The linear radical cation of cyanoacetylene, HC_{3}N^{+} (^{2}Π), is not only of astrophysical interest, being the, so far undetected, cationic counterpart of the abundant cyanoaceteylene, but also of fundamental spectroscopic interest due to its strong spin–orbit and Renner–Teller interactions. Here, we present the first broadband vibrational action spectroscopic investigation of this ion through the infrared pre-dissociation (IRPD) method using a Ne tag. Experiments have been performed using the FELion cryogenic ion-trap instrument in combination with the FELIX free-electron lasers and a Laservision optical parametric oscillator/optical parametric amplifier system. The vibronic splitting patterns of the three interacting bending modes (*ν*_{5}, *ν*_{6}, *ν*_{7}), ranging from 180 to 1600 cm^{−1}, could be fully resolved revealing several bands that were previously unobserved. The associated Renner–Teller and intermode coupling constants have been determined by fitting an effective Hamiltonian to the experimental data, and the obtained spectroscopic constants are in reasonable agreement with previous photoelectron spectroscopy (PES) studies and *ab initio* calculations on the HC_{3}N^{+} ion. The influence of the attached Ne atom on the infrared spectrum has been investigated by *ab initio* calculations at the RCCSD(T)-F12a level of theory, which strongly indicates that the discrepancies between the IRPD and PES data are a result of the effects of the Ne attachment.

## I. INTRODUCTION

The simplest cyanopolyyne, cyanoacetylene (HC_{3}N), is one of the most widespread polyatomic species in the interstellar medium (ISM) and has been observed in a variety of astronomical sources in the Milky Way and in external galaxies.^{1–3} It also plays an important role in the complex nitrogen chemistry of Titan, Saturn’s largest moon, being one of the most abundant nitrogen-bearing species detected in its atmosphere.^{4,5} Its highly reactive cationic counterpart, HC_{3}N^{+}, is efficiently produced by ionization of the neutral cyanoacetylene by solar vacuum-ultraviolet (VUV) radiation and may participate in Titan’s tholin formation.^{6} In the ISM, neutral HC_{3}N is readily ionized by cosmic rays or UV photons to form HC_{3}N^{+}.^{7} However, this cation has yet to be detected in the ISM, which is likely a result of lack of reference data.

Besides being astrochemically relevant, the cyanoacetylene radical cation (^{2}Π) is interesting on a fundamental spectroscopic level due to its open-shell linear character. The vibronic coupling effects that occur as a result of this character, such as Renner–Teller (RT)^{8} coupling, cause a breakdown of the Born–Oppenheimer (BO) approximation. Subsequent analysis of the complex splitting pattern then requires methods that go beyond the BO approximation, such as effective Hamiltonian analysis (for small couplings)^{9} or a full nonadiabatic description of the molecule.^{10,11}

Previous work on HC_{3}N^{+} includes several low- and high-resolution photoelectron spectroscopy (PES) studies.^{12–15} The high-resolution pulsed-field ionization zero kinetic energy (PFI-ZEKE) study by Dai *et al.*^{12} presented a sufficient experimental resolution to reveal an intricate RT- and spin–orbit (SO)-splitting pattern in the observed vibronic spectrum, which was analyzed based on diabatic calculations. The observed medium to weak coupling strengths make this ion an excellent candidate for an effective Hamiltonian analysis rendering experimental spectroscopic constants that can be used to benchmark *ab initio* calculations.

Vibrational spectroscopic work on HC_{3}N^{+} is limited to a Ne-matrix-assisted absorption spectroscopy study in the C–H stretching region,^{16} which does not contain any information on the three RT-affected vibrational bending modes. Gaining information on these modes through vibrational spectroscopy would be complementary to the earlier PES work due to the different selection rules at hand and would aid in a full understanding of this complex ion. Furthermore, these data could serve as a reference for future high-resolution studies and astronomical searches (e.g., with the James Webb Space Telescope operating in the infrared region).

Infrared pre-dissociation spectroscopy (IRPD) is an excellent method to obtain gas-phase vibrational spectra of molecular ions. Here, a messenger [usually a rare-gas (RG) atom] is weakly bound to the target ion at cryogenic temperatures, and its subsequent on-resonant dissociation is monitored by mass spectrometry. This messenger atom, also called tag, acts as a spectator, and in the case of RG atoms like He or Ne, its influence on the vibrational structure is generally rather small.^{17–19} This method is especially suited for small reactive cations, since other (tag-free) action spectroscopic methods are not suitable^{20} (e.g., infrared multi-photon dissociation is limited due to the small size of the ion,^{21,22} laser-induced reactions^{23} require a suitable endothermic reaction, and laser inhibition of complex growth^{24,25} does work only with cw lasers).

The goal of the present paper is to obtain the first broad-band gas-phase vibrational spectrum of the HC_{3}N^{+} cation covering all fundamental vibrational modes including the RT-perturbed bending modes to complement and extend the earlier PES studies. The spectrum was recorded by means of IRPD using Ne as a RG messenger atom carried out in a cryogenic ion trap interfaced with the widely tunable FELIX (Free Electron Laser for Infrared eXperiments)^{26} free electron lasers.^{27} The recorded spectrum is fitted with an effective Hamiltonian and compared with *ab initio* calculations of Dai *et al.*,^{12} and the results are discussed with an emphasis on the influence of the Ne atom used as a tag in the IRPD scheme.

