Gas-phase electronic spectra of HC_2n+1H⁺ (n = 2–6) chains

Abstract

Highly unsaturated carbon chains are generated in combustion processes and electrical discharges, and are confirmed constituents of the interstellar medium. In hydrogen-rich environments smaller carbon clusters tend to exist as linear chains, capped on each end by hydrogen atoms. Although the HC_2nH⁺ polyacetylene chains have been extensively characterized spectroscopically, the corresponding odd HC_2n+1H⁺ chains have received far less attention. Here we use two-colour resonance enhanced photodissociation spectroscopy to measure electronic spectra for HC_2n+1H⁺ (n = 2–6) chains contained in a cryogenically cooled quadrupole ion trap. The HC_2n+1H⁺ chains are formed either top-down by ionizing and fragmenting pyrene molecules using pulsed 266 nm radiation, or bottom-up by reacting cyclic carbon cluster cations with acetylene. Ion mobility measurements confirm that the HC_2n+1H⁺ species are linear, consistent with predictions from electronic structure calculations. The HC_2n+1H⁺ electronic spectra exhibit three band systems in the visible/near infrared spectral range, which each shifts progressively to longer wavelength by ≈90 nm with the addition of each additional C=C subunit. The strongest visible HC₁₁H⁺ band has a wavelength (λ = 545.1 nm) and width (1.5 nm) that match the strong λ 5450 diffuse interstellar band (DIB). However, other weaker HC₁₁H⁺ bands do not correspond to catalogued DIBs, casting doubt on the role of HC₁₁H⁺ as a carrier for the λ 5450 DIB. There are no identifiable correspondences between catalogued DIBs and bands for the other HC_2n+1H⁺ chains, allowing upper limits to be established for their column densities in diffuse interstellar clouds.

---

1 Introduction

Polyynes and highly unsaturated carbon chains (C_nH_m, C_nH_m⁺, C_nH_m⁻, with m = 1, 2) are confirmed constituents of the interstellar medium (ISM), particularly in molecular clouds such as the starless core TMC-1, where they have been detected through their microwave and infrared transitions.^1–5 Although carbon chain molecules are prevalent in molecular clouds, they are typically less abundant in hot molecular cores, translucent clouds, and diffuse clouds.⁶ Highly unsaturated carbon chains were also proposed as potential carriers of diffuse interstellar bands (DIBs), discrete spectral features occurring across the visible and near infrared spectral ranges that are presumed to arise from the absorptions of molecules in diffuse molecular clouds.^7–9 Although more than 500 DIBs have been identified, the only known carrier is the C₆₀⁺ fullerene, which is associated with five DIBs.^10,11

The widespread importance of carbon chains and the ease with which they can be generated has motivated several laboratory investigations of their infrared and electronic transitions.^6,12–17 Electronic spectra for bare carbon chains have been recorded for neutral,^18–21 cation,^22–24 and anion species.^25–27 For hydrogenated species, electronic spectra have been measured for neutral C_nH chains up to n = 16,²⁷ C_nH⁺ chains up to n = 17,²⁸ and C_nH⁻ chains up to n = 24.^14,29 Neutral C_nH (n = 2–9) chains have also been characterized through photoelectron spectra of anion C_nH⁻ chains.^30,31 Electronic spectra have been measured for polyacetylene cations as large as HC₁₄H⁺.^32,33 In work related to the current study, the A³Σ⁻_u ← X³Σ⁻_g electronic band systems of neutral HC_2n+1H (n = 3–6) chains have been measured using cavity ring down spectroscopy, confirming that these species have linear, centro-symmetric structures.³⁴ The visible electronic transitions of neutral HC_2n+1H (n = 3–6) chains have also been studied using resonance enhanced two-colour photoionization spectroscopy.³⁵

Here we examine the HC_2n+1H⁺ chains through their electronic absorptions in a cryogenically cooled ion trap, extending our earlier spectroscopic studies of bare and hydrogenated carbon clusters, including C_n⁺ cyclocarbons,^36,37 and singly hydrogenated carbon chains and rings C_2n+1H⁺.²⁸ As they lack a permanent dipole moment, the HC_2n+1H⁺ chains cannot easily be detected using microwave spectroscopy so their investigation in the ISM necessarily relies on either their electronic or infrared transitions. The only previously reported electronic spectra for the HC_2n+1H⁺ chains (n = 2–7) are for ions embedded in a neon matrix.^19,22,38 These matrix spectra exhibited several different band systems, that were interpreted with the aid of multireference electronic structure calculations.^39,40 Although the Ne matrix spectra are useful for identifying relevant transitions and assessing their relative intensities, the lines are broadened and shifted in wavelength due to ion-matrix interactions. Potentially, there are also contributions to the spectra from neutral molecules in the matrix. The higher resolution gas-phase spectra reported here provide the first opportunity for comparisons with astronomical data and for gauging the presence of HC_2n+1H⁺ chains in the ISM.

Although HC_2n+1H⁺ chains are yet to be detected in space, they may be produced in different regions through ion–molecule reactions. For example, flow tube studies show that C₅H₂⁺ is generated efficiently from the reaction of C₃H⁺ and C₄H₂.⁴¹ This and similar reactions may occur in protoplanetary disks in which diacetylene has recently been detected.⁴² Other formation mechanisms could involve protonation of neutral C_2n+1H molecules by H₃⁺ or HCO⁺. Laboratory studies show that HC_2n+1H⁺ ions are also produced through reactions of carbon cluster cations with acetylene, such as:⁴³

C_n⁺ + C₂H₂ → C_n−1H₂⁺ + C₃

C_n⁺ + C₂H₂ → C_n+2H₂⁺

The first reaction occurs for C₄⁺, C₆⁺ and C₈⁺, which are presumably mainly linear molecules, and is favoured by the stability of the C₃ co-fragment. The second reaction occurs over a size range for which the carbon cluster population consists of linear and cyclic isomers (C₇⁺, C₈⁺, C₉⁺) or only cyclic isomers (C₁₀⁺, C₁₁⁺, C₁₂⁺ C₁₃⁺, C₁₄⁺).⁴³ The reactivity of small cyclo-carbon cations with acetylene stands in contrast to their unreactivity with many other molecules, including H₂, D₂,⁴⁴ O₂,⁴⁴ CH₄,⁴⁴ N₂O,⁴⁵ and HCN.⁴⁶ Hydrogenated carbon chains might also be formed top-down through photodissociation of PAHs or other large hydrocarbon molecules.^47–49 In the current study, we mimic this route by generating HC_2n+1H⁺ chains by ionizing and dissociating pyrene molecules using pulsed 266 nm radiation.

