Open Access
Issue
A&A
Volume 684, April 2024
Article Number A41
Number of page(s) 25
Section Atomic, molecular, and nuclear data
DOI https://doi.org/10.1051/0004-6361/202348293
Published online 03 April 2024

© The Authors 2024

Licence Creative CommonsOpen Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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1 Introduction

Observational studies of molecular gas within the Milky Way have revealed a diverse zoo of about 300 molecules, from simple diatomics such as H2 and CO to ever more complex molecules such as NC4NH+ (Agúndez et al. 2023), NH2C(O)CH2OH (synglycolamide, Rivilla et al. 2023), 2-C9H7CN (2-Cyanoindene, Sita et al. 2022), and H2CCCHC3N (cyanoacetyleneallene, Shingledecker et al. 2021). Due to the cold temperatures of the interstellar medium (~ 10–50 K), much of the gas-phase chemistry is driven through ion-neutral initiated chemistry (Larsson et al. 2012; Tielens 2013; van Dishoeck 2014). In regions shielded from external radiation, highly energetic charged particles, known as cosmic rays, in particular, protons, secondary electrons, and photons, provide the primary ionization source (Umebayashi & Nakano 1981). Finally, chemical models of ices irradiated by energetic particles or X-ray radiation, such as interstellar dust grains and comets, necessarily need to include electron-impact ionization cross sections for as many molecules as possible in order to account for electron production and subsequent interactions (e.g. Shingledecker et al. 2020).

Astrochemical models describe the evolution of vast chemical networks over time for a wide range of physical environments, including the effects of gas density, temperature, atomic abundances, and radiation environments. Two key astrochem-ical databases contain the required reaction rate coefficients: KIDA1 (Wakelam et al. 2012) and UMIST2 (McElroy et al. 2013). However, for cosmic-ray ionization, there is a substantial paucity in reaction rate data, with KIDA and UMIST combined only reporting reaction rate coefficients for H, He, C, O, N, H2, N2, and CO. In astrochemical models, these reaction rates are expressed in the form ζm,T=cm,TζH2, T,${\zeta _{{\rm{m}},{\rm{T}}}} = {c_{{\rm{m}},{\rm{T}}}}{\zeta _{{{\rm{H}}_2},{\rm{T}}}},$(1)

where cm is a scaling factor that relates the rate of the reaction of interest to the total ionization rate, ζH2, T${\zeta _{{{\rm{H}}_2},{\rm{T}}}}$, of H2. The coefficients are computed with respect to some reference total H2 ionization rate, typically, ζH2,03×1017 s1${\zeta _{{{\rm{H}}_2},0}} \approx 3 \times {10^{ - 17}}{{\rm{s}}^{ - 1}}$, which is often referred to as the “canonical” or “fiducial” ionization rate.

The coefficients cmT in KIDA and UMIST date back to early studies from the 1970s and 1980s (Cravens et al. 1975; Cravens & Dalgarno 1978; Glassgold & Langer 1974; Black 1975). For molecules containing atoms beyond hydrogen and helium, the computed ionization rates often use scaling relations between the high-energy cross sections of the molecules and assumed a Voyager-like proton cosmic-ray spectrum. However, there has been substantial development in the theoretical calculation and experimental measurements of these cross sections. Furthermore, it has been demonstrated that the spectrum of secondary nonthermal electrons is highly sensitive to the local proton cosmic-ray spectral shape (Ivlev et al. 2021). Observations of the ionization rate have demonstrated that the H2 ionization rate within molecular clouds is more commonly around 10−16 s−1 (Caselli et al. 1998; van der Tak & van Dishoeck 2000; Neufeld et al. 2010; Indriolo & McCall 2012; Indriolo et al. 2015; Neufeld & Wolfire 2017; Sabatini et al. 2020; Luo et al. 2023a,b; Sabatini et al. 2023), with a decreasing trend toward an increasing column density. This is consistent with a combination of energy losses and possible diffusive transport (e.g. Padovani et al. 2009, 2018; Silsbee & Ivlev 2019; Phan et al. 2023).

A sizable number of studies have reported computed values for the total electron-impact ionization cross sections of a variety of molecules. Public databases such as PlasmaData3 (Zhong et al. 2021), BEAMDB4 (Marinković et al. 2017), the US-based National Institute Standards and Technology (NIST) Electron-Impact cross sections for Ionization and Excitation Database5, and the Japan-based National Institute for Fusion Science (NIFS) datatabase6 have compiled the cross sections for a wide range of molecules, often for use in plasma physics, medicine, or other industry applications. The latter of these databases comprises a significant number of experimental and evaluated ionization cross sections. Recently, Heathcote & Vallance (2018) and Zhou et al. (2019) presented large datasets of calculated and experimental cross sections, respectively, for a wide range of molecules of astrochemical interest. These datasets and databases represent a significant advancement in the availability of cross-section data for astrochemical use. However, they generally do not include molecules containing heavier atoms, and do not report on computed and recommended reaction rate coefficients either. Furthermore, databases comprising a large number of ionization cross sections often either do not use a standard evaluation process or do not provide the data in a readily accessible way.

A number of different electron-impact total ionization cross sections have been introduced in the literature (see e.g. Kim et al. 1997; Deutsch et al. 2000; Blanco et al. 2010). For large molecules, in which the computing electronic structure can become untenable, various different additivity rules have been introduced, which introduce a sort of algebra for adding atomic ionization cross sections and atomic or molecular orbits. One of the most important models is the binary-encounter dipole and its simplification, the binary-encounter Bethe (BEB) cross section (Kim & Rudd 1994; Kim et al. 1997), which we detail below in Sect. 2.2.3. It has performed well compared to experimental data (e.g., Kim et al. 1997; Zhou et al. 2019; Zhong et al. 2021) and has thus become a standard. All of the above databases also present the BEB cross sections as a major result.

