Semiclassical theory of resonant dissociative excitation of molecular ions by electron impact

We start from the basic expression for the differential cross section of DE of molecular ion per unit energy interval of heavy particles relative motion in the final state

$$\begin{align}\hfill \frac{\mathrm{d}{\sigma }_{vJ\to E}^{\text{de}}\left(\varepsilon \to {\varepsilon }^{\prime }\right)}{\mathrm{d}E}& =\frac{4{\pi }^{3}{g}_{\mathrm{f}}}{{k}^{2}{g}_{\mathrm{i}}}\hfill \\ \hfill & \quad {\times}\sum _{lm,{l}^{\prime }{m}^{\prime }}{\left\vert \left\langle {\chi }_{vJ}^{\left(\mathrm{i}\right)}\left(R\right)\vert {V}_{\mathrm{i},\varepsilon lm}^{\mathrm{f},{\varepsilon }^{\prime }{l}^{\prime }{m}^{\prime }}\left(R\right)\vert {\chi }_{EJ}^{\left(\mathrm{f} \right)}\left(R\right)\right\rangle \right\vert }^{2},\hfill \end{align} \tag{ 3 }$$

here ɛ = ℏ² k²/2m_e and ɛ' = ℏ²(k')²/2m_e = $\varepsilon -E-\left\vert {E}_{vJ}\right\vert$ are the incident electron energies in the initial and final channels, respectively; ℏk is its momentum in the initial channel; v J is a rovibrational state of the molecular ion, and E is the energy of the relative motion of heavy particles after the reaction. The ratio of the statistical weights is defined by the statistical weights of the electronic states of the molecular ion, ${g}_{\mathrm{f}}/{g}_{\mathrm{i}}\equiv {g}_{\text{de}}={g}_{{\mathrm{B}\mathrm{A}}^{+}\left(\mathrm{f} \right)}/{g}_{{\mathrm{B}\mathrm{A}}^{+}\left(\mathrm{i}\right)}$. χ_{v J} and χ_EJ are the radial nuclear wave functions. The electronic transition matrix element is

$$\begin{equation}{V}_{\mathrm{i},\varepsilon lm}^{\mathrm{f},{\varepsilon }^{\prime }{l}^{\prime }{m}^{\prime }}\left(\mathbf{R}\right)=\left\langle {\psi }_{{\varepsilon }^{\prime }{l}^{\prime }{m}^{\prime }}\left\vert \left\langle {\varphi }_{\mathrm{f}}\left({\mathbf{r}}_{k},\mathbf{R}\right)\left\vert V\right\vert {\varphi }_{\mathrm{i}}\left({\mathbf{r}}_{k},\mathbf{R}\right)\right\rangle \right\vert {\psi }_{\varepsilon lm}\right\rangle ,\end{equation} \tag{ 4 }$$

where ${\psi }_{\varepsilon lm}\left(\mathbf{r}\right)$ and ${\psi }_{{\varepsilon }^{\prime }{l}^{\prime }{m}^{\prime }}\left(\mathbf{r}\right)$ are the wave functions of the incident electron before and after the collision, respectively. Electronic wave functions of the molecular ion denoted by φ_i(r_k , R) and φ_f(r_k , R) correspond to the initial, U_i(R), and final, U_f(R) electronic terms of BA⁺.

Further we introduce an effective coupling parameter, ${{\Gamma}}_{\varepsilon \to {\varepsilon }^{\prime }}\left(R\right)$, in the form

$$\begin{align}\hfill {{\Gamma}}_{\varepsilon \to {\varepsilon }^{\prime }}\left(R\right)& =\sum _{lm,{l}^{\prime }{m}^{\prime }}{{\Gamma}}_{{\varepsilon }^{\prime }{l}^{\prime }{m}^{\prime },\varepsilon lm}\left(R\right)\hfill \\ \hfill & =2\pi \sum _{lm,{l}^{\prime }{m}^{\prime }}{\left\vert {V}_{\mathrm{i},\varepsilon lm}^{\mathrm{f},{\varepsilon }^{\prime }{l}^{\prime }{m}^{\prime }}\left(R\right)\right\vert }^{2}.\hfill \end{align} \tag{ 5 }$$

Next we adopt a semiclassical approximation for the non-adiabatic transition matrix elements. A more elaborate semiquantal method can be derived using the results of our works [39, 61]. Then (3) becomes

$$\begin{equation}\frac{\mathrm{d}{\sigma }_{vJ\to E}^{\text{de}}\left(\varepsilon \to {\varepsilon }^{\prime }\right)}{\mathrm{d}E}=\frac{2{\pi }^{2}{g}_{\text{de}}{{\Gamma}}_{\varepsilon \to {\varepsilon }^{\prime }}\left({R}_{\omega }\right)}{{k}^{2}{\Delta}{F}_{\mathrm{f}\mathrm{i}}\left({R}_{\omega }\right)}{R}_{\omega }^{2}{\left\vert {\chi }_{vJ}\left({R}_{\omega }\right)\right\vert }^{2},\end{equation} \tag{ 6 }$$

ΔF_fi(R) = |F_f − F_i| being the difference of the slopes, F_k = −dU_k /dR (k = i, f), of the electronic terms of BA⁺. The transition frequency ω is given by

$$\begin{equation}\hslash \omega =\varepsilon -{\varepsilon }^{\prime }=E+\left\vert {E}_{vJ}\right\vert ,\end{equation} \tag{ 7 }$$

while the internuclear separation R_ω is determined by the energy splitting of the electronic terms,

$$\begin{equation}\hslash \omega ={U}_{\mathrm{f}}\left({R}_{\omega }\right)-{U}_{\mathrm{i}}\left({R}_{\omega }\right)={\Delta}{U}_{\mathrm{f}\mathrm{i}}\left({R}_{\omega }\right).\end{equation} \tag{ 8 }$$

With the help of coordinate distribution function

$$\begin{align}\hfill {W}_{vJ}\left(R\right)& =\frac{1}{2J+1}\sum _{M=-J}^{J}{\left\vert {\psi }_{vJM}\left(\mathbf{R}\right)\right\vert }^{2}=\frac{1}{4\pi }{\left\vert {\chi }_{vJ}\left(R\right)\right\vert }^{2},\hfill \\ \hfill & {\int }_{0}^{\infty }{W}_{vJ}\left(R\right)4\pi {R}^{2}\mathrm{d}R=1\hfill \end{align} \tag{ 9 }$$

of a molecular ion with the given magnitudes of vibrational, v, and rotational, J, quantum numbers, we can rewrite expression (6) as

$$\begin{equation}\frac{\mathrm{d}{\sigma }_{vJ\to E}^{\text{de}}}{\mathrm{d}E}=\frac{2{\pi }^{2}{g}_{\text{de}}}{{k}^{2}}\frac{{{\Gamma}}_{\varepsilon \to {\varepsilon }^{\prime }}\left({R}_{\omega }\right)}{{\Delta}{F}_{\mathrm{f}\mathrm{i}}\left({R}_{\omega }\right)}{W}_{vJ}\left({R}_{\omega }\right)4\pi {R}_{\omega }^{2}.\end{equation} \tag{ 10 }$$