2 Experimental methods

The experimental setup has been described previously.^28,36,50 The apparatus, shown in Fig. 1, consists of several linked stages that include an ion source, an ion mobility drift tube, a hexapole ion guide/pretrap where the ions were accumulated, a quadrupole mass filter, an octopole ion guide, a cryogenically cooled quadrupole ion trap (QIT), and a time-of-flight mass spectrometer. The HC_2n+1H⁺ ions were generated in two ways, which are illustrated in Fig. 1 (Schemes 1 and 2). First, as shown in Fig. 1 (Scheme 1), the HC_2n+1H⁺ ions were produced by exposing vapour emanating from a solid pyrene sample (vapor pressure P ≈ 5× 10⁻⁶ torr at 300 K) with the fourth harmonic of a pulsed Nd:YAG laser (λ = 266 nm, 50 μJ per pulse, 100 Hz). The ions passed through a drift tube ion mobility spectrometer (IMS) containing He buffer gas at ≈3 torr, propelled by a ≈10 V cm⁻¹ electric field. The drift tube IMS served to separate the ions spatially and temporally according to their collision cross sections with He buffer gas. When required, an electrostatic Bradbury–Nielsen ion gate positioned at the end of the drift region was used to select ions with a particular range of collision cross sections. Following the drift region the ion packet was compressed radially by an RF ion funnel and passed through a 1 mm orifice in a differential wall and then into an RF hexapole ion guide. Ions were accumulated in the hexapole for 50 shots of the ablation laser, then released to travel through a quadrupole mass filter (QMF). Ions then passed through an octupole ion guide and into a three-dimensional quadrupole ion trap (QIT) connected to a cryohead, where they were cooled to ≈10 K through collisions with He gas introduced through a pulsed nozzle operating at 2 Hz.

Fig. 1 IMMS-cryotrap instrument for obtaining electronic spectra of HC_2n+1H⁺ chains. HC_2n+1H⁺ chains were formed in two different ways. Scheme 1: HC_2n+1H⁺ chains were produced by exposing pyrene vapour (P ≈ 5 × 10⁻⁶ torr) to pulsed 266 nm radiation, which induced ionization and fragmentation. Following formation, the ions passed through an ion mobility drift tube, which served to separate different isomers. At the end of the drift tube there is a Bradbury–Nielsen (B–N) ion gate that could be used to select a particular isomer. Ions were then accumulated in a hexapole ion guide. Following ejection from the hexapole, the ions passed through a quadrupole mass filter (QMF), and an octupole ion guide, before being trapped in a cryogenically cooled quadrupole ion trap (QIT). In the QIT the ions were exposed to tunable light from an OPO overlapped (spatially and temporally) with a fixed wavelength beam from a second OPO. Resulting photofragments were separated and detected using a time-of-flight mass spectrometer. Scheme 2: HC_2n+1H⁺ chains were produced by reacting carbon chains (C_n⁺), formed through laser ablation of a graphite disk, with acetylene gas at a partial pressure of ≈2 × 10⁻⁶ torr in the hexapole ion guide for ≈0.5 s. If required, the reactant C_n⁺ ions could be mobility-selected in the drift tube using the B–N ion gate. Following their creation in the hexapole, the HC_2n+1H⁺ ions were mass-selected and spectroscopically probed in the QIT as described for Scheme 1.

Electronic action spectra of HC_2n+1H⁺ ions were recorded using two-colour resonance enhanced photodissociation (REPD) by monitoring C_2n+1H⁺ and C_2n+1⁺ photofragments (corresponding to loss of H, 2H, or H₂), and also for HC₁₃H⁺ the C₁₁⁺ photofragment (loss of HCCH). After 400 ms in the QIT, ions were exposed to a single pulse of light from a tunable optical parametric oscillator (OPO, EKSPLA NT342B, 6 ns pulse, bandwidth ≈4 cm⁻¹) followed 10 ns later by a pulse of fixed-wavelength light from a second OPO (EKSPLA NT340, 6 ns pulse width, bandwidth ≈4 cm⁻¹). The spectra in the visible region were normalised with respect to the laser power at each wavelength. The wavelength of the OPO output was measured using a wavemeter (Ångstrom High Finesse). The wavelength of the second OPO output was chosen to maximise the resonant two-colour photofragment signal, while minimising the signal produced by the second OPO alone, and was 440 nm for HC₅H⁺, 500 nm for HC₇H⁺, 550 nm for HC₉H⁺, 450 nm for HC₁₁H⁺, and 235 nm for HC₁₃H⁺.

The HC_2n+1H⁺ chains were also synthesised by reacting carbon cluster cations with acetylene as shown in Fig. 1 (Scheme 2). The carbon cluster cations were generated by laser ablation of a graphite disk with the second-harmonic of the Nd:YAG ablation laser (λ = 532 nm) and passed through the IMS stage. Specific carbon cluster isomers were selected by their arrival time (t_a) using the Bradbury–Nielson ion gate. Ion mobility arrival time distributions were measured by scanning the delay between firing the ablation laser and opening the Bradbury–Nielsen ion gate. The mobility-selected ions then passed into the hexapole ion guide where they were stored for up to 500 ms. The hexapole region contained a 2% acetylene in argon gas mixture (P ≈ 2 × 10⁻⁴ torr, acetylene number density ≈10¹¹ cm⁻³). Ions exiting the hexapole passed through the QMF and octupole ion guide and into the QIT where they were probed spectroscopically as described above for Scheme 1.

It should be emphasized that the HC_2n+1H⁺ molecules do not dissociate directly following excitation to the lower electronic states (Ã, B̃ and C̃ states), but require absorption of an additional photon or photons from the second laser. Without the second laser pulse, there was essentially no photodissociation. This is consistent with bond dissociation energies (D₀) calculated for C₅H₂⁺ at the CCSD(T)/cc-pVTZ level of theory using optimized geometries found at the TPSSh/def2-TZVP level of theory. The calculated dissociation energies are:

(a) C₅H⁺(¹Σ⁺) + H(²S_1/2) 4.28 eV

(b) l-C₅⁺(²Σ⁺_u) + H₂(¹Σ⁺_g) 6.28 eV

(c) c-C₃⁺(²B₂) + C₂H₂ (¹Σ⁺_g) 6.65 eV

(d) l-C₃⁺(²Σ⁺_u) + C₂H₂ (¹Σ⁺_g) 6.99 eV

Therefore, the lowest energy dissociation channel (H-loss) should have a single-photon onset at around 290 nm, at a shorter wavelength than investigated in the current study. A similar situation should prevail for the larger HC_2n+1H⁺ chains for which the Ã, B̃ and C̃ states progressively shift to lower energies. The photodissociation yields did not depend on the delay between the first and second laser pulses (out to 10 μs) indicating that the molecules undergo rapid internal conversion following excitation to the Ã, B̃ and C̃ states and that the second photon delivers sufficient additional energy to take the vibrationally excited ions above the dissociation threshold.