In this paper, we present a new public database containing electron ionization cross sections and cosmic-ray ionization rate coefficients for over 200 molecules of astrochemical interest. We report the cross sections for low-energy electrons (10 eV ≤ Ee 5000 eV) for each of these molecules using both a screening-corrected additivity rule (SCAR) and the BEB model cross section, the molecular orbital binding and kinetic energies we used, and the recommended total reaction rate coefficients for molecular cloud environments. Crucially, the cross sections and coefficients are evaluated for nearly all molecules using the same procedure, providing homogeneous datasets to enable better comparisons and consistency.

2 Methods

We present here the methods for the different approaches of calculating the total electron-impact ionization cross sections. Then, quantum chemistry computations are performed, and the resulting ionization rates are calculated. In brief, we calculated three different models of the electron-impact total ionization rate: the SCAR, the BEB, and the damped-BEB (dBEB) model. In particular, we explored the accuracy of the SCAR method because it may become more applicable for larger molecules due to the computational expense required for accurately computing BEB-related cross sections. The latter models require knowledge of the electronic structure, while the former only requires knowledge of the geometry and a basis set of electron-atom ionization cross sections. For our molecule selection, we chose the primary molecules used in astrochemical networks with reaction rate data on the kinetics database KIDA.

We emphasize that the total ionization cross sections below are for single-ionization events. At high energies, multiple ionization events can occur, in particular, through Auger ionization. Nishimura et al. (1999) estimated the possible impact of multiple ionizations by doubling the contribution of the cross section from inner shell electrons and found that their inclusion shifted the peak by 5% and toward higher impact energies. The contributions of multiple ionization events are beyond the scope of this initial release, and they will be considered for future releases. Furthermore, molecules are assumed to be ionized from their ground state.

thumbnail Fig. 1

Electron ionization cross sections. The solid lines give the fits from Eq. (2), and the black points show the data from the NIFS database.

2.1 Electron-atom ionization cross sections

The semi-empirical method we use requires a basis of electron-atom ionization cross sections. We obtain these cross sections by fitting a polynomial to experimental data from the NIFS AMIDIS-ION5 database, where the electron-atom ionization cross section takes the form σi(E)={ 0 for Ee<IPia02(x1x)[ c02(lnxx)+k=15ckxk ] for EeIPi, ${\sigma _i}(E) = \{ \matrix{ 0 \hfill & {{\rm{for}}{E_{\rm{e}}} < {\rm{I}}{{\rm{P}}_i}} \hfill \cr {a_0^2\left( {{{x - 1} \over x}} \right)\left[ {c_0^2\left( {{{\ln x} \over x}} \right) + \mathop \sum \limits_{k = 1}^5 {{{c_k}} \over {{x^k}}}} \right]} \hfill & {{\rm{for}}{E_{\rm{e}}} \ge {\rm{I}}{{\rm{P}}_i},} \hfill \cr } $(2)

where α0 is the Bohr radius, Ee is the electron energy, IPi is the ionization potential of the atomic species i, x = Ee/IPi and ck are fitting coefficients. Figure 1 shows the experimental data and the fit cross sections. Table 1 shows the results of the fits of Eq. (2) for each atom.

2.2 Semi-empirical cross-section calculation

2.2.1 Additivity rule

The simplest semi-empirical cross section is the simple additivity rule (AR). Here, the total molecule ionization cross section of species, m, is σm,AR(E)=iσi(E),${\sigma _{{\rm{m}},{\rm{AR}}}}(E) = \mathop \sum \limits_i {\sigma _i}(E),$(3)

where the summation is over the different atoms, i. This ignores all possible overlap of the atomic cross sections and electronic structure and always provides an overestimate of the ionization cross section.

2.2.2 SCAR method

The cross sections are computed using the SCAR following Blanco & García (2003); Blanco et al. (2010). This method enables the quick computation of molecular cross sections using just a precomputed basis of electron-atom ionization cross sections. SCAR is a generalized AR that takes into account the effective geometrical overlap of the atomic electron cross sections. The cross section for a specific molecule, m, is given by σm,SCAR(E)=isiσi(E),${\sigma _{{\rm{m}},{\rm{SCAR}}}}(E) = \mathop \sum \limits_i {s_i}{\sigma _i}(E),$(4)

where the sum is carried out over all constituent atoms, and σi(E) is the electron-atom ionization cross section for an atom i. Under the simplest additivity rules, as shown above, si = 1. For the SCAR method, the additivity coefficients, si, are si=k=1Nk(1)k+1εi(k)k!,${s_i} = \mathop \sum \limits_{k = 1}^{{N_k}} {{{{( - 1)}^{k + 1}}\varepsilon _i^{(k)}} \over {k!}},$(5)

where Nk is the number of perturbation terms included. The coefficients, εi(k)$\varepsilon _i^{(k)}$, follow the recursive relation εi(1)=1$\varepsilon _i^{(1)} = 1$(6) εi(k)=Nkk+1Nk1j(i)σjεj(k1)αji,k=2,,N,$\varepsilon _i^{(k)} = {{{N_k} - k + 1} \over {{N_k} - 1}}\mathop {{{\mathop \sum \nolimits^ }^}}\limits_{{\rm{j}}( \ne {\rm{i}})} {{{\sigma _j}\varepsilon _j^{(k - 1)}} \over {{\alpha _{ji}}}},k = 2, \ldots ,N,$(7)

where αji=max(4πrij2,σi,σj)${\alpha _{ji}} = \max \left( {4\pi r_{ij}^2,{\sigma _i},{\sigma _j}} \right)$, and rij is the distance between atoms i and j. The terms in Eq. (5) amount to higher-order correction factors accounting for the overlap of the cross sections of all the individual items. Due to the recursive nature of the coefficients (Eq. (6)), including higher-order correction terms leads to an exponential increase in computational cost. However, this is alleviated by building a dictionary cache during the recursion to avoid recomputing the same screening terms. This leads to a sublinear increase in computing time against maximum k-atom screening correction included.