## II. METHODS

### A. Experimental methods

The vibrational spectrum of the cyanoacetylene cation (HC_{3}N^{+}) was recorded using the FELion cryogenic 22-pole ion trap instrument. A detailed account of the instrument and the employed action spectroscopic scheme, IRPD, of *in situ* RG-tagged cold molecular ions has been given previously,^{27} and here, we only give a brief account of details specific to the HC_{3}N^{+} ion. The ion is produced by direct electron impact ionization (28(2) eV) from the neutral precursor acrylonitrile (CH_{2}CHCN, $\u226599%$ purity, Sigma-Aldrich). The liquid precursor was evaporated into the ion source and diluted with helium in a 5:1 (He:CH_{2}CHCN) mixing ratio. A ∼100 ms long pulse of ions is extracted from the source, and the ions of interest, i.e., HC_{3}N^{+} with m/z 51, are mass-selected by a quadrupole mass filter before entering the 22-pole ion trap, which is held at a fixed temperature in the range 8–9 K. Around 10–15 ms before the ions enter the trap, an intense ∼80 ms long Ne:He pulse (1:3 mixing ratio and number density of $\u223c1015$ cm^{−3}) is admitted to the trap, leading to efficient collisional cooling of the ions close to the trap-ambient temperature and the formation of Ne-ion complexes by termolecular collisions. Under these conditions, around $10%$ of the primary ions form weakly bound complexes with Ne, see Fig. 1.

The ions are stored for several seconds in the ion trap (typically 1–3 s) and exposed to several FELIX IR laser pulses before extraction. An IRPD spectrum is recorded by mass selecting and detecting the Ne−HC_{3}N^{+} complex ions as a function of wavenumber. The following wavenumber ranges were covered in this study: (a) 130–270 cm^{−1}, (b) 310–2500 cm^{−1}, and (c) 3110–3270 cm^{−1}, using the free-electron IR lasers FEL-1 (a) and FEL-2 (b) of the FELIX Laboratory^{28} with a macropulse repetition rate of 10 Hz, a maximum pulse energy in the trap region of $<35$ mJ (at 1100 cm^{−1}), and linewidths (FWHM) of around 0.5% of the center wavenumber. Region (c) was covered using a Laservision OPO/OPA system ($\u223c1$ cm^{−1} FWHM, 10 Hz repetition rate) with a typical output power of $<20$ mJ.

A relative depletion $D=1\u2212NON(\nu )NOFF$ in the number of complex ions *N*_{ON}(*ν*) from the baseline value *N*_{OFF} is observed upon resonant vibrational excitation. To account for varying laser pulse energy *E*, pulse number *n*, and for saturation effects, the signal is normalized prior to averaging using $I=\u2212ln[NON(\nu )/NOFF]n\u22c5E$, giving the intensity *I* in units of cross-section per joule. After normalizing each individual spectrum in this way, the final spectrum is then obtained by averaging using statistical binning with a typical bin size of 2 cm^{−1}. Line parameters such as band positions, intensities, and line widths (FWHM) are then obtained by fitting a multi-component Gaussian function to the experimental data, also providing statistical errors of the line parameters.

### B. Theoretical approach

#### 1. Ab initio

To understand and describe the vibrational IRPD spectra and the influence of the attached Ne atom on the observed band positions, we performed *ab initio* quantum chemical calculations on the HC_{3}N^{+} cation and the Ne−HC_{3}N^{+} complex. Geometry optimization and subsequent harmonic wavenumber calculations on the bare ion were performed at the partially spin-restricted, explicitly correlated, coupled cluster level of theory, with single, double, and perturbative triple excitations, RCCSD(T)-F12a^{29} using cc-pVXZ-F12 (X = D,T,Q)^{30} basis sets, and for the Ne−HC_{3}N^{+}, using the cc-pVTZ-F12 basis set. Information on the perpendicular component of the dipole moment of the bare ion was obtained by the use of finite-field perturbation theory, where a finite dipole field (*F* = 0.005 a.u.) is added to the core energy and the one-electron Hamiltonian. The dipole moment *μ* is then obtained as

where *E*(*F*) is the energy as a function of the field.

To investigate the interaction of HC_{3}N^{+} with the Ne atom a one-dimensional cut of the potential energy surface was made by attaching the Ne atom to the middle carbon atom for fixed Ne–C–H angles while optimizing all other geometry parameters. All quantum chemical calculations were performed using the MOLPRO suite, version 2015.1.^{31}

#### 2. Effective Hamiltonian

The spin-vibronic energy levels of HC_{3}N^{+} were calculated with an effective Hamiltonian approach similar to the model of He and Clouthier^{9} following the nomenclature employed by Dai *et al.*^{12}. We ignore the effects of molecular rotation since its effects are too small to be seen with the experimental resolution of ∼0.5% of the center wavenumber: for the Ne−HC_{3}N^{+} complex, *B*_{e} ≈ 0.033 cm^{−1}, calculated at the RCCSD(T)-F12a/cc-pVTZ-F12 level of theory.

A Hund’s case (a) basis, $n$, was chosen with,

Here, quantum numbers Λ = ±1 and Σ = ±1/2 are the projections of the orbital and spin angular momenta on the molecular axis, respectively, *v*_{k} = 0, 1, … is the vibrational quantum number of mode *v*_{k}, and *l*_{k} is the projection of the vibrational angular momentum (*l*_{k} = −*v*_{k}, − *v*_{k} + 2, …, *v*_{k}). We only include the three-bending normal modes *v*_{5}–*v*_{7}. Quantum number *K* = Λ + *∑*_{k}*l*_{k} is the projection of the total angular momentum excluding electron spin, and *P* = *K* + Σ is the projection of the total angular momentum onto the molecular axis. In the case of strong vibronic coupling, such as RT coupling, Λ and *l*_{k} are ill-defined, but *P* is a good quantum number since we neglect the overall rotation. Furthermore, we only include diagonal spin–orbit coupling and first-order RT, see below, and hence, *K* is also a good quantum number. For a basis truncated at *v*_{tot} = *v*_{5} + *v*_{6} + *v*_{7} = 8, we find that energy levels are converged up to *v*_{tot} = 3.