To obtain a total cross section of the DE of a molecular ion in a given v J-state, one needs to perform an integration with respect to E:

$$\begin{equation}{\sigma }_{vJ}^{\text{de}}\left(\varepsilon \right)={\int }_{{E}_{\mathrm{min}}}^{{E}_{\mathrm{max}}}\frac{\mathrm{d}{\sigma }_{vJ\to E}^{\text{de}}}{\mathrm{d}E}\mathrm{d}E,\quad {E}_{\mathrm{min}}=0,\end{equation} \tag{ 11 }$$

E_max = $\varepsilon -\left\vert {E}_{vJ}\right\vert$ being the threshold condition. Using (7), we change the integration variable to ℏω with ℏω_min = $\left\vert {E}_{vJ}\right\vert \equiv {\Delta}{U}_{\mathrm{f}\mathrm{i}}\left({R}_{vJ}\right)$ and ℏω_max = $\varepsilon \equiv {\Delta}{U}_{\mathrm{f}\mathrm{i}}\left({R}_{\varepsilon }\right)$. Since E_max > 0, we imply that ${\sigma }_{vJ}^{\text{de}}\left(\varepsilon {< }\left\vert {E}_{vJ}\right\vert \right)$ = 0. Differentiation of (8) by R_ω yields d(ℏω) = ${\Delta}{F}_{\mathrm{f}\mathrm{i}}\left({R}_{\omega }\right)\mathrm{d}{R}_{\omega }$, and the cross section of process (2) becomes

$$\begin{equation}{\sigma }_{vJ}^{\text{de}}=\frac{8{\pi }^{3}{g}_{\text{de}}}{{k}^{2}}\underset{{R}_{\mathrm{min}}^{\text{de}}}{\overset{{R}_{\mathrm{max}}^{\text{de}}}{\int }}{{\Gamma}}_{\varepsilon \to {\varepsilon }^{\prime }}\left({R}_{\omega }\right){W}_{vJ}\left({R}_{\omega }\right){R}_{\omega }^{2}\mathrm{d}{R}_{\omega },\end{equation} \tag{ 12 }$$

with ${R}_{\mathrm{min}}^{\text{de}}={R}_{\varepsilon }$ and ${R}_{\mathrm{max}}^{\text{de}}={R}_{vJ}$. Note that in pure classical approximation for the nuclear particle relative motion we have ${R}_{\mathrm{min}}^{\text{de}}=\mathrm{max}\left\{{R}_{\varepsilon },{a}_{vJ}^{\mathrm{i}}\right\}$ and ${R}_{\mathrm{max}}^{\text{de}}=\mathrm{min}\left\{{R}_{vJ},{b}_{vJ}^{\mathrm{i}}\right\}$, where the ${a}_{vJ}^{\mathrm{i}}$ and ${b}_{vJ}^{\mathrm{i}}$ are the left and right classical turning points corresponding to a given v J state in the effective potential energy, ${U}_{\text{eff}}^{\left(J\right)}\left(R\right)={U}_{\mathrm{i}}\left(R\right)+{\hslash }^{2}{\left(J+1/2\right)}^{2}/\left(2\mu {R}^{2}\right)$, i.e., ${U}_{\text{eff}}\left({a}_{vJ}^{\mathrm{i}}\right)={U}_{\text{eff}}\left({b}_{vJ}^{\mathrm{i}}\right)=-\left\vert {E}_{vJ}\right\vert$, ${a}_{vJ}^{\mathrm{i}}{< }{b}_{vJ}^{\mathrm{i}}$. μ is the reduced mass of the nuclei of BA⁺ ion.

The rate constant, ${\alpha }_{vJ}^{\text{de}}$, of the DE of a molecular ion in a given v J-state is obtained by averaging of ${v}_{\varepsilon }{\sigma }_{vJ}^{\text{de}}\left(\varepsilon \right)$ over the electron velocity distribution. For brevity's sake we introduce the statistical weights of the unit volume of the free electrons at temperature T_e and the free heavy particles with reduced mass μ at gas temperature T:

$$\begin{equation}{g}_{{T}_{\mathrm{e}}}={\left(\frac{{m}_{e}{k}_{\mathrm{B}}{T}_{\mathrm{e}}}{2\pi {\hslash }^{2}}\right)}^{3/2},\quad {g}_{T}={\left(\frac{\mu {k}_{\mathrm{B}}T}{2\pi {\hslash }^{2}}\right)}^{3/2}.\end{equation} \tag{ 13 }$$

For Maxwellian distribution corresponding to electronic temperature T_e, one gets

$$\begin{align}\hfill {\alpha }_{vJ}^{\text{de}}\left({T}_{\mathrm{e}}\right)& =\frac{{g}_{\text{de}}}{\hslash {g}_{{T}_{\mathrm{e}}}}{\int }_{\left\vert {E}_{vJ}\right\vert }^{\infty }\mathrm{exp}\left[-\frac{\varepsilon }{{k}_{\mathrm{B}}{T}_{\mathrm{e}}}\right]\mathrm{d}\varepsilon \hfill \\ \hfill & \quad {\times}{\int }_{{R}_{\mathrm{min}}}^{{R}_{\mathrm{max}}}{{\Gamma}}_{\varepsilon \to {\varepsilon }^{\prime }}\left({R}_{\omega }\right){W}_{vJ}\left({R}_{\omega }\right)4\pi {R}_{\omega }^{2}\mathrm{d}{R}_{\omega }.\hfill \end{align} \tag{ 14 }$$

