Cross Sections and Rate Coefficients for Rotational Excitation of HeH⁺ Molecule by Electron Impact

Abstract

Cross sections for rotational excitation and de-excitation of the HeH⁺ ion by an electron impact are computed using a theoretical approach that combines the UK R-matrix code and the multi-channel quantum defect theory. The thermally-averaged rate coefficients derived from the obtained cross sections are fitted to an analytical formula valid for a wide range of temperatures.

1. Introduction

Cross sections for the electronic, rotational, and vibrational excitation of molecules in collisions with electrons are important for understanding and modeling various plasma environments, such as the interstellar medium (ISM), planetary ionospheres and exospheres, in plasma processing and de-pollution technologies, and others. Measuring cross sections experimentally for such processes is usually difficult and expensive. However, theoretical methods for electron–molecule scattering together with abundant computational resources have made it possible to obtain reliable cross sections numerically, at least for diatomic and small polyatomic molecules.

The excitation of rotational and vibrational states of molecular ions has been studied theoretically for several decades. In particular, in one of the earliest studies of this kind, Boikova and Ob’edkov [1] considered the process using the Coulomb–Born approximation for the low-energy region. The first-order perturbation theory was applied, and a general analytical formula was derived in which dipole and quadrupole moments of the target ion determine the cross section for rotational excitation, while derivatives of the moments with respect to nuclear distances determine the cross section for vibrational excitation. The direct non-resonant excitation mechanism of the molecules was assumed to maintain the validity of the Born–Oppenheimer approximation for such processes. Later, Chu and Dalgarno [2] applied the same Coulomb–Born approximation to compute rate coefficients for the rotational excitation $j=0\to j^{\prime }=1$ of the CH⁺ ion.

Flower [3] applied the semi-classical approximation and the time-dependent perturbation theory to the $j=0\to j^{\prime }=1$ transition for CH⁺ and HeH⁺. The rotation of the target molecule was quantised, while the motion of the incident electron was treated classically. The applicability of method is restricted to incident electron energies $E_{el}\gtrsim 2(E_{j^{\prime }}-E_j)$, where $E_j$ and $E_{j^{\prime }}$ are the energies of the rotational states of the target ion.

In a series of publications, Rabadan et al. [4,5,6] modified the method developed by Chu and Dalgarno [2] for diatomic molecules. In their approach, the scattering matrix is obtained from first principles using the R-matrix approach [7], rather than from the Coulomb–Born approximation. Similarly to Flower [3], cross sections for rotational excitation of CH⁺ and HeH⁺, as well as for NO⁺ [4,5] are computed. However, as in the other studies mentioned above, the method does not account for near-threshold effects and assumes that different rotational levels are degenerate. In particular, it does not account for Rydberg resonances associated with closed rotational states of the neutral molecule, such as rotational Rydberg resonances of the HeH molecule in the case of $e^{-}$-HeH⁺ collisions. As it was shown later [8,9], this assumption is not appropriate if the incident electron couples strongly different rotational states of the target ion. In such a situation, the near-threshold effects should be accounted for to produce accurate rate coefficients at temperatures below 150 K. To address this problem, in Reference [10] the authors extrapolated the numerically-obtained cross sections down to the threshold using Wigner’s threshold law, producing more accurate results compared to those obtained from a kinetic scaling of the previous -matrix study [6] on HeH⁺ and CH⁺. However, closed-channel effects associated with rotational resonances were still neglected in this study [10].

The theoretical method accounts for near-threshold effects, including rovibrational Rydberg resonances, and makes use of first-principle calculations (or experimental spectroscopic data if necessary), and is based on (1) the electron–molecule scattering matrix computed for fixed positions of nuclei (molecular-frame scattering matrix), (2) the idea of the rotational frame transformation [11], and (3) the molecular quantum defect theory (QDT) [12,13], which makes it possible to evaluate the scattering matrix in the laboratory frame (with respect to which the molecule rotates) and excitation cross sections. This method will be referred to below as the QDT method. It has been used in slightly different implementations in theoretical studies of rotational excitation for several molecular ions: ${\mathrm{H}}_2^{+}$ [14,15], ${\mathrm{H}}_3^{+}$ [16], HeH⁺ [17], and CH⁺ [18].

