Calculated electron impact dissociation cross sections for molecular chlorine (Cl₂)

The R-matrix method treats electron scattering from molecules by dividing the space of the problem into two separately-calculated regions [14], comprising an inner region containing the wavefunction of the molecular target along with the colliding electron, and an outer region in which only the incident, scattering electron is considered. The R-matrix calculation constructs and solves an electron-energy-independent wave equation for the inner region, whose solutions are then used to solve the much simpler, energy-dependent problem of the scattering electron in the outer region. By making the inner region of the problem independent of the colliding electron energy and only the outer region energy dependent, the outer region can be resolved on a very fine energy grid, showing all of the features and structure of the cross section.

The low-energy calculations reported in this work were all performed using the polyatomic implementation of the UK molecular R-matrix code UKRMol [15]. These calculations were performed using the Quantemol-N expert system [16] which runs the UKRMol codes. A full review of the molecular R-matrix method has been given in [17].

Established electron scattering theory provides a range of models for treating the interaction of the incident scattering electron with the bound molecular electrons as discussed by Tennyson [17] and the references therein. In the scattering calculations carried out here, the selected target states were included in the scattering wavefunction through the use of a close-coupling (CC) expansion. Here, the target states are represented using a complete active space (CAS) configuration interaction (CI) model [18] in which bound electrons from the highest (valence) occupied molecular orbitals (HOMOs) are excited to the lowest unoccupied molecular orbitals (LUMOs). This model can calculate cross sections for electronically inelastic processes while also accounting reliably for Feshbach resonances, which are temporary anion states in which the scattering electron is trapped following excitation of the target.

Dissociation occurs when molecules are excited to electronic states that are either unbound or have curve-crossings to unbound states. The total electron impact dissociation cross section can therefore be taken to be the sum of excitation cross sections to all unbound states:

$$\begin{eqnarray}&&{\sigma }_{{\rm{eid}}}^{{\rm{tot}}}=\displaystyle \sum _{i=1}^{\infty }{\sigma }_{{\rm{ex}}}^{i}\,,\end{eqnarray} \tag{ 1 }$$

where ${\sigma }_{{\rm{eid}}}^{{\rm{tot}}}$ is the total electron impact dissociation cross section and ${\sigma }_{{\rm{ex}}}^{i}$ is the electron impact excitation cross section to a specific unbound state i. To fully understand the dynamics of electrons in low-temperature plasmas, electron-impact dissociation cross-sections via specific excited states, with specific excitation energies, i.e. ${\sigma }_{{\rm{ex}}}^{i}$, should be known. In addition, it is important to know whether or not the atoms created by dissociation are created in excited states or the ground state, i.e. the branching ratio of the products. In this work, the nature of the orbitals populated by excitation collisions were used to ascertain whether or not an excitation process results in dissociation. The asymptotes of the potential energy curves for the dissociative excited states were used to determine the branching ratios of the dissociation products.

The R-matrix model is known to provide accurate cross section data in the low energy range, defined here as between 0 eV and the ionisation potential (IP). For higher energies, the inelastic cross sections in this work are scaled according to the specific nature of the transition. In the case of dipole-forbidden transitions, i.e. those that involve a spin change, the cross section is scaled as ${\textstyle \tfrac{1}{\epsilon }}$ , where ε is electron energy. In the case of dipole-allowed transitions, that is those with no spin change, the cross sections are scaled as ${\textstyle \tfrac{{\rm{l}}{\rm{n}}(\epsilon )}{\epsilon }}$. Where the calculated cross-sections showed non-physical structure at energies above the IP, this non-physical structure was assumed to be an artefact of the calculation, and smoothed. Such non-physical structure can arise in the calculations due to using only single geometry, incomplete continuum orbital sets, or due to pseudo-resonances. This method of scaling has previously been employed in [19].

