$\|\lambda-\lambda_{0}\|_{{\mathcal{P}}}\leq\frac{\delta}{2\Upsilon}$, we have

$\begin{split}\|\Theta_{\lambda}(x)-x_{0}\|&\leq\|\Theta_{\lambda}(x)-\Theta_{\lambda}(x_{0})\|+\|\Theta_{\lambda}(x_{0})-\Theta_{\lambda_{0}}(x_{0})\|\\\
&\leq\frac{1}{2}\|x-x_{0}\|+\Upsilon|\lambda-\lambda_{0}|\leq\delta.\end{split}$

It follows that $K_{\delta}=\\{x\in{\mathcal{B}}\;:\;\|x-x_{0}\|\leq\delta\\}$
is an invariant set for all $\Theta_{\lambda}$, such that
$\|\lambda-\lambda_{0}\|_{{\mathcal{P}}}\leq\frac{\delta}{2\Upsilon}$. In
addition, setting
$\delta_{1}:=\min\left\\{\delta,\frac{\delta}{2\Upsilon}\right\\}$, for all
$\lambda\in\\{\lambda\;:\;\|\lambda-\lambda_{0}\|_{\mathcal{P}}\leq\delta_{1}\\}=:\Delta$,

$\|\Theta_{\lambda}(x)-\Theta_{\lambda}(y)\|\leq\frac{1}{2}\|x-y\|.$

The existence and uniqueness of $x\in K_{\delta}$ such that
$\Theta_{\lambda}(x)=x$ follows then by the standard contraction fixed point
Theorem. We have obtained a function $g:=\Delta\to{\mathcal{B}}$ such that
$F(g(\lambda),\lambda)=0$.

It remains the question of regularity. Let
$\lambda,\lambda^{\prime}\in\Delta$. By (A.1)

$\begin{split}\|g(\lambda)-g(\lambda^{\prime})\|&\leq\|\Theta_{\lambda}(g(\lambda))-\Theta_{\lambda}(g(\lambda^{\prime}))\|+\|\Theta_{\lambda}(g(\lambda^{\prime}))-\Theta_{\lambda^{\prime}}(g(\lambda^{\prime}))\|\\\
&\leq\frac{1}{2}\|g(\lambda)-g(\lambda^{\prime})\|+\Upsilon|\lambda-\lambda^{\prime}|.\end{split}$

This yields the Lipschitz continuity of the function $g$ with Lipschitz
constant $2\Upsilon$. To obtain the differentiability we note that, by the
differentiability of $F$ and the above Lipschitz continuity of $g$, for
$\varepsilon\in{\mathcal{P}}$ small enough, we have

$\begin{split}\|F(g(\lambda+\varepsilon),\lambda+\varepsilon)-F(g(\lambda),\lambda)+\partial_{x}F&[g(\lambda+\varepsilon)-g(\lambda)]+\partial_{\lambda}F\varepsilon\|=o(\|\varepsilon\|_{\mathcal{P}})\\\
&+o(\|g(\lambda+\varepsilon)-g(\lambda)\|)=o(\|\varepsilon\|_{\mathcal{P}}).\end{split}$

Since $F(g(\lambda+h),\lambda+h)=F(g(\lambda),\lambda)=0$, it follows

$\lim_{\varepsilon\to
0}\|\varepsilon\|_{\mathcal{P}}^{-1}\|g(\lambda+\varepsilon)-g(\lambda)+[\partial_{x}F]^{-1}\partial_{\lambda}F\varepsilon\|=0$

which concludes the proof of the Theorem. ∎

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# High Resolution Rovibrational and Rotational Spectroscopy of H2CCCH+

Weslley Guilherme Dias de Paiva Silvaa, Divita Guptaa, Eline Plaara, José Luis
Doménechb, Stephan Schlemmera, Oskar Asvanya CONTACT: Oskar Asvany
<EMAIL_ADDRESS>a I. Physikalisches Institut, Universität zu Köln,
Zülpicher Str. 77, 50937 Köln, Germany;
b Instituto de Estructura de la Materia, IEM-CSIC, Serrano 123, 28006 Madrid,
Spain

