Theoretical Study of the Dissociative Recombination and Vibrational (De-)Excitation of HCNH⁺ and Its Isomers by Electron Impact

Abstract

Protonated hydrogen cyanide, HCNH⁺, is one of the most important molecules of interest in the astrophysical and astrochemical fields. This molecule not only plays the role of a reaction intermediary in various types of interstellar reactions but was also identified in Titan’s upper atmosphere. The cross sections for the dissociative recombination (DR) and vibrational (de-)excitation (VE and VDE) of HCNH⁺ and its ${\mathrm{CNH}}_2^{+}$ isomer are computed using a theoretical approach based on a combination of the normal mode approximation for the vibrational states of the target ions and the UK R-matrix code to evaluate electron-ion scattering matrices for fixed geometries of ions. The theoretical convoluted DR cross section for HCNH⁺ agrees well with the experimental data and a previous study. It was also found that the DR of the ${\mathrm{CNH}}_2^{+}$ isomer is important, which suggests that this ion might be present in DR experiments of HCNH⁺. Moreover, the ab initio calculations performed on the H₂CN⁺ isomer predict that this ion is a transition state. This result was confirmed by the study of the reaction path of the HCNH⁺ isomerization that was carried out by evaluating the intrinsic reaction coordinate (IRC). Finally, thermally averaged rate coefficients derived from the cross sections are provided for temperatures in the 10–10,000 K range. A comprehensive set of calculations is performed to assess the uncertainty of the obtained data. These results should help in modeling non-LTE spectra of HCNH⁺, taking into account the role of its most stable isomer, in various astrophysical environments.

1. Introduction

HCNH⁺ is an important species in astrophysical environments such as dark interstellar molecular clouds (Sgr B2 [1], TMC-1 [2]), proto-star (L483 [3]) or pre-stellar and mass starforming cores [4,5]. This molecule was also detected by the Ion and Neutral Mass Spectrometer (INMS) instrument aboard the Cassini probe in the upper atmosphere of Titan, Saturn’s largest moon [6,7]. HCNH⁺ could be an important precursor of the aerosols (Tholins) present on this satellite. The latter may themselves be the origin of molecules of prebiotic interest such as amino acids, nucleic acids, sugars, or even more complex molecules such as proteins [8,9]. HCNH⁺ is the simplest protonated nitrile, known as N-protonated HCN or protonated hydrogen cyanide. In the interstellar medium (ISM), it was often postulated that HCNH⁺ is at the origin of the thermochemically unrealistic HNC/HCN abundance ratio [10]. HCN is one of the most interesting molecules for cosmochemistry, as it is considered by some to be one of the first molecules present on the prebiotic earth [11] and could, following polymerization and contact with water and oxygen, give rise to more complex molecules known to be the building blocks of life [12]. Observations in cold dark clouds report values for the HNC/HCN isomer abundance ratio ranging from 0.015 to 5 [13,14], whereas a previous theoretical study indicated that this abundance ratio should have an upper limit of 1 [15].

Due to the relatively large abundance of electrons and HCNH⁺ in the ISM [10], collisions of HCNH⁺ with electrons play a significant role, in particular, leading to dissociation (DR—dissociative recombination), vibrational (de-)excitation (VE, VDE), and rotational (de-)excitation of HCNH⁺. The DR process leads to the formation of HCN or HNC, while VE and VDE compete with the latter. Recent studies have attempted to reproduce the observed HCNH⁺ abundance, also in response to controversies over the HNC/HCN abundance ratio, within dense cold regions by taking into account their chemical models not only in the DR process of HCNH⁺ [4], but also in other formation paths for HCNH⁺ (following NH₃ + C⁺, for instance) [5], or destruction paths of HCN (in collision with oxygen, for example) [16], or in the excitation of HCNH⁺ (in collision with H₂ and He, for example, accounting for the hyperfine structure of the target ion) [17]. In these aforementioned studies, the authors provided the updated HCNH⁺ abundances in better agreement with the observations. Despite these enhancements, discrepancies between observations and predictions remain unresolved, which requires a better understanding of the HCNH⁺ chemistry, in particular, the DR process occurring in HCNH⁺ and its isomers.