The molecular-frame scattering matrix in the QDT method can be evaluated in different ways. For example, in Reference [16], in calculations of the rotational excitation of ${\mathrm{H}}_3^{+}$, the matrix was obtained by extrapolating quantum defects extracted from ab initio calculations of excited electronic states of ${\mathrm{H}}_3$ for several internuclear geometries of the molecule. A similar method for the evaluation of the scattering matrix was used by Takagi et al. [19,20,21,22] in the study of the dissociative recombination of HeH⁺. Another way to obtain the molecular-frame scattering matrix is to perform electron scattering calculations directly, using first principles. In a recent study [10], Čurík and Greene employed the molecular scattering matrix computed directly using the UK R-matrix method [7] in the calculation of the rotational excitation cross sections in $e^{-}$-HeH⁺ collisions.

In the present study, we discuss a general theoretical approach for the determination of rotational excitation cross sections for collisions of electrons with molecular ions at low scattering energies. A detailed derivation of the theory is presented for symmetric-top and linear target ions. A generalization to asymmetric top polyatomic targets is straightforward. We apply the method to the benchmark $e^{-}$-HeH⁺ system and compute cross sections and rate coefficients for excitation and de-excitation of the ion from the five lowest rotational states. The HeH⁺ ion is one of the simplest molecular ions. It is present in helium-containing plasma, such as in fusion devices. It is thought to be the first molecule formed in the early Universe. It has also been suggested [23,24,25,26,27] that HeH⁺ could be detected in planetary atmospheres, white dwarfs, and the interstellar medium. At low energies, $e^{-}$-HeH⁺ collisions can lead to dissociative recombination, rotational excitation, or dissociation recombination. The dissociative recombination of HeH⁺ has been extensively studied both experimentally and theoretically [19,20,21,22,28]. There are no experimental measurements of cross sections for the rotational excitation of HeH⁺, but there are a few previous theoretical studies [1,3,10,17]. Therefore, the present results can be compared with the previous calculations.

In the present study, essentially the same theoretical method as in Reference [17] is used to represent the rotational excitation of HeH⁺, but the results for a larger number of rotational transitions in HeH⁺ are obtained and an analytical fit of the thermally-averaged rate coefficients is performed, allowing the use of the data in plasma modeling.

The next section of the article presents the theoretical approach. In Section 3 we discuss the cross sections and rate coefficients obtained for the $e^{-}$-HeH⁺ collisions. Section 4 summarizes the obtained results. Finally, in Appendix A, details of the theoretical derivation of the main formulas of Section 2 are provided.

2. Theoretical Approach

A detailed derivation of the present theory is given in Appendix A. Here, we discuss only the main steps in the implementation of the approach.

As the first step in the calculation, the body-frame $e^{-}$-HeH⁺ reactance matrix ${\widehat{K}}^{\Lambda }$ is determined. For rotational-excitation transitions without changing the vibrational state of the target, the reactance matrix can be evaluated using just one internuclear distance corresponding to the equilibrium position of the target ion, $R_{eq}=$ 1.445 bohr for HeH⁺. The ground electronic state of the HeH⁺ ion is $X^1{\Sigma }^{+}$ with the Hartree–Fock electronic configuration of $1{\sigma }^2$. The matrix is obtained numerically using the UK R-matrix code [7,29]. Performing the scattering calculations, the R-matrix sphere of radius 10 bohr is used. Initially, several basis sets, including 6-311G*, DZP, and cc-pVTZ, were tested to investigate the stability of target properties such as dipole moment and ground state energy. Finally, the cc-pVTZ basis set was selected to perform the final calculations. A multicentered configuration interaction wave function expansion was used in the inner region, including two target states. The $e^{-}$-HeH⁺ reactance matrix is smooth at low electron energies, and the lowest electronic resonance appears at about 4 eV at geometries near the equilibrium.

At electron energies below the first excited electronic state of the target ion, different channels of the body-frame reactance matrix are labeled with partial wave labels, including the angular momentum quantum number l and its projection $\lambda$ of the molecular axis. The matrix is block-diagonal, where each block corresponds to a given projection $\Lambda$ of the total angular momentum (ion+electron) of the system. Thus, in practice, the body-frame reactance matrix is calculated separately for each $\Lambda$. At low scattering energies, below 10 eV, the target ion can only be in the ground electronic state ¹Σ⁺. Therefore, in the present calculations, the projection of the electronic angular momentum in a given channel is equal to $\Lambda$.

