Issue 
A&A
Volume 654, October 2021



Article Number  A172  
Number of page(s)  6  
Section  Atomic, molecular, and nuclear data  
DOI  https://doi.org/10.1051/00046361/202141597  
Published online  27 October 2021 
Photoionisation of the CH radical using the Bspline Rmatrix method^{★}
School of Physics, Henan Normal University,
Xinxiang
453007,
PR China
email: wangkd@htu.cn; Yangchj@hotmail.com
Received:
20
June
2021
Accepted:
23
August
2021
Aims. The primary motivation for this paper is to provide accurate data for the photoionisation of the CH radical, including the absolute total photoionisation crosssection, partial crosssections, and photoelectron angular distribution. In addition, the near threshold features in the photoionisation curve (which are absent in previous studies) are produced with high precision.
Methods. A multichannel wavefunction based on the Rmatrix approach, which uses the configuration interaction (CI) method to describe electronic correlation, is carried out in the present calculations. A set of Bspline orbitals is employed to represent the accurate continuum. The distinctive feature of the present calculations allows us to generate a more accurate description of the bound and continuum states than those employed before.
Results. Total photoionisation crosssections from the ground state of CH radicals and partial crosssections corresponding to 1π, 3σ, and 2σ states of CH^{+} ions are presented for photon energies ranging from threshold to 80 eV. Extensive resonance structures, which are absent in previous studies, are observed for the first time near the ionisation threshold. The crosssection dataset obtained from the present calculations is expected to be sufficiently accurate and comprehensive for most current modelling applications involving the photon and CH radical scattering system.
Key words: molecular data / molecular processes / methods: numerical / ultraviolet: ISM
Data tables for crosssections are only available at the CDS via anonymous ftp to cdsarc.ustrasbg.fr (130.79.128.5) or via http://cdsarc.ustrasbg.fr/vizbin/cat/J/A+A/654/A172
© ESO 2021
1 Introduction
The response of atomic systems to ionising radiation is a dominant process in the Universe. It involves neutral atoms, molecules, and clusters, as well as their ions, and takes place in many physical systems including a variety of astrophysical objects, the upper atmosphere, fission, and fusion plasmas, and laserproduced plasma. In many cases, the role of the photoionisation process is central in determining the overall properties of a system. CH is one of the basic components in a variety of astronomical objects, including interstellar clouds (Smith 1992), comets (Wyckoff et al. 1988), and stars (Singh & Chaturvedi 1987). Because of its special importance in these aspects, the CH radical attracts considerable astrophysical interest. Photoionisation and photodissociation of CH, together with the dissociative recombination of CH^{+}, play an important role in modelling the relative abundances of CH^{+}∕CH in diffuse interstellar clouds (van Dishoeck 1987; Federman & Lambert 2002; Heays et al. 2017). However, the formation mechanism of CH^{+} is difficult to explain. One longrunning issue is that the observed interstellar medium (ISM) concentration of CH^{+} is persistently larger than that predicted by the theoretical models from Smith et al. (2014), Godard & Cernicharo (2013), and Myers et al. (2015). This problem was later considered in great detail by Faure et al. (2017) who showed that the proper treatment of nonlocal thermodynamic equilibrium effects are essential to modelling the production of CH^{+} ions in the ISM. Reliable crosssections for photoionisation dynamics are therefore needed for astrophysical model calculations.
Numerous experimental techniques have been employed to study this radical and its cation: absorption (Herzberg & Johns 1969; Davidson et al. 2001; Watson 2001; Pearson & Drouin 2006) and fluorescence (Zachwieja 1997; Ghosh et al. 1999; Truppe et al. 2014) spectroscopy over an extended energy range from the millimeter wave (MW) into the vacuum ultra violet (VUV) and resonanceenhanced multiphoton ionisation (REMPI) (Chen et al. 2000). These measurements have produced very precise descriptions of CH and CH^{+} in their ground and excited electronic states. Several related theoretical studies have also been carried out on the lowest electronic states of CH and CH^{+} (Baluja & Msezane 2001; Vázqueza et al. 2007; Hamilton et al. 2016; Chakrabarti et al. 2017, 2019).