The DR mechanism of HCNH⁺ was also subject to controversy. Indeed, Hickman et al. [18] reported that the direct dissociative recombination process (when a doubly excited state dissociating into neutral fragments crosses the ground state of the ion near its equilibrium geometry) could occur at low energy, while Ngassan and Orel [19] found that the direct DR cross section is lower than the experimental value. Later, Douguet et al. [20] demonstrated that the major contribution to the DR cross section at low electron collisional energies resulted from indirect mechanisms (electron captured into a vibrationally excited Rydberg state of the neutral molecule that couples to the doubly excited state dissociating into neutral fragments). In the study, the authors employed a theoretical approach based on the multichannel quantum defect theory (MQDT) [21,22]. After computing the ab initio potential energy surface (PES) of HCNH⁺ and its series of Rydberg energies, the quantum defects are obtained from energies of excited Rydberg states. In the present study, instead of the quantum defect, we employed the scattering matrix obtained from the UK molecular R-Matrix code (UKRMol) [23,24]. Thus, we revisited the DR cross section of HNCH⁺ and compared it with the available experimental studies of Semaniak et al. [25] carried out at the heavy-ion storage ring CRYRING. As the authors of this study cannot exclude a possible involvement of other isomers, we also provided the first calculations for the ${\mathrm{CNH}}_2^{+}$ isomer in its singlet state. This isomer was little studied, even by the scientific community interested in Titan. However, some authors such as Fortenberry et al. [26] showed that this molecule could be well present in a kinetically favorable potential. Moreover, ab initio calculations of the singlet ground state of the H₂CN⁺ isomer were found with one imaginary frequency, suggesting that the ion is unstable. This was confirmed by performing the ${\mathrm{HCNH}}^{+}$ isomerization reaction path. Finally, the four lowest triplet states for isomers of HCNH⁺ are not treated in the present study because these states are situated at very high energy, at least 5.3 eV above the singlet ground state of HNCH⁺ [25,27].

The objective of this study is to demonstrate that the DR cross section of ${\mathrm{CNH}}_2^{+}$ by electron impact is not negligible at low energy. Thus, this isomer could play an important role in the chemistry of HCNH⁺, and must therefore be taken into account in chemical models that attempt to explain the thermochemically unrealistic HNC/HCN abundance ratio in the interstellar medium. This article is organized in the following way. After the above Introduction, Section 2 describes the theoretical approach used in the present calculations. The obtained cross sections and the corresponding rate coefficients are displayed and discussed in Section 3, while Section 4 concludes the study.

2. Theoretical Approach

The basic formalism employed in our model is presented in detail in Refs. [28,29,30,31,32], we only highlight in this section its major ideas.

2.1. The Properties of HCNH⁺ and Its Isomers

2.1.1. HCNH⁺ and its ${\mathrm{CNH}}_2^{+}$ Isomer

HCNH⁺ is a closed-shell molecule, having the symmetry of the ${\mathrm{C}}_{\infty v}$ point group at equilibrium and ground state electronic configuration:

$$X^1\Sigma :1{\sigma }^{2}2{\sigma }^{2}3{\sigma }^{2}4{\sigma }^{2}5{\sigma }^{2}1{\pi }^4.$$

At low electron collisional energies, the ion can be characterized by five normal modes of vibration: three stretching modes ${\nu }_1$, ${\nu }_2$, and ${\nu }_3$, with the respective frequencies ${\omega }_1$, ${\omega }_2$, and ${\omega }_3$ and corresponding coordinates $q_1$, $q_2$, and $q_3$; and two doubly degenerate transverse modes ${\nu }_4$ and ${\nu }_5$, with lower frequencies ${\omega }_4$ and ${\omega }_5$ and coordinates $\left(q_{4x},q_{4y}\right)$ and $\left(q_{5x},q_{5y}\right)$. The normal coordinates and the related frequencies are obtained using the cc-pVTZ basis set centered on each atom and including s, p, and d orbitals. After performing calculations of Coupled Cluster Singles, Doubles and Triples (CCSD(T)) in the ${\mathrm{C}}_{2v}$ symmetry group using the Molpro suite of codes [33], we found an equilibrium geometry of the ion for values of bond lengths $\left(r_1,r_2,r_3\right)$ and bond angles $\left({\theta }_1,{\theta }_2,{\theta }_3\right)$ that are given in

Table 1. Bond lengths ($r_1$, $r_2$, and $r_3$ in Å) and bond angles (${\theta }_1$, ${\theta }_2$, and ${\theta }_3$ in degrees) at the equilibrium geometry of HCNH⁺ and its ${\mathrm{CNH}}_2^{+}$ isomer, both displayed in Figure 1. The total energies are given in atomic units. Data obtained in this study are compared with the calculations of Ref. [34].