The point group symmetry of equilibrium Cl₂ is ${{\rm{D}}}_{\infty v}$. Molecular symmetries can be taken advantage of to calculate the integrals between Gaussian-type orbital (GTO) basis functions, however, the Sweden-Molecule quantum chemistry codes [20], upon which the polyatomic UKRmol inner region codes used in this work are based, are limited to only using Abelian or commutative point groups. This is also true of MOLPRO [21], another quantum chemistry code which can be used to calculate the integrals between GTO basis functions used subsequently in the polyatomic UKRmol inner region codes. Due to this restriction, the non-Abelian symmetry ${{\rm{D}}}_{\infty v}$ is represented in the D_2h point group.

The Cl₂ target was represented using a Dunning cc-pVTZ GTO basis set. The ground state of Cl₂, X ${}^{1}{{\rm{\Sigma }}}_{g}^{+}$, has the configuration [1-5σ_g,1-2π_u,1-4σ_u,1-2π_g]³⁴. MOLPRO was used to calculate these orbitals. The target was represented using a CAS-CI treatment, freezing electrons of the lowest 13 orbitals with 8 electrons from the 2 π HOMOS active in these open orbitals and 5 valence orbitals: [1-5 σ_g,1 π_u,1-4 σ_u,1 π_g]²⁶[6 π_g,2-3 π_u,5-6 σ_u,2-3 π_g]⁸. All calculations were performed at the Cl₂ equilibrium geometry sourced from the NIST Computational Chemistry Comparison and Benchmark Database [22] and given in table 1.

The vertical excitation energies (VEEs) of the excited states of Cl₂ calculated from this model are given in table 2 along with a comparison to published calculated values. The VEEs calculated in this work compare well to the published VEEs. Rescigno [10] and Peyerimhoff and Buenker [23] identified the fourth excited state of Cl₂ as B ¹Π_g, whereas in this work we identified the fourth state as c ${}^{3}{{\rm{\Sigma }}}_{g}^{-}$, and the B ${}^{1}{{\rm{\Pi }}}_{g}$ state as the fifth excited state. The implications of the different symmetries of the c ${}^{3}{{\rm{\Sigma }}}_{g}^{-}$ state identified in this work and the c' ${}^{3}{{\rm{\Sigma }}}_{u}^{+}$ state identified in [10, 23] for the calculated cross sections will be discussed in section 4.

Table 2.  Comparison of vertical excitation energies for excited states of Cl₂ calculated in this work with those calculated by Rescigno [10], and Peyerimhoff and Buenker [23]. Energies are given for excited states up to and just above the measured IP of Cl₂ at 11.481 eV [24].

State	This work	[10]	[23]	State	This work	[23]
X ${}^{1}{{\rm{\Sigma }}}_{g}^{+}$	0.000	0.00	0.00	C ${}^{1}{{\rm{\Delta }}}_{g}$	7.790	8.12
a ${}^{3}{{\rm{\Pi }}}_{u}$	3.252	3.36	3.31	D ${}^{1}{{\rm{\Sigma }}}_{g}^{+}$	8.228	8.29
A ${}^{1}{{\rm{\Pi }}}_{u}$	4.348	4.30	4.05	E ${}^{1}{{\rm{\Sigma }}}_{u}^{-}$	8.982	9.43
b ${}^{3}{{\rm{\Pi }}}_{g}$	6.498	6.38	6.29	d ${}^{3}{{\rm{\Delta }}}_{u}$	9.113
c ${}^{3}{{\rm{\Sigma }}}_{g}^{-}$	7.257			e ${}^{3}{{\rm{\Sigma }}}_{u}^{+}$	9.219	9.74
c' ${}^{3}{{\rm{\Sigma }}}_{u}^{+}$		7.02	6.87	f ${}^{3}{{\rm{\Sigma }}}_{u}^{-}$	12.691
B ${}^{1}{{\rm{\Pi }}}_{g}$	7.537	7.01	6.83

The scattering calculation target used an R-matrix sphere of radius 10 a₀. The continuum basis was represented using GTOs with ℓ ≤ 4 (up to g orbitals) [25], which were orthogonalised to the target orbitals.

