Electron collision studies on the CH$_2^+$ molecular ion

Calculations are performed for electron collision with the methylene molecular ion CH$_2^+$ in its bent equilibrium geometry, with the goal to obtain cross sections for electron impact excitation and dissociation. The polyatomic version of the UK molecular R-matrix codes was used to perform an initial configuration-interaction calculation on the doublet and quartet states of the CH$_2^+$ ion. Subsequently, scattering calculations are performed to obtain electron impact electronic excitation and dissociation cross sections and, additionally, the bound states of the CH$_2$ molecule and Feshbach resonances in the $e$-CH$_2^+$ system.


I. INTRODUCTION
Many low temperature plasma environments have hydrocarbon molecular ions as important constituents.Collision of electrons with molecules and their ions in these environments are important processes that play a fundamental role in initiating chemistry, particle balance and transport.For example, although the present tendency in the magnetically controlled International Thermonuclear Fusion Reactor type fusion devices is to coat the reactor walls with beryllium or tungsten, hydrocarbon ions, in particular the methylene ion CH +  2 , are found in the edge and divertor plasmas of fusion devices which operate with graphite as plasma facing material [1].Another important context is that of the dusty plasmas, the CH 2 and CH + 2 being species involved in the chain of reactions resulting in the growths of the nano and micro-particles.In these situations, the cross sections for different electron induced process are necessary to model the plasma flow (see for example [2]), in particular electron impact dissociation and dissociative ionisation: These processes, leading to the destruction of the CH + 2 ions and to the C atom production [3,4], are highly significant for understanding the carbon redeposition.
In the interstellar medium (ISM), CH + 2 ions are synthesised in gas phase by collision of C + ions with hydrogen [5] and through hydrogen abstraction by CH + .On the other hand, CH +  2 ions can be removed by reactions such as (1), (2) and by dissociative recombination (DR), which is known to proceed via Feshbach resonances [6].
There is a considerable literature on CH + y hydrocarbon ions, and CH + 2 ions in particular, in the context of synthesis of hydrocarbons in the ISM [5,[7][8][9][10].Significant work on CH + y hydrocarbon ions have also been done relevant to plasmas for fusion [2-4, 11, 12].These works mainly focus on different electron impact cross sections relevant for plasma modeling.
Molecular structure calculations on CH + 2 have been reported by several authors [13][14][15][16][17][18].[18] obtained bending potential energy curves (PEC) and vertical excitation energies of CH + 2 for its bent C 2v and its linear D ∞h configurations.Accurate global potential energy surfaces (PES) were reported by [13] and [16].Apart from these, there are also many spectroscopic studies on the rovibrational states of CH + 2 [19][20][21][22][23].In a number of earlier studies [24][25][26][27] we have studied electron collision with the CH molecule and its positive ion CH + in considerable detail.In these works, we not only computed cross sections for different electronic processes, but also identified many new neutral valence states of CH that are relevant for the DR.The present article aims to continue and extend our work to more complex hydrocarbon ions.

A. R-matrix method
The R-matrix method, described in detail by [28] and [29], and its implementation in the polyatomic version of the UK molecular R-matrix codes (UKRmol) [30] is used in the present work.The method employs a division of space into an inner region, a sphere of radius a (chosen to be 10 a 0 in this work) called the R-matrix sphere whose purpose is to include within it all short range interactions, and an outer region exterior to this sphere which contains all the long range interactions.This division allows the treatment of the short range and, the more complicated, long range interactions separately using different techniques [28].
In the inner region, the wave function of the target (here CH + 2 ) and a single continuum electron, together having N + 1 electrons, is taken as where The inner region wave function Ψ k is then used with appropriate boundary conditions to obtain scattering information, the details of which are given in the following subsections.

