Dissociative electron attachment to the HNC$_3$ molecule

TL;DR

This paper models dissociative electron attachment to HNC$_3$, revealing a low-energy resonance that efficiently produces negative ions in space, suggesting DEA dominates over REA in interstellar chemistry.

Contribution

The study provides a first-principles theoretical analysis of DEA to HNC$_3$, quantifying the rate coefficient and proposing DEA as the main formation pathway for negative ions in space.

Findings

DEA rate coefficient at 300 K is 5×10⁻⁹ cm³/s

DEA is three orders of magnitude more efficient than REA in space

Negative ions in space likely originate from DEA, not REA

Abstract

Dissociative electron attachment (DEA) to the HNC$_3$ is modeled theoretically using a first-principles approach. In HNC$_3$+$e^-$ collisions, there is a low-energy resonance, which has a repulsive character along the H+NC$_3$ coordinate and becomes a bound electronic state of the HNC$_3^-$ anion near the equilibrium of HNC$_3$. The anion state dissociates without a potential barrier towards C$_3$N$^-$+H. The cross section and the rate coefficient of the process were computed. The obtained rate coefficient at low temperatures is $5\times 10^{-9}$cm$^3$/s at 300~K. Such a value of the DEA rate coefficient makes the DEA process by three orders of magnitude more efficient in producing negative molecular ions in the interstellar space than the radiative electron attachment (REA). It is suggested that negative molecular carbon-chain ions, observed in the interstellar medium, are produced by DEA rather than REA.

Tables

Table 1. Table 1: HNC 3 normal modes and their frequencies (in cm -1 ), obtained using the CCSD(T) method with the cc-pVQZ basis set.

irrep		type	frequency
1,	$A^{\prime }$	in-plane C-C-C bending	152
2,	$A^{\prime \prime }$	off-plane C-C-C bending	162
3,	$A^{\prime }$	C-N-H bending + N-H stretching	470
4,	$A^{\prime \prime }$	off-plane C-C-N bending	555
5,	$A^{\prime }$	in-plane C-C-N bending	574
6,	$A^{\prime }$	C-C-C-N symmetric stretching	943
7,	$A^{\prime }$	C2-C3 and C3-N stretching	1910
8,	$A^{\prime }$	C-C-C-N antisymmetric stretching	2251
9,	$A^{\prime }$	N-H stretching	3734

Equations

$$\Gamma (E_{\mathrm{r}})=a{E}_{\mathrm{r}}^b.$$

$$d{q}_i=-\frac{1}{|\vec{\nabla }U|}\frac{\partial U}{\partial q_i}ds,$$

$$\sigma (\epsilon )=\frac{2{\pi }^2}{k^2}\frac{\Gamma (s_{\epsilon })}{|U^{\prime }(s_E)|}|\zeta (s_E){|}^2,$$

$${\tau }_d/{\tau }_a\lesssim 0.2\pi \Gamma (s_{\epsilon })/\hslash {\omega }_5$$

$$\frac{[e^{-}][\mathrm{HN}{\mathrm{C}}_3]k_{\mathrm{DEA}}}{[e^{-}][{\mathrm{C}}_3\mathrm{N}]k_{\mathrm{REA}}}\approx 2000.$$

Full text

I Introduction

Dissociative electron attachment to polyatomic ions plays an important role in molecular plasma. Knowing rate coefficients and products of the reaction are important for fundamental science, such as for understanding processes in the interstellar medium and star formation, planetary atmospheres, and other applications, such as modeling the behavior of plasma in technological processes. Experiments can provide some of the required data, but they are expensive and often are not feasible, for example, if the molecule of interest is in a radical or an excited state, which is unstable in collisions with other species present nearby. Theory is now able to predict the rate coefficients with quite good accuracy and the product distribution of the DEA reaction for diatomic molecules. For polyatomic molecules, several approaches have been developed and applied to a number of molecules; however, these approaches are often limited to one particular molecule and are difficult to adapt to other molecules. In this study, we developed a quite general method to determine the DEA rate coefficients. The method is based on first principles only and can be applied to a wide range of polyatomic molecules.

