Kinetic rate coe ffi cients for electron-driven collisions with CH + : dissociative recombination and rovibronic excitation.

Cross sections and rate coe ffi cients for rovibronic excitation of the CH + ion by electron impact and dissociative recombination of CH + with electrons are evaluated using a theoretical approach combining an R-matrix method and molecular quantum defect theory. The method has been developed and tested, comparing the theoretical results with the data from the recent Cryogenic Storage Ring experiment. The obtained cross sections and rate coe ffi cients evaluated for temperatures from 1 K to 10,000 K could be used for plasma modeling in interpretation of astrophysical observations and also in technological applications where molecular hydrocarbon plasma is present.


INTRODUCTION
Hydrides and their ions are often used to trace various characteristics of the interstellar medium (ISM).Understanding their formation and destruction mechanisms is therefore necessary in using them as accurate tracers.The CH + ion, discovered in the interstellar medium by Douglas and Herzberg (1941) and Adams (1941), was originally thought to be formed primarily through the reaction The reaction is endothermic, requiring 0.398 eV (∼4620 K) to proceed (Hierl et al. 1997), but typical kinetic temperatures of diffuse clouds are roughly between 40 and 130 K (Shull et al. 2021).Hence, the reaction cannot explain the observed abundance of CH + in diffuse clouds (Godard and Cernicharo 2013).Observations of CH + in diffuse clouds have motivated many theoretical and experimental studies on the structure and reactivity of the ion.The structure of the ion has been well established for several decades, while there is still need for cross sections for processes involving collisions of CH + with electrons.In particular, the knowledge of cross sections for excitation of the ion by electron impact is important for interpretations of astrophysical observations.For example, rotational excitation of CH + by electron impact was found to be the dominant process producing CH + in the ISM Godard and Cernicharo (2013).
Besides astronomical applications, cross sections and rate coefficients for e -+ CH + collisions are important for the interpretation and modelling of hydrocarbon plasma behavior.Other than rotational excitation, mentioned above, one needs the data on vibrational and electronic excitation and dissociative recombination (DR).CH + is also a suitable candidate for benchmark theoretical studies of such ⋆ Email: j.forer@posteo.netprocesses.On one hand, this is because there are several experimental measurements (Amitay et al. 1996;Paul et al. 2022;Kálosi et al. 2022) available -of DR in particular.On the other hand, the ion has a relatively complex electronic structure, such that theoretical methods could be tested on this system and, if successful, be applied to other similar problems.
Our recent theoretical study (Forer et al. 2023) has demonstrated that theory can now accurately describe the DR process in CH + .The theoretical approach developed in that study (originated from several previous works (Hamilton and Greene 2002;Kokoouline and Greene 2003;Čurík et al. 2020)) with small modifications can also be used to obtain cross sections for rotational, vibrational, and even electronic excitation of the CH + ion by electron impact.Because the theoretical method was validated comparing the DR results with the experimental data, it is expected to provide reliable data on the electron-impact excitation processes as well.This study is devoted to the theoretical evaluation of cross sections and rate coefficients for rovibronic excitation of CH + ion by electron impact.The only available experimental result on excitation of CH + is from a recent study in the Cryogenic Storage Ring (CSR) (Kálosi et al. 2022), where the rate coefficient for rotational excitation from the ground rovibrational level v = 0, j = 0 of CH + to the first excited v = 0, j = 1 was measured.However, there have been several theoretical studies on rotational excitation using the Coulomb-Born approximation (Chu and Dalgarno 1974;Dickinson and Munoz 1977;Lim et al. 1999) and a semi-classical method (Flower 1979).More recently, vibronic excitation of CH + was studied by Jiang et al. (2019) using a combination of an R-matrix approach (Carr et al. 2012) and molecular quantum defect theory (Seaton 1983;Aymar et al. 1996).
The present study is based on a fully quantum description of e -+ CH + collisions and considers rotational and vibrational degrees of freedom of the target ion, as well as its electronic structure, including the three lowest electronic states and corresponding Rydberg resonances appearing in the CH + + e -collisional spectrum.The article is organized in the following way: section 2 gives an overview of the present theoretical approach and the differences between its application for DR and rovibronic excitation, section 3 presents our DR rate coefficients, section 4 describes the Coulomb-Born approximation and its application to the present rotational excitation calculations, section 5 is devoted to the discussion of obtained results on rovibronic excitation, and section 6 concludes our findings.