B. Target calculations
We used the cc-pVTZ Gaussian basis sets [31] centered on the C and H atoms to represent the target orbitals.These not only gave reasonably good target vertical excitation energies but also allows the inner region calculation to remain manageable with respect to computational resources.
The X 2 A 1 ground state of CH + 2 is known to be bent in C 2v symmetry and has the electronic configuration (1a 1 ) 2 (2a 1 ) 2 (1b 2 ) 2 (3a 1 ) 1 [32,33].In this work all calculations are reported at the equilibrium C-H bond length 2.066 a 0 and the H-C-H bond angle 140.1 • taken from [18], which are very close to those obtained by more sophisticated coupled cluster calculations using large basis sets [34].An initial Hartree-Fock (HF) calculation was first performed on the X 2 A 1 ground state of CH + 2 .The HF orbitals were then used in a configuration interaction (CI) calculation.
We considered two target models.In both models we kept two electrons frozen in the 1a 1 orbitals while the remaining five electrons were distributed in the CAS.In the first model (M1), the complete active space was defined by (1a , while in the second (M2) the CAS was chosen to be bigger with an additional 1a 2 orbital, namely (1a Table I shows the comparison of the vertical excitation  to the first 12 low lying excited states.Although the second model M2 appears to produce VEEs slightly in better agreement with the theoretical results of [18] and [35], calculations with this model required much longer time compared to the model M1.We therefore chose the model M1 for subsequent calculations as it was computationally more efficient.
From Table I, we see that apart from the VEE of the 4 2 A 1 state, which appears too high compared to [18], all other vertical excitation energies obtained by the target model M1 are in reasonably good agreement with the results of [18] and [35].Moreover, our dipole moment for the CH + 2 ground state with model M1 is 0.701 D which compares perfectly with the value 0.701 D obtained by [34] using coupled cluster calculation.The model M1 therefore provides a good description of the target for subsequent scattering calculations.

C. Scattering calculations
For the scattering calculations, we used 8 a 1 , 6 b 1 , 6 b 2 and 2 a 2 target orbitals allowing 2 virtual orbitals for each symmetry.Since the target CH + 2 is a positive ion, the continuum functions were represented by Coulomb functions which were obtained as a solution of the radial Coulomb equation for an isotropic Coulomb potential, and the solutions with l ≤ 4, and energy eigenvalue ≤ 5 Ryd were retained in the calculation.The Coulomb functions were then fitted to GTOs using the procedure outlined in [36].
The target and continuum orbitals must all be orthogonal to each other.To ensure the orthogonality, the target and continuum orbitals were first individually Schmidt orthogonalized and finally the full set of target and continuum orbitals were symmetric orthogonalized, retaining only those orbitals for which the eigenvalue of their overlap matrix was less than a deletion threshold, chosen here to be 5 × 10 −5 .The deletion threshold depends on the R-matrix radius and was adjusted to ensure that there was no linear dependence.An R-matrix was built at the boundary of the inner and outer region from the inner region solutions (Eq.( 4)).The R-matrix were then propagated to an asymptotic distance R asy = 70 a 0 in a potential which included the Coulomb potential and the dipole and quadrupole potentials of the target, where they were then matched to asymptotic functions obtained from a Gailitis expansion [39].This matching procedure yields the K-matrix from which all scattering observables can be obtained.A different route, however, is followed for obtaining bound states, as is outlined below.
For bound states, the R-matrix and wave functions were propagated using Runge-Kutta-Nystrom method [40] to an asymptotic distance R asy = 50 a 0 in a potential which included the Coulomb potential and the dipole and quadrupole potentials of target, and were matched to exponentially decreasing asymptotic functions [39].A searching algorithm [41] over a non-linear quantum defect based grid [42] was then used to find the bound states as roots of a determinant B(E) dependent on the energy.The details of the method are omitted and can be found in [41].

III. RESULTS
All scattering calculations in this work were performed at a single geometry, namely the equilibrium geometry of the CH cross sections for elastic scattering and electronic excitations.An 11 state scattering model including three 2 A 1 , three 2 B 1 , three 2 B 1 and two 2 A 2 target states in the close coupling expansion Eq. ( 4) were used for the singlet e+CH + 2 scattering close-coupling expansion, while a 15 state model which included the same 11 doublet states as in the singlet state model, together with the lowest each of the 4 A 1 , 4 B 1 , 4 B 2 , and 4 A 2 states was used in Eq. ( 4) for triplet symmetry close-coupling expansion.As will be shown below, this procedure provides a reliable scattering model which was tested by calculating the bound states of CH 2 .