3 Quantum chemical calculations

The HC_2n+1H⁺ chains were investigated computationally using the CAM-B3LYP density functional theory method⁵¹ including empirical dispersion (D3BJ)⁵² and the triple zeta def2-TZVP basis set.⁵³ These geometries were employed for electronic structure calculations using the complete active space self consistent field (CASSCF)⁵⁴ method with the def2-TZVP basis set taking spin–orbit coupling into account as implemented in the ORCA/5.0.2 program package.⁵⁵ CASSCF calculations were undertaken for HC₅H⁺, HC₇H⁺, and HC₉H⁺ using the active spaces recommended in ref. 39 with averaging over eight doublet and eight quartet states. We tested this approach by calculating the spin–orbit coupling constants of C₇⁻ and HC₄H⁺. For C₇⁻, the active space was comprised of nine electrons in two σ and eight π orbitals, which yielded spin–orbit constants (experimental value 27.4 cm⁻¹) and (experimental value 0.6 cm⁻¹).⁵⁶ For HC₄H⁺, the active space was comprised of seven electrons in eight π orbitals, which yielded spin–orbit constants (experimental value −31.1 cm⁻¹) and (experimental value −30 cm⁻¹).^33,57 The equilibrium geometry and ground state rotational constant of HC₅H⁺ were calculated using the CCSD(T) method⁵⁸ with the TZVP basis set⁵⁹ in the ORCA program package.⁵⁵

4 Results and discussion

4.1 Electronic structure of HC_2n+1H⁺ chains

Before reporting and discussing the electronic spectra it is worth briefly reviewing the electronic states and transitions of the HC_2n+1H⁺ linear molecules. As described previously,^38–40,60 the X̃²Π_u/g ground state of the linear HC_2n+1H⁺ molecules is described by the …(n/2)π⁴_u (n/2)π⁴_g (n/2 + 1)π¹_u electronic configuration for n even, and …((n − 1)/2)π⁴_g ((n + 1)/2)π⁴_u ((n + 1)/2)π¹_g configuration for n odd. These configurations give rise to X̃²Π_u and X̃²Π_g ground states for even and odd n, respectively.

There are three excited electronic states that are relevant for transitions in the 200–2000 nm range – the 1²Π_g/u, 2²Π_g/u and 3²Π_g/u states.³⁹ As explained in ref. 39, for HC_2n+1H⁺ chains with n even, the 1Π_g and 2Π_g states have predominantly …(n/2)π⁴_u (n/2)π³_g (n/2 + 1)π²_u configurations, whereas the 3Π_g state is an admixture of …(n/2)π⁴_u (n/2)π³_g (n/2 + 1)π²_u and …(n/2)π⁴_u (n/2)π³_g (n/2 + 1)π⁰_u (n/2 + 1)π¹_g configurations. For HC_2n+1H⁺ chains with n odd, the 1²Π_u and 2²Π_u states have predominantly …((n − 1)/2)π⁴_g ((n + 1)/2)π³_u ((n + 1)/2)π²_g configurations, whereas the 3²Π_u state is an admixture of …((n − 1)/2)π⁴_g ((n + 1)/2)π³_u ((n + 1)/2)π²_g and …((n − 1)/2)π⁴_g ((n + 1)/2)π⁴_u ((n + 1)/2)π⁰_g ((n + 3)/2)π¹_u configurations. In other words, the à and B̃ states, are best described as SOMO ← SOMO−1 excitations, whereas SOMO ← SOMO−1 and LUMO ← SOMO excitations are important for the C state. The 1²Π_g/u ← X²Π_u/g and 2²Π_g/u ← X²Π_u/g transitions are predicted to be relatively weak compared to the 3²Π_g/u ← X²Π_u/g transition.³⁹ There is also an adjacent ²Φ_g/u state that is not optically coupled to the ground state. However, transitions to this state potentially gain intensity through vibronic coupling with the excited Π_g/u states.³⁸ Previously Fulara et al. labelled the three lowest excited Π_u/g states as òΠ_u/g, B̃²Π_u/g and C̃²Π_u/g states,³⁸ a convention we retain in this paper.

4.2 Electronic spectra

The REPD spectra of HC_2n+1H⁺ molecules (n = 2–6) are presented in Fig. 2. Wavelengths (λ_air), widths and assignments for the bands are given in Tables S2 and S3 in the ESI.† The HC_2n+1H⁺ spectra are similar to Ne matrix absorption spectra measured by Maier and coworkers,^19,22,38 except that peaks in the gas-phase spectra are sharper and are displaced slightly to the blue from the Ne matrix peaks. Peak intensities in the REPD spectra depend on the photodissociation efficiency resulting from the 2-colour excitation scheme and are therefore probably less reliable than the relative intensities of peaks in the Ne matrix absorption spectra.

Fig. 2 Two-colour REPD spectra of HC_2n+1H⁺ chains recorded by monitoring C_2n+1H⁺ and C_2n+1⁺ photofragments. Indicated are the origin bands and extent of the à ← X̃ (red), B̃ ← X̃ (green) and C̃ ← X̃ (blue) band systems. Gaps in the spectra near 710 nm and 410 nm correspond to drops in the OPO output power due to crystal changes.

Maier and coworkers assigned most HC_2n+1H⁺ transitions in the visible and UV regions to òΠ_u/g ← X̃²Π_g/u, B̃²Π_u/g ← X̃²Π_g/u and C̃²Π_u/g ← X̃²Π_g/u band systems, with the latter system carrying the highest oscillator strength.³⁸ Complications in the spectra of HC₉H⁺, HC₁₁H⁺, HC₁₃H⁺ and HC₁₅H⁺ led them to invoke transitions to another electronic state, possibly the ²Φ_g/u state, which may derive intensity through vibronic coupling.³⁸ The gas-phase spectra are broadly consistent with this previous interpretation, although some of our assignments differ from those in ref. 38, particularly for HC₉H⁺, HC₁₁H⁺ and HC₁₃H⁺. In our interpretation, the spectra are affected by vibronic interactions, particularly between overtone vibrational levels in the à state manifold and the vibrationless level of the B̃ state. Vibronic mixing of zero-order òΠ_u/g and B̃²Π_u/g levels leads to sharing of intensity between the strong B̃²Π_u/g ← X̃²Π_g/u origin transition and weaker òΠ_u/g ← X̃²Π_g/u overtone transitions. Depending on the size of the chains, the consequence is that instead of the B̃²Π_u/g ← X̃²Π_g/u origin transition appearing as a single band it is manifested as several peaks. Furthermore, for some chain sizes, higher vibronic levels of the B̃²Π_u/g state are resonant with the lower levels of the C̃²Π_u/g state, further complicating the spectrum. This mixing is particularly evident for HC₁₁H⁺ for which the interacting zero-order levels are nearly resonant. The nature of the vibronic coupling between the A, B and C states is not clear at this stage, but, given that all three electronic states share the same symmetry, the coupling is probably mediated by totally symmetric CC or CH stretch vibrational modes.