It is convenient to rewrite (14) as follows,

$$\begin{align}\hfill {\alpha }_{vJ}^{\text{de}}\left({T}_{\mathrm{e}}\right)& =\frac{{g}_{\text{de}}}{\hslash {g}_{{T}_{\mathrm{e}}}}{\int }_{0}^{{R}_{\mathrm{max}}^{vJ}}{\left\langle {{\Gamma}}_{\varepsilon \to {\varepsilon }^{\prime }}\left({R}_{\omega }\right)\right\rangle }_{{T}_{\mathrm{e}}}\hfill \\ \hfill & \quad {\times}\mathrm{exp}\left[-\frac{{\Delta}{U}_{\mathrm{f}\mathrm{i}}\left({R}_{\omega }\right)}{{k}_{\mathrm{B}}{T}_{\mathrm{e}}}\right]{W}_{vJ}\left({R}_{\omega }\right)4\pi {R}_{\omega }^{2}\mathrm{d}{R}_{\omega },\hfill \end{align} \tag{ 15 }$$

$$\begin{equation}{\left\langle {{\Gamma}}_{\varepsilon \to {\varepsilon }^{\prime }}\left({R}_{\omega }\right)\right\rangle }_{{T}_{\mathrm{e}}}=\underset{{x}_{\mathrm{min}}}{\overset{{x}_{\mathrm{max}}}{\int }}{{\Gamma}}_{\varepsilon \to {\varepsilon }^{\prime }}\left({R}_{\omega }\right){e}^{\frac{{\Delta}{U}_{\mathrm{f}\mathrm{i}}\left({R}_{\omega }\right)-\varepsilon }{{k}_{\mathrm{B}}{T}_{\mathrm{e}}}}\mathrm{d}\left(\frac{\varepsilon }{{k}_{\mathrm{B}}{T}_{\mathrm{e}}}\right),\end{equation} \tag{ 16 }$$

where ${x}_{\mathrm{min}}={\Delta}{U}_{\mathrm{f}\mathrm{i}}\left({R}_{\omega }\right)/\left({k}_{\mathrm{B}}{T}_{\mathrm{e}}\right)$ and x_max = ∞.

Plasmas of rare gas mixtures considered herein typically feature molecular ions with small dissociation energies D₀ (i.e. D₀ varies from 13.1 to 171 meV for RgXe⁺, Rg = He, Ne, Ar, and Kr). As a result, even at room gas temperatures T  ≳ 300 K one often has k_B T  ≳  ℏω_e, where ℏω_e is the first vibrational quantum of the ion in the bound state. This leads to the population of a rather large number of rovibrational states, so that the total cross section of the DE process is determined by the contributions from the entire v J-quasicontinuum. To provide a reasonable description of the process, one needs to consider the cross sections, ${\sigma }_{T}^{\text{de}}\left(\varepsilon \right)$, averaged over the distribution of the molecular ions over the rovibrational levels. For the simplest case of Boltzmann distribution, ${\sigma }_{T}^{\text{de}}\left(\varepsilon \right)$ can be written as

$$\begin{align}\hfill {\sigma }_{T}^{\text{de}}\left(\varepsilon \right)& =\frac{1}{\mathfrak{s}{Z}_{\text{vr}}}\sum _{vJ}\left(2J+1\right)\mathrm{exp}\left[-\frac{{\mathcal{E}}_{vJ}}{{k}_{\mathrm{B}}T}\right]{\sigma }_{vJ}^{\text{de}}\left(\varepsilon \right),\hfill \\ \hfill {Z}_{\text{vr}}& ={\mathfrak{s}}^{-1}\sum _{vJ}\left(2J+1\right)\mathrm{exp}\left[-\frac{{\mathcal{E}}_{vJ}}{{k}_{\mathrm{B}}T}\right].\hfill \end{align} \tag{ 17 }$$

${Z}_{\text{vr}}\left(T\right)$ is the rovibrational partition function of BA⁺, and $\mathfrak{s}$ is a symmetry factor equal to 2 for homo-nuclear ions with U_i(R) being a Σ-term, and equal to 1 otherwise. Using the quasicontinuous spectrum approximation for rovibrational states, we have

$$\begin{equation}{\sigma }_{T}^{\text{de}}\left(\varepsilon \right)=\frac{{\mathfrak{s}}^{-1}}{{Z}_{\text{vr}}}\underset{0}{\overset{{v}_{\mathrm{max}}}{\int }}\mathrm{d}v\underset{0}{\overset{{J}_{\mathrm{max}}\left(v\right)}{\int }}2J\mathrm{d} J{\mathrm{e}}^{-\frac{{E}_{vJ}+{D}_{0}}{{k}_{\mathrm{B}}T}}{\sigma }_{vJ}^{\text{de}}\left(\varepsilon \right),\end{equation} \tag{ 18 }$$

here ${E}_{vJ}={\mathcal{E}}_{vJ}-{D}_{0}$ is the rovibrational energy referred to its dissociation limit, E_v=0,J=0 = −D₀. With the help of the Bohr–Sommerfeld relation, dv can be expressed as $\mathrm{d}v=\left({T}_{vJ}/\left(2\pi \hslash \right)\right)\mathrm{d}{E}_{vJ}$, so (18) becomes

$$\begin{align}\hfill {\sigma }_{T}^{\text{de}}\left(\varepsilon \right)& =\frac{{\left(\pi \hslash \mathfrak{s}\right)}^{-1}}{{Z}_{\text{vr}}}{\int }_{-{D}_{0}}^{0}\mathrm{d}E{\int }_{0}^{{J}_{\mathrm{max}}}{\sigma }_{v\left(E\right)J}^{\text{de}}\left(\varepsilon \right)\hfill \\ \hfill & \quad {\times}\mathrm{exp}\left[-\frac{E+{D}_{0}}{{k}_{\mathrm{B}}T}\right]{T}_{v\left(E\right)J}J\mathrm{d}J,\hfill \end{align} \tag{ 19 }$$

where E ≡ E_{v J} < 0 and the period T_v(E)J of rovibrational motion over the classically allowed region of internuclear separation R is given by

$$\begin{align}\hfill {T}_{v\left(E\right)J}& =2{\int }_{{a}_{v\left(E\right)J}^{\mathrm{i}}}^{{b}_{v\left(E\right)J}^{\mathrm{i}}}\mathrm{d}R/\left[{V}_{v\left(E\right)J}\left(R\right)\right],\hfill \\ \hfill {V}_{v\left(E\right)J}\left(R\right)& =\sqrt{\left(2/\mu \right)\left[E-{U}_{i}\left(R\right)-{\hslash }^{2}{J}^{2}/\left(2\mu {R}^{2}\right)\right]}.\hfill \end{align} \tag{ 20 }$$

Substitution of expression (12) into (19) yields

$$\begin{align}\hfill {\sigma }_{T}^{\text{de}}\left(\varepsilon \right)& =\frac{8{\pi }^{2}{g}_{\text{de}}}{\hslash {k}^{2}\mathfrak{s}{Z}_{\text{vr}}}{\int }_{-{D}_{0}}^{0}\mathrm{d}E{\int }_{0}^{{J}_{\mathrm{max}}}J\mathrm{d} J{\mathrm{e}}^{-\frac{E+{D}_{0}}{{k}_{\mathrm{B}}T}}{T}_{v\left(E\right)J}\hfill \\ \hfill & \quad {\times}{\int }_{{R}_{\mathrm{min}}^{\text{de}}\left(\varepsilon \right)}^{{R}_{\mathrm{max}}^{\text{de}}\left(v\left(E\right)J\right)}{{\Gamma}}_{\varepsilon \to {\varepsilon }^{\prime }}{W}_{v\left(E\right)J}{R}_{\omega }^{2}\mathrm{d}{R}_{\omega }.\hfill \end{align} \tag{ 21 }$$