Geometry	HCNH+		${\mathrm{CNH}}_2^{+}$
	This Study	Calc.	This Study	Calc.
$r_1$	1.0803	1.0804	1.2514	1.2514
$r_2$	1.1403	1.1403	1.0326	1.0327
$r_3$	1.0139	1.0140	1.0326	1.0327
${\theta }_1$	180	180	120.979	120.988
${\theta }_2$	180	180	120.979	120.988
${\theta }_3$	0	0	118.041	118.024
Total energy	−93.557075	−93.557076	−93.475788	−93.475788

. The table below shows a comparison of the results obtained in the present calculation with theoretical data, while the upper panel of Figure 1 shows normal displacements for each mode of HCNH⁺ with the bond lengths and bond angles of Table 1 depicted for the first normal mode.

N-protonated hydrogen isocyanide ${\mathrm{CNH}}_2^{+}$ is an isomer of HCNH⁺ belonging to the ${\mathrm{C}}_{2v}$ point group at equilibrium geometry. Its ground state electronic configuration is

$$1^1{\mathrm{A}}_1:1a_1^{2}2a_1^{2}3a_1^{2}4a_1^{2}1b_2^{2}1b_1^{2}5a_1^2.$$

This isomer has six non-degenerate normal modes ${\nu }_i$ with respective frequencies ${\omega }_i$ and corresponding coordinates $q_i$$(i=1,2,3,4,5,6)$. Analogously, the normal coordinates and the related frequencies are obtained using the CCSD(T) method and cc-pVTZ basis set. Bond lengths and bond angles are given in Table 1 and normal displacements for each mode are shown in Figure 1.

Table 2. Vibrational frequencies (ω_i in cm⁻¹) obtained in this study for HCNH⁺ and its ${\mathrm{CNH}}_2^{+}$ isomer are compared with previous data available in the literature (experimental or theoretical data).

HCNH+
Normal Mode, ${\mathit{\nu }}_{\mathit{i}}$	Symmetry	Normal Coordinate, ${\mathit{q}}_{\mathit{i}}$	Frequency, ${\mathit{\omega }}_{\mathit{i}}$
			This Study	Exp. [34]
NH stretch, ${\nu }_1$	$\Sigma$	$q_1$	3645.07	3482.8
CH stretch, ${\nu }_2$	$\Sigma$	$q_2$	3316.36	3187.9
CN stretch, ${\nu }_3$	$\Sigma$	$q_3$	2179.51	2155.7
HCN bend, ${\nu }_4$	$\Pi$	$q_{4x},q_{4y}$	805.33	801.6
HNC bend, ${\nu }_5$	$\Pi$	$q_{5x},q_{5y}$	647.86	645.9
${\mathrm{CNH}}_2^{+}$
Normal Mode, ${\mathit{\nu }}_{\mathit{i}}$	Symmetry	Normal Coordinate, ${\mathit{q}}_{\mathit{i}}$	Frequency, ${\mathit{\omega }}_{\mathit{i}}$
			This Study	Calc. [34]
${\nu }_1$	A1	$q_1$	3317	3318
${\nu }_2$	A1	$q_2$	1723	1724
${\nu }_3$	A1	$q_3$	1394	1394
${\nu }_4$	B1	$q_4$	723	725
${\nu }_5$	B2	$q_5$	3405	3405
${\nu }_6$	B2	$q_6$	630	627

lists the obtained vibrational frequencies for both ions and compares them with previous data. The calculations agree well with the data available in the literature.