We approximate the total effective Hamiltonian by

where $H\u0302vib$ represents the harmonic vibrational energy,

with *ω*_{k} the harmonic frequencies of the bending modes *ν*_{5}–*ν*_{7}. The spin–orbit Hamiltonian is given by^{32}

where we take the SO constant *A*_{SO} = −44 cm^{−1} independent of the vibrational mode,^{12} and $L\u0302z$ and $S\u0302z$ are the molecule-fixed components of the electronic orbital and spin angular momenta operators, respectively. The effective RT Hamiltonian is^{33}

where h.c. stands for Hermitian conjugate, and the operator |Λ = −1⟩⟨Λ = 1| couples the two diabatic electronic states. The constants *g*_{k} are related to the dimensionless RT constants *ɛ*_{k} as

and the coupling parameters *g*_{kl} are related to the dimensionless intermode RT couplings *ɛ*_{kl} by

The spherical normal mode operators *q*_{k,±} are related to the Cartesian normal modes *q*_{k,x} and *q*_{k,y} by

The wave function is expanded in the basis

and the expansion coefficients *u*_{n,i} are determined variationally by solving the matrix eigenvalue problem given in Eq. (11) below, using the free and open source numerical software SCILAB version 6.1.1.^{34}

Here, *E*_{i} are the eigenvalues, and ** H** is the Hamiltonian matrix of which the matrix elements are given in Ref. 35. All parameters, excluding

*A*

_{SO}and the

*g*

_{57}intermode RT coupling parameter, were obtained from a nonlinear least squares fit to 14 lines of the experimental spectrum. We use the lsqrsolve Levenberg–Marquardt algorithm implemented in SCILAB, starting from the calculated spectroscopic parameters of Dai

*et al.*

^{12}. For this purpose, we write the Hamiltonian matrix

**as**

*H*where the *p*_{j}’s are the parameters to be fitted {*ω*_{5}, *ω*_{5}, *ω*_{7}, *g*_{5}, *g*_{6}, *g*_{7}, *g*_{56}, *g*_{67}} and *H*_{0} contains the spin–orbit Hamiltonian and *ν*_{5}–*ν*_{7} intermode RT coupling, which are kept constant. The fitting algorithm employs the Jacobian matrix ** J** of the derivatives of the transition energies

*E*

_{i}−

*E*

_{0}with respect to the parameters,

We obtain the derivatives of the energies with respect to the parameters as expectation values of the Hamiltonian matrices *H*_{j} for the normalized eigenvectors *u*_{i},

where the *T* indicates the transpose of the column vector *u*_{i}.

The fitting error (*σ*_{j}) in parameter *p*_{j} is approximated by

where the covariance matrix ** C** is related to the Jacobian matrix

**and the root-mean-squares (rms) error in the transition energies (**

*J**r*) by

Finally, the intensities are computed by

where the dipole operator $\mu \u0302\xb1$ is approximated by

The perpendicular dipole moments $\mu k\u22a5$ are given in Sec. III B.

#### 3. The HC_{3}N^{+} ion

The linear HC_{3}N^{+} ion exhibits a ^{2}Π_{3/2} electronic ground state known from experimental PES work and calculations.^{12–14} As discussed in earlier work,^{12–15} the *ν*_{1}–*ν*_{4} modes represent the C–H, C≡N, C≡C, and C–C stretches (of *a*_{1} symmetry), and the *ν*_{5}–*ν*_{7}, H–C≡C, C–C≡N, and C≡C–C in- (*b*_{1}) and out-of-plane (*b*_{2}) bendings, respectively. Here, the plane is defined with respect to the molecular orbitals. For closed-shell species, these bending modes are degenerate, but since this ion is open-shell and linear, they are RT perturbed.

Already within the Born–Oppenheimer approximation, the degeneracy is lifted, and the RT coupling causes a complicated splitting pattern in the vibrational structure. *Ab inito* spectroscopic parameters and experimental results from the earlier PFI-ZEKE work of Dai *et al.*^{12} show that *ν*_{5} has the largest RT perturbance (*ɛ*_{5} ≈ 0.18), whereas *ν*_{6} (*ɛ*_{6} ≈ −0.05) and *ν*_{7} (*ɛ*_{7} ≈ −0.06) are only minimally affected. Some differences between the vibrational IRPD spectrum and the PES work may, however, be expected because of the different selection rules at hand; photoelectron spectroscopy is subjected to Franck–Condon overlap, and vibrational spectroscopy to the Δ*K* = ±1 and Δ*P* = ±1 selection rules for the RT-perturbed bending modes and Δ*P* = 0 for the stretching modes. Furthermore, the HC_{3}N^{+} ion is cooled to its vibrational and SO ground state (*P* = 3/2, *A*_{SO} = −44 cm^{−1}), so that one of the two SO components is predominantly observed (*P* = 3/2 for the stretching modes and *P* = 1/2 or *P* = 5/2 for the bending modes). The population of the other SO level should be limited to approximately $\u223c4.5$%, based on a Boltzmann distribution calculated with 44 cm^{−1} energy level separation and an estimated ion temperature of 20 K.

## III. RESULTS AND DISCUSSION

Harmonic vibrational calculations were performed on the RCCSD(T)-F12a/cc-pVXZ-F12 (X = D,T,Q) level of theory both for the bare ion and the ion–Ne complex (see Sec. III C). The calculated equilibrium geometries are in good agreement with previous calculations^{12,13,15} and are shown for the sake of completeness in Table I.