A. Bound states
Table II shows the vertical excitation energies for the bound states of CH 2 form its X 3 B 1 ground state to few of its low lying excited states.For a comparison, we also did a quantum chemistry-style CI calculation on CH 2 at its equilibrium geometry (C-H bond length 2.0314 a 0 and H-C-H bond angle 133.8 • ) using a (1a 1 ) 2 (2 − 6a 1 , 1 − 4b 1 , 1 − 4b 2 ) 6 CAS-CI model.The vertical excitation energies are then compared with the multi reference double excitation (MRD-CI) calculation of [37] and the coupled cluster results of [38].Form Table II, it is clear that the vertical excitation energies obtained by the R-matrix method are in very good agreement with all others.The fact that the calculated bound state energies are consistent and accurate enough indicates that our scattering model is reasonably good.

B. Resonances at equilibrium
For resonance calculation, the R-matrix was propagated to 70 a 0 .Resonances were detected by the characteristic change in sign of the second derivative of the Cross section (a FIG. 2: Cross sections for electronic excitation form the X 2 A1 ground state to the first four excited states of doublet symmetry as given in Table I and indicated in each panel. eigenphase sum δ(E) given by where K ii are the diagonal elements of the K matrix.They were then fitted to a Breit-Wigner profile [43] with an energy grid 0.005 eV to obtain the resonance energy E and width Γ.
Figure 1 shows the plot of two typical eigenphase sums for 1 A 1 and 3 B 1 symmetries.Characteristic of electron collision with ions, the figure shows numerous resonances some of which are tabulated in Table III according to their parent state.As seen from the table, many of the resonances appear to be in Rydberg series which can be identified by their effective quantum numbers.The relatively small gap between the ground and first excited electronic state means that the resonances start with relatively high effective quantum numbers, ν ≈ 4, and then closely space.
A full set of fits to the resonances covering all overall scattering symmetries is given in the supplementary material.

C. Electron impact excitation
Our calculated cross sections for electron impact excitation of CH + 2 from the ground state to four of its lowest doublet states are shown in Figure 2. As in the elastic Expt.fitted and summed FIG.3: Cross section for the electron impact dissociation of the CH + 2 ion.Thin curve: present R-matrix results (see text).Thick (red) curve: present result after smoothing with a Gaussian function.Green circles with error bars: experimental results for C + ion production [12].Blue circles with error bars: experimental results for CH + ion production [12].Dashed curve: experimental results for C + and CH + ion production summed after suitable fitting (see text).
cross section, the excitation cross sections show highly resonant behaviour due to temporary captures into resonant states.Particularly, the X 2 A 1 → 1 2 B 1 excitation cross section shows evidence of a large resonance near threshold.Referring to Figure 1, this is likely due to the large 3 B 1 resonance near 1 eV as seen from the plot of the 3 B 1 eigenphase sum.Generally, the excitation cross sections decrease with increasing incident energy and the cross sections from the higher lying excited states are much smaller.