2.1.2. The H₂CN⁺ Isomer

Another singlet state of the HCNH⁺ isomer, known as the hydrocyanonium cation H₂CN⁺, with hydrogen atoms next to carbon, was reported in the literature [27,35,36]. H₂CN⁺ belongs to the ${\mathrm{C}}_{2v}$ point group at equilibrium geometry and its ground state electronic configuration is

$${}^1\mathrm{A}_1:1a_1^{2}2a_1^{2}3a_1^{2}4a_1^{2}1b_2^{2}5a_1^{2}1b_1^2,$$

with a total energy of −93.440025 (atomic units). This isomer also has six non-degenerate normal modes ${\nu }_i$ with respective frequencies ${\omega }_i$ and corresponding coordinates $q_i$$(i=1,2,3,4,5,6)$. By performing the Molpro calculations, the normal coordinates and the related frequencies were obtained using the CCSD(T) method and cc-pVTZ basis set.

Table 3. Vibrational frequencies (${\omega }_i$ in cm⁻¹) obtained in this study for the H₂CN⁺ isomer are compared with previous calculations available in the literature. Note that the normal mode ${\nu }_6$ has an imaginary frequency ${\omega }_6$. The table below gives bond lengths (in Å) and bond angles (in degree). A sketch of the employed coordinates is given in the lower panel of Figure 2.

Mode, ${\mathit{\nu }}_{\mathit{i}}$	Symmetry	Normal Coordinate, ${\mathit{q}}_{\mathit{i}}$	Frequency, ${\mathit{\omega }}_{\mathit{i}}$
			This Study	Calc. [34]
${\nu }_1$	A1	$q_1$	2862	2859
${\nu }_2$	A1	$q_2$	1843	1843
${\nu }_3$	A1	$q_3$	1034	1025
${\nu }_4$	B1	$q_4$	810	804
${\nu }_5$	B2	$q_5$	2897	2892
${\nu }_6$	B2	$q_6$	ı437	ı456
	Geometry	This Study	Calc. [34]
	$r_{\mathrm{CN}}$	1.2089	1.2089
	$r_{\mathrm{CH}1}$	1.1169	1.1168
	$r_{\mathrm{CH}2}$	1.1169	1.1168
	${\theta }_1$	119.950	119.927
	${\theta }_2$	119.950	119.927
	${\theta }_3$	120.099	120.145

compares the results with data available in the literature [34]. As expected, we found that the H₂CN⁺ isomer has the normal mode ${\nu }_6$ with imaginary frequency ${\omega }_6$ corresponding to the torsional movement of the H atoms and the N-H stretch. This result suggests that the isomer is unstable with respect to isomerization in the HCNH⁺ linear form.

To verify the nature of the eventual transition state obtained, we determined the reaction path throughout the intrinsic reaction coordinate (IRC) by invoking the Quadratic Steepest Descent Reaction Path method (QSDPATH) implemented in Molpro [33]. The IRC is defined similarly to the minimum energy path (MEP), but instead of the steepest-descent path on the potential energy surface, the IRC follows the maximum instantaneous acceleration from the transition state (TS) down towards a local minimum. IRC is the solution of a differential equation of the mass-weighted Cartesian coordinates with respect to the coordinate along the IRC. See, for example, Ref. [37] for more details.

Starting from the equilibrium geometry of H₂CN⁺, obtained after optimization (see Table 3), we performed IRC calculations and found that the linear structure HCNH⁺ is predicted to lie lower by 3.18 eV. The lower panel of Figure 2 shows the total energy (in atomic units) and the upper panel shows the bond lengths and angles along the IRC (in atomic units). Following the positive direction in the reaction path (blue arrow in the lower panel of the figure), the migration of a hydrogen atom (here, H1) from carbon to nitrogen led to the formation of the more stable linear isomer HCNH⁺ with the bond length and angle characteristics given in Table 1. Analogously, the migration of the second hydrogen atom (H2) to nitrogen, in case of a negative IRC direction (violet arrow in the lower panel of the figure ), gives the same molecular ion configuration. The sketch in the lower panel of that figure displays both migration processes with a color code according to the bond length and angle curves. Thus, the reaction path of the HCNH⁺ isomerization confirms that the H₂CN⁺ isomer is a transition state, which could explain why it has not yet been identified in interstellar space.