###### Abstract

The rovibrational spectrum of the molecular ion H2CCCH+ was investigated in a
4 K cryogenic ion trap instrument employing the leak-out spectroscopy method.
Transitions within the fundamental $\nu_{1}$ (C-H stretch) and the combination
band $\nu_{3}$+$\nu_{5}$ (C-C stretches) were detected, the search aided by
high level quantum chemical calculations. The analysis of the rovibrational
measurements enabled us to predict the rotational structure of the ground
state. Using a rotational-vibrational double-resonance scheme, 14 pure
rotational transitions were measured. This, in turn, led to the radio
astronomical detection of H2CCCH+ in the interstellar medium, as recently
reported (Silva et al., Astron. Astrophys. 676, L1, 2023).

###### keywords:

ion traps; rovibrational spectroscopy; rotational spectroscopy; molecular
cations

††articletype: Special Issue

##  1 Introduction

Astrochemical models (see, e.g., [1]) show that C3H${}_{3}^{+}$ cations play
an important role in the chemical evolution of the interstellar medium (ISM).
They are considered to undergo dissociative recombination with electrons,
leading to the formation of neutral C3H2 and C3H, which are well known
molecules in space [2, 3, 4, 5]. The two most stable isomers of
C3H${}_{3}^{+}$ are the cyclopropenyl (_c_ -C3H${}_{3}^{+}$, _D_ 3h) and the
open chain propargyl (H2CCCH+, _C_ 2v) cation. Although H2CCCH+ is 27.9
kcal/mol less stable than _c_ -C3H${}_{3}^{+}$ [6], spectroscopic signals from
both isomers have been previously observed in the laboratory, and also, both
are predicted to exist in space.

Initial reports on C3H${}_{3}^{+}$ focused on its structural characterization
by infrared (IR) spectroscopy in neon matrices and solutions of salts [7, 8].
Later, the gas-phase IR photodissociation (IRPD) spectra of weakly bound
complexes of C3H${}_{3}^{+}$ with several tags, including Ne, Ar, H2, O2, N2
and CO2 have been reported both experimentally and theoretically [9, 10, 11,
12, 13, 14, 15]. Additional knowledge on the isomers of C3H${}_{3}^{+}$ comes
from photoelectron [16, 17] and electronic [17] spectroscopic studies. While
the vibrational frequencies obtained from IRPD spectra of such complexes
provide useful structural information, astronomical identification often
requires high resolution studies of the bare species. Such information was
first provided by Zhao et al. [18], who investigated the rovibrational
spectrum of the $\nu_{4}$ antisymmetric C-H stretch of _c_ -C3H${}_{3}^{+}$.
Their spectral identification was supported by high level quantum chemical
calculations [6, 19]. But due to the absence of a permanent dipole moment in
_c_ -C3H${}_{3}^{+}$, its detection by radio astronomy is prohibited.

With an eye on a potential radio astronomical detection, we investigated those
isomers and isotopologues of C3H${}_{3}^{+}$ which possess a permanent dipole
moment, and studied the rovibrational and rotational spectra of the singly
deuterated c-C3H2D+ [20] and the open chain H2CCCH+ cation. These
investigations culminated in the detection of H2CCCH+ in the cold molecular
cloud TMC-1, whereas only upper limits could be given for the column density
of c-C3H2D+ in the same source [21]. In this work, we would like to give a
thorough account on the rovibrational spectroscopy of H2CCCH+, for which the
$\nu_{1}$ and $\nu_{3}$+$\nu_{5}$ bands have been measured in high resolution.
A brief description of the double resonance scheme, which enabled the
measurement of pure rotational lines, is also given here.