THEORETICAL APPROACH
The method described in this section combines fixed-nuclei electron-scattering calculations with the R-matrix method, rovibrational frame transformation, and multichannel quantum defect theory (MQDT).Only the main elements of the approach will be presented here.We use the same method to calculate scattering matrices as in our previous study of the dissociative recombination (DR) of CH + (Forer et al. 2023).The theory and computational details have only two main differences: the formula for obtaining the rovibronic excitation (RVE) cross sections (instead of DR cross sections) and the vibrational Hamiltonian used in the vibrational frame transformation of the electronic S-matrix.
We perform electron-scattering calculations with the R-Matrix method implemented in UKRMol (Carr et al. 2012;Tennyson 2010), accessed via the Quantemol-N interface (Tennyson et al. 2007).The K-matrices for the e -+ CH + system are represented in a basis of electronic scattering channels, indexed by n, l, and λ.The quantum numbers l and λ correspond to the magnitude of the orbital angular momentum of the incident electron and its projection on the molecular axis, respectively.The electronic states of the target, CH + , are indexed by n.K-matrices are obtained for several values of R, the internuclear distance of CH + .We then transform the K-matrices into the S-matrix, via the intermediary matrix of phase shifts, δ(R).The matrices are formally related by R) . (2) The reasons for this transformation are twofold: the S-matrix is a smooth function of R, which is necessary to have a more accurate vibrational integral in the frame transformation, and the S-matrix is used in the formula to compute the cross sections.
After the electronic S-matrices are obtained for each value of R, we proceed with the vibrational frame transformation.The first difference in this approach with our study of DR (Forer et al. 2023) is that the vibrational Hamiltonian is Hermitian.To study DR, we used a complex absorbing potential to represent absorption by discretizing the continuum.Here, we only need to obtain vibrational wave functions for bound states.The vibrational frame transformation proceeds as where v indexes a vibrational level.The superscript Λ indicates that the S-matrices are block diagonal with respect to Λ, the projection of the total angular momentum on the molecular axis.The Hermitian Hamiltonian implies, of course, real eigenvalues.The channel energies in the case of DR are complex, with a nonzero imaginary part for continuum states.Here, all channel energies and vibrational wave functions ϕ nv (R) are real-valued.The S-matrix on the left-hand-side of (3) is the vibronic S-matrix and, unlike in the case of DR, unitarity is defined with the usual spectral norm, i.e., S J S J † = I.Following the vibrational frame transformation, we perform the rotational frame transformation on the vibronic S-matrix to obtain the rovibronic S-matrix, i.e., where the total angular momentum of the ion-electron system is ⇀ j is the total angular momentum of the ion, and µ is the projection of j on the molecular axis.The S-matrix, now expressed in a basis of rovibronic channels, block diagonal over J.
For each scattering energy, we then partition S J into blocks corresponding to open (o) and closed (c) channels and construct the diagonal matrix β for closed channels, where E i is the energy of the i th channel and is real, and E tot is the total energy of the ion-electron system.We proceed with the closedchannel elimination procedure, borrowed from MQDT, to reduce the S-matrix to only open channels.
The physical S-matrix, S J,phys , is then used to calculate the total RVE cross section from some initial channel |nv jµ⟩ to some final channel where m e is the mass of an electron and E el is the incident electron energy.It is also possible to calculate vibronic excitation (VE) cross sections, i.e., not including the rotational structure, by simply skipping the rotational frame transformation (4).The closed-channel elimination procedure remains identical, except that the S-matrices are block diagonal over Λ and not J.The total VE cross section from some initial channel |nv⟩ to some final channel |n ′ v ′ ⟩ is then The cross sections obtained from ( 7) and ( 8) only describe a single scattering event.To better describe conditions in the ISM, kinetic rate coefficients are needed, which rely on the above cross sections.State-selected kinetic rate coefficients, for DR, RVE, or VE, are obtained following where k is the Boltzmann factor and σ is a cross section.In practice, these integrals are carried out numerically.Additionally, one can average the state-selected rate coefficients obtained from (9) by where i indexes starting channels where the total angular momentum quantum number of the ion is j i .The rate coefficient (DR or (R)VE), starting from some channel indexed by i and obtained with (9), is given by α i k (T ).If α i k (T ) is a rate coefficient without rotational resolution, j i can be taken to be zero in (10).
The precision of theoretical cross sections are only limited by the numerical precision of the calculations.Experimental measurements have much larger uncertainties, so comparisons are often made by convolving theoretical results with experimental parameters.The convolution function differs for every experimental setup, but Gaussian functions are fairly common: The parameter γ is the convolution width (in the same energy units as the electron energy grid).The prefactor in ( 12) is the analytic expression of the denominator in (11).Because calculations are performed numerically on predetermined grids of scattering energies, ( 12) and ( 11) may give different results near the endpoints.