2.2. Fixed-Geometry Scattering Matrix

In our model, the fixed-nuclei reactance matrix (K-matrix) is employed to describe the e-HCNH⁺ isomer collisions. It is obtained numerically for each geometry configuration of the target molecule using the UK molecular R-Matrix code (UKRMol) [23,24] with the Quantemol-N expert system [38].

R-matrix calculations are performed in the ${\mathrm{C}}_1$ point group for a given ion in its ground electronic state. The four $1a^{2}2a^2$ core electrons are frozen and ten electrons are kept distributed in the active space, including $3-11a$ molecular orbitals. For each ion, a total number of 5292 configuration state functions (CSFs) are used for the ground state. All the generated states up to 10 eV were retained in the final close-coupling calculation. We employed an R-matrix sphere of radius 12 bohrs and a partial-wave expansion with continuum Gaussian-type orbitals up to $l\le 4$. In the following, this calculation with the cc-pVTZ basis set and the complete active space (${\mathrm{CAS}}_1$) described above will be referred to as Model 1.

K-matrices are obtained from the R-matrix calculations for a geometry configuration of the ion specified by the normal coordinates $\mathbf{q}=\left\{q_1,q_2,\cdots ,q_n\right\}$ with n being the number of normal modes. $K\left(\mathbf{q}\right)$ is transformed into the scattering matrix as $S\left(\mathbf{q}\right)=(1+\iota K\left(\mathbf{q}\right)){(1-\iota K\left(\mathbf{q}\right))}^{-1}$. At low collisional energies, $S\left(\mathbf{q}\right)$ depends only weakly on energy while a sharper energy dependence is observed at certain relatively high energies, corresponding to positions of Rydberg states attached to the excited electronic states of the ion. The eigenphase sum is a convenient way to identify the weak or strong energy dependence of the scattering matrix. Figure 3 shows the eigenphase sum for the equilibrium geometry ($q_0=0.01$) and displacement $q_i=0.1$ along each normal mode ${\nu }_i$ of both ions. The variation in the eigenphase sums is smooth for energies below 1 eV and 0.3 eV for HCNH⁺ and ${\mathrm{CNH}}_2^{+}$, respectively. Above these values, a sharp energy dependence at certain energies (at 3.4 eV for HCNH⁺ and 0.46 eV for ${\mathrm{CNH}}_2^{+}$, for instance) is observed due to the presence of electronic Rydberg resonances attached to the closed ionization limits.

2.3. Formulas of the Dissociative Recombination and Vibrational (de-)Excitation Cross Sections

The following assumptions are employed in the present model: (i) the rotation of the molecular ions is neglected, (ii) the cross section is averaged over the autoionizing resonances, (iii) the autoionization lifetime is assumed to be much longer than the predissociation lifetime, and (iv) the harmonic approximation is used to describe the vibrational state of the core ion. For more details, see Ref. [28].

Combining the above assumptions (i)–(iv) and applying the frame transformation, the vibrational excitation (VE) and (de-)excitation (VDE) cross sections are given, in terms of expanded scattering matrix elements to the first order of the normal coordinates, as follows:

$$\begin{array}{c}\hfill {\sigma }_i^{VE}\left(E_{el}\right)={\displaystyle \frac{\pi {\hslash }^2}{2m{E}_{el}}}{P}_i\Theta (E_{el}-\hslash {\omega }_i)\end{array}$$

(1)

and

$$\begin{array}{c}\hfill {\sigma }_i^{VDE}\left(E_{el}\right)={\displaystyle \frac{\pi {\hslash }^2}{2m{E}_{el}}}{P}_i,\end{array}$$

(2)

where

$$\begin{array}{cc}\hfill P_i& {\displaystyle =\frac{g_i}{2}\sum _{l{l}^{\prime }\lambda {\lambda }^{\prime }}{\left|\frac{\partial S_{l\lambda l^{\prime }{\lambda }^{\prime }}}{\partial q_i}\right|}_{q_0}^2}\end{array}$$

(3)

is a quantity that can be interpreted as the probability of excitation of the vibrational mode ${\nu }_i$. Above, $q_i$, $\hslash {\omega }_i$, and $g_i$$(i=1-n)$ are, respectively, the dimensionless coordinate, the energy, and the degeneracy of the mode ${\nu }_i$, where n stands for the number of normal coordinates. Again, $q_0=0.01$ is the equilibrium geometry of the target ion. For the linear molecular ion HCNH⁺, $n=5$, with a degeneracy of $g_4=2$ and $g_5=2$ for bending modes 4 and 5, and $g_{1-3}=1$ for the stretching modes 1, 2, and 3. In the case of the ${\mathrm{CNH}}_2^{+}$ isomer, $n=6$ with a degeneracy of $g_{1-6}=1$ (see Table 2).