Several negative molecular ions have been observed in interstellar and circumstellar clouds, CnN-($n=1,3,5$), CmH-($m=4,6,8$) and, possibly, \chC10H- or \chC9N-, [1, 2, 3, 4, 5, 6, 7, 8, 9]. Following an earlier theoretical prediction [10, 11, 12, 13] that such ions can be formed by the process of radiative electron attachment (REA) and indeed could be observed in the interstellar medium (ISM), it was generally accepted by the astronomical community since the first observations of the anions in the ISM that they are formed by REA. However, accurate first-principles calculations [14, 15, 16, 17, 18, 19, 20] from two independent groups have demonstrated that the REA process is so slow that the observed relative abundances of the anions and the corresponding neutral molecules exclude the REA as the main path of anion formation. This led to the idea, suggested in several studies, that molecular anions could instead be formed in the ISM via DEA, a process generally much faster than REA at low collision energies, if it is allowed energetically. For example, Petrie and Herbst [21] have suggested that C3N- could be formed in the ISM by DEA to the HNC3 molecule. DEA is also considered as the main mechanism of molecular anion formation in Titan’s atmosphere [22, 23]. Despite this and later suggestions that DEA could be the mechanism of the molecular anion formation in the ISM, there have been no significant attempts to calculate or measure DEA rate coefficients for molecules that could produce the anions observed in the ISM. Our study addresses this gap by providing a first-principles modeling of DEA to HNC3, observed in the ISM [24, 25, 26], which forms the C3N- ion at low temperatures.

This article is organized as follows. Section II gives the electronic structure of HNC3 and HNC${}_{3}^{-}$, and Section III sketches the employed theoretical approach to compute the DEA cross section and rate coefficient. Section IV discusses the obtained results. Section V concludes this study.

II Electronic structure of HNC3 and HNC${}_{3}^{-}$

The HNC3 molecule in its ground electronic state is planar. Its electronic state is of the $A^{\prime}$ irreducible representation (irrep) of the C${}_{\text{s}}$ point group with the electronic configuration of 11 $a^{\prime}$ orbitals and 2 $a^{\prime\prime}$ orbitals. Its equilibrium geometry has been determined by Botschwina [27], shown in the top left panel of Fig. 1. In this study, we have also performed the electronic structure calculation using a slightly larger basis set than that of  Botschwina [27] and two different methods: a coupled-cluster method CCSD(T), and the multireference configuration interaction (MRCI) method, both employing the MOLPRO suite [28]. We have obtained results that are essentially identical to those of Ref. [27]. The normal mode harmonic energies of the molecule have also been computed in this study and are given in Table 1. Animated representations of the modes are provided in the supplementary material.

Considering the HNC3+$e^{-}$ system at the geometry of the HNC3 equilibrium, there is an electronic resonance of the $A^{\prime}$ and of the $A^{\prime\prime}$ irreps at 0.142 eV and 1.064 eV (scattering energy), respectively. The lowest dissociation limit of the $A^{\prime}$ resonant state corresponds to C3N-+H. Figure 2 demonstrates energies of the HNC3 molecule and the HNC${}_{3}^{-}$ anion as a function of the distance $r_{\mathrm{HN}}$ between the H and N atoms. As evident from the figure, the geometry where the HNC${}_{3}^{-}$ PES approaches and crosses the PES of HNC3 is near $r_{\mathrm{HN}}=2.06$ $a_{0}$ (here and below $a_{0}$ is the Bohr radius, the atomic unit of length). Figure 1 shows how the highest singly occupied orbital of HNC${}_{3}^{-}$ changes when the distance $r_{\mathrm{HN}}$ increases. It changes from an $a^{\prime}$ orbital that is delocalized along the molecule near the HNC3 equilibrium to the 1$s$ orbital that is localized on the hydrogen atom at large values of $r_{\mathrm{HN}}$.

To model the DEA process, we have modified the approach developed by Yuen et al. [29], which is based on the DEA theory for diatomic molecules [30]. The approach requires the knowledge of the HNC3 PES near its equilibrium position, the HNC3 vibrational modes, as well as the PES and autodetachment widths of the anion near the HNC3 equilibrium and along the path to the dissociation of HNC${}_{3}^{-}$. The HNC3 PES and its vibrational modes were calculated as discussed above. Similarly to Ref. [29], for determination of the anion PES and widths, the UK R-matrix [31] and the Quantemol [32] codes were employed to perform $e^{-}$-HNC3 calculations for several geometries of the target molecule near the HNC3 equilibrium and towards the dissociation limit of HNC${}_{3}^{-}$. The $A^{\prime}$ and $A^{\prime\prime}$ resonances, mentioned above, are visible in Fig. 3 (upper panel), showing the $A^{\prime}$ and $A^{\prime\prime}$ eigenphase sums. The $A^{\prime}$ resonance becomes a bound electronic state of HNC${}_{3}^{-}$ for small displacements, typically, involving a stretching of the H-N bond. Figure 2 shows the energy of the bound state as a function of the H-N bond length, while all other internuclear distances are configured such that the HNC3 energy is minimized. The $A^{\prime}$ resonance plays a crucial role in the electron attachment at low collision energies and following dissociation, while the higher-energy $A^{\prime\prime}$ resonance is not active in the process at temperatures below 1000 K.