RATE COEFFICIENTS FOR DISSOCIATIVE RECOMBINATION
Fig. 1 compares state-selected kinetic DR rate coefficients obtained with the present method (Forer et al. 2023).DR and RVE are related in the sense that they are competing processes.During a collision between an ion and an electron, if the initial channel is not the only open channel, both processes may take place.This is why we take the DR probability to be the probability that no RVE occurs.Therefore, DR results exhibiting a certain level of agreement with experimental results would suggest that RVE is described, overall, with similar accuracy.The present method produces much more accurate kinetic DR rate coefficients than the previous theoretical results of Mezei et al. (2019) when compared to recent measurements made at the Cryogenic Storage Ring (Paul et al. 2022) over the astrophysically relevant temperature range for diffuse clouds (∼40 K -130 K (Shull et al. 2021)).

ON THE COULOMB-BORN APPROXIMATION FOR ROTATIONAL AND VIBRATIONAL (DE-)EXCITATION
The dipole moment of CH + -present in the molecular center-ofmass about which the molecule rotates -couples partial waves of ∆l = ±1, which reduces the accuracy of our partial wave basis (l=0-2) for such a long-range process as rotational excitation.We include the effect of higher partial waves in the Coulomb-Born approximation (Boikova and Ob'edkov 1968;Gailitis 1976;Chu and Dalgarno 1974), similar to the method described in the work of Rabadán and Tennyson (1998), by calculating three different cross sections: cross sections obtained from our R-matrix method (σ R-mat , calculated according to 7), total cross sections obtained in the Coulomb-Born approximation representing the contribution of all partial waves (σ TCB , 16), and partial cross sections obtained in the Coulomb-Born approximation representing the contribution of the partial waves included in our basis (σ PCB , 14).The final rovibrational excitation cross sections are then a sum of the R-matrix cross sections and the difference between the total and partial Coulomb-Born cross sections, i.e., Lower partial-wave scattering is typically not well described by the Coulomb-Born approximation because the electron is too close to the molecule for the dipole interaction to be considered a perturbation.This is especially true for s-wave scattering.Therefore, we replace the l = 0 -2 partial wave contribution from the Coulomb-Born approximation with those in our R-matrix calculations.The partial Coulomb-Born cross sections are given by where l max = 2 because our R-matrix calculations only include up to l = 2 partial waves.The dipole moment function is given by Q ξ (R) and the matrix elements M ξ ll ′ are given by where F l (η, r) is the regular radial Coulomb function, η = −1/k, and For an approach that does not treat vibration, the integral ⟨v ′ |Q ξ (R)|v⟩ in ( 14) can be replaced with the dipole moment at the equilibrium geometry of the ion.Considering the dipolar coupling (ξ = 1), the partial Coulomb-Born cross sections (14) converge to the following as l max → ∞: where The function 2 F 1 (a, b; c; z) is the Gaussian hypergeometric function defined as: It should be noted that we only include the Coulomb-Born correction to ∆ j = ±1 transitions; our ∆ j = ±2 transitions are obtained purely from our R-matrix method.For further detail in computing (17), we invite the reader to read the work of Chu and Dalgarno (1974).Additionally, the work of Rabadán and Tennyson (1998) 10 -14 10 -13 10 -12 10 -11 10 -10 10 -9 10 -8 10 2 10 3 10 4 State-selected kinetic VE rate coefficients within the ground electronic state of CH + .Solid lines represent rate coefficients from the present calculations, dashed lines are taken from a previous calculation (Jiang et al. 2019).The cross sections are obtained according to (8) and the kinetic rates are obtained according to (9).