##  2 Experimental

The rovibrational spectrum of H2CCCH+ was recorded in the 4 K 22-pole ion trap
instrument COLTRAP, which has been described in detail previously [22, 23].
The H2CCCH+ ions were created in a storage ion source (SIS) via electron
impact ionization ($E_{e}\approx 25-30$ eV) of the precursor gas allene
(C3H4), which was admitted to the source at a pressure of about $10^{-5}$
mbar. The created ions were then extracted from the source, mass selected in a
first mass filter for mass 39 u, which corresponds to all possible isomers of
C3H${}_{3}^{+}$ produced in the SIS and other possible species with the same
mass, and injected into the trap. The trap is mounted on a Sumitomo cold head
and maintains an ambient temperature of 4 K. To ensure thermalization of the
ions to the cryogenic temperature, the trap is constantly filled with He
buffer gas ($n$ $\approx$ 1013 cm-3).

Once trapped, the rovibrational transitions of H2CCCH+ were measured using the
novel leak-out spectroscopy (LOS) method [24]. LOS is a universal action
spectroscopic approach, which is based on the escape of ions from the trap due
to collision-induced transfer of (a fraction of) the ion’s internal energy
into kinetic energy. Specifically, after the ions are loaded into the trap,
they are cooled for 40 ms via collisions with He, after which they are
irradiated for 500 ms by an IR beam traversing the trap. In the case of
resonance, the ions are elevated into vibrationally excited states. These
vibrationally excited ions can then collide with noble gas atoms present in
the trap. We used here Ne as the collision partner and it was pulsed
additionally into the trap as a 1:3 Ne:He mixture via a piezoelectrically
actuated valve. When an ion encounters a Ne atom in the trap, a substantial
part of the vibrational energy can be transferred into kinetic energy of the
colliding pair. With a neutral-to-ion mass ratio of 20:39 a maximum kinetic
energy of $[20/(20+39)]\cdot 3200$ cm-1= 1085 cm-1 $\approx 0.13$ eV can be
gained by the ion. By keeping the electrostatic potential barrier at the exit
electrode of the trap below 130 mV, fast H2CCCH+ ions may leak out from the
trap. These escaping ions are then mass selected in a second mass filter, and
counted in an ion detector. By repeating these trapping cycles at a rate of 1
Hz, and counting the escaping ions as a function of the laser wavenumber, the
rovibrational spectrum of H2CCCH+ is recorded.

The IR beam was supplied by a continuous wave optical parametric oscillator
(cw-OPO, Aculight Argos Model 2400, Module C) operating in the 3 $\mu$m
spectral region. The beam entered the vacuum environment through a diamond
window (0.6 mm thickness, Diamond Materials GmbH), irradiated the ions by
crossing the 22-pole trap, and exited the instrument via a CaF2 window, after
which it was absorbed by a power meter. The measured power was on the order of
a few hundred mW. The frequency of the IR radiation was measured by a Bristol
wavemeter (Model 621A), which has an accuracy of about 0.001 cm-1.

The pure rotational transitions of H2CCCH+ are recorded by using a double-
resonance spectroscopy scheme based on LOS [25]. For this, the IR beam is
overlapped with a mm-wave beam outside the apparatus, and both steered into
the 22-pole trap simultaneously. The diamond window mentioned above guarantees
transparency for both frequency regimes. The mm-wave radiation is produced by
a microwave synthesizer (Rohde & Schwarz SMF 100A) that drives an
amplifier/multiplier chain source (Virginia Diodes Inc. or Radiometer Physics
GmbH). The synthesizer is locked to a Rb atomic clock with a typical stability
of $10^{-11}$.

##  3 Results

###  3.1 Identification of carrier of spectroscopic signal

Figure 1:  Rovibrational spectra of the $\nu_{3}+\nu_{5}$ (3184.915 cm-1,
green) and $\nu_{1}$ (3235.334 cm-1, red) bands of H2CCCH+ recorded using the
novel LOS method. The top portion shows the experimental spectra, while the
simulated spectra are given at the bottom. The simulations are obtained at a
rotational temperature of $T_{rot}=10$ K, using the final fitted parameters
given in Table 1. In the simulations, the para ($K_{a}=0$) and ortho
($K_{a}=1$) transitions are distinguished by the gray and black colours,
respectively, with an ortho-para ratio of 3:1. The inset shows the
rovibrational line 212 $\leftarrow$ 313 in the $\nu_{3}+\nu_{5}$ band. With a
FWHM of 42 MHz, the kinetic temperature of the ions is estimated as
$T_{kin}=15$ K.