The process of electron impact dissociation occurs when energy gained from the scattering electron promotes the molecule into an electronic state, which subsequently dissociates. This occurs when the dissociating bond is weakened by the transfer of electrons from bonding molecular orbitals, to orbitals away from the bonding region, which do not enforce the bond, and pull the nuclei apart. These orbitals are referred to as anti-bonding orbitals, orbitals that, if occupied, contribute to a reduction in the cohesion between the two atoms and raise the energy of the molecule above that of separated atoms [26].

As discussed by Peyerimhoff and Buenker [23], the σ_u orbital has anti-bonding character. As a result, excitation processes that populate σ_u or ${\sigma }_{u}^{2}$ orbitals result in dissociation. Of the states listed in table 2, the following were found to fulfil this criteria by Peyerimhoff and Buenker [23] and dissociate to form two ground state Cl atoms: ${{\rm{a}}}^{3}{{\rm{\Pi }}}_{u}$, A ${}^{1}{{\rm{\Pi }}}_{u}$, b ${}^{3}{{\rm{\Pi }}}_{g}$, c' ${}^{3}{{\rm{\Sigma }}}_{u}^{+}$, B ${}^{1}{{\rm{\Pi }}}_{g}$, C ${}^{1}{{\rm{\Delta }}}_{g}$, D ${}^{1}{{\rm{\Sigma }}}_{g}^{+}$ and e ${}^{3}{{\rm{\Sigma }}}_{u}^{+}$. The orbital movement for the c ${}^{3}{{\rm{\Sigma }}}_{g}^{-}$ state identified in this work is assumed to be the same as that of the c' ${}^{3}{{\rm{\Sigma }}}_{u}^{+}$ state identified in [23]. The d ${}^{3}{{\rm{\Delta }}}_{u}$ and f ${}^{3}{{\rm{\Sigma }}}_{u}^{-}$ states were not identified in the calculations of [23], as such, orbital movement information and potential energy curves are not available for these states. As a result, we cannot be certain of their dissociation pathway, so they are excluded from the remaining discussion on dissociation. In the work of [23] two ${}^{1}{{\rm{\Sigma }}}_{u}^{-}$ states were identified; one dissociating to form two ground state Cl atoms, and the other a Rydberg state dissociating into one ground state Cl atom and Cl⁺. Given that we cannot be certain which of the two states is identified in our calculations, we also exclude the E ${}^{1}{{\rm{\Sigma }}}_{u}^{-}$ from further discussion on dissociation. The excluded states have significantly smaller cross sections than the lower lying states (at least 2–3 orders of magnitude lower than that of the ${{\rm{a}}}^{3}{{\rm{\Pi }}}_{u}$ state for all energies), therefore these exclusions have very little effect on the total dissociation cross sections presented later.

The nature of the excited states can be better understood through the Cl₂ potential energy curves shown in figure 1, taken from Peyerimhoff and Buenker [23], which also show the states of the Cl atoms produced by dissociation of the different excited states of Cl₂. All of the dissociative states considered in the results section of this work lead to the formation of two ground state Cl atoms.

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Figure 2 presents our calculated cross sections for excitation from ground state Cl₂ to the dissociative Π states (with various spins and symmetries), along with those calculated by Rescigno [10]. Figure 2(a) shows the data over the energy range originally calculated by Rescigno [10] on a linear scale. Figure 2(b) shows the data on log axes to enable comparison over a wider energy range. Above 30 eV the cross-sections from Rescigno [10] are scaled as proposed by Grégorio and Pitchford [7], based on the experimental dissociation cross section of Cosby [12]. The agreement between the cross sections from this work and those from Rescigno [10] is very good, particularly at energies up to around 15 eV. Our calculations show a few sharp peaks in the cross sections which can be associated with resonances. These features are too narrow to be resolved at the resolution of the experiments (shown in figure 4) or the calculations of Rescigno; however, the inclusion of vibrational motion, neglected in the present calculations, would be expected to broaden these resonance features. The precise role of these resonances in the electronic excitation of Cl₂ awaits further study.