For the radial wave function ${\chi }_{vJ}\left(R\right)$ of the rovibrational motion of nuclei in the classically allowed region we use the JWKB approximation [62]

$$\begin{align}\hfill {\chi }_{vJ}\left(R\right)& =\frac{{C}_{vJ}\enspace \mathrm{cos}\enspace {\Phi}\left(R\right)}{R\sqrt{{V}_{vJ}\left(R\right)}},\quad {C}_{vJ}=\frac{2}{\sqrt{{T}_{vJ}}},\hfill \\ \hfill {\Phi}\left(R\right)& =\frac{1}{\hslash }{\int }_{{a}_{1}\left(vJ\right)}^{R}\mu {V}_{vJ}\left(R\right)\mathrm{d}R-\frac{\pi }{4}.\hfill \end{align} \tag{ 22 }$$

Then (9) becomes

$$\begin{equation}{W}_{v\left(E\right)J}\left(R\right)=\frac{{\left(\pi \right)}^{-1}\enspace {\mathrm{cos}}^{2}\enspace {\Phi}\left(R\right)}{{T}_{v\left(E\right)J}{R}^{2}{V}_{v\left(E\right)J}\left(R\right)},\end{equation} \tag{ 23 }$$

where ${a}_{v\left(E\right)J}^{\mathrm{i}}{\leqslant}R{\leqslant}{b}_{v\left(E\right)J}^{\mathrm{i}}$. Using standard approximation $\left\langle {\mathrm{cos}}^{2}\enspace {\Phi}\left(R\right)\right\rangle =1/2$ for the oscillating part of ${W}_{v\left(E\right)J}\left(R\right)$, the Boltzmann-averaged cross section of DE can be written as

$$\begin{align}\hfill {\sigma }_{T}^{\text{de}}\left(\varepsilon \right)& =\frac{2\pi {g}_{\text{de}}}{\hslash {k}^{2}\mathfrak{s}{Z}_{\text{vr}}\left(T\right)}{\int }_{-{D}_{0}}^{0}\mathrm{d}E{\int }_{0}^{{J}_{\mathrm{max}}}2J\mathrm{d} J{\mathrm{e}}^{-\frac{E+{D}_{0}}{{k}_{\mathrm{B}}T}}\hfill \\ \hfill & \quad {\times}{\int }_{{R}_{\mathrm{min}}^{\text{de}}\left(\varepsilon \right)}^{{R}_{\mathrm{max}}^{\text{de}}\left(v\left(E\right)J\right)}\frac{{{\Gamma}}_{\varepsilon \to {\varepsilon }^{\prime }}\left({R}_{\omega }\right)\mathrm{d}{R}_{\omega }}{{V}_{v\left(E\right)J}\left({R}_{\omega }\right)}.\hfill \end{align} \tag{ 24 }$$

On replacing the order of integration, we obtain

$$\begin{align}\hfill {\sigma }_{T}^{\text{de}}\left(\varepsilon \right)& =\frac{2\pi {g}_{\text{de}}}{\hslash {k}^{2}\mathfrak{s}{Z}_{\text{vr}}}\underset{-{D}_{0}}{\overset{0}{\int }}{\mathrm{e}}^{-\frac{E+{D}_{0}}{{k}_{\mathrm{B}}T}}\mathrm{d}E\underset{{R}_{\mathrm{min}}^{\text{de}}\left(\varepsilon \right)}{\overset{{R}_{\mathrm{max}}^{\text{de}}\left(E\right)}{\int }}{{\Gamma}}_{\varepsilon \to {\varepsilon }^{\prime }}\mathrm{d}{R}_{\omega }\hfill \\ \hfill & \quad {\times}{\int }_{0}^{{J}_{\mathrm{max}}}\frac{2J\mathrm{d} J}{\sqrt{\left(2/\mu \right)\left[E-{U}_{i}\left(R\right)-{\hslash }^{2}{J}^{2}/\left(2\mu {R}^{2}\right)\right]}},\hfill \end{align} \tag{ 25 }$$

where the maximal value ${J}_{\mathrm{max}}\left(E,R\right)$ of the rotational quantum number is to be found from the relation ${\hslash }^{2}{J}_{\mathrm{max}}^{2}/\left(2\mu {R}^{2}\right)=E-{U}_{\mathrm{i}}\left(R\right)$ and ${R}_{\mathrm{max}}^{\text{de}}\left(E\right)$ is defined by equation ${\Delta}{U}_{\mathrm{f}\mathrm{i}}\left({R}_{\mathrm{max}}^{\text{de}}\left(E\right)\right)=\vert E\vert$. Since

$$\begin{align}\hfill & \frac{{\left(2\pi \right)}^{-2}}{\hslash {R}^{2}}{\int }_{0}^{{J}_{\mathrm{max}}}\frac{2J\mathrm{d} J}{\sqrt{\left(2/\mu \right)\left[E-{U}_{\mathrm{i}}\left(R\right)-{\hslash }^{2}{J}^{2}/2\mu {R}^{2}\right]}}\hfill \\ \hfill & \quad ={W}_{E}\left(R\right)=\frac{{\mu }^{3/2}}{{\pi }^{2}{\hslash }^{3}\sqrt{2}}\sqrt{E-{U}_{\mathrm{i}}\left(R\right)},\hfill \end{align} \tag{ 26 }$$

with E_min ⩽ E < 0, a(E) ⩽ R ⩽ b(E) ($a\left(E\right)$ and $b\left(E\right)$ are the left and right turning points for the potential energy curve ${U}_{\mathrm{i}}\left(a,b\right)=E$), the DE cross section takes the form

$$\begin{align}\hfill {\sigma }_{T}^{\text{de}}\left(\varepsilon \right)& =\frac{8{\pi }^{3}{g}_{\text{de}}}{{k}^{2}\mathfrak{s}{Z}_{\text{vr}}}{\int }_{-{D}_{0}}^{0}\mathrm{exp}\left(-\frac{E+{D}_{0}}{{k}_{\mathrm{B}}T}\right)\mathrm{d}E\hfill \\ \hfill & \quad {\times}{\int }_{{R}_{\mathrm{min}}^{\text{de}}\left(\varepsilon \right)}^{{R}_{\mathrm{max}}^{\text{de}}\left(E\right)}{{\Gamma}}_{\varepsilon \to {\varepsilon }^{\prime }}\left({R}_{\omega }\right){R}_{\omega }^{2}{W}_{E}\left({R}_{\omega }\right)\mathrm{d}{R}_{\omega }.\hfill \end{align} \tag{ 27 }$$