H2CCCH+ is a closed-shell molecule of point group symmetry $C_{2v}$ with an
electronic ground state of ${}^{1}A_{1}$. It is a near-prolate asymmetric top
with an asymmetry parameter $\kappa=-0.9976$, calculated from the rotational
constants given by Huang et al.[6]. The rotationally resolved spectra of the
$\nu_{1}$ and $\nu_{3}+\nu_{5}$ parallel bands (both $a_{1}$ symmetry) are
depicted in Fig. 1 in red and green traces, respectively. A simulation of the
spectral features using the PGOPHER program [26] is also provided as black and
gray sticks in Fig. 1 for illustration. The identification of these features
as being the $\nu_{1}$ acetylenic C-H stretch and the $\nu_{3}+\nu_{5}$
combination band of both C-C stretches is supported by high level quantum
chemical calculations [6] and earlier observations from IRPD experiments [15].
Also, based on these works, we exclude fundamental or combination/overtone
bands of c-C3H${}_{3}^{+}$ to be responsible for the observed spectral
features. Specifically, measurements of these particular bands from Ne-tagging
experiments [9] locate the $\nu_{3}+\nu_{5}$ band at 3184 cm-1 and the
$\nu_{1}$ band at 3238 cm-1. The latter is the assignment given later by
Botschwina et al. [12]. Ar-tagging experiments [27] observe the
$\nu_{3}+\nu_{5}$ band at 3182 cm-1, also re-assigned by Botschwina et al.
[12], and the $\nu_{1}$ band at 3238 cm-1. These assignments are also in
agreement with the high level _ab initio_ calculations from Huang et al. [6],
namely $\nu_{3}+\nu_{5}$ and $\nu_{1}$ at 3193 and 3228 cm-1, respectively. In
a more recent Ne-tagging experiment, Marimuthu et al. [15] calculate the
$\nu_{1}$ band to be at 3230 cm-1, but neither $\nu_{1}$ nor $\nu_{3}+\nu_{5}$
were observed in their Ne-C3H${}_{3}^{+}$ IRPD spectrum (see their Fig. 4).

###  3.2 Spectral Analysis

With the lines of both bands in Fig. 1 being _a_ -type transitions (i.e.
$\Delta K_{a}=0,\Delta K_{c}=\pm 1$), we anticipated observing transitions
with $K_{a}=1\leftarrow 1$ and $K_{a}=0\leftarrow 0$, whereas transitions with
$K_{a}\geq 2$ were not expected due to the large $A$ rotational constant of
H2CCCH+ and the low temperature conditions in the trap. The most dominant
features in the spectra were the series of $K_{a}=1\leftarrow 1$ doublets,
which were readily identified in both bands reported here. Preliminary fits of
the rotational constants $B$ and $C$ from these observations agreed well with
those predicted by high level _ab initio_ calculations [6], confirming further
the unequivocal detection of H2CCCH+. Although peaks with $K_{a}=0\leftarrow
0$ were expected to be relatively intense, it was very challenging to identify
these transitions in the $\nu_{1}$ band because these lines (i) turned out to
be somewhat weaker (please see Fig. 1 and discussion in Section 4), (ii)
exhibited perturbations, (iii) were embedded in a plethora of unidentified
spectral features (Fig. 1), and, in hindsight, (iv) the calculated $A_{1}$
rotational constant (i.e., the rotation-vibration interaction term
$\alpha_{1}^{A}$) turned out to be misleading in sign and magnitude, see Table
1.