The lower panel of Fig. 3 shows the eigenphase sums for different displacements along mode 1 near the equilibrium geometry of HNC3. Similarly to Ref. [29], dimensionless normal mode coordinates are used, i.e. coordinates in units of the harmonic oscillator length $\sqrt{\hbar/m\omega}$, where $m$ is the mass and $\omega$ is the frequency of the oscillator. As evident from the figure, energies and widths of the two resonances do not change significantly in the interval $q_{1}\in[-0.5;0.5]$ – compared, for example, to mode 3, which is demonstrated in the upper panel of Fig. 4. At displacement $q_{3}\sim 0.15$, the ${}^{2}A^{\prime}$ resonance becomes a bound HNC${}_{3}^{-}$ state. Performing the calculations for all 9 normal modes, it was found that for modes 2 and 4 the energy and width of the ${}^{2}A^{\prime}$ resonance also do not change. In contrast, for all other modes 5-9, the dependence is significant and should be accounted for in the DEA process. In the following discussion, modes 1, 2, and 4 will be referred to as inactive modes, and modes 3, 5-9 as active modes.

There are resonant and potential scattering (non-resonant) contributions to the scattering phases and the eigenphase sums. The resonant contribution changes significantly with $q$ for active modes, while the potential scattering one does not. One can separate the resonant contribution, subtracting the eigenphase sum for a geometry where the resonant contribution is not present, i.e. where the resonance becomes a bound state. The lower panel of Fig. 4 shows the same data as the upper panel with the potential contribution subtracted for ${}^{2}A^{\prime}$ symmetry. The subtracted potential contribution is the total eigenphase sum, evaluated at displacement $q_{3}=0.5$.

Figure 5 summarizes energies and widths of the $A^{\prime}$ resonance as a function of the displacements for the active modes. Symbols mark the values obtained from the eigenphase sums, while the lines are analytical fits to the power law [29]

[TABLE]

Together with the PES of the neutral molecule, represented by the quadratic potential along the normal modes, the data shown in figure represent the PES surface $U(\vec{q})$ and its autodetachment width in the region near HNC3 minimum and where the PES’s of HNC3 and HNC${}_{3}^{-}$ cross each other.

III Cross section and rate coefficient

Similarly to Ref. [29], to account for the multidimensional nature of the nuclear motion, one determines the dissociative coordinate $\vec{s}$ pointing in the direction of the steepest descent along the surface $U(\vec{q})$: $\vec{s}=-{\vec{\nabla}U}/{|\vec{\nabla}U|}$. In the present DEA model, the scattering electron, incident on HNC3, is captured in the ${}^{2}A$ resonant state. Once it is captured, the nuclear motion proceeds along the $U(\vec{q})$ PES in the direction of $\vec{s}$.

For a displacement $ds$ along $\vec{s}$, normal coordinate displacements $dq_{i}$ change as

[TABLE]

allowing us to compute the displacements $dq_{i}$, the normal and Cartesian coordinates of the geometries for motion on $U(\vec{q})$ along the direction $\vec{s}$.

The process of DEA is schematically shown in Fig. 6, representing the one-dimensional cut along the coordinate of the steepest descent $s$. In the figure, $E$ is the total energy of the $e^{-}+$HNC3 system. The initial vibrational state $\zeta(s)$ of HNC3  with energy $E_{0}$ is shown with the green line, while the dissociating state $U(s)$ of HNC${}_{3}^{-}$  is shown by blue circles (calculated) and the dashed line (analytical fit).

The electron, incident on HNC3  with asymptotic energy $\varepsilon=E-E_{0}$, is captured in the ${}^{2}A^{\prime}$ resonant state. After the electron capture, the nuclear motion proceeds along the $U(\vec{q})$ PES in the direction of $\vec{s}$, as shown in Fig. 5. The DEA cross section is computed similarly to Refs. [30, 29] as

[TABLE]

where the classical turning point $s_{E}$ for $U(s)$ and the Frank-Condon point $s_{\varepsilon}$ are obtained by solving $U(s_{E})=E$ and $E_{\mathrm{r}}(s_{\varepsilon})=\varepsilon$, respectively. Notice that $U(\vec{q})=V(\vec{q})+E_{\mathrm{r}}(\vec{q})$ for any geometry $\vec{q}$.

The resulting DEA cross section, obtained using Eq. 3, and the corresponding thermally averaged rate coefficient are plotted in Fig. 7. The cross section does not change significantly, varying between 4 and 5.6 Å2 in the interval of scattering energies from 1 meV to 1 eV. As a result, the rate coefficient is almost an exact square root function of temperature: $\tilde{k}\sqrt{T/\textrm{K}}$ with $\tilde{k}=3.2(\pm 0.1)\times 10^{-10}$cm3/s.