The lowest energy òΠ_g/u ← X̃²Π_u/g band systems exhibit sharp peaks, whose widths are limited by the bandwidth of the OPO (≈5 cm⁻¹). The à ← X̃ transitions are predicted to be relatively weak with calculated oscillator strengths of 0.0005, 0.0002, 0.0012, 0.0009, and 0.0019 for HC₅H⁺, HC₇H⁺, HC₉H⁺, HC₁₁H⁺ and HC₁₃H⁺, respectively.³⁹ The band shift induced by the Ne matrix for the à ← X̃ origin transition is −110 cm⁻¹ for HC₅H⁺, −71 cm⁻¹ for HC₇H⁺, −9 cm⁻¹ for HC₉H⁺, −37 cm⁻¹ for HC₁₁H⁺, and −33 cm⁻¹ for HC₁₃H⁺. The à ← X̃ band systems of the HC_2n+1H⁺ chains exhibit a progression in the acetylenic C≡C stretch mode, extending to one or two quanta, with a spacing of ≈2000 cm⁻¹. For HC₅H⁺, a progression in even quanta of the ν₈ mode (the lowest frequency π_g mode) is also observed. As shown in Fig. 3, the wavelength of the òΠ_g/u ← X̃²Π_u/g origin transition increases linearly with the number of carbon atoms in the chain (by ≈94 nm for each additional C₂ subunit), akin to other carbon chain series.^13,14,16,17

Fig. 3 Wavelengths for the origin transitions of òΠ_u/g ← X̃²Π_g/u, B̃²Π_u/g ← X̃²Π_g/u and C̃²Π_u/g ← X̃²Π_g/u band systems for HC_2n+1H⁺ chains plotted against the number of carbon atoms (2n + 1). Wavelengths for the progression members in the acetylenic C≡C stretch mode are also plotted for the òΠ_u/g ← X̃²Π_g/u band system. Note that as the chains become longer, overtone vibrational levels (2 quanta of the acetylenic C≡C stretch mode) associated with the òΠ_u/g state come into resonance with the vibrationless level of the B̃²Π_u/g manifold leading to complications in the spectra. The average energy for transitions in the region of the B̃²Π_u/g ← X̃²Π_g/u origin transition are plotted for HC₉H⁺, HC₁₁H⁺ and HC₁₃H⁺.

Our assignments for the origins of the B̃²Π_g/u ← X̃²Π_u/g and C̃²Π_g/u ← X̃²Π_u/g band systems for HC₅H⁺ and HC₇H⁺ agree with those of Fulara et al.³⁸ However, for HC₉H⁺, HC₁₁H⁺ and HC₁₃H⁺, the bands originally assigned to the C̃²Π_g/u ← X̃²Π_u/g origin transitions are reassigned as origins of the B̃²Π_g/u ← X̃²Π_u/g band systems, while strong peaks following the linear trend between wavelength and chain length established for the C̃²Π_g/u ← X̃²Π_u/g origin transitions for HC₅H⁺ and HC₇H⁺ are assigned as the C̃²Π_g/u ← X̃²Π_u/g origins for HC₉H⁺, HC₁₁H⁺ and HC₁₃H⁺. As noted above, it seems that overtones and combinations of vibrational modes associated with the òΠ_g/u state interact with the vibrationless level of the B̃²Π_g/u state, leading to mixing and sharing of oscillator strength. These interactions particularly affect the spectrum of HC₁₁H⁺, and are responsible for a cluster of peaks near 600 nm. Assuming that the zero-order oscillator strength of the B̃²Π_g/u ← X̃²Π_u/g origin transition greatly exceeds the oscillator strengths for the òΠ_g/u ← X̃²Π_u/g overtone transitions, one can estimate the deperturbed energies for the B̃²Π_g/u state from the centre-of-gravity of the transitions in the origin region (the corresponding wavelengths are plotted in Fig. 3). If this is done, the wavelengths for the deperturbed origin transitions of the three band systems follow linear trends with chain size (see Fig. 3). Significantly, with this new set of assignments there is no need to invoke an additional electronic state to explain the spectra of HC₉H⁺, HC₁₁H⁺ and HC₁₃H⁺ in the visible region as was done in ref. 38.

A weak band appears approximately 27 cm⁻¹ to the red of the òΠ_u/g ← X̃²Π_g/u origin transitions in the HC₅H⁺, HC₇H⁺, and HC₉H⁺ spectra. The two transitions can be assigned to the ²Π_1/2 ← ²Π_1/2 sub-band (strong) and ²Π_3/2 ← ²Π_3/2 sub-band (weak), which are separated by the difference in the spin–orbit splittings for the ground and excited states . Using the SOC + CASSCF/def2-TZVP method, we calculated spin–orbit splittings for HC₅H⁺, HC₇H⁺, and HC₉H⁺. The calculated ΔA_SO values agree with the observed peak separations for HC₅H⁺ (26 cm⁻¹), HC₇H⁺ (29 cm⁻¹), and HC₉H⁺ (27 cm⁻¹). The spin–orbit coupling constants for the HC_2n+1H⁺ ions are similar to those measured for the X̃²Π and Ã²Π states of the isoelectronic C₇⁻ anion .⁵⁶ The relative intensity for the ²Π_3/2 ← ²Π_3/2 sub-band compared to the ²Π_1/2 ← ²Π_1/2 sub-band (10–20%) is consistent with predicted thermal populations of the ²Π_3/2 and ²Π_1/2 levels for molecules in the cryotrap (12% at T = 20 K, assuming a Boltzmann distribution). An alternative assignment of the lower energy peak as a vibrational hot band is less convincing as the vibrational modes should be effectively cooled in the cryotrap. For example, the lowest frequency π_u bend mode (ν₁₁) for HC₅H⁺ has a calculated frequency of 128 cm⁻¹;³⁹ at T = 30 K less than 0.4% of the molecules would have a single quantum of ν₁₁.