Table 1

Electron-atom ionization cross-section fit coefficients for Eq. (2).

2.2.3 BEB cross sections

The semi-empirical BEB cross section was developed as a simplification of the binary-encounter dipole cross section (Kim & Rudd 1994; Hwang et al. 1996). This cross section has been found to provide a reasonable match with experimental data, and it only requires knowledge of the molecular orbital energies. The BEB cross sections are the base of numerous cross section databases, in particular, the NIST database. The BEB cross section is defined as σm,BEB=[ St+(u+1) ] [ lnt2(11t2) ${\sigma _{{\rm{m}},{\rm{BEB}}}} = \mathop {{{\mathop \sum \nolimits^ }^}}\limits_\ell \left[ {{{{S_\ell }} \over {{t_\ell } + \left( {{u_\ell } + 1} \right)}}} \right][{{\ln {t_\ell }} \over 2}\left( {1 - {1 \over {t_\ell ^2}}} \right)$(8) +(11tlntt+1) ],$ + \left( {1 - {1 \over {{t_\ell }}} - {{\ln {t_\ell }} \over {{t_\ell } + 1}}} \right)],$(9)

where the sum is over orbitals, indexed ℓ, t = Ee/B, u = U/B and S=4πa02n(RB)2${S_\ell } = 4\pi a_0^2{n_\ell }{\left( {{{{R_\infty }} \over {{B_\ell }}}} \right)^2}$. Here, B and U are the orbital binding and orbital kinetic energies of the ejected electron, respectively, n is the orbital occupation number, and R is the Rydberg constant. The parameters B and U are generally computed using electronic structure methods.

We also propose a modified version of the BEB cross section (dBEB) to dampen the impact of orbitals with binding energies greater than the ionization potential. In dBEB cross section formulation, the B is scaled by an exponential function such that the modified binding energy of orbital is given by B=Be(1B/IP).$B_\ell ^\prime = {B_\ell }{{\rm{e}}^{ - \left( {1 - {B_\ell }/{\rm{IP}}} \right)}}.$(10)

As we demonstrate below, this prevents the BEB cross section from overestimating with respect to experimental values, which are generally upper limits. In general, BEB cross sections underpredict experimental results at higher energies since they only account for single ionizations, while experimental data, in which the measured signal is generally the ion current, include contributions from multiple ionization events that contribute more to the measured signal per event than single ionizations.

2.3 Quantum chemistry computations

We selected a set of 156 neutral species including atoms and molecules to calculate the BEB and dBEB cross sections. The initial structures were taken from NIST Computational Chemistry Comparison and Benchmark DataBase7 (CCCBDB) database (Johnson 2022) or ChemSpider8. The equilibrium geometries were obtained at MP2/aug-cc-pVTZ method and basis set. To validate the accuracy of this level of theory, we also optimized a group of 50 structures using the highly accurate DF-CCSD(T)-F12/cc-pVDZ-F12 (U-CCSD(T)-F12/cc-pVDZ-F12 for radical species). The set of electron binding energy (eBE), B, taken as the negative of the energy of orbital , was computed by means of electron propagator theory (EPT) in its P3+ implementation along with the aug-cc-pVTZ basis. In EPT, the electron correlation is taken into account, and it thus provides a more accurate value for B than Hartree-Fock (HF) canonical orbital energies. To compute the BEB cross sections, we used the EPT-corrected orbital energies when the related pole strength was greater than 0.8, otherwise, we used the HF canonical orbital energies as an estimate of the B. Since the lowest B, namely the IP, according to Koopmans’ theorem, is the predominant contribution to the BEB cross section, the accuracy of the eBE plays a significant role.

We computed vertical IPs for 148 molecules in our sample using the density functional theory (DFT) method CAM-B3LYP/aug-cc-pVQZ, employing the D3 empirical dispersion. Furthermore, to validate the accuracy of the IPs we compared the DFT results to highly accurate values that we computed at CCSD(T) paired with complete basis set (CBS) extrapolation using cc-pVXZ (X=D,T) basis functions for a subset of 56 species. This subset consists of molecules with two to four atoms.

The computations were performed using the software packages GAUSSIAN169 (Frisch et al. 2016), MOLPRO10 (Werner et al. 2012, 2020), and PSI411 (Smith et al. 2020b, interfaced with the QCFRACTAL12 infrastructure; see Smith et al. 2020a). Details of the different electronic structure methods are presented in Table 2.

Finally, we included molecular geometries and orbitals from Heathcote & Vallance (2018), which comprise 141 molecules, including 6 anions and 18 cations. These were computed at the HF/aug-cc-pVTZ level, with a small subset at the MP2/aug-cc-pVTZ level (C4H, C5H, C5N, and C6H; see Heathcote & Vallance 2018 for details). H2 was computed at the CCSD(T)/aug-cc-pVTZ level. We also included the unpublished calculated data for cations and anions, which were previously excluded because there is a lack of experimental data for these molecules against wich to benchmark, and the method of calculation therefore could not be confirmed. We therefore excluded the anions and cations from the reaction rate plots, although they are included in the full database.