For the ionisation process, to our knowledge, the only currently available experimental observations for photonionisation of CH radicals over the energy range from threshold to 12 eV originate from the measurements by Gans et al. (2016), who produce the radicals by successive hydrogenatom abstractions on methane by fluorine atoms in a continuous microwave discharge flow tube. On the theoretical side, photoionisation spectra of CH radicals were reported by several groups. A study by Walker & Kelly (1972) based on manybody perturbation theory, predicted total photoionisation crosssections of CH over a broad photoenergy range. However, these data suggested substantially larger values for the crosssections than the later results calculated by Barsuhn & Nesbet (1978). Barsuhn & Nesbet (1978) predicted the photoionisation crosssection using the Stieltjesimaging method and a configurationinteraction target wave function. Subsequently, based on the Schwinger variational method using multipletspecific HartreeFock potentials (MSHF) and numerical continuum orbitals, Lee et al. (1990) predicted crosssections and photoelectron angular distributions for the 3σ and 1π levels of CH. Finally,a molecularadapted quantum defect calculation was reported by Lavín et al. (2009), who used semiempirical model potential methods. These authors only considered photoionisation crosssections leading to the ground state of the CH^{+} ion. Recently, using measured and calculated crosssections from several different databanks, Heays et al. (2017) compiled photoabsorption and photoionisation crosssections of CH radicals and modelled their photodissociation and photoionisation rates.
The above discussion shows that the experimental result is available in the relatively narrow range of incident energies up to 12 eV. All the calculations predicted sparse spectra, and no study provided significant details near the ionisation threshold. The goal of the present work is therefore to provide an extensive dataset of crosssections for the photoionisation of CH radicals. The present calculations were performed with the Rmatrix method (Mašín et al. 2020) employing the configuration interaction (CI) method to describe electronic correlation. The distinctive feature of the method is the use of Bspline orbitals to generate accurate descriptions of the continuum. This feature was clearly illustrated in detail in our recent benchmark calculations for photoionisation of OH radicals (Wang et al. 2021). This is very important for an accurate description of resonances and autoionisation processes, especially when they are located very close to the thresholds.
This paper is organized as follows. Section 2 describes the computational method for the structure and the photoionisation processes. This is followed by our results and their discussion in Sect. 3, and finally our conclusions in Sect. 4.
2 Computational details
2.1 Theoretical method
The differential crosssections for photoionisation of a molecule in the length gauge are represented as (Tennyson et al. 1986; Harvey et al. 2014): (1)
where α is the fine structure constant, a_{0} is the Bohr radius, ω is the photon energy in atomic units, ϵ is the polarisation vector of the ionising light in the molecular frame, and d_{if} (k_{f}) is the molecular frame transition dipole between the initial state, i, and a single continuum state, j, as a function of the ejected electron momentum, k_{f}.
If the molecular frame cannot be recovered, Eq. (1) must be orientationally averaged and in the case of linearly polarised laser field one obtains (2)
where β is the asymmetry parameter, σ_{if} is the partial photoionisation crosssection, P_{2} is the secondorder legendre polynomial, and θ is the electron ejection angle between the photoelectron emission direction and photon polarisation direction in the case of linear polarisation.
In the present paper, the bound and continuum wavefunctions in Eq. (1) are presented with an Rmatrix calculation. The key feature of the Rmatrix method (Burke 2011) is the division of space into two separate regions in which different approximations can be made. The usual division is into an inner, outer, and asymptotic region. The boundary between the inner and outer region is defined so as to fully contain the bound wavefunctions of the molecule. In the inner region, the electronelectron effects, exchange, polarisation, and correlation are fully accounted for in a manner analogous to a quantum chemistry calculation, and a flexible basis is constructed to describe thescattering or bound wavefunction in the inner region. The Rmatrix is defined on the boundary and relates the radial part of the continuum electron wavefunction to its derivative. In the outer region, exchange is neglected and the continuum electron moves in the longrange multipole potential of the molecule. The Rmatrix is propagated through the outer region and matched to known asymptotic solutions, allowing for the calculation of observables and giving the expansion coefficients of the full multielectron wavefunction in terms of the inner region basis.