D. Electron impact dissociation
Experimental cross sections for electron impact dissociation were obtained by [12] for the production of CH + and C + fragments.Although model calculations exist for the electron impact dissociation of CH + 2 ions (see for example [2]), to the best of our knowledge, the 2007 experiments have never been described by any ab initio calculation.
The dissociation of CH + 2 into the lowest dissociation channels, namely e+ CH + + H and e+ C + + H 2 , proceed via direct dissociative excitation and have thresholds 6.08 eV and 5.62 eV respectively [12].In deriving the dissociation cross section, we assumed that electronic excitations to all states above the respective dissociation thresholds lead to dissociation.In our calculation, we have included the states 1 2 B 1 , 1 2 B 2 , 1 2 A 2 , and the 2 2 A 2 excited states all of which lie close to one another at the CH + 2 equilibrium.Moreover, except the 1 2 B 1 state, which has both valence and Rydberg character, all the other three are of valence character at equilibrium (see for example [18]) and hence are likely to dissociate to the e+C + +H 2 (5.62 eV) or the e+CH + +H (6.08 eV) dissociation limits, which are the most relevant in the energy range considered.Since the dissociation channels cannot be separated in our calculations, our cross section in Figure 3 represents a sum over these channels.For better comparison, we have also shown our cross sections after smoothing by using a Gaussian function.The experimental data from [12] are available for each of the dissociation channels mentioned above.The energy grid of the experimental cross sections for these two dissociation channels, however, are neither same nor uniform.Therefore, to sum the experimental data, we first spline interpolated each set over the same and uniform energy grid.This allowed us to sum the cross section for the experimental data.In Figure 3, this sum is shown as the dashed line and it agree fairly well with our computed cross section.In particular, the peak near 8 eV agrees quite well with the spline interpolated summed experimental curve.However, we note that since the experimental data is non uniformly spaced, the interpolated curve may not often reflect the actual trend.For example, we suspect the agreement of the interpolated curve with our calculation would have been much better between 11 eV -15 eV had there been more experimental points available in this region.Similarly, below 6 eV the interpolated experimental curve, appears to diverge for our Gaussian fitted curve.This is a result of the fitting procedure, the interpolated experimental curve actually follows the trend of our raw cross section curve.

IV. CONCLUSION
As mentioned in the introduction, CH + 2 is a very important constituent in low temperature plasma environments.However, despite its importance, electron collisions studies on CH + 2 have been rare.In fact, we could not find any ab initio calculation for the cross sections included in the present work.The only cross section result available, seems to be the total ionisation cross section of CH + 2 calculated within the Binary-Encounter-Bethe (BEB) model by [44].
In this work, we have presented a reliable set of cross sections for the electronic excitation, and electron impact dissociation of the CH + 2 ion.In fact, none of these cross sections have ever been reported before by any ab initio calculation.Additionally, we have also calculated and have given the position and width for some of the Feshbach resonances in the e-CH + 2 system.These Feshbach resonances, as is well known, are the routes to dissociative recombination of the CH + 2 ion.However, its treatment requires a more comprehensive calculation of the resonance energies, widths and the potential energy surfaces of the CH + 2 states.This is the subject of an ongoing project.and LabEx EMC3 via the projects PTOLEMEE, Bioengine, the Institute for Energy, Propulsion and Environment (FR-IEPE),and ERASMUS-plus conventions between Université Le Havre Normandie and University College London.We are indebted to Agence Nationale de la Recherche (ANR) via the project MONA, Centre National de la Recherche Scientifique via the Programme National 'Physique et Chimie du Milieu Interstellaire' (PCMI).JZsM thanks the financial support of the National Research, Development and Innovation Fund of Hungary (NKFIH), under the K18 and FK19 funding schemes with project no.K128621 and FK132989.JZsM and IFS are grateful for the support of the NKFIH-2019-2.1.11-TÉT-2020-00100 and Campus France-Programme Hubert Curien-BALATON-46909PM projects.

TABLE I :
Comparison of the vertical excitation energies (in eV) from the X 2 A1 ground state to 12 low lying excited states of CH + 2 for different target models.The target models used are the following: M1

TABLE II :
Comparison of the vertical excitation energies (in eV) from the ground X 3 B1 state of CH2 to some of its low lying excited states.CH2 state This work CI a Romelt b Yamaguchi c (2a1 − 6a1, 1b1 − 4b1, 1b2 − 4b2) 6 CAS-CI model. ,e,f Absolute energies of the ground states are respectively −38.985233 d Hartree, −38.966950 e Hartree and −39.06034 f Hartree.

TABLE III :
Resonance positions and widths (in Ryd) and effective quantum numbers at the CH + 2 equilibrium for states of 3 B1 and 1 A1 symmetry of the e-CH +