Table 2

Summary of the quantum chemistry methods and references.

2.4 Low-energy electron spectrum and ionization rate

Low-energy electrons in molecular clouds are primarily generated as secondary particles produced by ionization of H2 by primary cosmic-ray protons or electrons. We obtained depth-dependent electron spectra, for example, je(Ee,NH2)${j_{\rm{e}}}\left( {{E_{\rm{e}}},{N_{{{\rm{H}}_2}}}} \right)$, where NH2${N_{{{\rm{H}}_2}}}$ is the H2 column density in the cloud following the new rigorous prescription in Ivlev et al. (2021), with corrections for H2 excitation and new H2 electron cross sections (Padovani et al. 2022).

The depth-dependent electron ionization rate of a molecule, m, is given by adding the contributions of primary protons, ζp,m(NH2)${\zeta _{{\rm{p}},{\rm{m}}}}\left( {{N_{{{\rm{H}}_2}}}} \right)$, using the approximation meEpmpEe, ζp,m(NH2)=2πjp(Ep,NH2)σm[ (me/mp)Ep ]dEp,${\zeta _{{\rm{p}},{\rm{m}}}}\left( {{N_{{{\rm{H}}_2}}}} \right) = 2\pi \mathop \smallint \nolimits^ {j_p}\left( {{E_{\rm{p}}},{N_{{{\rm{H}}_2}}}} \right){\sigma _{\rm{m}}}\left[ {\left( {{m_{\rm{e}}}/{m_{\rm{p}}}} \right){E_{\rm{p}}}} \right]{\rm{d}}{E_{\rm{p}}},$(11)

primary electrons, ζe,m(NH2)=2πje(Ee,NH2)σm(Ee)dEe,${\zeta _{{\rm{e}},{\rm{m}}}}\left( {{N_{{{\rm{H}}_2}}}} \right) = 2\pi \mathop \smallint \nolimits^ {j_{\rm{e}}}\left( {{E_{\rm{e}}},{N_{{{\rm{H}}_2}}}} \right){\sigma _{\rm{m}}}\left( {{E_{\rm{e}}}} \right){\rm{d}}{E_{\rm{e}}},$(12)

and secondary electrons from both primary protons and electrons, ζse,m(NH2)=4πjse(Ee,NH2)σm(Ee)dEe,${\zeta _{{\rm{se}},{\rm{m}}}}\left( {{N_{{{\rm{H}}_2}}}} \right) = 4\pi \mathop \smallint \nolimits^ {j_{se}}\left( {{E_{\rm{e}}},{N_{{{\rm{H}}_2}}}} \right){\sigma _{\rm{m}}}\left( {{E_{\rm{e}}}} \right){\rm{d}}{E_{\rm{e}}},$(13)

where se represents the secondary electrons. The factors of 2π and 4π account for fluxes, which are assumed to be plane parallel and isotropic, respectively. The total ionization rate for a molecule is then ζm = ζpm + ζe,m + ζse,m. The factor of 4π comes from treating the secondary electrons as an isotropic local source.

We used the “High” primary proton and proton-induced secondary electron spectra as a function of column density from Padovani et al. (2022), who computed the electron spectrum down to 1 eV following the recent more rigorous theory of Ivlev et al. (2021). The High proton spectrum was calibrated to match diffuse gas observations of the H2 ionization rate, which are not reproduced with a Voyager-like spectrum (Ivlev et al. 2015; Padovani et al. 2018). We also used an interstellar primary electron spectrum from Padovani et al. (2018) and their induced secondary electrons. The secondary electron computation assumes that the gas is fully molecular and includes energy losses from Coloumb interactions, as well as for H2 ionizations and electronic and rovibrational excitations. We defined our column-dependent total ionization rate coefficient for species m, cm,T(NH2)${c_{{\rm{m}},T}}\left( {{N_{{{\rm{H}}_2}}}} \right)$ from ζm,T(NH2)=cm,T(NH2)ζH2, T(NH2),${\zeta _{{\rm{m}},T}}\left( {{N_{{{\rm{H}}_2}}}} \right) = {c_{{\rm{m}},T}}\left( {{N_{{{\rm{H}}_2}}}} \right){\zeta _{{{\rm{H}}_2},{\rm{T}}}}\left( {{N_{{{\rm{H}}_2}}}} \right),$(14)

where ζH2,T(NH2)${\zeta _{{{\rm{H}}_2},{\rm{T}}}}\left( {{N_{{{\rm{H}}_2}}}} \right)$ is the total H2 ionization rate including all primary, secondary, and tertiary processes. We report the column-density average coefficient, c¯m,T${\bar c_{{\rm{m}},{\rm{T}}}}$, for the cloud column density range, NH210201023${N_{{{\rm{H}}_2}}} \approx {10^{20}} - {10^{23}}$ cm−2, although we note that the coefficients only marginally scale with column density. Hereafter, we denote c¯m,T${\bar c_{{\rm{m}},{\rm{T}}}}$ as cm,T, due to the marginal scaling with column density. In general, since the BEB cross sections perform better than the SCAR data in comparisons against experimental data, we report here only the coefficients using the BEB cross section, although all cross sections are available in the database.

Table 3

Unique molecules included in this database (202), sorted by their number of constituent atoms.