Then we rewrite (27) as

$$\begin{align}\hfill {\sigma }_{T}^{\text{de}}\left(\varepsilon \right)& =\frac{2{\pi }^{2}{g}_{\text{de}}}{{k}^{2}{Z}_{\text{vr}}}\mathrm{exp}\left(-\frac{{D}_{0}}{{k}_{\mathrm{B}}T}\right)\underset{{R}_{\mathrm{min}}^{\text{de}}\left(\varepsilon \right)}{\overset{\infty }{\int }}{{\Gamma}}_{\varepsilon \to {\varepsilon }^{\prime }}\left({R}_{\omega }\right)\hfill \\ \hfill & \quad {\times}{W}_{T}^{\text{de}}\left({R}_{\omega }\right)4\pi {R}_{\omega }^{2}\mathrm{d}{R}_{\omega },\hfill \end{align} \tag{ 28 }$$

here ${W}_{T}^{\text{de}}\left(R\right)$ is the Boltzmann averaged coordinate distribution function taking into account the integral contribution of all discrete rovibrational states,

$$\begin{equation}{W}_{T}^{\text{de}}\left(R\right)=\frac{{\mu }^{3/2}/\sqrt{2}}{\mathfrak{s}{\pi }^{2}{\hslash }^{3}}{\int }_{{E}_{\mathrm{min}}}^{0}\sqrt{E-{U}_{\mathrm{i}}\left(R\right)}{\mathrm{e}}^{-\frac{E}{{k}_{\mathrm{B}}T}}\mathrm{d}E,\end{equation} \tag{ 29 }$$

${E}_{\mathrm{min}}=-\left\vert \mathrm{min}\left\{{U}_{\mathrm{i}}\left(R\right),\enspace 0\right\}\right\vert$. The integral over E in (29) is reduced to the incomplete gamma function $\gamma \left(3/2,z\right)={\int }_{0}^{z}{t}^{1/2}{\mathrm{e}}^{-t}\mathrm{d}t$, so that ${W}_{T}^{\text{de}}$ can be written as (U_i(R) < 0)

$$\begin{equation}{W}_{T}^{\text{de}}\left(R\right)=\frac{1}{\mathfrak{s}}{\left(\frac{\mu {k}_{\mathrm{B}}T}{2\pi {\hslash }^{2}}\right)}^{3/2}{e}^{-\frac{{U}_{\mathrm{i}}\left(R\right)}{{k}_{\mathrm{B}}T}}\frac{\gamma \left(\frac{3}{2},\frac{\left\vert {U}_{\mathrm{i}}\left(R\right)\right\vert }{{k}_{\mathrm{B}}T}\right)}{{\Gamma}\left(3/2\right)}.\end{equation} \tag{ 30 }$$

The partition function ${Z}_{\text{vr}}\left(T\right)$ is evaluated using semiclassical expression [63]

$$\begin{align}\hfill {Z}_{\text{vr}}\left(T\right)& =\frac{1}{\mathfrak{s}}{\left(\frac{\mu {k}_{\mathrm{B}}T}{2\pi {\hslash }^{2}}\right)}^{3/2}{e}^{-\frac{{D}_{0}}{{k}_{\mathrm{B}}T}}{\int }_{{R}_{0}}^{\infty }{e}^{-\frac{{U}_{\mathrm{i}}\left(R\right)}{{k}_{\mathrm{B}}T}}\hfill \\ \hfill & \quad {\times}\left[1-\frac{2}{\sqrt{\pi }}{\Gamma}\left(\frac{3}{2},\frac{\left\vert {U}_{\mathrm{i}}\left(R\right)\right\vert }{{k}_{\mathrm{B}}T}\right)\right]4\pi {R}^{2}\mathrm{d}R.\hfill \end{align} \tag{ 31 }$$

The rate constant of the electron-impact DE of molecular ions, taking into account the contributions from all rovibrational states, is obtained by averaging of ${v}_{\varepsilon }{\sigma }_{T}^{\text{de}}\left(\varepsilon \right)$ over the electron velocity distribution corresponding to a given T_e. In case of Maxwellian distribution one has

$$\begin{align}\hfill {\alpha }^{\text{de}}\left({T}_{\mathrm{e}},T\right)& ={\left\langle {v}_{\varepsilon }{\sigma }_{T}^{\text{de}}\left(\varepsilon \right)\right\rangle }_{{T}_{\mathrm{e}}}=\frac{\sqrt{8/\left(\pi {m}_{\mathrm{e}}\right)}}{{\left({k}_{\mathrm{B}}{T}_{\mathrm{e}}\right)}^{3/2}}\underset{0}{\overset{\infty }{\int }}\varepsilon {\sigma }_{T}^{\text{de}}\left(\varepsilon \right)\hfill \\ \hfill & \quad {\times}\mathrm{exp}\left(-\varepsilon /\left({k}_{\mathrm{B}}{T}_{\mathrm{e}}\right)\right)\mathrm{d}\varepsilon .\hfill \end{align} \tag{ 32 }$$

It is convenient to use formula (28) for the cross section and switch to the integration over ℏω,

$$\begin{align}\hfill {\alpha }^{\text{de}}\left({T}_{\mathrm{e}},T\right)& =4\pi \frac{{k}_{\mathrm{B}}{T}_{\mathrm{e}}}{\hslash }\frac{{\tilde {g}}_{\text{de}}}{{Z}_{\text{vr}}}\mathrm{exp}\left(-\frac{{D}_{0}}{{k}_{\mathrm{B}}T}\right){\left(\frac{\mu T}{{m}_{\mathrm{e}}{T}_{\mathrm{e}}}\right)}^{3/2}\hfill \\ \hfill & \quad {\times}\underset{0}{\overset{\infty }{\int }}{e}^{-\frac{\varepsilon }{{k}_{\mathrm{B}}{T}_{\mathrm{e}}}}\mathrm{d}\left(\frac{\varepsilon }{{k}_{\mathrm{B}}{T}_{\mathrm{e}}}\right)\underset{0}{\overset{\mathrm{min}\left(\varepsilon ,{\Delta}{U}_{\mathrm{f}\mathrm{i}}\left({R}_{0}\right)\right)}{\int }}{{\Gamma}}_{\varepsilon \to {\varepsilon }^{\prime }}\left({R}_{\omega }\right)\hfill \\ \hfill & \quad {\times}\mathrm{exp}\left(\frac{-{U}_{\mathrm{i}}\left({R}_{\omega }\right)}{{k}_{\mathrm{B}}T}\right)\frac{\gamma \left(\frac{3}{2},\frac{\left\vert {U}_{\mathrm{i}}\left({R}_{\omega }\right)\right\vert }{{k}_{\mathrm{B}}T}\right)}{{\Gamma}\left(3/2\right)}\frac{{R}_{\omega }^{2}\hslash \mathrm{d}\omega }{{\Delta}{F}_{\mathrm{i}\mathrm{f}}\left({R}_{\omega }\right)},\hfill \end{align} \tag{ 33 }$$