4.2.1 Formation of HC₁₁H⁺ from reaction of C₉⁺ and C₂H₂. The spectra shown in Fig. 2 clearly show that C_2n+1H₂⁺ chains are generated through the ionization and decomposition of pyrene vapour. From the work of McElvany, it is known that carbon cluster cations react with acetylene to form hydrogenated carbon molecules by the C_n⁺ + C₂H₂ → C_n+2H₂⁺ association reaction.⁴³ At low pressure (≤2 × 10⁻⁷ torr), this reaction occurs at bimolecular collision rates over a size range for which the carbon cluster population exists as mixed linear and cyclic isomers (C₇⁺, C₈⁺, C₉⁺) or only cyclic isomers (C₁₀⁺, C₁₁⁺, C₁₂⁺ C₁₃⁺, C₁₄⁺).⁴³ McElvany proposed that the C_n+2H₂⁺ products are formed through radiative association and are essentially C_n⁺…C₂H₂ complexes consisting of an acetylene molecules attached to an intact cyclic carbon cluster cation. To explore this hypothesis we investigated C_2n+1H₂⁺ molecules produced by reacting cyclocarbons with acetylene, focussing on C₁₁H₂⁺ products from the reaction of cyclic C₉⁺ and acetylene. To do this, we selected cyclic C₉⁺ molecules using the Bradbury–Nielsen ion gate situated at the end of the IMS tube. These mobility-selected C₉⁺ ions reacted with acetylene gas introduced into the hexapole ion guide to form C₁₁H₂⁺ ions, which were subsequently mass-selected by the QMF and introduced into the QIT where they were spectroscopically probed. The scheme is explained in more detail in Section S4 of the ESI.† As shown in Fig. S2 in the ESI,† the C₁₁H₂⁺ cations formed by reacting C₉⁺ with acetylene and C₁₁H₂⁺ cations generated by exposing pyrene to 266 nm radiation have nearly identical spectra, proving that they share the same linear structure.

To better understand the formation of linear HC₁₁H⁺ through the bimolecular reaction of C₉⁺ and acetylene we constructed a potential energy surface describing the most important reaction steps (Fig. 4). Following the initial encounter, the C₉⁺ + C₂H₂ reaction proceeds through a series of bicyclic and cyclic intermediates without any prohibitive barrier. The last step involves ring opening to yield the HC₁₁H⁺ chain, which, in the absence of collisions would have an internal energy above the Ã, B̃ and C̃ excited states. Therefore, it is possible that the nascent HC₁₁H⁺ product ions are stabilized under low pressure conditions through inverse internal conversion and radiative emission from an excited electronic state (Poincaré fluorescence),⁶¹ or alternatively through emission of infrared radiation. Although we believe that Fig. 4 illustrates a credible mechanism for C₁₁H₂⁺ chain formation directly from a bimolecular collision, we cannot discount the possibility that a nascent C_n⁺…C₂H₂ complex is initially formed by a three body collision in the hexapole, and is collisionally transformed into a linear C₁₁H₂⁺ chain either in the heaxapole, en route to the QIT, or in the QIT itself. It is also unclear at this stage whether any of the intermediates shown in Fig. 4 are formed in the hexapole and stabilized by collisions with Ar gas (P ≈ 2 × 10⁻⁴ torr), making their way, along with the linear C₁₁H₂⁺ chains, into the QIT.

Fig. 4 Proposed formation mechanism for HC₁₁H⁺ from c-C₉⁺ + C₂H₂ reaction. Ground state energies (black bars) are calculated using DFT (ωB97X-D/def2-SVP). Excited state energies (purple bars) for HC₁₁H⁺ are determined from the spectra shown in Fig. 2. Putative deactivation of HC₁₁H⁺ product ions by fluorescence from excited electronic states is indicated by a blue arrow. All values are in units of eV.

4.3 Astronomical implications

It is relevant to consider briefly whether any of the HC_2n+1H⁺ chains are DIB carriers. We first consider the relatively narrow à ← X̃ transitions, which occur in the visible or near infrared regions for HC_2n+1H⁺ (n = 2–6). The stronger B̃ ← X̃ and C̃ ← X̃ origin transitions either do not occur in the visible (C₅H₂⁺), or are broad (C₇H₂⁺, C₉H₂⁺, C₁₃H₂⁺). The exception is the intense C̃ ← X̃ origin transition of C₁₁H₂⁺, which we discuss later.

We first consider HC₅H⁺, noting that its strongest band in the visible region, the òΠ_1/2 ← X̃²Π_1/2 origin transition at 498.0 ± 0.2 nm, is reasonably close to DIBs at 497.961 nm and 498.214 nm in spectra for HD 204827.⁶² To investigate the possible correspondence between the HC₅H⁺ absorption and these nearby DIBs, we recorded the à ← X̃ origin band at higher resolution using a pulsed dye laser (bandwidth 0.04 cm⁻¹). The spectrum exhibits partial resolution of rotational substructure with an R-branch maximum at 498.02 nm (see Fig. 5). The band was best simulated assuming a temperature of 12 K, a calculated ground state rotational constant B′′ = 0.07477 cm⁻¹ (computed using CCSD(T)/TZVP), and an excited state rotational constant of B′ = 0.07193 cm⁻¹ that was adjusted to give the best match between the simulated and measured spectra. It is apparent that the HC₅H⁺ band does not correspond to either of the adjacent DIBs and that the match would not improve at higher temperatures. The origin bands for the à ← X̃ transitions of HC₇H⁺, HC₉H⁺ and HC₁₁H⁺, which occur across the visible region, also do not match tabulated DIBs for HD 183143 or HD 204827 sight-lines.^62,63

Fig. 5 REPD spectrum of the HC₅H⁺ òΠ_1/2 ← X̃²Π_1/2 band origin measured using a dye laser (bandwidth 0.04 cm⁻¹). The spectrum is compared with a simulation generated using B′′ = 0.07477 cm⁻¹, B′ = 0.07193 cm⁻¹, and T = 12 K. The vertical arrow indicates the position of the 497.961 nm DIB (data from ref. 62).

The absence of à ← X̃ origin transitions for HC₅H⁺, HC₇H⁺, HC₉H⁺ and HC₁₁H⁺ in the HD 183143 or HD 204827 spectra allows one to estimate upper limits for column densities for these sight-lines using the relationship:⁶⁴

where ε₀ is the permittivity of free space, m_e is the electron mass, c is the speed of light, e is the electron charge, λ is the transition wavelength, f is the transition's oscillator strength and W_λ is the equivalent width. Assuming a minimum detectable equivalent width of W_λ = 10 mÅ, implies upper limits for column densities reported in Table 1. Values are given assuming both calculated oscillator strengths (f_calc) and experimental oscillator strengths determined from Ne matrix measurements (f_exp).^38,39 Unfortunately, there are large differences between the calculated and experimental oscillator strengths, which are reflected in differences in the upper limits for column densities. The upper limits for column densities estimated from the à ← X̃ transitions are relatively large because the oscillator strengths for these transitions are small.