In the previous equations, $S_{l\lambda ,l^{\prime }{\lambda }^{\prime }}$ is an element of the fixed-nuclei scattering matrix for electron-ion collisions with the initial channel ($\lambda l$) and the exit channel (${\lambda }^{\prime }{l}^{\prime }$), where l is the electron angular momentum and $\lambda$ is its projections on the molecular axis. Finally, m is the reduced mass of the electron-ion system and $E_{el}$ is the incident energy of the electron. $\Theta$ in Equation (1) stands for the Heaviside step function.

In the present theoretical approach, the initial state of a given ion is its ground vibrational level, so the electron can only be captured into the first excited vibrational state of each normal mode of the ion. Formulas of Equations (1) and (2) give the VE and VDE cross sections for changing one quantum in each normal mode. Based on the propensity rule, the (de-)excitation process changing two or more quanta is neglected in this study because their contributions in the cross sections are small.

As for the dissociative recombination (DR) process, the cross section is obtained [28] as

$$\begin{array}{c}\hfill 〈{\sigma }^{DR}\left(E_{el}\right)〉={\displaystyle \frac{\pi {\hslash }^2}{2m{E}_{el}}}\sum _{i=1}^n{P}_i\Theta (\hslash {\omega }_i-E_{el}),\end{array}$$

(4)

where the bracket stands for the temporary captures in all the accessible Rydberg states. The present model suggests that the electron scattering energy is not sufficient to excite the ion and then to leave it. The probability of excitation $P_i$ of the ion by the electron is described by the same physics: The electron is captured in a Rydberg resonance attached to the vibrational state excited by the electron. In such a situation, the system electron-ion will most likely dissociate (DR process), rather than autoionize (VE process of Equation (1)).

Finally, to calculate the excitation probabilities, $P_i$—the derivative of the scattering matrix with respect to the normal coordinate, $q_i$—the scattering matrix is evaluated for two values of $q_i$, $q_i=0.01$, and $q_i=0.1$, keeping the other normal coordinates fixed at the equilibrium geometry, i.e., $q_0=0.01$.

3. Results and Discussions

3.1. Cross Sections

In the theoretical model described above, we assumed that the excitation probabilities are energy-independent. Figure 4 shows the weak dependence of $P_i$ of Equation (3) on energy. As demonstrated in Figure 3 for the eigenphase sums, those quantities are constants at low energies and therefore could be used in the calculations of cross sections of Equations (1), (2), and (4), as well as for the thermally averaged rate coefficients, given in the next section.

Table 4. Partial derivatives with respect to the normal coordinates of the largest scattering matrix element for HCNH⁺ and ${\mathrm{CNH}}_2^{+}$ molecular ions at $E_{el}=0.01$ eV.

Normal Mode, ${\mathit{\nu }}_{\mathit{i}}$	HCNH+		${\mathrm{CNH}}_2^{+}$
	Electronic States $\mathit{l}\mathit{\lambda }-{\mathit{l}}^{\prime }{\mathit{\lambda }}^{\prime }$	${\left|{\displaystyle \frac{\partial {\mathit{S}}_{\mathit{l}\mathit{\lambda }{\mathit{l}}^{\prime }{\mathit{\lambda }}^{\prime }}}{\partial {\mathit{q}}_{\mathit{i}}}}\right|}_{{\mathit{q}}_{\mathit{0}}}$	Electronic States $\mathit{l}\mathit{\lambda }-{\mathit{l}}^{\prime }{\mathit{\lambda }}^{\prime }$	${\left|{\displaystyle \frac{\partial {\mathit{S}}_{\mathit{l}\mathit{\lambda }{\mathit{l}}^{\prime }{\mathit{\lambda }}^{\prime }}}{\partial {\mathit{q}}_{\mathit{i}}}}\right|}_{{\mathit{q}}_{\mathit{0}}}$
1	$p\sigma -d\sigma$	0.4598	$s\sigma -s\sigma$	0.2345
2	$p\sigma -d\sigma$	0.4793	$d\delta -d\delta$	0.2149
3	$p\sigma -p\sigma$	0.3446	$d\delta -d\delta$	0.2963
4	$d\pi -d\pi$	0.3392	$d\delta -p\pi$	0.2308
5	$d\pi -d\pi$	0.2206	$p\pi -d\delta$	0.3322
6	-	-	$p\pi -p\sigma$	0.1672