where ${\tilde {g}}_{\text{de}}$ = ${g}_{\text{de}}/\mathfrak{s}$. Then we change the order of integration and make use of (16). The final expression for the rate constant of the DE follows from changing to the integration over R_ω

$$\begin{align}\hfill {\alpha }^{\text{de}}\left({T}_{\mathrm{e}},T\right)& ={\left(\frac{\mu T}{{m}_{\mathrm{e}}{T}_{\mathrm{e}}}\right)}^{3/2}\frac{{k}_{\mathrm{B}}{T}_{\mathrm{e}}}{\hslash }\frac{4\pi {\tilde {g}}_{\text{de}}}{{Z}_{\text{vr}}}\mathrm{exp}\left(-\frac{{D}_{0}}{{k}_{\mathrm{B}}T}\right)\hfill \\ \hfill & \quad {\times}{\int }_{{R}_{0}}^{\infty }{\left\langle {{\Gamma}}_{\varepsilon \to {\varepsilon }^{\prime }}\left({R}_{\omega }\right)\right\rangle }_{{T}_{\mathrm{e}}}\left(\gamma \left[\frac{3}{2},\frac{\left\vert {U}_{\mathrm{i}}\left({R}_{\omega }\right)\right\vert }{{k}_{\mathrm{B}}T}\right]/{\Gamma}\left(3/2\right)\right)\hfill \\ \hfill & \quad {\times}\mathrm{exp}\left(-\frac{{U}_{\mathrm{i}}\left({R}_{\omega }\right)}{{k}_{\mathrm{B}}T}\right)\mathrm{exp}\left(-\frac{{\Delta}{U}_{\mathrm{f}\mathrm{i}}\left({R}_{\omega }\right)}{{k}_{\mathrm{B}}{T}_{\mathrm{e}}}\right){R}_{\omega }^{2}\mathrm{d}{R}_{\omega }.\hfill \end{align} \tag{ 34 }$$

As it was shown above, the cross sections and rate constants of the DE process depend on the magnitude of the coupling parameter, ${{\Gamma}}_{\varepsilon \to {\varepsilon }^{\prime }}^{\text{de}}\left({R}_{\omega }\right)$, at the transition point. In general case, determining the coupling parameter is a complicated problem involving extensive many-electron calculations done using quantum chemistry methods. While the formulae derived can be used in detailed description of the DE of the molecular ions by electron impact, here we are primarily interested in statistically averaged cross sections and rate constants owing to the physical conditions considered. To provide a reasonable description of these quantities one can often make use of simplified models for the calculation of the coupling parameter.

For homo-nuclear ions the dipole approximation for the interaction potential [48, 49] is known to produce results in good agreement with more elaborate studies. Dipole approximation for ${{\Gamma}}_{\varepsilon \to {\varepsilon }^{\prime }}\left({R}_{\omega }\right)$ yields

$$\begin{equation}{{\Gamma}}_{\varepsilon \to {\varepsilon }^{\prime }}\left({R}_{\omega }\right)=\frac{4}{3\sqrt{3}}G\left(\varepsilon ,\varepsilon -\hslash \omega \right)\frac{{d}_{\mathrm{f}\mathrm{i}}^{2}\left({R}_{\omega }\right)}{{\left(e{a}_{0}\right)}^{2}},\end{equation} \tag{ 35 }$$

here a₀ is the Bohr radius, $G\left(\varepsilon ,\varepsilon -\hslash \omega \right)$ is the Gaunt factor equal to unity in the framework of Kramers approximation, and ${d}_{\mathrm{f}\mathrm{i}}\left({R}_{\omega }\right)$ is the transition matrix element of the dipole moment evaluated at R = R_ω . Explicit formulae for G can be found in, e.g., [64, 65].

For hetero-nuclear rare gas ions considered herein the situation is even more complex. Due to strong influence of the spin–orbit coupling, no detailed studies of the coupling parameter were published to date. However, as was demonstrated in our recent work [39] on recombination, a simple semi-analytic approach based on vacancy model [66, 67] can be sufficient to reproduce the general features of the cross sections and rate constants. The approach reduces the electronic shell of the ion to the effective one-electron vacancy ('hole') shell interacting with the incident electron. The resulting expression for Γ_ɛ→ɛ' is given by

$$\begin{align}\hfill {{\Gamma}}_{\varepsilon \to {\varepsilon }^{\prime }}& =\frac{{n}^{3}{{\Gamma}}_{\varepsilon \to n}}{2Ry},\quad {{\Gamma}}_{\varepsilon \to n}=\sum _{l=0}^{n-1}\left(2l+1\right){{\Gamma}}_{nl\to \varepsilon },\hfill \\ \hfill {{\Gamma}}_{nl\to \varepsilon }& =2\pi {\nu }_{l}/\left[\left(2l+1\right){n}_{{\ast}}^{3}\right],\quad {{\Gamma}}_{\varepsilon \to {\varepsilon }^{\prime }}=\pi {\nu }_{l}/Ry,\hfill \end{align} \tag{ 36 }$$

where n_* is the effective principal quantum number of A(n) atom, ν_l = 2∑_l' γ_ll'/25 and γ_ll' is the radial integral [66, 67] depending on the wave functions of the Rydberg electron of A(n) and of the incident electron. In the first-order approximation of the vacancy model (see [39, 66, 67]), the R-dependence of γ_ll' can be neglected. The value Γ_ɛ→ɛ' for the free–free electron transitions, according to (36), is expressed through the autoionization width, Γ_n→ɛ , of Rydberg level.