The reactance matrix is used to compute the body-frame scattering matrix

$$\begin{array}{c}\hfill {\widehat{S}}^{\Lambda }=\frac{\widehat{1}+i{\widehat{K}}^{\Lambda }}{\widehat{1}-i{\widehat{K}}^{\Lambda }},\end{array}$$

(1)

where $\widehat{1}$ is the identity matrix. The laboratory-frame scattering matrix is then obtained by the transformation [30]

$$\begin{array}{c}\hfill {\mathcal{S}}_{j^{\prime }{\mu }^{\prime }{l}^{\prime };j\mu l}^J=\sum _{\lambda {\lambda }^{\prime }}{(-1)}^{l^{\prime }+{\lambda }^{\prime }+l+\lambda }{C}_{l^{\prime }-{\lambda }^{\prime }J{\Lambda }^{\prime }}^{j^{\prime }{\mu }^{\prime }}{C}_{l-\lambda J\Lambda }^{j\mu }{S}_{l^{\prime }{\lambda }^{\prime };l\lambda }^{\Lambda },\end{array}$$

(2)

where J is the total angular momentum of the $e^{-}$-HeH⁺ system, $j,\mu$ and $j^{\prime },{\mu }^{\prime }$ are the angular momenta with their projections on the molecular axis of the target before and after the rotational excitation of HeH⁺, and $C_{l^{\prime }-{\lambda }^{\prime }J{\Lambda }^{\prime }}^{j^{\prime }{\mu }^{\prime }}$ and $C_{l-\lambda J\Lambda }^{j\mu }$ are Clebsch–Gordan coefficients.

The total energy E of the system is the sum $E=E_{el}+E_{j\mu }$ of the relative kinetic energy $E_{el}$ and the energy $E_{j\mu }$ of the initial state of the target. At a given energy E, the size $N_o$ of the scattering matrix should be equal to the number of open channels with energies below E. Therefore, $N_o$ varies with the energy depending on how many rotational states are open for a given E. However, the scattering matrix ${\mathcal{S}}^J$ in Equation (2) does not contain information about which channels are open or closed. The actual scattering matrix $S^J$ with the correct energy dependence is obtained from ${\mathcal{S}}^J$ using the procedure of the closed-channel elimination [31,32] according to Equation (A15) of Appendix A. The resulting “physical” scattering matrix $S^J$ represents properly rotational resonances associated with the closed rotational levels of the target.

Having the “physical” scattering matrix $S^J$, the cross section for the rotational excitation or de-excitation of the linear molecule by an electron impact is given as

$$\begin{array}{c}\hfill {\sigma }_{j^{\prime }{\mu }^{\prime }\leftarrow j\mu }\left(E_{el}\right)=\frac{1}{2j+1}\frac{\pi }{k_j^2}\sum _{J,l,l^{\prime }}(2J+1){\left|e^{i(l\pi /2+{\sigma }_l)}{S}_{j^{\prime }{\mu }^{\prime }{l}^{\prime };j\mu l}^J{e}^{-i(l^{\prime }\pi /2+{\sigma }_{l^{\prime }})}\right|}^2,\end{array}$$

(3)

assuming that the initial $j\mu$ and final $j^{\prime }{\mu }^{\prime }$ rotational states are different (an inelastic process) and that the vibrational state is not changed during the process. In the above formula, ${\sigma }_l$ is the Coulomb phase shift (see Equation (A5) in Appendix A).