presents the largest vibronic interactions in both molecules and along each normal coordinate. The couplings are given in the form of partial derivatives, with respect to the normal coordinates, of the scattering matrix. Several observations can be made from the values and forms of the couplings. It appears that the indirect DR cross section of HCNH⁺ will be larger than that of the isomer. As expected for linear polyatomic ions, the vibronic interactions mediated by the molecular bending are responsible for the indirect DR mechanism in HCNH⁺. However, the present results also show that the contribution from the vibronic interactions induced by stretching modes of the ion is also important, as reported in Ref. [28] for HCO⁺ and N₂H⁺. Furthermore, the most interesting point is certainly the unexpectedly high values of the vibronic couplings in the ${\mathrm{CNH}}_2^{+}$ isomer. This result implies a large cross section for that ion.

The theoretical VE, VDE, and DR cross sections are displayed in Figure 5 for HCNH⁺ (left panel) and ${\mathrm{CNH}}_2^{+}$ (right panel). Values of Table 2 and

Table 5. Parameters of Equations (1)–(4) calculated at $E_{el}=0.01$ eV collision energy.

Normal Mode, ${\mathit{\nu }}_{\mathit{i}}$	HCNH+	${\mathrm{CNH}}_2^{+}$
	${\mathit{P}}_{\mathit{i}}$	${\mathit{P}}_{\mathit{i}}$
1	0.4877205	0.2213105
2	0.4882937	0.1519490
3	0.1729784	0.2175728
4	0.4469524	0.1332123
5	0.3781402	0.3158785
6	-	0.1075744

were employed to produce these cross sections. At low energies, VE and DR cross sections are featureless and behave simply as $1/E_{el}$ following the Wigner law. For energies higher than 0.1 eV, the cross section drops in a stepwise manner because the scattering electron excites the vibrational level of the ion by one quantum.

Figure 6 shows the theoretical cross section in comparison with the experimental results by Semaniak et al. [25] and previous calculations by Douguet et al. [20]. The cross section was convoluted according to Equation (2) of Ref. [39] with a parallel electron energy of 0.1 meV and a transverse energy spread of 2 meV. One interesting feature for HCNH⁺ is the double drop in its theoretical DR cross section corresponding to the two transverse normal modes ${\nu }_4$ and ${\nu }_5$, with very different asymmetrical elongations of the hydrogen atoms (see Table 2). There is a good agreement between the present result and previous theoretical and experimental data. The figure demonstrates that the DR cross section in the ${\mathrm{CNH}}_2^{+}$ isomer is also important, presenting a drop at about 90 meV, which corresponds to the vibrational thresholds of ${\nu }_4$ and ${\nu }_6$ normal modes.

3.2. Rate Coefficients

Thermally averaged rate coefficients are evaluated from the general expression of Maxwell–Boltzmann averaging (see Equation (7) of Ref. [40], for instance). Due to the simple analytical forms of the cross sections (1), (2), and (4), thermally averaged rate coefficients take the following expressions:

$$\begin{array}{c}\hfill {\alpha }_i^{VE}\left(T\right)=\sqrt{{\displaystyle \frac{2\pi }{k_{b}T}}}{\displaystyle \frac{{\hslash }^2}{m^{3/2}}}{P}_{i}exp\left(-{\displaystyle \frac{\hslash {\omega }_i}{k_{b}T}}\right),\end{array}$$

(5)

$$\begin{array}{c}\hfill {\alpha }_i^{VDE}\left(T\right)=\sqrt{{\displaystyle \frac{2\pi }{k_{b}T}}}{\displaystyle \frac{{\hslash }^2}{m^{3/2}}}{P}_i,\end{array}$$