Table 1 Upper limits for column densities (N_max) for HC_2n+1H⁺ chains in diffuse interstellar clouds derived by considering their à ← X̃ transitions in the visible region and astronomical data for HD 183143.⁶³N_max values are based on a 10 mÅ minimum detectable equivalent width (W_λ), and use calculated (f_calc) and experimental (f_exp) oscillator strengths from ref. 39 and ref. 38, respectively. Band positions and widths are taken from the current work. N_max values based on f_calc are given first and those based on f_exp are given second in parentheses

Species	λ nm	f calc (fexp)	N max 10^12 cm^−2
HC5H^+	498.0	0.00046 (0.00075)	100(60)
HC7H^+	597.1	0.00022 (0.0018)	140(18)
HC9H^+	694.3	0.00118 (0.004)	20(6)
HC11H^+	786.8	0.00089 (0.0046)	20(4)

---

4.3.1 HC₁₁H⁺. The strong C̃²Π_u ← X̃²Π_g origin transition of HC₁₁H⁺ corresponds in position and width to the λ 5450 DIB measured in several astronomical surveys. This can be seen in Fig. 6(a) where the HC₁₁H⁺ REPD spectrum is plotted together with a synthetic spectrum generated using data for HD 183143.⁶³ Assuming for the moment that the C̃²Π_u ← X̃²Π_g origin transition is responsible for the λ 5450 DIB, a column density of ≈8 × 10¹² cm⁻² is estimated using eqn (1) with λ = 5450 Å, W_λ = 0.3595 Å (from ref. 63) and f_calc = 0.18 (from ref. 39). A somewhat larger column density of ≈5 × 10¹³ cm⁻² is estimated employing the Ne matrix oscillator strength (f_matrix = 0.025).

Fig. 6 REPD spectra for HC₁₁H⁺ in the visible/NIR range for: (a) C̃²Π_u ← X̃²Π_g origin transition; (b) òΠ_u ← X̃²Π_g origin transition. In each case, a synthetic astronomical DIB spectrum for HD 183143 generated using data from ref. 63 is shown above the REPD spectrum. Arrows indicate DIBs close to HC₁₁H⁺ absorptions.

If HC₁₁H⁺ is responsible for the λ 5450 DIB, then other weaker HC₁₁H⁺ bands should also appear in astronomical spectra. The nearby HC₁₁H⁺ transition at 542.3 nm may contribute to the broad DIB at 542.001 nm (see Fig. 6(a)). Comparisons for other transitions over the 600–700 nm range are difficult because the HC₁₁H⁺ bands are broad and there is a high density of DIBs in this region. However, the òΠ_u ← X̃²Π_g origin transition at 786.8 nm is relatively narrow and occurs in a region where DIBs are sparse, with the nearest DIB at 786.243 nm (see Fig. 6(b)).⁶³ The òΠ_u ← X̃²Π_g transition is estimated to be weaker than the C̃²Π_u ← X̃²Π_g transition by a factor of five based on the matrix oscillator strengths,³⁸ or by a factor of 200 based on the calculated oscillator strengths.³⁹ If the matrix oscillator strengths are correct, then the òΠ_u ← X̃²Π_g origin transition would have an equivalent width of W_λ = 70 mÅ, making it 4–5 times more intense than the λ 7862 DIB in the HD 183143 spectrum reported in ref. 63 (W_λ = 16 mÅ). In this case the transition should certainly be apparent in existing astronomical spectra. However, if the calculated oscillator strengths are correct, then the òΠ_u ← X̃²Π_g origin transition would have an equivalent width of W_λ = 2 mÅ, probably making it difficult to detect.

If HC₁₁H⁺ exists in the ISM one might expect similar or greater abundances of HC₅H⁺, HC₇H⁺ and HC₉H⁺, although the relative amounts may depend sensitively on their formation mechanisms and the stability of their precursors. If the Ne matrix oscillator strengths are correct, then their column densities should probably exceed 5 × 10¹³ cm⁻², which, according to the upper limits reported in Table 1, would imply that the òΠ_u ← X̃²Π_g origin transitions of HC₇H⁺, HC₉H⁺ and perhaps HC₅H⁺ should be apparent in existing astronomical spectra. On the other hand, if the calculated oscillator strengths are correct, the column densities would exceed 8 × 10¹² cm⁻² and the òΠ_u ← X̃²Π_g origin transitions of HC₅H⁺, HC₇H⁺, HC₉H⁺ may be too weak to detect.

On balance, the correspondence between the C̃ ← X̃ origin transition of HC₁₁H⁺ and the λ 5450 DIB should probably be viewed as a coincidence. Confirmation would require the discovery of other weaker HC₁₁H⁺ bands in astronomical spectra. As well, one might also expect to observe visible bands for the smaller HC₅H⁺, HC₇H⁺ and HC₉H⁺ species, which would presumably be formed by similar chemical pathways to HC₁₁H⁺.

5 Conclusions

Electronic spectra of HC₅H⁺, HC₇H⁺, HC₉H⁺, HC₁₁H⁺, and HC₁₃H⁺ have been measured in a cryogenically cooled ion trap. The spectra exhibit sharp bands over the NIR, visible and UV ranges, associated with the òΠ_u/g ← X̃²Π_g/u, B̃²Π_u/g ← X̃²Π_g/u and C̃²Π_u/g ← X̃²Π_g/u band systems. Regular red-shifts of 70–94 nm in wavelength for the origin transitions of these band systems accompanying addition of each C=C unit are disrupted and obscured by vibronic resonances, particularly for the B̃²Π_u/g ← X̃²Π_g/u transitions of HC₉H⁺, HC₁₁H⁺, and HC₁₃H⁺.

We find no correspondences between catalogued DIBs and the electronic absorption bands of HC₅H⁺, HC₇H⁺, HC₉H⁺ and HC₁₃H⁺. However, the strongest HC₁₁H⁺ band matches the broad λ 5450 DIB. There are two points counting against HC₁₁H⁺ being a carrier for the λ 5450 DIB. First, less intense bands in the 600–900 nm range in the HC₁₁H⁺ REPD spectrum are absent from astronomical spectra, although this may be because these bands are much weaker than the 545.1 nm band. Second, there is no evidence in the astronomical spectra for the visible bands of HC₅H⁺, HC₇H⁺, HC₉H⁺ which would presumably have similar or greater abundances than HC₁₁H⁺. The most promising HC₁₁H⁺ band for future astronomical detection is probably the A ← X origin transition at 786.8 nm, which, if HC₁₁H⁺ is the carrier, should be detectable for sight lines that give a strong signal for the λ 5450 DIB. Better estimates for upper limits of the column densities for HC₅H⁺, HC₇H⁺, HC₉H⁺ and HC₁₁H⁺ require reliable oscillator strengths for the electronic transitions. Hopefully, fresh experiments or calculations will help resolve the discrepancies between computed and measured oscillator strengths.