3 Results

In total, we have computed the structure, orbitals, and cross sections for 202 unique (by composition, not counting isomers) neutral molecules ranging in size from two to 24 atoms, shown in Table 3, including data augmented by the results from Heathcote & Vallance (2018). Figure 2 shows the distribution of the number of atoms for the molecules in our database. While most of the molecules have fewer than 6 atoms, we include some with up to 24 atoms (C14H10). When KIDA does not specify an isomer but multiple isomers exist, we took the most stable isomer. We detail a summary of the results below, with all of the data available online in the public database13.

3.1 Ionization potentials, electron binding energies, and electron kinetic energies

We present here the ionization potentials computed at both the DFT (CAM-B3LYP) and CCSD(T)/CBS levels, and we compare them to the experimental value recommended in NIST. Table A.1 shows the resulting ionization potentials in eV for the molecules. Figure 3 shows a comparison between the different calculations and the NIST database. In general, our calculations for molecules with fewer than seven atoms are consistent with the values presented in the NIST database. For molecules with higher ionization potentials, our calculated IPs are generally higher. The agreement between the CAM-B3LYP and CCSD(T) calculations is relatively good, but there is a general trend that CAM-B3LYP produces smaller IPs compared to CCSD(T) for molecules with IP <10 eV.

As part of the database, we also present the molecular orbital binding and kinetic energies for each molecule. An example result of this for CO and H2O is presented in Table 4 and Table 5. The tables show the eBEs computed from the canonical HF orbitals and those from the EPT theory. The eBE computed with the latter method are smaller than those obtained with HF since they are corrected to include the instantaneous electron–electron repulsion (electron correlation) in its energy, which increases the energy of the orbitals, and thus facilitates detachment for an electron. According to Koopmans’ theorem, the lowest eBE is equal to the ionization potential of the molecule. The values of the EPT-eBE and the IP computed at our best level of theory (CCSD(T)/CBS) agree excellently. The discrepancies are about 0.2 eV and are much improved with respect to HF-eBEs. Given that the lowest eBE carries the most weight in the BEB cross section, it is paramount to ensure that it is accurate. Our results not only exhibit a marked improvement over canonical HF values, but also suggest that the error related to the electronic structure is very small. However, for the deep-lying orbitals, no reliable EPT-eBEs can be computed. We therefore chose to use the HF values for these orbitals.

Our HF-BE data and the NIST electron-impact cross-section database agree well. The database uses data from Hwang et al. (1996) computed at the HF/6-311-G level for the eBE. The agreement is expected because the only difference between the NIST and our HF orbital binding energies is a larger atomic orbital basis set in the latter case. However, the incorporation of EPT-BE into our database enhances the overall data quality compared to the existing NIST data. We thus conclude that our calculations of orbital binding energies are derived from a robust theoretical framework, offering the required level of accuracy given the underlying assumptions in our cross-section calculations.

Finally, we also included the optimized geometries at the MP2/aug-cc-pVTZ and DF-CCSD(T)-Fl2/cc-pVDZ-Fl2 levels presented here and the sample from Heathcote & Vallance (2018) in the database. The geometries are not appreciably different from each other. All geometries that represent minimum energy structures were verified by ensuring that no imaginary frequencies in the diagonalized Hessian matrix are present.

thumbnail Fig. 2

Distribution of the number of atoms in each molecule for this data release.

thumbnail Fig. 3

Comparison of the ionization potentials presented here with the NIST database. Left: NIST database vs. CCSD(T) and CAM-B3LYP DFT calculations (solid and empty squares, respectively). Right: CCSD(T) vs. CAM-B3LYP DFT ionization potentials. The dashed line represents the one-to-one ratio.

Table 4

Molecular orbitals of CO computed at the MP2/aug-cc-pVTZ level.

Table 5

Same as Table 4, but for H2O.

3.2 Low-energy electron cross sections

As described earlier, we have computed the cross sections using three different approaches: the SCAR, the BEB model, and the new dBEB.

The SCAR method is inherently recursive, so that care must be taken to avoid exponential increases in computational time with the number of atoms. We used recursive caching to speed the calculations up, and we only computed to a maximum of the 10-atom screening correction. Figure 4 shows the individual screening corrections, the magnitude of the correction, and the total time as a function of the maximum screening terms kept. The screening terms past the 8-atom screening correction are negligible contributions to the total cross section, which is dominated by the first few correction factors.

We show results for a subset of the molecules, ranging from simple to complex, to compare the difference between the SCAR and BEB cross sections. Figure 5 shows them for a subsample of ten molecules, ranging in size from CO to c-C6H6. We find that the SCAR approach generally overestimates the cross section, although the impact is most pronounced in c-C6H6 because the molecular orbitals overlap. However, given the speed with which these cross sections were computed, they may be useful for first investigations when the optimized molecule geometry is known a priori. While we encourage the use of BEB cross sections, we include a Python script in the database to compute the SCAR cross section from a provided optimized geometry.

We investigated the model predictions for a subset of these molecules in more detail for which experimental data are available. Figure 6 shows a comparison with experimental data for a subset of molecules, including the simple molecule CO2, symmetric ring c-C6H6, prebiotic species of interest NH2CHO, and carbon chains. Experimental and theoretical cross section data are known to deviate quite significantly (see the discussion in Zhou et al. 2019). In particular, experimental cross sections include double ionizations and Auger ionizations, and they therefore overestimate the single-ionization cross section. The cross sections we present here are for single-ionization events. Multiple ionizations are left for future work. With the data we computed, the BEB slightly overesitmates the cross section in low-energy regions and underestimates it in higher-energy regions. Nonetheless, the values from the BEB calculation agree far better with experimental results than the SCAR or the AR model. Because the experimental curve should serve as an upper limit for the theoretical BEB model, and noting that even with highly accurate eBEs it still overestimates the cross section, we suggest a modified BEB model in which the contribution from deeper-lying electron ionizations are scaled, to better balance its weight in the total cross section. As expected, we find that the dBEB corss-section underpredicts the experimental value. Future calculations using the more complicated binary-encounter dipole model (Kim & Rudd 1994) will be investigated for future releases. In our further analysis, we use the BEB cross sections to calculate the reaction rates because toward higher energies, they tend to agree better with the data.