. | H–C . | C≡C . | C–C . | C≡N . |
---|---|---|---|---|

RCCSD(T)-F12a/cc-pVQZ-F12 | 1.078 | 1.244 | 1.339 | 1.186 |

RCCSD(T)-F12a/cc-pVTZ-F12 | 1.078 | 1.244 | 1.339 | 1.186 |

RCCSD(T)-F12a/cc-pVDZ-F12 | 1.078 | 1.245 | 1.340 | 1.187 |

RCCSD(T)-F12a/cc-p(c)VTZ-F12^{a} | 1.078 | 1.244 | 1.339 | 1.186 |

CASPT2/AVTZ^{b} | 1.067 | 1.237 | 1.328 | 1.188 |

CCSD(T)/AVTZ^{c} | 1.072 | 1.213 | 1.352 | 1.155 |

PBE0/AVTZ^{c} | 1.079 | 1.233 | 1.333 | 1.179 |

. | H–C . | C≡C . | C–C . | C≡N . |
---|---|---|---|---|

RCCSD(T)-F12a/cc-pVQZ-F12 | 1.078 | 1.244 | 1.339 | 1.186 |

RCCSD(T)-F12a/cc-pVTZ-F12 | 1.078 | 1.244 | 1.339 | 1.186 |

RCCSD(T)-F12a/cc-pVDZ-F12 | 1.078 | 1.245 | 1.340 | 1.187 |

RCCSD(T)-F12a/cc-p(c)VTZ-F12^{a} | 1.078 | 1.244 | 1.339 | 1.186 |

CASPT2/AVTZ^{b} | 1.067 | 1.237 | 1.328 | 1.188 |

CCSD(T)/AVTZ^{c} | 1.072 | 1.213 | 1.352 | 1.155 |

PBE0/AVTZ^{c} | 1.079 | 1.233 | 1.333 | 1.179 |

Figure 2 shows the recorded IRPD spectrum of HC_{3}N^{+} using Ne as a messenger atom in the wavenumber ranges 130–250 cm^{−1} (FEL-1), 350–2500 cm^{−1} (FEL-2), and 3110–3270 cm^{−1} (OPO/OPA). The obtained line positions are shown in Table II together with previous experimental^{12–16} and computational^{12,14} work. To gain accurate line positions of the weaker bands, the relative depletion spectrum (not power corrected) was fitted with a Gaussian profile as described above (Sec. II A). A full list of the obtained frequencies, relative intensities, and their uncertainties is given in supplementary material, Table 1. The provided relative intensities were estimated from the power-normalized spectrum. For clarity, we treat the assignment of the well-behaved stretching modes (Sec. III A) separately from the analysis of the RT and SO splitting patterns of the bending modes (Sec. III B).

. | ν_{1}
. | ν_{2}
. | ν_{3}
. | ν_{4}
. | ν_{5}
. | ν_{6}
. | ν_{7}
. |
---|---|---|---|---|---|---|---|

IRPD (this work) | 3184(1) | 2171(1) | 1845(1) | 957(1) | 626(1)–846(1) | 439(1)–490(1) | 189(1)–238(1) |

PFI-ZEKE^{a} | ⋯ | 2176(4) | ⋯ | ⋯ | ⋯ | 445(5) | 198(5) |

TPES^{b} | 3123(20) | 2177(20) | 1855(30) | 829(30) | 648(40) | 422(20) | 203(40) |

IR-matrix^{c} | 3196.47 | 2175.79 | 1852.82 | ⋯ | ⋯ | ⋯ | ⋯ |

SPES^{d} | 3105 | 2185 | 1830 | ⋯ | ⋯ | 411 | ⋯ |

PFI-ZEKE^{e} | 3121 | 2171 | ⋯ | ⋯ | 628–873 | 438–488 | 190–236 |

RCCSD(T)-F12a^{f} | 3318 | 2224 | 1870 | 910 | [767,646] | [427,410] | [183,179] |

RCCSD(T)-F12a^{g} | 3317 | 2222 | 1868 | 908 | [771,644] | [462,444] | [196,186] |

RCCSD(T)-F12a^{h} | 3316 | 2217 | 1864 | 907 | [763,638] | [452,436] | [194,186] |

RCCSD(T)-F12a^{i} | 3322 | 2228 | 1872 | 912 | [843,699] | [474,449] | [204,198] |

CASPT2^{j} | 3467 | 2270 | 1881 | 951 | [853,687] | [501,468] | [222,215] |

. | ν_{1}
. | ν_{2}
. | ν_{3}
. | ν_{4}
. | ν_{5}
. | ν_{6}
. | ν_{7}
. |
---|---|---|---|---|---|---|---|

IRPD (this work) | 3184(1) | 2171(1) | 1845(1) | 957(1) | 626(1)–846(1) | 439(1)–490(1) | 189(1)–238(1) |

PFI-ZEKE^{a} | ⋯ | 2176(4) | ⋯ | ⋯ | ⋯ | 445(5) | 198(5) |

TPES^{b} | 3123(20) | 2177(20) | 1855(30) | 829(30) | 648(40) | 422(20) | 203(40) |

IR-matrix^{c} | 3196.47 | 2175.79 | 1852.82 | ⋯ | ⋯ | ⋯ | ⋯ |

SPES^{d} | 3105 | 2185 | 1830 | ⋯ | ⋯ | 411 | ⋯ |

PFI-ZEKE^{e} | 3121 | 2171 | ⋯ | ⋯ | 628–873 | 438–488 | 190–236 |

RCCSD(T)-F12a^{f} | 3318 | 2224 | 1870 | 910 | [767,646] | [427,410] | [183,179] |

RCCSD(T)-F12a^{g} | 3317 | 2222 | 1868 | 908 | [771,644] | [462,444] | [196,186] |

RCCSD(T)-F12a^{h} | 3316 | 2217 | 1864 | 907 | [763,638] | [452,436] | [194,186] |

RCCSD(T)-F12a^{i} | 3322 | 2228 | 1872 | 912 | [843,699] | [474,449] | [204,198] |

CASPT2^{j} | 3467 | 2270 | 1881 | 951 | [853,687] | [501,468] | [222,215] |

^{a}

From Ref. 14.

^{b}

Threshold Photoelectron Spectroscopy (TPES) from Ref. 13.

^{c}

From Ref. 16.

^{d}

Slow Photoelectron Spectroscopy (SPES) from Ref. 15.

^{e}

From Ref. 12.

^{f}

Using cc-pVQZ-F12 (this work).

^{g}

Using cc-pVTZ-F12 (this work).

^{h}

Using cc-pVDZ-F12 (this work).

^{i}

Using cc-p(c)VQZ-F12 basis from Ref. 12.

^{j}

Using CASPT2/AVTZ/CAS(9,9) from Ref. 13.

### A. Stretching modes

The bands at 3184, 2170, and 1846 cm^{−1} can be readily assigned to the *ν*_{1}, *ν*_{2}, and *ν*_{3} modes representing the C–H, C≡N, and C≡C stretches, respectively, and they are in good agreement with *ab initio* calculations and earlier experimental work presented in Table II. It is noteworthy that the *ν*_{1} mode is ∼60 cm^{−1} blue shifted compared with earlier PES works, which is likely a direct consequence of the Ne attachment (see Sec. III C). This hypothesis is strengthened by comparing our experimentally derived wavenumber with the earlier Ne-matrix-assisted vibrational spectroscopic measurements by Smith-Gicklhorn *et al.*^{16} exhibiting a similar blue shift.