First we present the results obtained for RgXe⁺ ions which are the systems the paper is focused on. Plotted in figure 2 are the cross sections of the resonant electron-impact dissociation σ^de(ɛ, T) (cm²) of KrXe⁺ 2(a), ArXe⁺ 2(b) and NeXe⁺ 2(c) at gas temperatures of T = 300, 500 and 900 K as functions of the incident electron energy, ɛ. The dependences exhibit qualitatively different behaviour for the systems considered, especially at low ɛ, which results from the substantial differences in the dissociation energies (see table 1). At low T for ions with relatively high and moderate D₀, like KrXe⁺ and ArXe⁺, σ_de(ɛ, T) has a distinct threshold-like ɛ-dependence as the nuclei are primarily localized in the vicinity of the equilibrium distance, R_e, of U_i(R). The position of the cross section maximum can be estimated from the condition that the point R_ɛ of the resonant ɛ → ɛ' transition with ɛ' ≈ 0 coincides with the equilibrium distance so that ΔU_fi(R_e) = ɛ. The increase of T smears out the localization of the nuclei and allows for DE process to occur at ɛ → 0. For ArXe⁺ the threshold is already suppressed at T = 900 K. At high ɛ the cross sections do not depend on T as the transitions may occur at any R with U_i(R) < 0 so that the dissociation is not affected by the features of the vibrational state population. When D₀ of the molecular ion is low, as in NeXe⁺, σ^de(ɛ, T) does not have a maximum and exhibits weak T-dependence as indicated by the inset to figure 2(c).

Table 1.  Constants of molecular rare gas ions considered.

Molecular Ion	${R}_{\mathrm{e}}\enspace \left(\mathrm{\AA}\right)$	D0 (eV)	ℏωe (meV)	Ref.
NeXe+	3.05	0.033	6.9	[68]
ArXe+	3.1	0.171	11	[68]
KrXe+	3	0.4	13	[68]
${\mathrm{X}\mathrm{e}}_{2}^{+}$	3.1	0.98	17	[69]
${\mathrm{A}\mathrm{r}}_{2}^{+}$	2.4	1.34	40	[70]

The rate constants of the resonant DE, α^de(T_e, T), of KrXe⁺ + e are presented in figure 3(a). It is seen that for a given gas temperature α^de(T_e, T) first increases and then decreases with the increase of T_e, which directly follows from the ɛ-dependence of the cross sections (see figure 2). For KrXe⁺ + e, which has moderate D₀, unlike the cross sections, the rate constants vary substantially with the gas temperature change, especially for T_e ≲ 5000 K. Also plotted in figure 3 are the integral rate constants of the DR resulting in the population of atomic (xenon) Rydberg states, ${\alpha }_{\text{Ry}}^{\text{dr}}\left({T}_{\mathrm{e}},T\right)$. The rate constants were evaluated by the summation of the respective rates of DR reaction channels leading to B + A(n) (A = Xe, B = Kr) final state, over the Rydberg Xe states with principal quantum numbers n ⩾ 8, ${\alpha }_{\text{Ry}}^{\text{dr}}\left({T}_{\mathrm{e}},T\right)={\sum }_{n{\geqslant}8}\;{\alpha }_{n}^{\text{dr}}\left({T}_{\mathrm{e}},T\right)$. Here ${\alpha }_{n}^{\text{dr}}\left({T}_{\mathrm{e}},T\right)$ denotes the rate constant of the direct mechanism of DR process, BA ⁺ + e → B + A(n), which we determined using recently developed theoretical approach [39]. One can see that ${\alpha }_{\text{Ry}}^{\text{dr}}\left({T}_{\mathrm{e}},T\right)$ decreases monotonically with the increase of the electronic temperature and has a weak dependence on T caused by the slow increase of probability of dissociative capture resulting in the population of Rydberg states. This process realizes via the non-adiabatic electron transitions which occur far from the equilibrium distance. The comparison of the rate constants of DE and DR processes shows that for KrXe⁺ + e the electron capture predominates at low T_e, particularly when gas temperatures, T, are not high. Under such conditions one can neglect the role of DE. In contrast to that, at T_e  ≳  10⁴ K the molecular ion breakup mainly proceeds via process (2). It should be noted that elevated T may change the situation significantly: α^de(T_e, T) may become comparable with ${\alpha }_{\text{Ry}}^{\text{dr}}\left({T}_{\mathrm{e}},T \right)$ even at T_e = 3000 K. The maximal values of α^de(T_e, T) go up to 10⁻⁷ cm³ s⁻¹, so the process is rather efficient and should be accounted for in the plasma kinetic models.

For weakly-bound molecular cations process (2) is competitive with DR even at temperatures below 300 K. Figure 3(b) shows that for NeXe⁺ at T = 100 K the rates, α^de(T_e, T) and ${\alpha }_{\text{Ry}}^{\text{dr}}\left({T}_{\mathrm{e}},T\right)$, become equal at T_e = 310 K. Such behaviour is a result of the low efficiency of DR caused by small probability density of the cation nuclei at high R required for the transition to occur. One can see from the results of RD process calculations (dashed–dotted curve in figure 3(b)) that even at very low T the value of α^rd deviates from that of ${\alpha }_{\text{Ry}}^{\text{dr}}$ so the contribution from DE cannot be neglected for ions with small D₀. It is important to note that α^de(T_e, T) has weak T and T_e dependences and has even higher values as compared to the case of ions with moderate D₀. Thus, DE process should always be included in the self-consistent description of electronic transitions in plasmas containing molecular ions with small D₀.

In this section we apply the method developed to the study of the resonant electron-impact DE of several homo-nuclear molecular ions. As compared to the hetero-nuclear RgXe⁺ ions considered above, homo-nuclear cations possess substantially higher dissociation energies, so the process of electron-impact DE of these ions has some specific features.

The cross sections of the resonant electron-impact dissociation of ${\mathrm{A}\mathrm{r}}_{2}^{+}$ and ${\mathrm{X}\mathrm{e}}_{2}^{+}$ calculated at k_B T = 0.03, 0.06 and 0.09 eV are shown in figure 4. Data for electronic terms and dipole matrix elements were obtained from [69, 70]. We used dipole approximation for the coupling parameter, so there are two allowed types of the non-adiabatic transitions from the ground I(1/2)_u electronic state leading to the dissociation: I(1/2)_u → I(1/2)_g and I(1/2)_u → II(1/2)_g (see figure 1(b)). The analysis suggests that for ${\mathrm{A}\mathrm{r}}_{2}^{+}$ the dipole transition matrix element square of I(1/2)_u → I(1/2)_g transition in the vicinity of R_e is two orders of magnitude smaller than that of I(1/2)_u → II(1/2)_g , and can be neglected. In ${\mathrm{X}\mathrm{e}}_{2}^{+}$ both these transitions have comparable efficiencies at R = R_e and should be accounted for.

As both systems have high D₀ (see table 1), the cross sections exhibit the typical threshold behaviour. Similarly to RgXe⁺ ions, the positions of the maxima can be estimated from the condition stating that ɛ → ɛ' transitions with ɛ' = 0 should occur in the vicinity of R = R_e. Rather large energy threshold ɛ  ≳  2 eV makes the process (2) inefficient at T_e ≲ 5000 K. One should also note that due to high D₀, the behaviour of the cross sections does not change even at T ≲ 2000 K.