In the inner region, both the continuum and the boundstate wavefunctions are given in terms of the basis functions, , as (3) (4)
where and B_{ik} are energydependent expansion coefficients determined from matching the wavefunctions Eqs. (3) and (4) to the wellknown asymptotic solutions of the system, and x_{i} stands for the spacespin coordinates of the ith electron. The Rmatrix basis functions in turn are written in the closecoupling (CC) form (5)
Here, η_{ij} are the continuum orbitals orthogonalised with respect to the target orbitals, and is an antisymmetrisation operator. Coefficients a_{ijk} and b_{kp} are variational parameters determined by the matrix diagonalisation. The summation in the second term of Eq. (5) runs over configurations χ_{p}, where all electrons are placed in targetoccupied and virtual molecular orbitals. The choice of appropriate χ_{p} is crucial. These are L^{2} configurations and are needed to account for polarisation and for correlation effects arising from excitations in the neutral molecule.
2.2 Target model
The UKRMol+ package (Mašín et al. 2020) is employed to investigate photoionisation for the ground state of CH radicals. The multichannel close coupling (CC) calculations were performed at the equilibrium geometry with R_{CH} = 1.120 Å for the X^{2}Π ground state (Huber & Herzberg 1979). The electronic configuration for the ground state of CH radicals is 1σ^{2} 2σ^{2}3σ^{2}1π^{1}. The selfconsistent field HartreeFock molecular orbitals were generated with the 6311G^{**} basis set using the Molpro programs (Werner et al. 2012). The complete active space (CAS) configuration interaction (CI) method was used to deal with photoionisation of the CH radical. In the CASCI model, the orbitals of CH are represented as a sum of frozen and active orbitals as and , respectively. For the CH^{+} target, the two frozen electrons will remain in the same orbital configuration and the other four electrons are shared among 17 active orbitals.
The calculated vertical ionisation potentials (IPs) for the first ten lowlying states are listed in Table 1, together with the previous experimental results (Herzberg & Johns 1969; Gans et al. 2016) for comparison. As shown in the table, the IP values are in good agreement with the experimental results (Herzberg & Johns 1969; Gans et al. 2016). The principal configuration state functions (CSFs) of the ion states were also listed in Table 1. It is clearly evident that removal of a single electron gives rise to various states. Some of the configurations have the same primary configuration and differ by their spin symmetry.
Vertical ionisation potentials (in eV) for CH radicals obtained with the 6311G** basis set.
2.3 Inner and outer regions
In the inner region, the continuum orbital is constructed from BTOs centered on the center of mass of the system. Bspline basis functions allow calculations with higher kinetic energies of the free electron and the use of greatly enlarged inner regions allowing both large targets and targets with more diffuse electronic states to be studied. The order of Bsplines and the maximum number of radical Bsplines were taken to be k = 6 and N = 20 in the present calculations. The first two radial Bsplines are not included in the basis because these have a nonzero first derivative at the starting point. These continuum orbitals are orthogonalised to the target orbitals, and the continuum orbitals with an overlap of less than 10^{−5} were removed (Morgan et al. 1997). The value of the Rmatrix radius taken to enclose total charges of the target inside the inner region was 15a_{0}. The maximum angular momentum l_{max} = 4 was used to perform partial wave expansions.
In the present calculations, the L^{2} configurations in Eq. (5) can be written in two classes, (core)^{2}(CAS)^{4}(virtual)^{1} and (core)^{2}(CAS)^{5}. Here the active space is composed of 2σ–12σ, 1π–5π, and 1δ orbitals. The inclusion of a large number of target states is necessary to converge the close coupling expansion and to avoid any unphysical pseudoresonances that may otherwise appear at higher energies related to target states left out of the expansion. The number of target states employed in the present CC expansion is 80, which is enough to obtain the convergence of the CC expansion.
Fig. 1 Total photoionisation crosssections from the X^{2}Π ground state of CH radials calculated with our best model in the energy range (a) 10.60–11.80 eV, (b) 1080 eV, the inset panel shows the same crosssections for the energy range 10.50–14.0 eV. Comparison with the available results. 
3 Results and discussion
The total photoionisation crosssections from the X^{2}Π ground state of CH radials in the energy region between 10.60 eV and 11.80 eV are shown in Fig. 1a. We used a fine mesh for photon energies in steps of 0.04 eV to scan and properly separate the possible resonance structures. The relative photoionisation spectra of CH radials measured by Gans et al. (2016) using a double imaging photoelectron/photoion coincidence spectrometer are also shown in Fig. 1a. For comparison, the ionisation threshold has been shifted to the empirical values. Our total crosssection shows numerous sharp peaks originating from two or many electron excited resonances as well as Rydberg resonances associated with the excited electronic states of the CH^{+} ion. In addition to the minimum at 11.6 eV, there are two peaks at around 10.88 and 10.96 eV, which are in good agreement with the experimental results of Gans et al. (2016). In the higherenergy region, the positions for autoionisation peaks predicted by us are slightly different from the experimental results, and the experimental results show more peaks in the crosssection. This is due to the measured results including vibrational effects which are not considered in our present calculations where the fixednuclei approximation is used.