thumbnail Fig. 4

Benchmark of the number of k-atom screening corrections included using C5H10. Left: k-atom screening corrections as a function of electron energy for C5H10. Right: time to compute σSCAR(Ee) for a given maximum k-atom screening correction (black) and the magnitude of the maximum correction (blue) as a function of the maximum k.

thumbnail Fig. 5

Sample of the total electron ionization cross sections for a subsample of the molecules. The solid line denotes cross sections calculated using the BEB method, and the dotted line shows the cross sections calculated with the SCAR rule.

3.3 Molecular cloud rate coefficients

We discuss here the molecular cloud reaction rate coefficients, scaled to the total ionization rate (see Sect. 2.4). Figure B.1 show the coefficients cm,T for our database sample. There is a general trend of an increase in cm,T with the size of the molecule. However, for a given number of atoms, there is still substantial scatter due to the geometry and composition. Our reaction rate is marginally greater than that reported by UMIST for CO, originally from Black (1975), where they report cCO,T = 3 whereas we find cCO,T = 3.476. We report the mean coefficients in Table A.2, where the mean is taken of the coefficients for total hydrogen nucleus column densities ranging from 1020– 1023 cm−2. We note that these reaction rates are tailored for molecular cloud-like environments, such as those whose prescribed external cosmic-ray spectrum matches the “High” model from Padovani et al. (2022) (see also Ivlev et al. 2015) and whose total hydrogen nucleus column densities lie between 1020NH ≤ 1023 cm−2.

There is a substantial difference in the reaction rates found with the SCAR and BEB methods. While the additivity rules account for the geometric overlap of the atomic cross sections, it does not account for the structure of the molecule orbitals nor for differences in eBEs due to molecular bonds. Therefore, we do not recommend the use of additivity rules to compute the cross sections for use in astrochemical modeling, unless it is not practical to compute molecular orbitals and use the BEB cross section.

Figure 7 shows the mean cm,T coefficient as a function of the number of atoms. We find a general increase in the reaction rate with the number of constituent atoms. The figure also shows a best-fit relation, cm,T=1.19Natom +1.84,${c_{{\rm{m}},{\rm{T}}}} = 1.19{N_{{\rm{atom}}}} + 1.84,$(15)

where Natom is the number of constituent atoms. The relation reproduces the general trend for Natom 20. We also fit the coefficients as a function of the number of valence electrons, which produces a slightly better fit, cm,T=0.45Nv,elec0.64,${c_{{\rm{m}},{\rm{T}}}} = 0.45{N_{{\rm{v}},{\rm{elec}}}} - 0.64,$(16)

where Nv,elec is the number of valence electrons in the molecule. This fit was only constrained for molecules with 9 ≤ Nv,elec≤ 70. In both cases, there is significant scatter of approximately a factor of 2 around the fit trends, so that caution should be used when these are used for molecules that are not listed in the database.

thumbnail Fig. 6

Electron ionization cross sections computed with the AR (dotted), SCAR (solid), and BEB (dash-dotted) and dBEB (dashed) methods compared with experimental data from the NIFS database (blue dots).

thumbnail Fig. 7

Total ionization rate coefficient, cm,T, as a function of the constituent number of atoms in each molecule. The color denotes the energy at the cross section maximum.

3.4 Database description and file format

We include a number of different formatted files for the total electron-impact cross sections, ionization potentials, molecular and atomic orbitals, and reaction rate coefficients. Cross sections are found in the ion_xs/ directories for the SCAR, HF, MP2, and CCSD(T) calculations and atoms. The cross-section files are formatted as in the following example, co. xs

# CO cross section
256
1.    00000e+01    0.00000e+00    0.00000e+00
1.    04618e+01    0.00000e+00    0.00000e+00
1.    09450e+01    0.00000e+00    0.00000e+00
1.    l4505e+01    0.00000e+00    0.00000e+00
1.    19793e+01    0.00000e+00    0.00000e+00
1.    25325e+01    0.00000e+00    0.00000e+00
1.    31113e+01    0.00000e+00    0.00000e+00
1.    37169e+01    0.00000e+00    0.00000e+00
           {more data}
9.    13659e+05    8.52230e-03    6.30407e-03
9.    55855e+05    8.17497e-03    6.04801e-03
1.    00000e+06    7.84171e-03    5.80155e-03

where the first row gives the number of electron energy bins. The later rows give the data in (two) three columns. For BEB cross sections, the columns provide electron impact energy (eV), BEB cross section in units of σ02$\sigma _0^2$ and, if available, the damped-BEB cross section in units of σ02$\sigma _0^2$. For SCAR cross sections, the three columns provide energy (eV) and SCAR cross section in units of σ02$\sigma _0^2$.