RATE COEFFICIENTS FOR ROVIBRONIC (DE-)EXCITATION
Fig. 2 shows the VE rate coefficients obtained by the present method.
The corresponding state-selected kinetic rate coefficients agree well with those of Jiang et al. (2019), as shown in Fig. 2. In Fig. 3, we compare the present VE rate coefficients to the same work, starting from the ground vibronic state to the first excited electronic state.The agreement is worse for excitation between electronic states, possibly due to our improved treatment of channels attached to the excited electronic states.Fig. 4 shows the present RE cross sections convolved with a Gaussian distribution according to (12,11) with widths (γ) of 1 meV and 5 meV to demonstrate the smoothing out of resonances and to compare their overall magnitude.The ∆ j = ±1 transition is the largest over all displayed electron energies (< 1 eV), as expected for such a strongly dipolar system as CH + .Fig. 5 shows 10 -15 10 -14 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 cross section (cm 2 ) electron scaering energy (eV) the individual cross sections of ( 13), which show an increasing correction as a function of the incident electron energy.The σ RVE and σ R-mat cross sections are not convolved, unlike Fig. 4. Fig. 6 compares the present rotational excitation rate coefficients with those obtained by Hamilton et al. (2015), who included the Coulomb-Born correction for ∆ j = ±1, ±2 transitions.We only include this correction for ∆ j = ±1 transitions, but the agreement between the results is good overall.Hamilton et al. (2015) also use an R-matrix approach, but use the adiabatic-nuclei-rotation approximation to obtain rotational excitation (RE) cross sections and rate coefficients, while we use a frame transformation (4) to describe the rotational structure of the ion.Fig. 7 illustrates the difference between the state-selected RE rates with and without the Coulomb-Born approximation, compared to the experimentally determined RE rate coefficient at the CSR (Kálosi et al. 2022) and again to the results of Hamilton et al. (2015).Our present results with the Coulomb-Born correction show the best overall agreement with the CSR measurements, although all theoretical rates are within the provided 1-σ uncertainty for most of the kinetic temperatures shown (10-140 K).

CONCLUSIONS
This paper presents kinetic state-selected DR, VE, and RE rate coefficients obtained with our DR method (Forer et al. 2023), which also allows us to calculate RVE cross sections and rate coefficients with little extra effort.The DR rate coefficients agree well overall with recent experimental measurements made at the Cryogenic Storage Ring (Paul et al. 2022) and better than previous theoretical treatments.Our VE rate coefficients, compared to the work of Jiang et al. (2019), agree well for vibrational excitation within the ground electronic state of CH + .However, the results differ by up to an order of magnitude for vibronic excitation to the first excited state of CH + , which we attribute to a more accurate description of channels attached to excited electronic states (Forer et al. 2023).
We also obtain RE rate coefficients within the ground electronic state of CH + , which we compare to the work of Hamilton et al. (2015) which is an R-matrix method that describes rotational excitation with the adiabatic-nuclei-rotation approximation and does not treat vibration.They correct their ∆ j = ±1, ±2 transitions with the Coulomb-Born approximation, while we only do so for ∆ j = ±1 transitions.Results between our approaches agree well over the pre- ) electron scaering energy (eV) Figure 5.Comparison of cross sections for rotational excitation j = 0 → j ′ = 1 obtained using the R-matrix approach with s, p, and d partial waves (σ R-mat ) and the closed-form total Coulomb-Born approximation (σ TCB ).
The figure shows also the partial Coulomb-Born cross section obtained with s, p, and d partial waves (σ PCB ) and the cross section where the R matrix data is combined with the total Coulomb-Born cross section accounting for partial wave with l > 2 (σ RVE ).sented kinetic temperatures.Compared to j = 0 → j ′ = 1 rate coefficients recently measured at the CSR (Kálosi et al. 2022), our theoretical results using the Coulomb-Born correction agree better over all plotted kinetic temperatures than our theoretical results without the Coulomb-Born correction, and slightly better than the recent theoretically determined rate coefficients of Hamilton et al. (2015) over most kinetic temperatures between 10 K and 140 K.However, all theoretically determined rates under 100 K are within the experimental uncertainty.

Figure 1 .
Figure1.State-selected DR cross sections from the ground vibronic state of CH + .The solid line was obtained with (9) using the theoretical DR cross sections obtained with the current method(Forer et al. 2023).The dashed line with an error-curve represents recent experimental data(Paul et al. 2022).The dot-dashed line represents previous calculations(Mezei et al. 2019).

Figure 4 .
Figure 4. Rotational excitation cross sections within the ground vibronic state of CH + .The cross sections are obtained according to (7), and convolved with a Gaussian as per (11) with γ = 1 meV (thin lines) and γ = 5meV (thick lines).

Figure 6 .
Figure 6.State-selected kinetic RE rate coefficients within the ground vibronic state of CH + .Solid lines represent the present theory, dashed lines represent the results of Hamilton et al. (2015).