In Fig. 1b we show the total photoionisation crosssections in the energy region from 10 to 80 eV. Three theoretical studies, namely those performed by Walker & Kelly (1972) based on manybody perturbation theory, by Barsuhn & Nesbet (1978) using the Stieltjesimaging method and a configurationinteraction target wave function, and by Lee et al. (1990) using MSHF potentials, are also plotted in Fig. 1b for comparison. With increasing photo energies, discrepancies between all these three calculations diminish. In the resonant region below 18 eV, MSHF results are clearly lower than our crosssections. This is because the correlation is not considered in the approach and photoionisation of the 2σ is not includedin their calculations. In general, the total crosssections obtained by Walker & Kelly (1972) are in good agreement with our results, although their results are slightly higher in the energy region from 23 to 40 eV. Below 14 eV, the theoretical results from Barsuhn & Nesbet (1978) agree with our results, but for the photon energy range from 14 to 17 eV, their crosssections become lower. It should be noted that Barsuhn & Nesbet (1978) performed CI calculations on single excitation series and their photoionisation crosssections are extracted from the discrete CI spectrum, whereas we used a CASCI package of programs.
Our calculated crosssection values are also compared to those available in databases dedicated to astrophysical models. For instance, the Leiden database (Heays et al. 2017) proposes CH photoionisation crosssection values using the results of Barsuhn & Nesbet (1978). The resulting curve is presented in Fig. 1b in orange. In general, this curve agrees with our results in the lowenergy region from the threshold to 11.60 eV and in the other energy region from 11.90 to 13.70 eV. With the photon energy between 11.60 and 11.90 eV, the curve exhibits a peak centered at 12.81 eV with a width of about 0.85 eV, and is higher than our results.
Figure 2 presents the partial crosssections and asymmetry parameters for photoionisation from the ground state of CH leading to the X^{1}Σ^{+} state of CH^{+} ions. A comparison is made between our results and the available theoretical work of Barsuhn & Nesbet (1978), Lee et al. (1990), and Lavín et al. (2009). In general, the present crosssections are consistent with the other theoretical results. Among these results, the data of Lavín et al. (2009) show the best agreement with our results, except for the lack of resonant structures near thresholds in their crosssection, as can be seen in the picture. Our results for the photoelectron asymmetry parameters for this channel shown in Fig. 2 are compared with the data of Lee et al. (1990). The inclusion of correlation effects causes their asymmetry parameters to show a loss of intensity at the energy below 40 eV. Similar to the crosssections, our asymmetry parameters show numerous sharp peaks, while other available results are very smooth. With respect to the behaviour of the crosssections, the asymmetry parameters are more affected by electron correlation effects, as can be judged from the large differences between our results and the MSHF results. The similar correlation effect on photoionisation observables was investigated in our previous paper (Wang et al. 2021).
Figure 3 shows crosssections and asymmetry parameters for ionisation from the ground state of CH leading to the a^{3} Π state of CH^{+} ions with the previous theoretical results. The theoretical data obtained by Barsuhn & Nesbet (1978) are in agreement with those obtained here in the lowenergy region below 14 eV, but are quite different from our results in the energy region between 14 and 17 eV. The results of Lee et al. (1990) display smaller crosssections in the energy region from the threshold to 20 eV, but show good agreement with our results in the energy region between 20 and 45 eV. For the photon energy beyond 45 eV, there is no available experimental or theoretical data for comparison. In contrast with the results of Lee et al. (1990), our photoelectron asymmetry parameters in a^{3} Π, presented in Fig. 3, show a similar trend to those found in the X^{1} Σ^{+} channel presented in Fig. 2.