IV Discussion

Equation (3) for the DEA cross section does not include the autodetachment factor, which decreases the cross section. However, in the present case the autodetachment probability is negligible because (1) the time $\tau_{d}$, needed for the system to slide along the coordinate $s$ to the geometries where autodetachment is forbidden, is much smaller than the autodetachment time $\tau_{a}=\hbar/\Gamma(s_{\varepsilon})$ and (2) the dissociation is energetically open for any scattering energy (see Fig. 2). Time $\tau_{d}$ can be estimated from the the distance $\Delta s$ between the Franck-Condon point $s_{\varepsilon}$ and the point where the PESs of the ion and the neutral molecule cross. Inspecting Fig. 6, one estimates $\Delta s$ to be about 0.1 of the oscillator length, i.e. the time needed for the nuclei to move this distance is of the order of 0.1 of the period of oscillations (or smaller), $\tau_{d}\lesssim 0.2\pi/\omega$. Taking an average harmonic frequency $\omega_{5}$, we obtain the ratio

[TABLE]

For scattering energy 25 meV, corresponding to the room temperature, the ratio is about 0.04, i.e. the autodetachment probability is about 0.04. It is below the expected uncertainty of the present model and, therefore, can be ignored.

The obtained DEA cross section and the rate coefficient are significantly larger, by about 4 orders of magnitude, than the values obtained for H2CN [29]. It is because in the DEA of H2CN, the PESs of the ion and the neutral molecule cross far from the equilibrium geometry of H2CN and widths of the H2CN- resonant state are by about an order of magnitude smaller than the HNC${}_{3}^{-}$ widths.

In the present theoretical approach, the rotational motion of the target molecule was neglected. Although accurate calculations have not been performed yet, it is expected that the rotational excitation of HNC3 during the process should increase and, probably, significantly the DEA cross section. The target molecule has a relatively large dipole moment of 2.4 $ea_{0}$. Therefore, the cross section for rotational excitation by electron impact is expected to be significant at scattering energies above the rotational excitation energy. If the scattering energy is below the energy threshold for rotational excitation of the target, the scattering electron will not autoionize immediately and will orbit the molecule, approaching it another or several more times, thereby increasing the DEA cross section by the factor corresponding to the number of approaches during such a rotational resonance. Our preliminary estimates show that the factor is significant, but more accurate calculations are needed. A dedicated study of the role of rotational resonances in the DEA of molecules, leading to the formation of the anions observed in the ISM, will follow.

The DEA cross section and the rate coefficient, obtained above without accounting for the rotational motion, should be considered as a lower-bound estimate. Nevertheless, it is useful to compare the efficiency of formation of C3N- by DEA to HNC3 with radiative electron attachment (REA) to C3N using the obtained DEA rate coefficient. Taking the value of 500 for the ratio of C3N to HNC3 abundances derived from the IRC+10216 source [1] and the rate coefficients for formation of C3N- by REA and DEA, $k_{\mathrm{REA}}=5\times 10^{-15}$cm3/s [16] and $k_{\mathrm{DEA}}=5\times 10^{-9}$cm3/s (Fig. 7), respectively, at 300 K, we obtain the following ratio of formation efficiencies:

[TABLE]

Thus, the DEA mechanism is estimated to be about 2000 times more efficient in producing C3N- than REA at 300 K in IRC+10216.

V Conclusion

In this study, we have considered dissociative electron attachment to the HNC3. The approach combines the electron-scattering calculations and the O’Malley theory of dissociative attachment generalized to polyatomic targets. In HNC3+$e^{-}$ collisions, there is a low-energy resonance, which has a repulsive character along the H+NC3 coordinate and becomes a bound electronic state of the HNC${}_{3}^{-}$ anion near the equilibrium of HNC3. The anion state dissociates without a potential barrier. The present ıab initio calculations, performed without accounting for the rotational motion of HNC3 have demonstrated that the DEA is by several orders of magnitude more efficient in formation of C3N- anion in the ISM than the REA, previously considered by the astronomical community as the main mechanics of anion formation in the ISM. Other molecular ions observed in the ISM could also be formed by DEA to the corresponding neutral molecules.

Acknowledgements

VK and JF acknowledge the support from the US National Science Foundation, grants 2409570 and 2303895 respectively. The study was also partially supported by the Transatlantic Mobility Program and Chateaubriand Fellowship of the Office for Science and Technology of the Embassy of France in the United States, Programme National “Physique et Chimie du Milieu Interstellaire” (PCMI) of CNRS/INSU, the program “Accueil des chercheurs étrangers” of CentraleSupélec and from French State aid under France 2030 (QuanTEdu-France) bearing the reference ANR-22-CMAS-0001 and PEPR - SPLEEN Plasma-N-Act.