et al. 2021) and Bachau et al. (1990). The overall agreement between our calculated energies and available data is excellent, and at worst the deviation equals to −1.46% for the doubly excited autoionizing states (O⁴⁺(1s²3d4d¹G^e)).

The cross sections reported in the following have been compared with those from a smaller basis set on the O⁶⁺ center built from 53 GTOs (10 for l = 0; 7 × 3 for l = 1; 3 × 5 for l = 2; and 1 × 7 for l = 3). The total direct SEC (processes in Equation (1a)) cross sections from these two sets agree with each other within 10%. The cross sections for the dominant channels (SEC into O⁵⁺(1s²3ℓ and 1s²4ℓ)) from these two sets agree with each other within 30% at lower- and higher-impact energies, while they agree within about 20% in the intermediate-energy region. For the ADC processes, the cross sections differ by less than 3% at low energies, by less than 9% at intermediate energies, and by less than 20% at high-impact energies. For the DEC processes, the cross sections are more than 3 orders of magnitude smaller than the ADC ones. The convergence was found to be better than 50% at the lowest energy (0.1 keV u⁻¹), to be about 30% at intermediate energies, and within 12% at high-impact energies.

3.1. Total SEC and DEC Cross Sections

In Figure 2, our total SEC cross sections for O⁶⁺ ion in collisions with H₂ molecule are presented as a function of impact energy, together with available experimental data(Crandall et al. 1977, 1979; Phaneuf et al. 1982; Dijkkamp et al. 1985; Machacek et al. 2014; Han et al. 2021). The present results are also listed in Table 2. It is shown that the present results of direct SEC, i.e., including only the three processes listed in Equation (1a), decrease monotonically for increasing impact energies. Compared with the available experimental data, our results lie systematically lower than the experimental ones in the whole overlapping energy region. Before commenting on the discrepancy, we will compare our results with the existing data for two-electron processes. We display in Figure 3 the present DEC and ADC cross sections, compared with available experimental data for DEC (Crandall et al. 1977; Machacek et al. 2014; Han et al. 2021). The experimental results do not agree with each other, showing disagreement of a factor of 3 around 3 keV u⁻¹. Our cross sections for the DEC process are more than 1 order of magnitude smaller than the experimental ones, while the present ADC results are about a factor of 3 larger than the experimental data.

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Table 2. Present Total and n ℓ-resolved SEC, DEC, and ADC Cross Sections (in Units of 10⁻¹⁶ cm²) for O⁶⁺ + H₂ Collisions