The orbital information is stored in two directories containing the orbitals in the NIST format, NIST_orbitals/ and including all orbitals, full_orbitals/. An example of the NIST formatted file, co.norb

#CO NIST Orbitals
#Orbital   B   U   N   Q
1   562.416986   794.326705   2   1
2   309.246847   436.294142   2   1
3    41.332626    78.262641   2   1
4    21.888866    71.857734   2   1
5    17.015000    53.945104   2   1
6    17.015000    53.945104   2   1
7    14.266000    42.788163   2   1

where the B column gives the eBE (eV), U gives the average electronkineticenergy(eV), N gives theorbital occupationnum-ber and Q is a scaling factor to include higher-order ionization effects. We also show an example of the full orbital file, co.forb

#CO Full Orbitals
#Alpha orbitals
#Orb.   KE   BE-HF   BE-P3+   Pole
1   794.32671   562.41699      ***     ***
2   436.29414   309.24685      ***     ***
3    78.26264    41.33263      ***     ***
4    71.85773    21.88887      ***     ***
5    53.94510    17.35920   17.015   0.903
6    53.94510    17.35920   17.015   0.903
7    42.78816    15.11922   14.266   0.910

where the file will give the information for both alpha and beta orbitals for open-shell molecules that were computed with an unrestricted formalism. The KE and BE-HF columns give the average electron kinetic energy and eBE (eV) computed by the population analysis in GAUSSIAN 16 following geometry optimization. The latter two columns include the results from EPT, where *** denotes orbitals where EPT was not computed. For orbitals where it was computed, we include the EPT-corrected HF canonical orbital energies (at P3+ level) results and the pole strength (PS).

We include the computed ionization potentials in the ips/ directory for the compiled NIST and computed CAM-B3LPY and CCSD(T) molecules. These are two-column files with the molecule and ionization potential in eV. We also include two network files containing all the new molecular and atomic ionization rate coefficients in the networks/ directory. These network files are in the KIDA and UMIST formats, alecs.kida.in and alecs.umist.d, respectively. We recommend users to only include ionization rates for molecules in which there are associated recombination rates for the ion. In our rate files, we assume that all ionizations occur in the manner AB + e→ AB+ + e + e, and we emphasize that potential users should check their chemical networks for other branches such as AB + e→ A + B+ + e + e. We leave these differences for future database releases. Finally, in the geoms/ folder, we include all molecule geometries computed at the HF, MP2, and CCSD(T) levels. The MP2 and CCSD(T) geometries are formatted as PDB files and the HF geometries as XYZ.

Table 6

Initial abundances as a function of the total H nuclei abundance, i.e., nH,tot (Hincelin et al. 2011).

3.5 Astrochemical modeling

We include the new calculations in a model that uses the KIDA reactions framework, providing a zero-dimensional model as in Sect. 3.1 of Wakelam et al. (2015). It is important to note that in our context, this test has not been designed to quantify the impact of the new rates in an astrophysical environment, but to show that the new rate equations are compatible with previously established formats. Therefore, the cross sections may play a more important role in ice chemistry, disks, planetary atmospheres, exomoons, or cometary environments, for instance. However, this analysis is beyond the aims of the present paper, and it will be discussed in a forthcoming work.

We modeled a gas with total hydrogen density nH,tot = 2 × 104 cm−3, gas temperature T = 10 K, initial conditions as in Table 6, and no dust (e.g., see Hincelin et al. 2011; Loison et al. 2014). We computed the ionization rate ζ(NH2)$\zeta \left( {{N_{{{\rm{H}}_2}}}} \right)$ including protons, primary and secondary electrons, and the secondary from primary electrons. The ionization rate is a function of the column density N (see Eq. (14)), and we assumed that the visual extinction (necessary for the photochemistry) is AV = 1.0638 × 10−21 N, where N is in units of cm−2. To model the time-dependent evolution of the chemical abundances for 10 Myr, we made use of KMARX (Grassi in prep., commit 58f6ac9), a Python-based database that allowed us to solve the chemical ordinary differential equations with a standard BDF solver14 (Hindmarsh 1983). We include the HTML output produced by the code in the database as a zip file in the chem_models/ folder, where the reactions present in this work are listed under the class CRReactionAdv.

When using our chemical network, we replaced the ionization reactions from KIDA where possible (i.e., H, He, N, O, H2, and CO) and added the missing ionization rates that affect the molecules listed in Table 6. We did not add the species that create sinks or sources in the chemical network, that is, species that only appear in the products or in the reactants. We also note that for the sake of comparison, the cosmic-ray reaction rates from KIDA were scaled so that the H2 ionization matches the ionization rate of molecular hydrogen in our database. The aim of this scaling is to avoid discrepancies determined by different assumptions in the cosmic-ray spectra employed.

Figure 8 shows the results obtained with the KIDA database reaction rates (solid lines) and calculated with the rates present in this work (dashed lines). For the sake of clarity, we only plot the species that show a difference larger than one-tenth of an order of magnitude and reach at least n(t) = 10−8nH,tot during their evolution. Only a few species present a negligible discrepancy between the two databases. The extent of these variations is very small because the critical cosmic-ray-driven reactions are very similar in ALeCS and in KIDA, but the results might be less interchangeable in different environments. Moreover, we note that we assumed that the constituents of the spectra match, that is, that both include protons, electrons, and secondary processes, hence the scaling mentioned above.

thumbnail Fig. 8

Comparison of the chemical evolution using KIDA reaction rates (solid lines) and our rates (dashed lines). We report the species that present a difference of at least half an order of magnitude and reach at least n(t) = 10−8nH,tot. The ionization rate of H2 is 1.64 × 10−16 s−1, and the rate for model “High” and with the assumptions described in the text corresponds to Av = 10.6 and N = 1022 cm −2.