Also plotted in figure 4 are the results of the calculations from [48]. While our results agree qualitatively with those of [48], there are significant differences in the positions and the magnitudes of the maxima, and in the asymptotic behaviour of the cross sections of I(1/2)_u → I(1/2)_g transitions in ${\mathrm{X}\mathrm{e}}_{2}^{+}$. Numerical analysis indicates that the discrepancies stem from more accurate treatment of the statistical sums done in the present work, and from the use of more reliable data and methods for the evaluation of the electronic terms and transition matrix elements.

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The respective rate constants α^de(T_e, T) are presented in figure 5 for T ∼ 300–2000 K and T_e ∼ 1000–70 000 K. As compared to figure 3, α^de(T_e, T) retain the threshold behaviour even at T = 2000 K. The maximal values are reached at rather high T_e ≈ 50 000 K for ${\mathrm{A}\mathrm{r}}_{2}^{+}$ and T_e ≈ 30 000 K for ${\mathrm{X}\mathrm{e}}_{2}^{+}$. There is a noticeable difference between the T-dependences of the results obtained for both ions at high T_e. Another important feature of the electron-impact dissociation of these cations is the very slow α^de(T_e, T) decrease at very high electronic temperatures. Thus, one can expect that for homo-nuclear ions DE process will dominate over DR at high enough T_e. Our calculations suggest that the contributions from the processes become comparable at T_e ≈ 10 000 K.

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To confirm the validity of our approach to the description of the electron-impact DE of the rare gas ions we compared our results with experimental data [47, 71]. Both these works considered the total rates of molecular ion dissociation, α^rd(T_e, T), which include the contributions from the DR. As the experiments were performed at T = 300 K and T_e = 300–20 000 K, the rates of the DR process have a major fraction of the capture to low-n states. To account for this, we represent the rate of the DR as ${\tilde {\alpha }}^{\text{dr}}\left({T}_{\mathrm{e}},T \right)$ = ${\tilde {\alpha }}_{\text{low}}^{\text{dr}}\left({T}_{\mathrm{e}},T \right)$+ ${\alpha }_{\text{Ry}}^{\text{dr}}\left({T}_{\mathrm{e}},T \right)$, where ${\tilde {\alpha }}_{\text{low}}^{\text{dr}}\left({T}_{\mathrm{e}},T \right)$ is an effective rate constant of the dissociative capture to low-lying xenon states. ${\tilde {\alpha }}_{\text{low}}^{\text{dr}}\left({T}_{\mathrm{e}},T\right)$ was dissected from the experimental data using a recently developed approach [60] based on the effective state model and the least square interpolation in the low T_e ≲ 1000 K range.

The results were compared for T_e = 300–20 000 K (see figure 6). At low T_e the molecular ion dissociation is dominated by the contribution from ${\tilde {\alpha }}_{\text{low}}^{\text{dr}}\left({T}_{\mathrm{e}},T \right)$. In the range of 2000 ≲ T_e ≲ 4000 K the RD rate can be well approximated by the sum of ${\tilde {\alpha }}_{\text{low}}^{\text{dr}}\left({T}_{\mathrm{e}},T \right)$ and ${\alpha }_{\text{Ry}}^{\text{dr}}\left({T}_{\mathrm{e}},T \right)$, where the latter is described by (37), so that the electron capture to Rydberg states becomes important. For T_e  ≳ 4000 K it is crucial to take the electron-impact DE process into account. It can be seen that the inclusion of α^de(T_e, T) allows for a reasonable description of the experimental data at high T_e. Process (2) becomes dominant at T_e ≈ 15 000 K. Overall, our results agree well with experimental data [47, 71]. The method correctly reproduces the ranges of T_e corresponding to significant contributions from all molecular ion breakup mechanisms considered.

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Below we demonstrate that our method is not limited to molecular ions of rare gases and present the results of calculations of the cross sections of the resonant DE (2) of ${\mathrm{H}}_{2}^{+}$ ions in comparison to experimental data [41, 43]. The experiments were carried out using merged beams setup. The electron kinetic energy in center of mass system varied in the range of 0.05–35 eV. Due the way ${\mathrm{H}}_{2}^{+}$ ions were produced, Boltzmann distribution over the vibrational levels did not hold. To account for this, we first determined the cross sections of process (2) for a given v J state, and then performed the averaging over J: ${\sigma }_{v}^{\text{de}}\left(\varepsilon \right)={\sum }_{J=0}^{{J}_{\mathrm{max}}}{\sigma }_{vJ}^{\text{de}}\left(\varepsilon \right)\left[\left(2J+1\right)/{J}_{\mathrm{max}}^{2}\right]$. Cross-section σ^de(ɛ) was then obtained by the averaging over the distribution of the vibrational states specific to the experimental setups, ${\sigma }^{\text{de}}\left(\varepsilon \right)={\sum }_{v=0}^{{v}_{\mathrm{max}}}{\sigma }_{v}^{\left(\mathrm{d}\right)}\left(\varepsilon \right){f}_{v}^{\text{sp}}$.

The comparison of our results with the experimental data is presented in figure 7. The calculations were carried out using two types of ${f}_{v}^{\text{sp}}$ distributions. First one (von-Busch and Dunn distribution) was obtained in [72] by the analysis of the calculated and measured cross sections of photodissociation. Second one is the Franck–Condon distribution deduced in [73] using the analysis of the ionization processes and the Franck–Condon principle. One can see that the results obtained using both distributions provide a reasonable description of the experimental data. The results are higher for Franck–Condon distribution, which was also obtained in [41] on the base of estimations done within Born approximation. At low energies ɛ ≲ 1 eV the cross sections decrease rapidly with the increase of ɛ. Then at ɛ  ≳ 1 eV the decrease slows down, which results in a plateau and even a small rise of σ^de(ɛ). Such behaviour is caused by the fact that for ɛ in the range discussed the non-adiabatic transitions primarily occur near the equilibrium distance. Thus, the dissociation becomes dominated by low-v states, for which the magnitudes of ${f}_{v}^{\text{sp}}$ are very high. At high ɛ  ≳ 10 eV the transitions occur at any R so that the cross sections no longer depend on the specific vibrational state distribution. σ^de(ɛ) then follow the standard ɛ⁻¹ power law.

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The cross sections do not exhibit the threshold behaviour. This is a feature of the specific vibrational state distributions of ${\mathrm{H}}_{2}^{+}\left(v\right)$ ions produced by collisions with electrons. The distributions have a significant fraction of high-v state population. Such states dissociate efficiently under the impact of electrons with small ɛ which lead to large σ^de(ɛ) at low ɛ.

Finally, we note that despite the limited applicability of our method at low ɛ, it gives reliable results in good agreement with the experimental data for both vibrational state distributions used.