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Figure 3 shows the calculated cross sections for excitation into dissociative Δ and Σ states. The cross-section to the c' ${}^{3}{{\rm{\Sigma }}}_{u}^{+}$ state, calculated in [10] but not identified in this work, is also shown. Here, it is clear that the excitation cross-sections to the dissociative Δ and Σ states are much smaller in magnitude than the cross-sections to the Π states (shown in figure 2), but similar in shape. Furthermore, the cross-section for excitation of the c' ${}^{3}{{\rm{\Sigma }}}_{u}^{+}$ state calculated in [10] is significantly larger than (but similar in shape to) that of the ${{\rm{c}}}^{3}{{\rm{\Sigma }}}_{g}^{-}$ state identified in our work, despite their similar threshold energies. This is likely to be a result of the different symmetries of the states in the two calculations. According to Goddard et al [27] the electron scattering cross sections for ${{\rm{\Sigma }}}_{g}^{+}$ ↔${{\rm{\Sigma }}}_{g}^{-}$ transitions in a linear molecule must approach zero for scattering angles of 0° and 180°, since the reflection symmetry of the molecule (+↔−) cannot change during forward or backward collisions. In contrast, electron impact scattering cross sections for ${{\rm{\Sigma }}}_{g}^{+}$ ↔${{\rm{\Sigma }}}_{u}^{+}$ transitions, i.e. from the ground state to the c' ${}^{3}{{\rm{\Sigma }}}_{u}^{+}$ state as identified by Rescigno [10], are not constrained in this way, and therefore exhibit a larger integrated cross section.

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Figure 4 shows a comparison of the sum of the excitation cross sections leading to dissociation calculated in this work, those calculated by Rescigno [10], and the measured dissociation cross section of Cosby [12]. As before, (a) shows the electron energy range originally calculated by Rescigno with linear axes, while (b) shows a comparison over a wider energy range with log axes. The summed dissociation cross section from Rescigno is larger above 15 eV and exhibits better agreement with the measured total cross section of Cosby [12] in this range. However, it is important to emphasise that in the measurements of Cosby [12] the Cl₂ ground state target molecules were in a distribution of vibrational states due to the experimental technique used, whereas for the calculations in this work and Rescigno [10] only Cl₂(v = 0) was considered. As a result, the measurements and calculations are not directly comparable. Furthermore, calculations for electron-impact dissociation of vibrationally excited H₂, O₂ and N₂ molecules have shown that the magnitude of the dissociation cross section increases strongly as the vibrational level is increased [28–30]. As a result, we believe our calculations to be consistent with the data of Cosby, although further calculations of vibrational-state resolved electron-impact dissociation cross sections would be required to confirm this.

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In fluid and global plasma simulations, electron impact cross sections are typically incorporated in the form of rate coefficients, k, which are derived from the electron energy distribution function f through the relation:

$$\begin{eqnarray}&&k({T}_{{\rm{eff}}})={\left(\displaystyle \frac{2e}{{m}_{e}}\right)}^{1/2}{\int }_{0}^{\infty }\sigma (\epsilon ){\epsilon }^{1/2}f(\epsilon ){\rm{d}}\epsilon .\end{eqnarray} \tag{ 2 }$$

Here, T_eff is the effective electron temperature, e is the electron charge, m_e the electron mass and the electron energy. In low-temperature plasmas, the shape and effective temperature of the electron energy distribution function is highly variable, depending on parameters such as the nature of the plasma source, the operating pressure [31–33], the voltage/current [34, 35] the driving frequency [36–38] and the gas or gas mixture [39, 40]. The shape and temperature of the distribution function can also vary strongly in space and time within the same plasma source [41–46]. To understand how the cross sections calculated in this work affect the corresponding rate coefficients for electron impact dissociation we follow the approach of Gudmundsson [47] and Toneli et al [48] and define a general expression for the electron energy distribution function:

$$\begin{eqnarray}&&f(\epsilon )={c}_{1}{\epsilon }^{1/2}{\rm{\exp }}(-{c}_{2}{\epsilon }^{x})\end{eqnarray} \tag{ 3 }$$