Figure 4 shows photoionisation crosssections and asymmetry parameters for ionisation from the ground state of CH leading to the A^{1}Π state of CH^{+} ions, together with other theoretical results, included for comparison. Overall, our present crosssections are in agreement with other available theoretical data in the resonant region below 20 eV. Above 20 eV, our crosssections are 0.5 Mb lower than those found by Lee et al. (1990). We also note that the photoelectron asymmetry parameters in A^{1} Π presented in Fig. 4 are almost identical to those obtained in the a^{3} Π channel presented in Fig. 3.
In astrochemistry, photoionisation crosssections can be converted to photoionisation rates if the photon flux is known in a given region of space. By quantitatively constraining the rates of photoprocesses, other chemical and physical parameters can be more reliably obtained, thus facilitating the interpretation of observations. The photoionisation rate of a molecule exposed to an ultraviolet radiation fields is (6)
where σ(λ) is the photoionisation crosssection, and I(λ) is the photonbased radiation intensity summed over all incidence angles. The rate of photoionisation of CH radicals in seven different astrophysically relevent ultraviolet radiation fields is shown in Table 2, together with the available rate values from the Leiden database (Heays et al. 2017) for comparison. Our rate is lower than those shown in the Leiden database for interstellar radiation field (ISRF) from Draine (1978), Galactic radiation field made by Mathis et al. (1983), and blackbody radiation field. But in both solar (Curdt et al. 2001) and TWHya (France et al. 2014) radiation fields, our rate is higher. Considering the more precise photoionisation crosssections used in our calculation, the present result should be more accurate than those found before.
Fig. 2 Photoionisation crosssection and asymmetry parameter for ionisation from the ground state of CH leading to the X^{1} Σ^{+} state of CH^{+} ions. 
Fig. 3 Photoionisation crosssection and asymmetry parameter for ionisaiton from the ground state of CH leading to the a^{3} Π state of CH^{+} ions. 
Fig. 4 Photoionisation crosssection and asymmetry parameter for ionisation from the ground state of CH leading to the A^{1} Π state of CH^{+} ions. 
Photoionisation rates (in s^{−1}) for CH radicals in different radiation fields.
4 Summary
We describe an extensive calculation of the photoionisation of the ground state of CH radicals and present both total and partial crosssections, as well as asymmetry parameters. The present calculations were performed with the multichannel Rmatrix method, where a Bspline basis is employed to represent the continuum orbitals and the configuration interaction method is used to describe electronic correlation. This distinctive feature of the present calculations allowed us to generate a more accurate description of the bound and continuum states than those employed before.
The total photoionisation crosssections from the ground state of CH radicals and partial crosssections corresponding to 1π, 3σ, and 2σ states of CH^{+} ions are presented. The predicted photoionisation observables are compared with available results. Our experience with such calculations leads us to suggest that the present results are by far the most accurate available today. An extensive resonance structure near threshold, which is absent in previous studies, is observed for the first time.
Experimental measurements of the photoionisation dynamical parameters are challenging, in particular for free radicals. Therefore, they are often missing or are found in the literature within a wide range. This is the first theoretical calculation in such a highenergy spectral region with highly accurate values of valence state photoionisation crosssections. We hope that the dearth of data on the photodynamics involving CH radicals encourages further experimental and theoretical investigations.
Acknowledgements
This work was supported by the Natural Science Foundation of China under grant nos. U1504109 and 11604085 and of Henan Province under grant no. 212300410054, the Program for Science and Technology Innovation Talents in the Universities of Henan Province under grant no. 19HASTIT018, and the Science Foundation of Henan Normal University under grants nos. YQ201601 and 2021PL14.
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All Tables
Vertical ionisation potentials (in eV) for CH radicals obtained with the 6311G** basis set.
All Figures
Fig. 1 Total photoionisation crosssections from the X^{2}Π ground state of CH radials calculated with our best model in the energy range (a) 10.60–11.80 eV, (b) 1080 eV, the inset panel shows the same crosssections for the energy range 10.50–14.0 eV. Comparison with the available results. 

In the text 
Fig. 2 Photoionisation crosssection and asymmetry parameter for ionisation from the ground state of CH leading to the X^{1} Σ^{+} state of CH^{+} ions. 

In the text 
Fig. 3 Photoionisation crosssection and asymmetry parameter for ionisaiton from the ground state of CH leading to the a^{3} Π state of CH^{+} ions. 

In the text 
Fig. 4 Photoionisation crosssection and asymmetry parameter for ionisation from the ground state of CH leading to the A^{1} Π state of CH^{+} ions. 

In the text 
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