(6)

$$\begin{array}{c}\hfill {\alpha }^{DR}\left(T\right)=\sqrt{{\displaystyle \frac{2\pi }{k_{b}T}}}{\displaystyle \frac{{\hslash }^2}{m^{3/2}}}\sum _{i=1}^n{P}_i\left[1-exp\left(-{\displaystyle \frac{\hslash {\omega }_i}{k_{b}T}}\right)\right],\end{array}$$

(7)

where $k_b$ is the Boltzmann coefficient, m is the reduced mass of the electron-ion system, and T is the temperature. Figure 7 shows the obtained rate coefficients for VE, VDE, and DR (from values of Table 2 and Table 5) as functions of temperature. For low temperatures, $T<500$ K, the VDE and DR rate coefficients behave as $1/\sqrt{T}$, while for the VE, $T^{-0.5}exp\left(-{\displaystyle \frac{\hslash {\omega }_i}{k_{b}T}}\right)$. At higher temperatures, the DR rate coefficient decreases faster than $1/\sqrt{T}$ because the vibrational excitation becomes more probable.

3.3. Assessment of Uncertainties

The main identifiable source of uncertainty is the scattering model used in the calculation. To assess the associated uncertainty, we performed a complete calculation of the VE, VDE, and DR cross sections using different basis sets and orbital spaces in the electron-scattering calculations. The main scattering model (Model 1) is described in Section 2.2. In the second set of calculations (Model 2), the electronic basis was reduced from cc-pVTZ to DZP. In Model 3, the complete active space (CAS) in the configuration interaction calculations was reduced with respect to Model 1 by two orbitals, i.e., $3a$ and $4a$.

Figure 8 shows the obtained results. The difference between the DR cross section produced in the three calculations is about 10% for HCNH⁺ and 15% for the ${\mathrm{CNH}}_2^{+}$ isomer. This latter result implies that the DR cross section of ${\mathrm{CNH}}_2^{+}$ is also important, which suggests that this ion might be present in DR experiments of HCNH⁺ [25].

4. Conclusions

To summarize the results of the present study. We computed cross sections and rate coefficients for VE, VDE, and DR of HCNH⁺($X^1\Sigma$) and its stable isomer ${\mathrm{CNH}}_2^{+}$(1 ¹A₁) by electron impact using a theoretical approach that combines the normal mode approximation for the vibrational states of the target ions, the vibrational frame transformation, and the UK R-matrix code.

The convoluted DR cross section for HCNH⁺ agrees well with the experimental data and a previous study. Another interesting feature is the importance of the DR cross section of ${\mathrm{CNH}}_2^{+}$ isomer, which suggests that this ion could be present in the DR experiment of HNCH⁺. To confirm these findings, a comprehensive set of calculations was performed to assess the uncertainty of the obtained cross sections.

Since the cross sections and thermally averaged rate coefficients have simple analytical forms, they can be readily used in the modeling of non-LTE spectra of HCNH⁺, involving the ${\mathrm{CNH}}_2^{+}$ isomer, in various astrophysical environments. These results demonstrate that ${\mathrm{CNH}}_2^{+}$ must be taken into account in chemical models that attempt to explain the HCNH⁺ abundance and HNC/HCN abundance ratio observed in the interstellar medium.

Moreover, the ab inito calculations performed on the lowest singlet state of the H₂CN⁺ isomer provided an imaginary frequency for one of its normal modes, which suggests that the ion is likely unstable. A study of the HCNH⁺ isomerization reaction path was performed by determining the intrinsic reaction coordinates (IRCs). The linear structure HCNH⁺ was found to lie lower, by 3.18 eV. This proves that the H₂CN⁺ isomer is a transition state, probably explaining why it has not yet been identified in interstellar space.

Finally, the rotational structure of the target ions and the neutral molecules was neglected in the present approach. Hence, the obtained cross sections and rate coefficients should be viewed as averaged over initial rotational states and summed over final rotational states corresponding to the initial and final vibrational levels (for VE and VDE) or dissociative states (for DR). Discrepancies between the computed results and the experimental measurements observed at low electron scattering energy may be due to neglecting the rotational structure in the present model. Rotationally resolved cross sections, i.e., without changing the vibrational state, will be the subject of a further study.