In the *ν*_{1} mode, a clear substructure is observed with peaks at 3174.0, 3182.9, 3184.7, 3185.7, 3192.9, and 3195.2 cm^{−1}. The predicted rotational structure (calculated with PGOPHER^{36} at 20 K and using *B* = 0.033 cm^{−1} for Ne−HC_{3}N^{+}, see supplementary material, Fig. 1) has a FWHM due to the unresolved rotational structure of ∼4 cm^{−1} and cannot explain all observed peaks. We hypothesize that the 3183, 3184, and 3185 cm^{−1} peaks are the P, Q, and R branches of the C–H stretch, although the observed strong Q-band intensity remains a mystery. The 3192 and 3195 cm^{−1} bands may be attributed to the P-R branches of a combination band of the C–H stretch with one of the low-lying modes involving the Ne atom, which is linearly attached to the hydrogen atom (see Sec. III C).

Less trivial is the assignment of the *ν*_{4} C–C stretching mode, which we attribute to the band observed at 950 cm^{−1}. This value agrees with the harmonic vibrational wavenumber calculations that predict a band between 908 and 951 cm^{−1}, depending on the level of theory, but is significantly different from the 829 cm^{−1} reported in the earlier Threshold PES (TPES) work of Desrier *et al.*^{13} The authors speculated that this red shift is a result of anharmonic coupling of the polyad involving the *ν*_{4}, *ν*_{6}, and *ν*_{7} vibrational modes. No other works claim to have detected the *ν*_{4} stretching mode, and in the high-resolution ZEKE work of Dai *et al.*,^{12} it was not mentioned in their analysis. They, however, do observe two bands at 873 and 920 cm^{−1}, which they attribute to the 5^{1}*κ*Σ fundamental and the 5^{1}7^{2}Π_{1/2} combination band of the RT- and SO-affected *ν*_{5} and *ν*_{7} vibrational modes. We propose that the bands at 873 and 920 cm^{−1} observed in the ZEKE study are in fact the two SO components of the *ν*_{4} stretching mode and the band at 829 cm^{−1} observed in the TPES work to be one of the vibronic splitting components (also observed by Dai *et al.*^{12}). In our work, however, only one of the two SO components is observed for all bands, which is a direct result of the cooling of the ions to their vibrational and SO ground state ($P=32$, with a population of 96% based on the Boltzmann distribution at 20 K). Since only the Δ*P* = ±1 transitions are allowed for the bending modes and Δ*P* = 0 for the stretching modes, we indeed expect to observe only one of the two SO components. The relative blue shift of the *ν*_{4} stretching observed here may be a result of the Ne attachment similar to the effect on the *ν*_{1} C–H stretching mode (see Sec. III C).

### B. Vibronic coupling effects

To test our effective Hamiltonian model, we first computed the energy levels of the bending modes based on the *ab initio* spectroscopic parameters of Dai *et al.*^{12} (see supplementary material, Table 2), and the obtained energies fully agree with the earlier work.

Based on our calculations, we assign the energy level at 739.9 cm^{−1} to 5^{1}*κ*Σ and the level at 875.4 cm^{−1} to 5^{1}Π_{3/2}. Note that in Table 3 of Dai *et al.*,^{12} these assignments were reversed. We also computed the vibrational transition intensities, where we may expect the selection rules Δ*K* = ±1 and Δ*P* = ±1 since the employed Hamiltonian does not include any mixing terms with stretching modes or between the *P* and *K* levels. The dipole moments were approximated by finite-field calculations in the *xy*-plane (perpendicular to the molecular *z*-axis) at a normal mode displacement of each of the three bending modes: $\mu 5\u22a5=0.21$ a.u., $\mu 6\u22a5=\u22120.016$ a.u., and $\mu 7\u22a5=\u22120.0087$ a.u.

Based on the calculated wavenumber positions, intensities, and the proposed selection rules, we could safely assign nine bands corresponding to the *μ*Σ, Δ_{5/2}, and *κ*Σ fundamentals of each mode (see Table III). These fundamentals explain the most intense peaks of the spectrum, but several weaker bands remain unassigned. Based on the *ab initio* calculations, these unassigned bands could be attributed to combination bands, overtones, or transitions that violate the selection rules of our model. Since the density of states in the higher wavenumber region (>800 cm^{−1}) is rather large and all transitions are of very low or zero predicted intensity, their assignment is nontrivial. To gain more clarity regarding the assignment of the weak features, we performed a nonlinear least squares fit of the fundamental bands to the effective Hamiltonian and iteratively included newly assigned bands. The final fit included the 14 bands marked with a star in Table III. In order to check the validity of this fitting routine as well as the quality of the *ab initio* parameters, the experimental results of Dai *et al.*^{12} were also fitted using this model. The resulting spectroscopic parameters are compared in Table IV.

Obs. [int] . | Ab initio [int]
. | Calc. fit [int] . | Assignment . |
---|---|---|---|