The HC_2n+1H⁺ chains may also be detectable in the ISM through their infrared transitions, particularly given the recent advances in IR astronomy with the James Webb Space Telescope. The requisite laboratory infrared spectra should be obtainable using various mass-selective action spectroscopy techniques, including He droplet ion spectroscopy,^65–67 rare gas tagging,⁶⁷ or leak-out spectroscopy.⁶⁸

The current study demonstrates that laser ionization/fragmentation of PAHs is an effective strategy for forming highly unsaturated carbon chains. Although C₁₃H₂⁺ is the largest chain generated from pyrene, even longer species will presumably be produced by ionizing and dissociating larger PAHs. Ultimately, it will be interesting to see whether there is a progressive transition from linear to monocyclic C_2n+1H₂⁺ structures with increasing size, as occurs for C_2n+1H⁺ ions, for which cyclic isomers dominate for species larger than C₁₅H⁺.²⁸ Ion mobility measurements for hydrogenated carbon ions generated through laser ablation of graphite in a He/H₂ supersonic expansion suggest that linear and monocyclic isomers are present in roughly equal proportions for C₁₅H₂⁺ and C₁₇H₂⁺, and that the monocyclic isomer dominates for C₁₉H₂⁺ and C₂₁H₂⁺.⁶⁹ Another avenue for future research involves spectroscopically characterizing carbon chains terminated with O or N atoms that are generated using appropriate heterocyclic or substituted PAHs as precursors.

Finally, we note that the strategy of reacting isomer-selected carbon cluster cations with other neutral molecules in the hexapole ion guide has promise for generating exotic molecular ions that can be spectroscopically probed in the cryo-QIT. In the current work the viability of this approach was demonstrated by producing HC_2n+1H⁺ chains as products of the reaction between cyclic carbon cluster cations and acetylene molecules. Although, cyclocarbon cations are relatively stable and do not react with most molecules, the smaller linear carbon chains react with a range of molecules, including H₂, D₂,⁴⁴ O₂,⁴⁴ CH₄,⁴⁴ N₂O,⁴⁵ and HCN.⁴⁶ It should be relatively straightforward to investigate product ions from these reactions, some of which have astrophysical relevance, using electronic spectroscopy.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This research was supported under the Australian Research Council's Discovery Project funding scheme (Project Numbers DP150101427 and DP160100474). The authors thank Richard Mathys from the Science Faculty Workshop for his contributions to the design and construction of the apparatus used in this study.