Direct SEC
E (keV u−1)	O5+(1s23s)	O5+(1s23p)	O5+(1s23d)	O5+(1s24s)	O5+(1s24p)	O5+(1s24d)	O5+(1s24f)	Total SEC
0.10	0.0160	0.0526	0.1535	9.6127	24.3754	3.4349	2.5248	43.3512
0.20	0.0344	0.0898	0.2339	14.6673	13.2150	3.7049	4.6816	39.4840
0.40	0.0799	0.1175	0.1839	17.8953	9.6162	2.3984	5.8566	38.2961
0.60	0.0892	0.1449	0.1692	18.1693	8.9535	2.3636	5.9265	38.0161
1.00	0.1590	0.2082	0.2046	16.9163	7.7414	2.7683	6.6700	36.8370
1.56	0.1926	0.3080	0.2473	14.8537	7.1690	3.2604	7.5361	35.9066
2.63	0.2227	0.4738	0.3343	10.8433	6.6308	4.4566	8.4080	33.6321
4.00	0.2352	0.6716	0.3868	7.7306	6.2688	5.7376	9.0684	32.2791
6.00	0.2831	0.6122	0.5035	4.9464	5.2889	6.3282	10.2174	30.8031
9.00	0.2119	0.5480	0.5890	2.6611	3.7867	6.7511	11.4367	29.4548
12.00	0.1596	0.5040	0.5662	1.5721	2.7879	6.4309	11.8308	27.4867
15.00	0.1775	0.4648	0.5804	0.9777	2.1032	5.8730	11.6577	25.4244
19.50	0.2073	0.4724	0.6073	0.5480	1.4943	5.0388	10.7792	22.5032
25.00	0.2268	0.4975	0.6048	0.3098	1.0122	4.0400	9.3645	19.2025
50.00	0.1765	0.4609	0.5201	0.0894	0.2574	1.1261	4.6686	10.6768
72.25	0.0984	0.3358	0.5164	0.0436	0.1564	0.3406	2.4628	7.3190
100.00	0.0380	0.2030	0.4483	0.0165	0.1101	0.1414	1.2382	5.1876
	DEC into Bound States				ADC
E (keV u−1)	O4+($1{s}^{2}2l3l^{\prime}$)	O4+($1{s}^{2}2l4l^{\prime}$)	O4+($1{s}^{2}2{lnl}^{\prime} ;n\geqslant 5$)	Total DEC	O4+($1{s}^{2}3l3l^{\prime}$)	O4+($1{s}^{2}3l4l^{\prime}$)	O4+($1{s}^{2}3l5l^{\prime}$)	Total ADC
0.10	0.0046	0.0114	0.0180	0.0351	4.7245	13.7343	1.0683	20.4485
0.20	0.0036	0.0099	0.0147	0.0297	5.8560	10.2587	0.9634	18.0987
0.40	0.0027	0.0086	0.0195	0.0313	6.4443	8.2578	0.9155	16.6945
0.60	0.0039	0.0081	0.0140	0.0274	6.7941	7.2339	0.7960	16.0875
1.00	0.0025	0.0080	0.0138	0.0254	6.5141	6.4484	0.9172	15.2058
1.56	0.0033	0.0100	0.0158	0.0301	6.2405	5.5603	0.8071	14.0299
2.63	0.0015	0.0077	0.0155	0.0250	5.8572	4.7677	0.8096	13.0394
4.00	0.0025	0.0110	0.0223	0.0360	5.0143	4.3449	0.8938	11.9547
6.00	0.0041	0.0147	0.0191	0.0379	3.9045	4.3832	1.0903	11.1342
9.00	0.0064	0.0166	0.0182	0.0412	2.9597	4.1733	1.3973	10.4397
12.00	0.0097	0.0207	0.0167	0.0472	2.4884	4.0649	1.7946	10.4928
15.00	0.0122	0.0290	0.0183	0.0595	2.1830	3.7056	2.0870	10.3975
19.50	0.0169	0.0398	0.0210	0.0778	2.1119	3.1998	2.2909	10.3152
25.00	0.0254	0.0444	0.0229	0.0930	2.2129	2.7897	2.2939	10.3466
50.00	0.0583	0.0481	0.0242	0.1324	1.8445	1.6305	0.9618	8.5183
72.25	0.0534	0.0431	0.0247	0.1249	0.8840	0.8684	0.3275	5.8566
100.00	0.0359	0.0327	0.0184	0.0925	0.3336	0.3450	0.1088	3.2052

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It should be noted that the doubly excited states populated by the ADC process were widely believed to completely autoionize (Roncin et al. 1993; Ali et al. 2016), in which only one electron remains on the projectile, thus contributing to the measured SEC cross sections (Machacek et al. 2014; Han et al. 2021). However, a number of experiments for multiply charged ions colliding with the neutral target have reported the stabilization of the doubly excited states populated by ADC, that both electrons could remain on the projectile with a rather high probability (Roncin et al. 1991; Gaboriaud et al. 1993; Martin et al. 1994, 1995; Chen et al. 1996; Hasan et al. 1999; Ali et al. 2016). Possible stabilization mechanisms were also proposed, including the two-step uncorrelated DEC followed by postcollisional autotransfer to Rydberg states (ATR; Bachau et al. 1992; Roncin et al. 1993), feeding doubly excited states with ($n\ll n^{\prime}$) from symmetric ($n=n^{\prime}$) or quasisymmetric ($n\approx n^{\prime}$) ones; the two-step correlated transfer excitation(Stolterfoht et al. 1986; Winter et al. 1987); the one-step correlated double-electron transfer (Chen & Lin 1993); and the two-step transfer excitation involving nucleus–electron interaction(Flechard et al. 1997). Therefore, the ADC processes listed in Equation (3) could also contribute to the measured DEC cross sections.