3. Cross Sections and Rate Coefficients

Cross sections for electron impact transitions between the lowest five $j,j^{\prime }=$ 0–4 rotational states of HeH⁺ were computed. Note that the only allowed projection $\mu$ in Equation (3) for HeH⁺ in its ground electronic state is zero. Therefore, for scattering energies below the first excited ionization threshold of HeH⁺, $\mu ={\mu }^{\prime }=0$ in Equation (3). Some examples are given in Figure 1. Solid lines in the figure represent the results obtained using the complete theoretical approach described in the previous section. The cross sections exhibit a strong resonant character. The resonances are produced by closed rotational states of the target. These resonances are washed out when thermally-averaged rate coefficients are computed. Therefore, in the calculation of the rate coefficients, one can use cross sections averaged over the resonances. Such averaged cross sections can be computed directly from the energy-independent scattering matrix ${\mathcal{S}}^J$, replacing in Equation (3) the matrix elements $S_{j^{\prime }{\mu }^{\prime }{l}^{\prime };j\mu l}^J$ of the physical scattering matrix with the corresponding elements ${\mathcal{S}}_{j^{\prime }{\mu }^{\prime }{l}^{\prime };j\mu l}^J$ of the energy-independent matrix. The cross sections calculated using the energy-independent scattering matrix are shown by dotted lines in Figure 1. Due to the overall $1/E_{el}$ dependence of the cross sections as a function of the collision energy, it is convenient to see the products $k_j^2{\sigma }_{j^{\prime }{\mu }^{\prime }\leftarrow j\mu }$, which could be viewed as excitation probabilities. They are shown in Figure 2. It is evident that the excitation probabilities obtained in the full treatment, including the closed-channel elimination, oscillate near the averaged value obtained in the treatment without considering the closed channels (i.e., from the energy-independent scattering matrix ${\mathcal{S}}^J$).

The obtained averaged rate coefficients are shown in Figure 3. They are compared with recently published data for the $j=0\leftrightarrow j^{\prime }$ transitions: dotted lines are the calculations by Hamilton et al. [10] and the dashed lines are those of Čurík and Greene [17]. The agreement between the three sets of calculations are perfect to the $0\leftrightarrow 2$ transitions, while for the $0\leftrightarrow 1$ transitions the rate coefficients of Reference [10] are somewhat larger than the present result and the one from Reference [17]. For the $0\leftrightarrow 3$ transitions, the coefficients of Reference [17] are somewhat larger than the present result and the one from Reference [10]. For the $0\leftrightarrow 4$ transitions, the coefficients from the two other calculations agree with each other and are slightly larger than the present result.

The thermally averaged coefficients at low temperatures are sensitive to exact positions and widths of the lowest resonances, because the averaging integral over thermal velocities at low temperatures T is determined only by the small collision energies, $E_{el}\sim k_{B}T$. For example, the actual value of the cross sections for the $j=1\to j^{\prime }=0$ and $j=2\to j^{\prime }=1$ transitions in Figure 1 depends strongly on the position and the widths of the lowest resonances: at very low energies (below 2 meV), the cross sections are very different from the averaged ones shown with blue dotted lines. Therefore, the closed-channel elimination procedure is essential at low temperatures. Computationally, the procedure is not expensive if the number of channels is not very large (e.g., less than a thousand), and therefore, one can use the cross sections with all resonances in calculation of the rate coefficients for all temperatures.

Due to the general $1/E_{el}$-dependence of cross sections ${\sigma }_{j^{\prime }\leftarrow j}$, the calculated rate coefficients behave as $1/\sqrt{T}$ for de-excitation and $exp\left(-{\Delta }_{j^{\prime }j}/T\right)/\sqrt{T}$ for excitation transitions, where ${\Delta }_{j^{\prime }j}=E_{j^{\prime }{\mu }^{\prime }}-E_{j\mu }$ (with $\mu ={\mu }^{\prime }=0$ for the present case) is the excitation energy. Therefore, similarly to References [16,33], for convenience of use, the rate coefficients are fitted to the formula

$${\alpha }_{j^{\prime }\leftarrow j}^{fit}\left(T\right)=\frac{1}{\sqrt{T}}{e}^{-\frac{{\Delta }_{j^{\prime }j}}{T}}{P}_{j^{\prime }j}^{fit}\left(x\right),$$

(4)

where $P_{j^{\prime }j}\left(T\right)$ are smooth functions of temperature and represented by the quadratic polynomial

$$P_{j^{\prime }j}^{fit}\left(x\right)=a_0+a_{1}x+a_2{x}^2\mathrm{and}x=\ln \left(T\right),$$

(5)

where ${\Delta }_{j^{\prime }j}$ is the threshold energy defined as

$${\Delta }_{j^{\prime }j}=\left\{\begin{array}{cc}{E}_{j^{\prime }}-E_j>0\hfill & \mathrm{for} \mathrm{excitation},\hfill \\ 0\hfill & \mathrm{for} \text{(de-)excitation}.\hfill \end{array}\right.$$

(6)