The parameter x defines the shape of f(). x = 1 represents a Maxwellian distribution function, x = 2 resembles a Druyvesteyn distribution (similar to distribution functions found at higher gas pressures) and x = 0.5 gives a concave distribution function that is highly populated at low electron energies, while also having a pronounced high energy tail (similar to distribution functions found at low gas pressures). The parameters c₁ and c₂ are given by the expressions [47, 48]

$$\begin{eqnarray}&&{c}_{1}=\displaystyle \frac{x}{\langle \epsilon {\rangle }^{3/2}}\displaystyle \frac{{[{\rm{\Gamma }}({\xi }_{2})]}^{3/2}}{{[{\rm{\Gamma }}({\xi }_{1})]}^{5/2}},\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&{c}_{2}=\displaystyle \frac{1}{\langle \epsilon {\rangle }^{x}}{\left[\displaystyle \frac{{\rm{\Gamma }}({\xi }_{2})}{{\rm{\Gamma }}({\xi }_{1})}\right]}^{x}.\end{eqnarray} \tag{ 5 }$$

Here, $\langle \epsilon \rangle$ is the mean electron energy $\langle \epsilon \rangle =3/2{T}_{{\rm{eff}}},{\rm{\Gamma }}$ denotes a gamma function, ${\xi }_{1}\,=\,3/2x$ and ${\xi }_{2}\,=\,5/2x$.

Figure 5(a) shows the form of f() for different values of x with T_eff = 3 eV. Figure 5(b) shows a comparison between the dissociation rate coefficients derived from the total cross section calculated in this work and that calculated in [10] for varying T_eff and with x = 0.5, 1 and 2. As x is decreased from 2 to 0.5 the rate coefficient for dissociation increases for a given effective electron temperature because a greater proportion of electrons populate the part of the distribution function above the threshold energy for excitation of dissociative states. This is the case for both the cross sections calculated in this work and those calculated in [10]. For a given value of x the dissociation rate coefficients derived from the cross sections calculated in this work and those from [10] differ to varying degrees. These differences are greatest at low values of T_eff and when x = 2, under which conditions the difference between the rate coefficients can be several orders of magnitude. This is also true (but to a lesser extent) when x = 1, but is significantly less important when x = 0.5. These differences have the potential to be important when modelling low T_eff plasmas, of interest for atomic layer etching and deposition, such as those produced by electron beams which have been shown to exhibit Maxwellian electron energy distribution functions with T_eff as low as 0.4 eV [49–51]. As T_eff increases to around 3 eV and above, the difference in rate coefficient becomes less pronounced, and therefore the impact of using the different dissociation cross section sets for modelling of plasmas will be less significant.

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The differences in dissociation rate coefficients between the cross sections calculated in this work and those calculated in [10] in the low T_eff range are primarily a result of the higher energy resolution of our calculations around the excitation threshold, as shown in figures 2(b) and 4(b). These differences are particularly pronounced for the dominant dissociation cross section, i.e. that going via the a ${}^{3}{{\rm{\Pi }}}_{u}$ state. Above T_eff = 3 eV, the rate coefficients derived from the cross-sections calculated in this work are generally lower than those from the cross sections calculated in [10]. This is a result of the lower magnitude of our cross sections at electron energies above 15 eV. The discrepancy in rate coefficients between the two cross section sets at higher T_eff reaches a maximum of a factor of 1.5 – 1.6 for all three values of x when T_eff = 10 eV. The strong differences between the rate coefficients for dissociation at low T_eff emphasises the importance of the near-threshold region of electron impact dissociation cross sections in low-temperature plasmas. This further emphasises the advantages of using theoretical calculations for the derivation of such cross sections, as they are capable of providing the required high resolution and are not limited by experimental detection limits.