In Figure 2, we also present the results for the sum of the direct SEC cross sections and 70% of the ADC ones (dashed red line). Our results agree well with available experimental data in the overlapping energy region though they lie slightly lower than the one from Han et al. (2021) for energies above 6 keV u⁻¹. On the other hand, it can be observed from Figure 3 that our results for the sum of DEC cross sections and 30% of the ADC ones (dashed–dotted black line) are in excellent agreement with the most recent measured DEC cross sections. The above comparison suggests that about 70% of the ADC may contribute to the measured SEC cross sections through postcollisional autoionization, while the stabilization of 30% of doubly excited states populated by ADC contributes to the measured DEC cross sections. The latter stabilization is mainly by way of the ATR mechanism since the symmetric ($n=n^{\prime}$) and quasisymmetric ($n\approx n^{\prime}$) doubly excited states are dominantly populated in ADC cross sections (see Figure 6(b)). It should be noted that, although our cross sections were obtained in the ab initio calculations, the contributions of 70% and 30% of ADC in the measured SEC and DEC cross sections were estimated from comparison with experimental measurements (Han et al. 2021).

3.2. nℓ-resolved SEC Cross Section

We now investigate n ℓ-resolved SEC cross sections. Such detailed information on the final-state distribution of captured electrons is of particular interest both in astrophysics and in plasma diagnostics research since it determines the characteristics of the emitted radiation.

Our calculated n ℓ-resolved SEC cross sections for the production of O⁵⁺(1s²4ℓ) are presented in Figure 4, together with the available experimental results from Dijkkamp et al. (1985) and Lubinski et al. (2000). The present results are also displayed in Table 2. Our results show that electron capture to O⁵⁺(1s²4f) is dominant above 3.5 keV u⁻¹. Below this energy, electron capture to O⁵⁺(1s²4s) takes over until about 0.2 keV u⁻¹, at which O⁵⁺(1s²4p) starts to dominate. Comparing our results with the experimental data, overall good agreements can be obtained, except for SEC into O⁵⁺(1s²4d) below 0.2 keV u⁻¹ and SEC into O⁵⁺(1s²4p) at 7 keV u⁻¹, where our results lie slightly higher than experimental data. Furthermore, we extend the cross sections to impact energies higher than 8 keV u⁻¹, for which no data exists.

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Figure 5 shows our calculated cross sections for SEC into O⁵⁺(1s²3ℓ) in comparison with the only available experimental data from Dijkkamp et al. (1985) for the production of O⁵⁺(n = 3). The present results are also listed in Table 2. It is shown that SEC to O⁵⁺(1s²3d) is the dominant capture channel for energies below 1 keV u⁻¹ and above 10 keV u⁻¹, while in between, the dominance of O⁵⁺(1s²3p) channel is seen. As can be seen in Figure 5, our cross sections for the SEC into O⁵⁺(n = 3) are smaller than the only available experimental data by a factor of 2 or worse. We emphasize that the convergence of our calculations has been checked and is better than 20% for SEC into O⁵⁺(1s²3ℓ). The validity of our results is also supported by the overall good agreements with available experimental data shown in Figure 4 for SEC into O⁵⁺(1s²4ℓ) since the lower excited states are obviously better described than the higher excited states with our GTO basis set.

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Note finally that our calculations indicate that the SEC into O⁵⁺(1s²3ℓ and 1s²4ℓ) follows approximately the statistical ℓ distribution at high-impact energies, for which the electron is therefore mainly captured into subshells of the maximum ℓ.