189(1)* [0.1]^{a} | 191 [0.042] | 190 [0.040] | 7^{1}μΣ |

200(3)* [0.1] | 196 [0.044] | 195 [0.017] | 7^{1}Δ_{5/2} |

208(2) [0.1] | ⋯ | ⋯ | 7^{1}Δ_{5/2} + ν_{Ne}?^{b} |

231(1) [0.1] | ⋯ | ⋯ | ⋯ |

238(1)* [0.224] | 237 [0.132] | 240 [0.130] | 7^{1}κΣ |

384(1)* [0.057] | 382 [0.000] | 381 [0.000] | 7^{2}Π_{1/2} |

439(1)* [0.786] | 446 [0.073] | 440 [0.073] | 6^{1}μΣ |

454(1)* [0.645] | 458 [0.081] | 452 [0.079] | 6^{1}Δ_{5/2} |

490(1)* [0.164] | 497 [0.385] | 491 [0.354] | 6^{1}κΣ |

552(2) [0.040] | ⋯ | ⋯ | ⋯ |

572(1)* [0.048] | 577 [0.005] | 574 [0.006] | 7^{3}Δ_{5/2} |

626(1)* [0.861] | 630 [0.717] | 627 [0.735] | 5^{1}μΣ |

630(1)* [0.367] | 627[0.0088] | 633 [0.013] | 7^{3}Δ_{5/2} |

688(1)* [1] | 698 [1] | 687 [1] | 5^{1}Δ_{5/2} |

704(1) [0.326] | ⋯ | ⋯ | 5^{1}Δ_{5/2} + ν_{Ne}? |

846(1)* [0.322] | 875 [0.547] | 847 [0.563] | 5^{1}κΣ |

926(1) [0.041] | ⋯ | ⋯ | ν_{4} stretch |

957(1) [0.147] | ⋯ | ⋯ | ν_{4} stretch |

981(1) [0.035] | ⋯ | ⋯ | ν_{4} stretch |

1097(1)* [0.013] | 1105 [0.003] | 1097 [0.002] | 6^{2}7^{1}Δ_{5/2} |

1243(1) [0.072] | 1257 [0.004] | 1256 [0.000] | 5^{2}Π_{3/2}? |

1253(1) [0.067] | 1259 [0.001] | 1256 [0.000] | 5^{2}Π_{3/2}? |

1331(1)* [0.039] | 1339 [0.001] | 1331 [0.001] | 6^{3}Σ |

1595(1) [0.072] | 1616 [0.001] | 1594 [0.000] | 5^{2}Π_{3/2}? |

Obs. [int] . | Ab initio [int]
. | Calc. fit [int] . | Assignment . |
---|---|---|---|

189(1)* [0.1]^{a} | 191 [0.042] | 190 [0.040] | 7^{1}μΣ |

200(3)* [0.1] | 196 [0.044] | 195 [0.017] | 7^{1}Δ_{5/2} |

208(2) [0.1] | ⋯ | ⋯ | 7^{1}Δ_{5/2} + ν_{Ne}?^{b} |

231(1) [0.1] | ⋯ | ⋯ | ⋯ |

238(1)* [0.224] | 237 [0.132] | 240 [0.130] | 7^{1}κΣ |

384(1)* [0.057] | 382 [0.000] | 381 [0.000] | 7^{2}Π_{1/2} |

439(1)* [0.786] | 446 [0.073] | 440 [0.073] | 6^{1}μΣ |

454(1)* [0.645] | 458 [0.081] | 452 [0.079] | 6^{1}Δ_{5/2} |

490(1)* [0.164] | 497 [0.385] | 491 [0.354] | 6^{1}κΣ |

552(2) [0.040] | ⋯ | ⋯ | ⋯ |

572(1)* [0.048] | 577 [0.005] | 574 [0.006] | 7^{3}Δ_{5/2} |

626(1)* [0.861] | 630 [0.717] | 627 [0.735] | 5^{1}μΣ |

630(1)* [0.367] | 627[0.0088] | 633 [0.013] | 7^{3}Δ_{5/2} |

688(1)* [1] | 698 [1] | 687 [1] | 5^{1}Δ_{5/2} |

704(1) [0.326] | ⋯ | ⋯ | 5^{1}Δ_{5/2} + ν_{Ne}? |

846(1)* [0.322] | 875 [0.547] | 847 [0.563] | 5^{1}κΣ |

926(1) [0.041] | ⋯ | ⋯ | ν_{4} stretch |

957(1) [0.147] | ⋯ | ⋯ | ν_{4} stretch |

981(1) [0.035] | ⋯ | ⋯ | ν_{4} stretch |

1097(1)* [0.013] | 1105 [0.003] | 1097 [0.002] | 6^{2}7^{1}Δ_{5/2} |

1243(1) [0.072] | 1257 [0.004] | 1256 [0.000] | 5^{2}Π_{3/2}? |

1253(1) [0.067] | 1259 [0.001] | 1256 [0.000] | 5^{2}Π_{3/2}? |

1331(1)* [0.039] | 1339 [0.001] | 1331 [0.001] | 6^{3}Σ |

1595(1) [0.072] | 1616 [0.001] | 1594 [0.000] | 5^{2}Π_{3/2}? |

^{a}

Bands marked with * were included in the fit.

^{b}

Tentatively assigned bands are marked with ?.