4 Conclusions

We presented the initial data release for the Astrochemistry Low-energy electron cross-section (ALeCS) database. In this release, we include the total ionization cross sections and ionization rate coefficients for over 200 neutral molecules of astrochemical interest calculated using three different semi-empirical methods: the SCAR (Blanco & García 2003; Blanco et al. 2010), the BEB model (Kim & Rudd 1994; Hwang et al. 1996), and a new dBEB presented here. The last model dampens orbitals deep within the potential well and was demonstrated to help prevent the BEB model from overestimating the ionization cross section when compared with experimental results. We also presented the ionization rate coefficients and molecular ionization rates scaled to a reference total H2 ionization rate.

The database is fully public and will include the ionization data described above, along with the molecule orbitals and optimized geometries. In this current release, we only include the semi-empirical ionization cross sections and reference experimental data. Future releases will include more sophisticated ionization cross-section calculations along with excitation and momentum transfer. Finally, the database will include open-source software tools necessary to couple these processes to astrochemical codes.

Acknowledgements

We thank the referee, Jonathan Tennyson, for their useful comments which improved this work. The MP2 and DFT calculations presented here were performed on the VERA computing cluster, managed by the Chalmers Center for Computational Science and Engineering. BALG and PG are supported by the Chalmers Initiative on Cosmic Origins as Cosmic Origins Postdoctoral Fellows. TG acknowledges support of the DFG (German Research Foundation) Excellence Cluster ORIGINS – EXC-2094 – 390783311. S.B. is financially supported by ANID Fondecyt Regular (project #1220033), and the ANID BASAL projects ACE210002 and FB210003. CV and DH would like to thank the UK Engineering and Physical Sciences Research Council for funding via Programme Grants EP/V026690/1 and EP/T021675/1. S.V.G. is financially supported by VRID project 20220005071NV. GMB gratefully acknowledges support from ANID Beca de Doctorado Nacional 21200180. This is University of Fexas Center for Planetary Systems Habitability (CPSH) contribution number #0073.

Appendix A Additional tables

Table A.1

Ionization potentials in eV, computed at the CCSD(T)/CBS, CAM-B3LYP/aug-cc-pVQZ and from NIST.

Table A.2

Chemical reaction rates, cm,T

Appendix B Additional figures

thumbnail Fig. B.1

Reaction rate coefficients following Equation 14 for all neutral molecules in the database. The boxes show the minimum and maximum coefficients in the column density range 1020 cm−2 NH21023$ \le {N_{{{\rm{H}}_2}}} \le {10^{23}}$ cm-2, and plus signs show the mean. The cyan boxes denote molecules using the MP2 calculations, and the red boxes denote calculations from Heathcote & Vallance (2018).

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All Tables

Table 1

Electron-atom ionization cross-section fit coefficients for Eq. (2).

Table 2

Summary of the quantum chemistry methods and references.

Table 3

Unique molecules included in this database (202), sorted by their number of constituent atoms.

Table 4

Molecular orbitals of CO computed at the MP2/aug-cc-pVTZ level.

Table 5

Same as Table 4, but for H2O.

Table 6

Initial abundances as a function of the total H nuclei abundance, i.e., nH,tot (Hincelin et al. 2011).

Table A.1

Ionization potentials in eV, computed at the CCSD(T)/CBS, CAM-B3LYP/aug-cc-pVQZ and from NIST.

Table A.2

Chemical reaction rates, cm,T

All Figures

thumbnail Fig. 1

Electron ionization cross sections. The solid lines give the fits from Eq. (2), and the black points show the data from the NIFS database.

In the text
thumbnail Fig. 2

Distribution of the number of atoms in each molecule for this data release.

In the text
thumbnail Fig. 3

Comparison of the ionization potentials presented here with the NIST database. Left: NIST database vs. CCSD(T) and CAM-B3LYP DFT calculations (solid and empty squares, respectively). Right: CCSD(T) vs. CAM-B3LYP DFT ionization potentials. The dashed line represents the one-to-one ratio.

In the text
thumbnail Fig. 4

Benchmark of the number of k-atom screening corrections included using C5H10. Left: k-atom screening corrections as a function of electron energy for C5H10. Right: time to compute σSCAR(Ee) for a given maximum k-atom screening correction (black) and the magnitude of the maximum correction (blue) as a function of the maximum k.

In the text
thumbnail Fig. 5

Sample of the total electron ionization cross sections for a subsample of the molecules. The solid line denotes cross sections calculated using the BEB method, and the dotted line shows the cross sections calculated with the SCAR rule.

In the text
thumbnail Fig. 6

Electron ionization cross sections computed with the AR (dotted), SCAR (solid), and BEB (dash-dotted) and dBEB (dashed) methods compared with experimental data from the NIFS database (blue dots).

In the text
thumbnail Fig. 7

Total ionization rate coefficient, cm,T, as a function of the constituent number of atoms in each molecule. The color denotes the energy at the cross section maximum.

In the text
thumbnail Fig. 8

Comparison of the chemical evolution using KIDA reaction rates (solid lines) and our rates (dashed lines). We report the species that present a difference of at least half an order of magnitude and reach at least n(t) = 10−8nH,tot. The ionization rate of H2 is 1.64 × 10−16 s−1, and the rate for model “High” and with the assumptions described in the text corresponds to Av = 10.6 and N = 1022 cm −2.

In the text
thumbnail Fig. B.1

Reaction rate coefficients following Equation 14 for all neutral molecules in the database. The boxes show the minimum and maximum coefficients in the column density range 1020 cm−2 NH21023$ \le {N_{{{\rm{H}}_2}}} \le {10^{23}}$ cm-2, and plus signs show the mean. The cyan boxes denote molecules using the MP2 calculations, and the red boxes denote calculations from Heathcote & Vallance (2018).

In the text

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