Notes and references

1. J. Pety, P. Gratier, V. Guzmán, E. Roueff, M. Gerin, J. R. Goicoechea, S. Bardeau, A. Sievers, F. Le Petit, J. Le Bourlot, A. Belloche and D. Talbi, Astron. Astrophys., 2012, 548, A68.
2. S. Brünken, L. Kluge, A. Stoffels, O. Asvany and S. Schlemmer, Astrophys. J., Lett., 2014, 783, L4.
3. J. Cernicharo, M. Agúndez, C. Cabezas, R. Fuentetaja, B. Tercero, N. Marcelino, Y. Endo, J. Pardo and P. de Vicente, Astron. Astrophys., 2022, 657, L16.
4. J. Cernicharo, A. M. Heras, A. Tielens, J. R. Pardo, F. Herpin, M. Guélin and L. Waters, Astrophys. J., 2001, 546, L123.
5. M. McCarthy, C. Gottlieb, H. Gupta and P. Thaddeus, Astrophys. J., 2006, 652, L141–L144.
6. K. Taniguchi, P. Gorai and J. C. Tan, arXiv, 2023, preprint, arXiv:2303.15769 [astro-ph.GA] DOI:10.48550/arXiv.2303.15769.
7. A. E. Douglas, Nature, 1977, 269, 130–132.
8. J. P. Maier, G. A. Walker and D. A. Bohlender, Astrophys. J., 2004, 602, 286.
9. The Diffuse Interstellar Bands, ed. A. G. G. M. Tielens and T. P. Snow, Kluwer, Dordrecht, 1995.
10. E. K. Campbell, M. Holz, D. Gerlich and J. P. Maier, Nature, 2015, 523, 322–323.
11. H. Linnartz, J. Cami, M. Cordiner, N. Cox, P. Ehrenfreund, B. Foing, M. Gatchell and P. Scheier, J. Mol. Spectrosc., 2020, 367, 111243.
12. A. Van Orden and R. J. Saykally, Chem. Rev., 1998, 98, 2313–2357.
13. E. B. Jochnowitz and J. P. Maier, Proc. Int. Astron. Union, 2008, 4, 395–402.
14. R. Nagarajan and J. P. Maier, Int. Rev. Phys. Chem., 2010, 29, 521–554.
15. M. A. Duncan, J. Phys. Chem. A, 2012, 116, 11477–11491.
16. C. Rice and J. Maier, J. Phys. Chem. A, 2013, 117, 5559–5566.
17. L. N. Zack and J. P. Maier, Chem. Soc. Rev., 2014, 43, 4602–4614.
18. P. Freivogel, J. Fulara, D. Lessen, D. Forney and J. P. Maier, Chem. Phys., 1994, 189, 335–341.
19. J. Fulara, P. Freivogel, D. Forney and J. P. Maier, J. Chem. Phys., 1995, 103, 8805–8810.
20. D. Forney, P. Freivogel, M. Grutter and J. P. Maier, J. Chem. Phys., 1996, 104, 4954–4960.
21. X. Chen, M. Steglich, V. Gupta, C. A. Rice and J. P. Maier, Phys. Chem. Chem. Phys., 2014, 16, 1161–1165.
22. P. Freivogel, J. Fulara, D. Lessen, D. Forney and J. P. Maier, Chem. Phys., 1994, 189, 335–341.
23. E. K. Campbell and P. W. Dunk, Rev. Sci. Instrum., 2019, 90, 1–7.
24. J. E. Colley, D. S. Orr and M. A. Duncan, J. Chem. Phys., 2022, 157, 121102.
25. D. W. Arnold, S. E. Bradforth, T. N. Kitsopoulos and D. M. Neumark, J. Chem. Phys., 1991, 95, 8753–8764.
26. P. Freivogel, J. Fulara, M. Jakobi, D. Forney and J. P. Maier, J. Chem. Phys., 1995, 103, 54–59.
27. D. A. Kirkwood, M. Tulej, M. V. Pachkov, M. Schnaiter, F. Güthe, M. Grutter, M. Wyss, J. P. Maier and G. Fischer, J. Chem. Phys., 1999, 111, 9280–9286.
28. S. J. Marlton, J. T. Buntine, C. Liu, P. Watkins, U. Jacovella, E. Carrascosa, J. N. Bull and E. J. Bieske, J. Phys. Chem. A, 2022, 126, 6678–6685.
29. H. Linnartz, T. Motylewski and J. P. Maier, J. Chem. Phys., 1998, 109, 3819–3823.
30. T. R. Taylor, C. Xu and D. M. Neumark, J. Chem. Phys., 1998, 108, 10018–10026.
31. E. Garand, T. I. Yacovitch, J. Zhou, S. M. Sheehan and D. M. Neumark, Chem. Sci., 2010, 1, 192–201.
32. J. Fulara, M. Grutter and J. P. Maier, J. Phys. Chem. A, 2007, 111, 11831–11836.
33. A. Dzhonson, E. B. Jochnowitz and J. P. Maier, J. Phys. Chem. A, 2007, 111, 1887–1890.
34. C. D. Ball, M. C. McCarthy and P. Thaddeus, J. Chem. Phys., 2000, 112, 10149–10155.
35. H. Ding, T. W. Schmidt, T. Pino, A. E. Boguslavskiy, F. Güthe and J. P. Maier, J. Chem. Phys., 2003, 119, 814–819.
36. J. T. Buntine, M. I. Cotter, U. Jacovella, C. Liu, P. Watkins, E. Carrascosa, J. N. Bull, L. Weston, G. Muller, M. S. Scholz and E. J. Bieske, J. Chem. Phys., 2021, 155, 214302.
37. S. J. Marlton, J. T. Buntine, P. Watkins, C. Liu, U. Jacovella, E. Carrascosa, J. N. Bull and E. J. Bieske, J. Phys. Chem. A, 2023, 127, 1168–1178.
38. J. Fulara, A. Nagy, I. Garkusha and J. P. Maier, J. Chem. Phys., 2010, 133, 024304.
39. J. Zhang, X. Guo and Z. Cao, Int. J. Mass Spectrom., 2010, 290, 113–119.
40. M. Mühlhäuser, J. Haubrich and S. D. Peyerimhoff, Int. J. Quantum Chem., 2004, 100, 53–58.
41. S. Dheandhanoo, L. Forte, A. Fox and D. K. Bohme, Can. J. Chem., 1986, 64, 641–648.
42. B. Tabone, G. Bettoni, E. F. van Dishoeck, A. M. Arabhavi, S. Grant, D. Gasman, T. Henning, I. Kamp, M. Güdel and P. O. Lagage, et al. , Nat. Astron., 2023, 1–10.
43. S. W. McElvany, J. Chem. Phys., 1988, 89, 2063–2075.
44. S. W. McElvany, B. I. Dunlap and A. O'Keefe, J. Chem. Phys., 1998, 86, 715–725.
45. M. S. Resat, J. N. Smolanoff, I. B. Goldman and S. L. Anderson, J. Chem. Phys., 1994, 100, 8784–8794.
46. D. C. Parent and S. W. McElvany, J. Am. Chem. Soc., 1989, 111, 2393–2401.
47. B. West, F. Useli-Bacchitta, H. Sabbah, V. Blanchet, A. Bodi, P. M. Mayer and C. Joblin, J. Phys. Chem. A, 2014, 118, 7824–7831.
48. H. R. Hrodmarsson, J. Bouwman, A. G. Tielens and H. Linnartz, Int. J. Mass Spectrom., 2022, 476, 116834.
49. H. R. Hrodmarsson, J. Bouwman, A. G. Tielens and H. Linnartz, Int. J. Mass Spectrom., 2023, 485, 116996.
50. J. T. Buntine, E. Carrascosa, J. N. Bull, U. Jacovella, M. I. Cotter, P. Watkins, C. Liu, M. S. Scholz, B. D. Adamson, S. J. P. Marlton and E. J. Bieske, Rev. Sci. Instrum., 2022, 93, 043201.
51. M. Saitow, U. Becker, C. Riplinger, E. F. Valeev and F. Neese, J. Chem. Phys., 2017, 146, 164105.
52. S. Grimme, S. Ehrlich and L. Goerigk, J. Comput. Chem., 2011, 32, 1456–1465.
53. F. Weigend and R. Ahlrichs, Phys. Chem. Chem. Phys., 2005, 7, 3297–3305.
54. B. O. Roos, P. R. Taylor and P. E. Siegbahn, Chem. Phys., 1980, 48, 157–173.
55. F. Neese, Wiley Interdiscip. Rev.: Comput. Mol. Sci., 2022, 12, e1606.
56. N. M. Lakin, M. Pachkov, M. Tulej, J. P. Maier, G. Chambaud and P. Rosmus, J. Chem. Phys., 2000, 113, 9586–9592.
57. U. Jacovella and F. Merkt, Phys. Chem. Chem. Phys., 2017, 19, 23524–23531.
58. K. Raghavachari, G. W. Trucks, J. A. Pople and M. Head-Gordon, Chem. Phys. Lett., 1989, 157, 479–483.
59. A. Schäfer, C. Huber and R. Ahlrichs, J. Chem. Phys., 1994, 100, 5829–5835.
60. M. Mühlhäuser, J. Haubrich, G. Mpourmpakis, A. Mavrandonakis and G. E. Froudakis, Internet Electron. J. Mol. Des., 2003, 2, 578–588.
61. A. Léger, P. Boissel and L. d'Hendecourt, Phys. Rev. Lett., 1988, 60, 921.
62. L. M. Hobbs, D. G. York, T. P. Snow, T. Oka, J. A. Thorburn, M. Bishof, S. D. Friedman, B. J. McCall, B. Rachford and P. Sonnentrucker, Astrophys. J., 2008, 680, 1256.
63. L. M. Hobbs, D. G. York, J. A. Thorburn, T. P. Snow, M. Bishof, S. D. Friedman, B. J. McCall, T. Oka, B. Rachford, P. Sonnentrucker and D. E. Welty, Astrophys. J., 2009, 705, 32.
64. T. Motylewski, H. Linnartz, O. Vaizert, J. P. Maier, G. A. Galazutdinov, F. A. Musaev, J. Krełowski, G. A. H. Walker and D. A. Bohlender, Astrophys. J., 2000, 531, 312–320.
65. C. J. Moon, S. Erukala, A. J. Feinberg, A. Singh, M. Y. Choi and A. F. Vilesov, J. Chem. Phys., 2023, 158, 224307.
66. S. Erukala, A. Feinberg, A. Singh and A. F. Vilesov, J. Chem. Phys., 2021, 155, 084306.
67. S. Brünken, F. Lipparini, A. Stoffels, P. Jusko, B. Redlich, J. Gauss and S. Schlemmer, J. Phys. Chem. A, 2019, 123, 8053–8062.
68. P. C. Schmid, O. Asvany, T. Salomon, S. Thorwirth and S. Schlemmer, J. Phys. Chem. A, 2022, 126, 8111–8117.
69. S. Lee, N. Gotts, G. von Helden and M. T. Bowers, J. Phys. Chem. A, 1997, 101, 2096–2102.

---

Footnote

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d4cp00625a

---