. | Ab initio^{a}
. | Fit PES^{b}
. | Fit IRPD (this work) . |
---|---|---|---|

ω_{5} (cm^{−1}) | 713.25 | 705(2) | 699(1) |

g_{55} | 129.50 | 132(4) | 115(2) |

g_{56} | −54.13 | −63(4) | −50(3) |

ω_{6} (cm−^{1}) | 462.05 | 460(2) | 455(1) |

g_{66} | −24.77 | −26(4) | −23(2) |

g_{57}^{c} | −34.72 | [−34.72] | [−34.72] |

ω_{7} (cm^{−1}) | 198.18 | 197(3) | 198(2) |

g_{77} | −12.14 | −14(2) | −14(2) |

g_{67} | 26.81 | 29(14) | 15(10) |

rms (cm^{−1}) | 2.8 | 1.8 |

. | Ab initio^{a}
. | Fit PES^{b}
. | Fit IRPD (this work) . |
---|---|---|---|

ω_{5} (cm^{−1}) | 713.25 | 705(2) | 699(1) |

g_{55} | 129.50 | 132(4) | 115(2) |

g_{56} | −54.13 | −63(4) | −50(3) |

ω_{6} (cm−^{1}) | 462.05 | 460(2) | 455(1) |

g_{66} | −24.77 | −26(4) | −23(2) |

g_{57}^{c} | −34.72 | [−34.72] | [−34.72] |

ω_{7} (cm^{−1}) | 198.18 | 197(3) | 198(2) |

g_{77} | −12.14 | −14(2) | −14(2) |

g_{67} | 26.81 | 29(14) | 15(10) |

rms (cm^{−1}) | 2.8 | 1.8 |

We found that the *g*_{57} intermode RT-coupling parameter was ill defined for both fits, likely because of its large covariance with the *g*_{67} and *g*_{56} intermode RT-coupling terms. Therefore, we decided to exclude the *g*_{57} parameter in the fit and kept it fixed at the *ab inito* value, which drastically improved the errors on the estimated parameters. Overall, a reasonable agreement was found between both fits and the *ab initio* values, and the rms values are close to the respective experimental uncertainties (3 cm^{−1} for PES and 1 cm^{−1} for IRPD). We note, however, that the *g*_{67} intermode parameter has a large error for both fits, indicating that it is not well-defined within our parameter space, which is likely caused by the interdependence with the g_{57} term. Furthermore, we notice that the *ω*_{5}, *g*_{55}, and *g*_{56} parameters are significantly lower for the IRPD work compared with the PES values. A possible explanation of this is the effect of the RG attachment, which is discussed in Sec. III C.

The fitted spectroscopic parameters were, in turn, used to predict the vibrational band positions and intensities. Figure 3 shows the predicted spectrum overlaid with the experimental one, and the predicted line positions with their (scaled) intensities are given in supplementary material, Table 3. By iteratively including new assignments in the fit, we could assign several more bands with a reasonable certainty (e.g., bands at 384, 572, 630, 1097, and 1331 cm^{−1}), although the large density of states >800 cm^{−1} leads to only tentative assignments of the bands at 1243, 1246, and 1595 cm^{−1}. Table III summarizes the observed bands together with the *ab initio* values and the predictions based on the final fit including the 14 assigned bands (marked with a star).

All of the newly assigned bands are, however, of zero or very low predicted intensity. We suggest three reasons why this may be happening: First, the employed model excludes coupling between stretching and bending modes, but the mixing of these terms could potentially result in an intensity gain of these low intensity-bending modes. Second, mode *ν*_{5} has a reasonably large RT parameter that may necessitate the inclusion of higher-order terms in the effective Hamiltonian, which would in turn result in a mixing of the *K* states and with it relax the selection rule Δ*K* = ±1. Finally, by attaching the Ne atom, another RT-affected bending mode is generated (see III C), which could couple to the bending modes of the bare HC_{3}N^{+}, affecting their intensity and line positions.

### C. Influence of the rare-gas tag

One of the main disadvantages of the IRPD method is that the RG-ion complex is taken as a proxy for the spectrum of the bare ion. For larger, closed-shell molecular ions, the attachment or RG atoms like Ne or He typically only results in minimal shifts of the vibrational frequencies,^{17,18,37–39} but symmetry-breaking effects were observed, e.g., in the case of cyclic $C3H3+$.^{40,41} The influence of the tag on the bending modes of RT-affected open-shell species has, however, not yet been investigated. When comparing the observed splitting pattern of the Ne−HC_{3}N^{+} with that of the earlier ZEKE work^{12} of the bare HC_{3}N^{+}, we note two key differences: First, due to the different selection rules, we have, on the one hand recorded several features that have not been observed previously, such as the 6^{1}Δ_{5/2} fundamental, but on the other hand, we failed to see several combination bands and overtones, such as the bands at 1414 and 1460 cm^{−1} (see supplementary material, Table 4 for a full comparison between the bands observed with ZEKE^{12} and IRPD). Second, we only see one of the two SO components, since ions are cooled down to their vibrational and SO ground state (Π_{3/2}), and the selection rule Δ*P* = ±1 for the bending modes must be obeyed. Finally, some of the bands that were observed by both methods are shifted compared with each other. To capture this effect in a reliable way, the fitted spectroscopic parameters that were presented before in Table IV were compared. Although the parameters agree fairly well, the largest deviation is seen in the RT constant of mode *ν*_{5} (*ɛ*_{5}), which represents the C–C–H bending. We hypothesize that this discrepancy is a result of the Ne attachment.

To investigate the influence of the Ne attachment, a scan of the Ne−HC_{3}N^{+} potential energy surface was made. The Ne atom was attached to the middle C-atom of the HC_{3}N^{+} and moved around the ion, with all geometry parameters relaxed except for the angle θ, see inlay in Fig. 4. For all optimized Ne−HC_{3}N^{+} geometries of the ground state (A′), the bare ion structure remained linear, so that a symmetry plane for the Ne–ion complex could be defined (here xz). The wavefunction can then be either symmetric, A′, or antisymmetric, A″, with regard to this plane. For the A′ state, the configuration is (*π*_{x})^{1}(*π*_{y})^{2}. and for the A″ state, it is (*π*_{x})^{2}(*π*_{y})^{1}. Figure 4 shows the calculated counter-poise-corrected interaction energy as a function of the Ne angle for both A′ and A″ symmetries. For both symmetries, the global minimum is located at *θ* = 0°, which represents a linear attachment of the Ne on the H atom, although both states show fairly different potential energy surfaces. Generally, the A′ state is lower in energy than the A″ state, which is likely due to the lower electron density in the xz-plane for A′ compared with A″. In order to explain the shape of the curves, we can look at the Mulliken charges of the bare HC_{3}N^{+}. The charges on the H, C_{1}, C_{2}, C_{3}, N are +0.37, +0.02, +0.61, +0.09, and −0.08, respectively. Since binding strength with the Ne is mainly determined by dispersion and induction interactions, the binding will be stronger for a larger positive charge. Furthermore, Pauli exchange repulsion of the *π*_{x} orbital must be taken into account. For the A″ state, this orbital is completely filled so that we only see minima at the H (*θ* = 0°) and C_{2} (*θ* = 100°) positions, where the positive charge is the largest. For the A′, the *π*_{x} orbital is only partially filled, lowering the Pauli exchange repulsion and allowing the Ne to come closer to the ion thus increasing its interaction energy. For this state, a second minimum can then be distinguished at a bent geometry, with *θ* = 60° and a $\u223c40$ cm^{−1} barrier to linearity. Since the Ne atom attaches on the H atom, we expect the largest impact to be on the modes that involve this hydrogen, so the C–H stretch (*ν*_{1}) and the C–C–H bend (*ν*_{5}). Harmonic wavenumber calculations (see Table V) indeed show that these modes are most affected. Here, modes *ν*_{1}–*ν*_{7} correspond to the vibrations of the bare ion, *ν*_{8} to the Ne–H stretch and *ν*_{9} to the Ne–H bending. The fact that the bending modes *ν*_{9} are not fully degenerate indicates that also this mode is Renner–Teller affected and may couple to the bending modes of the bare HC_{3}N^{+}. Although in the IRPD experiment, the C–H stretching wavenumber is about 60 cm^{−1} blue shifted, harmonic wavenumber calculations actually predict a red shift. We hypothesize that this blue shift could be a result of a restriction of the H-stretching amplitude due to the Ne attachment and a subsequent reduction of the anharmonicity of this mode, resulting in a relative blue shift. Regarding the C–H bending mode, it is beyond the scope of this paper to calculate the effect of the Ne attachment on the vibronic splitting patterns: In principle, the attachment leads to a six-atom linear open-shell species, introducing an additional degenerate bending mode that likely interacts with the three bending modes of the bare ion discussed above and is expected to show large-amplitude vibrational characteristics. However, the calculated position of the Ne attachment could explain the relatively large deviation of the RT constant of mode *ν*_{5} with respect to the earlier PES work.