The numerical parameters given in

Table 1. Parameters $a_0$, $a_1$, and $a_2$ of the polynomial $P_{j{j}^{\prime }}^{fit}\left(x\right)$ of Equations (4) and (5) for several pairs of initial and final rotational states for de-excitation $j\leftarrow j^{\prime }$ of HeH⁺, with $j<j^{\prime }$. The probabilities $P_{j^{\prime }j}^{fit}\left(x\right)$ for the opposite (excitation) process, $j\to j^{\prime }$, are obtained from $P_{j{j}^{\prime }}^{fit}\left(x\right)$ multiplying them with the factor $(2j^{\prime }+1)/(2j+1)$ (see Equation (7)). For convenience, we also specify (the second column) the threshold energy ${\Delta }_{j^{\prime }j}$ in units of temperature (K) for the excitation process of the corresponding pair. For the de-excitation processes, ${\Delta }_{j^{\prime }j}=0$.

$\mathit{j}\mathbf{\leftarrow }{\mathit{j}}^{\mathbf{\prime }}$	${\mathbf{\Delta }}_{{\mathit{j}}^{\mathbf{\prime }}\mathit{j}}$ (K)	${\mathit{a}}_{\mathbf{0}}$	${\mathit{a}}_{\mathbf{1}}$	${\mathit{a}}_{\mathbf{2}}$
$0\leftarrow 1$	96	$0.41\times {10}^{-5}$	$-0.18\times {10}^{-6}$	$-0.15\times {10}^{-8}$
$0\leftarrow 2$	289	$0.37\times {10}^{-5}$	$-0.14\times {10}^{-7}$	$0.85\times {10}^{-9}$
$0\leftarrow 3$	578	$0.57\times {10}^{-7}$	$0.33\times {10}^{-9}$	$-0.96\times {10}^{-10}$
$0\leftarrow 4$	964	$0.44\times {10}^{-9}$	$-0.20\times {10}^{-11}$	$0.77\times {10}^{-12}$
$1\leftarrow 2$	192	$0.73\times {10}^{-5}$	$-0.83\times {10}^{-6}$	$0.37\times {10}^{-7}$
$1\leftarrow 3$	482	$0.48\times {10}^{-5}$	$-0.93\times {10}^{-8}$	0.38 $\times {10}^{-9}$
$1\leftarrow 4$	868	0.75 $\times {10}^{-7}$	0.35 $\times {10}^{-9}$	−0.98 $\times {10}^{-10}$
$2\leftarrow 3$	289	0.57 $\times {10}^{-5}$	−0.19 $\times {10}^{-6}$	−0.10 $\times {10}^{-7}$
$2\leftarrow 4$	675	0.53 $\times {10}^{-5}$	0.61 $\times {10}^{-9}$	−0.39 $\times {10}^{-9}$
$3\leftarrow 4$	385	0.32 $\times {10}^{-5}$	0.56 $\times {10}^{-6}$	−0.61 $\times {10}^{-7}$

are such that when used in Equations (4) and (5) along with the temperature T expressed in kelvin, the obtained numerical value of the rate coefficient in Equation (4) will be in units of cm³/s.

Due to the detailed balance principle, the probabilities for the direct $P_{j^{\prime }j}^{fit}\left(x\right)$ ($j^{\prime }\leftarrow j$) and the inverse $P_{j{j}^{\prime }}^{fit}\left(x\right)$ ($j\leftarrow j^{\prime }$) processes are related to each other by the relative degeneracy factor

$$P_{j^{\prime }j}^{fit}\left(x\right)=\frac{2j^{\prime }+1}{2j+1}{P}_{j{j}^{\prime }}^{fit}\left(x\right).$$

(7)

The coefficients $a_i$ ($i=0,1,2$) are obtained numerically for each pair of transitions $j^{\prime }\leftrightarrow j$ and are given in Table 1.

4. Conclusions

In this study, cross sections and thermally-averaged rate coefficients for electron impact rotational transitions in HeH⁺ are computed for the five lowest rotational levels of HeH⁺ using the UK R-matrix method combined with the multichannel quantum defect theory (MQDT). Our improved channel elimination procedure removes this ambiguity to evaluate accurate results at low energy (<0.01 eV). This and our previous study [33] make us believe that once the collisional excitation cross section data is available, the analysis of the intensities of infra-red and microwave regions can provide information for the diagnostics of tokamak, as well as the study of planetary atmospheres and of the interstellar medium.

With certain modifications accounting for a different threshold behavior of the cross section at low collision energies, the present theoretical approach can be extended for collisions between an electron and a neutral molecule. These developments will be published later.