3.3. Shell-resolved DEC and ADC Cross Sections

For completeness, Figures 6(a) and (b), as well as Table 2, show our calculated shell-resolved cross sections for DEC and ADC, respectively. The cross sections for DEC into bound states are more than 2 orders of magnitude smaller than the ADC ones. Our results show that DEC into higher excited states of O⁴⁺($1{s}^{2}2{\ell }n{\ell }^{\prime} ;$ n ≥ 5) is preferred at low energies, while the productions of O⁴⁺($1{s}^{2}2{\ell }4{\ell }^{\prime}$) and O⁴⁺($1{s}^{2}2{\ell }3{\ell }^{\prime}$) become dominant for higher-impact energies. The present total DEC cross sections present a maximum around 60 keV u⁻¹ and decrease about 3 times at low energies; this behavior is due to the increasing importance of the productions for symmetric O⁴⁺($1{s}^{2}3{\ell }3{\ell }^{\prime}$) and quasisymmetric O⁴⁺($1{s}^{2}3{\ell }4{\ell }^{\prime}$) doubly excited states in the two-electron capture processes, as shown in Figure 6(b).

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For the ADC process, it can be observed from Figure 6(b) that the productions for symmetric O⁴⁺($1{s}^{2}3{\ell }3{\ell }^{\prime}$) and quasisymmetric O⁴⁺($1{s}^{2}3{\ell }4{\ell }^{\prime}$) doubly excited states are of comparable importance and dominant in the whole considered energy region. Both of them display oscillatory energy-dependence structures but with opposite phases over the energy range 0.1–50 keV, which tends to indicate that they are strongly coupled. The cross sections for ADC to O⁴⁺($1{s}^{2}3{\ell }5{\ell }^{\prime}$) are about 1 order of magnitude smaller than the ones for ADC to O⁴⁺($1{s}^{2}3{\ell }3{\ell }^{\prime}$) and O⁴⁺($1{s}^{2}3{\ell }4{\ell }^{\prime}$) at lower energies, while they get of comparable importance for energies higher than 15 keV u⁻¹. However, without any other data, it is clear that further experimental and theoretical efforts are required to draw definite conclusions.

In this paper, we have investigated one- and two-electron processes occurring in collisions of O⁶⁺ ions and H₂ molecules over a wide collision energy domain ranging from 0.1 to 100 keV u⁻¹. We used a two-active-electron, three-center, SCASCC method with a large basis set to reach controlled reasonable convergence. To date, our close-coupling description of the collision is the most elaborate one in terms of accounting for electron correlation, molecular structures, and active channels. For total SEC and DEC cross sections, our results are, in general, in good agreement with the experimental ones. The comparison between the present calculations and available experimental data suggests that about 70% of the ADC may contribute to the measured SEC cross sections through postcollisional autoionization, while the stabilization of 30% of doubly excited states populated by ADC via the ATR mechanism contributes to the measured DEC ones. For the n ℓ-resolved SEC cross sections, our results for SEC into dominant channels, i.e., O⁵⁺(1s²4ℓ), are in good agreement with available experimental data. It is shown that the SEC into O⁵⁺(1s²3ℓ and 1s²4ℓ) follows approximately the statistical ℓ distribution at high-impact energies, for which the electron is therefore mainly captured into subshells of the maximum ℓ. Furthermore, we also provide the first set of shell-resolved DEC and ADC cross sections.

Our work provides new data of total and n ℓ-resolved SEC, DEC, and ADC cross sections for O⁶⁺ + H₂ collisions, which are essential to improve the understanding of this relevant collision system and will be helpful for modeling astrophysical X-ray emissions induced by charge exchange.

This work is supported by the National Natural Science Foundation of China (grant No. 12004350), the startup project from Hangzhou Normal University, and the Key Laboratory of Computational Physics of Institute of Applied Physics and Computational Mathematics (grant No. 6142A05QN21001).