. | ν_{1}
. | ν_{2}
. | ν_{3}
. | ν_{4}
. | ν_{5}^{a}
. | ν_{6}^{a}
. | ν_{7}^{a}
. | ν_{8}
. | ν_{9}^{a}
. |
---|---|---|---|---|---|---|---|---|---|

HC_{3}N^{+} | 3317 | 2222 | 1868 | 908 | [771,644] | [461,445] | [196,187] | ⋯ | ⋯ |

Ne−HC_{3}N^{+} | 3306 | 2221 | 1869 | 910 | [785,662] | [463,447] | [203,196] | 68 | [28,26] |

. | ν_{1}
. | ν_{2}
. | ν_{3}
. | ν_{4}
. | ν_{5}^{a}
. | ν_{6}^{a}
. | ν_{7}^{a}
. | ν_{8}
. | ν_{9}^{a}
. |
---|---|---|---|---|---|---|---|---|---|

HC_{3}N^{+} | 3317 | 2222 | 1868 | 908 | [771,644] | [461,445] | [196,187] | ⋯ | ⋯ |

Ne−HC_{3}N^{+} | 3306 | 2221 | 1869 | 910 | [785,662] | [463,447] | [203,196] | 68 | [28,26] |

^{a}

The *x*- and *y*-components of the bending frequencies are given between brackets.

## IV. CONCLUSIONS

In this work, we have investigated the vibrational structure of the HC_{3}N^{+} ion with IRPD. The combination of a cryogenic ion trap with the wide wavenumber coverage of the FEL-1 and FEL-2 free electron lasers allowed to probe the low-lying RT-disturbed bending modes directly, giving complementary information to earlier PES work.^{12} The obtained spectrum was fitted with an effective Hamiltonian, and the resulting spectroscopic parameters are in reasonable agreement with the *ab initio* and experimental data of Dai *et al.*^{12} The largest deviations were found in the parameters describing the H–C≡C bending mode *ν*_{5}, which has the largest RT coupling of the three bending modes. We hypothesize that this discrepancy is a direct result of the Ne attachment, which was calculated to bind linearly on the H atom. This hypothesis is strengthened by the large blue shift (60 cm^{−1}) we observe for the C–H (*ν*_{1}) stretching mode compared with the other stretches.

This relatively large impact of the Ne on the HC_{3}N^{+} raises the question of whether the IRPD method may be suitable to investigate these RT affected ions, but currently, no alternative tag-free methods are available. To overcome the problems of the RG attachment, one might look into a way to elucidate the rare-gas effect on these complex open-shell species systematically by using different rare-gas tags (e.g., Ar or N_{2}) or attaching multiple tags to the same ion. This would not only help to extrapolate to the bare-ion spectrum but also give insight into weakly bound system interactions. Furthermore, these data might act as a theory benchmark for future research combining the large-amplitude motion of the tag with the vibronic coupling effects of the ion.

## SUPPLEMENTARY MATERIAL

See the supplementary material for the IRPD spectrum in the C–H stretching region, and for tables with observed band centers, intensities, and their respective errors in the Ne-IRPD spectrum of HC_{3}N^{+}, calculated transition frequencies from the effective Hamiltonian approach based on *ab initio* values and a fit to the present experimental work, and a comparison of previous and present experimental band center frequencies.

## ACKNOWLEDGMENTS

This work was part of the research program “ROSAA” with Project No. 740.018.010 and “HFML-FELIX: a Dutch Center of Excellence for Science under Extreme Conditions” (with Project No. 184.035.011) of the research program “Nationale Roadmap Grootschalige Wetenschappelijke Infastructuur,” which are partly financed by the Netherlands Organisation for Scientific Research (NWO). We thank the Cologne Laboratory Astrophysics group for providing the FELion ion trap instrument for the current experiments and the Cologne Center for Terahertz Spectroscopy funded by the Deutsche Forschungsgemeinschaft (DFG, Grant No. SCHL 341/15-1) for supporting its operation.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

K.S. and A.N.M. contributed equally to this work.

**Kim Steenbakkers**: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). **Aravindh N. Marimuthu**: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Software (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). **Britta Redlich**: Funding acquisition (equal); Project administration (equal); Resources (equal); Supervision (equal); Writing – review & editing (equal). **Gerrit C. Groenenboom**: Conceptualization (equal); Formal analysis (equal); Methodology (equal); Resources (equal); Software (equal); Supervision (equal); Writing – review & editing (equal). **Sandra Brünken**: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Supervision (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available within the article and its supplementary material and are openly available in the Radboud data repository at https://doi.org/10.34973/259w-3n81, Ref. 42.

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