Fortunately, the combination band $\nu_{3}+\nu_{5}$ contained only few
unidentified lines, and exhibited only small perturbations, allowing the
$K_{a}=0\leftarrow 0$ transitions to be readily identified. The assignment of
the $K_{a}=0\leftarrow 0$ subband within $\nu_{3}+\nu_{5}$ was helped by a
small perturbation in the upper $J_{KaKc}=6_{06}$ quantum level, leading to a
splitting of 0.064 cm-1 which could be spotted in the corresponding
transitions in the P- and R-branches and are indicated by small arrows in Fig.
1. The split lines are also given in the line lists in the supplementary
material. These assignments allowed first measurements of pure rotational
lines with $K_{a}=0\leftarrow 0$ using double resonance spectroscopy, as
outlined in the following section, and thus a complete determination of the
ground vibrational state of H2CCCH+ was obtained.

Due to the difficulties mentioned above, the search for the $K_{a}=0\leftarrow
0$ subband of $\nu_{1}$ was more challenging. To identify these transitions,
we used our knowledge gained of the ground state rotational structure, and
compared it to combination differences (CDs, see, e.g., references [28, 29,
30]) formed blindly in the $\nu_{1}$ state. First, we removed all assigned
$K_{a}=1\leftarrow 1$ transitions from the $\nu_{1}$ spectrum and formed all
possible CDs with the remaining lines. With about $N=130$ unassigned lines, we
obtained $N(N-1)/2=8385$ CDs. This list of CDs must contain those of the
ground state, the so-called GSCDs, with $K_{a}=0$. From our mm-wave
measurements, these GSCDs are constrained to kHz accuracy. Based on this
knowledge, one can select pairs of lines from all possible CDs, one in the
P-branch and one in the R-branch, namely $P(J+2)$ and $R(J)$, which are
connected by a common rotational quantum level $J+1$ in the upper $\nu_{1}$
state and form the GSCD $\approx B_{0}(4J+6)$. These pairs of lines with the
proper GSCD are then potential members of the $K_{a}=0\leftarrow 0$ subband.
This approach is illustrated by a table of GSCDs in the supplementary
material. Our final assignments of the $K_{a}=0\leftarrow 0$ lines in
$\nu_{1}$ were then confirmed also with double resonance spectroscopy
measurements, as shown in the following subsection.

A detailed list of all assigned lines within $\nu_{1}$ and $\nu_{3}+\nu_{5}$
is given in the supplementary material. Overall, the observed rovibrational
lines in both bands exhibit narrow Doppler widths, given the low temperature
of the ion trap. The line 212 $\leftarrow$ 313 in the $\nu_{3}+\nu_{5}$ band
depicted in the inset of Fig. 1, for example, has a full width at half maximum
(FWHM) of about 42 MHz which corresponds to a kinetic temperature of 15 K. The
obtained temperature is slightly higher than the nominal trap temperature due
to known heating effects [31].

###  3.3 Double-resonance rotational measurements

Rotational-vibrational double resonance methods are a superb tool to measure
rotational fingerprints of molecular ions investigated in cold ion traps [32].
The particular double resonance approach which involves the LOS action
spectroscopic scheme has only recently been developed [25]. The application of
this rotational method to H2CCCH+, and in particular the subsequent detection
of this ion in the cold molecular cloud TMC-1 have already been documented
[21]. In that publication, the 14 laboratory mm-wave transitions as well as
ground state rotational and centrifugal parameters are listed for H2CCCH+.
Additionally, a summary of all measured rotational transitions as well as a
comparison to a simulation is given in Fig. S1 of the supplementary material.

In this paper, we would like to highlight the ability of the double resonance
method to establish the connectivity of transitions and thus to confirm
assignments. As outlined above, these assignment checks were crucial for the
transitions with $K_{a}=0\leftarrow 0$, in particular within the heavily
perturbed $\nu_{1}$ band. As an example, Fig. 2 shows the rotational line
$8_{08}\leftarrow 7_{07}$ at 152118.074 MHz, detected with the IR laser fixed
at 3230.208 cm-1, the $7_{07}\leftarrow 8_{08}$ transition in the $\nu_{1}$
band.

Figure 2:  Pure rotational transition $J^{\prime}_{Ka^{\prime}Kc^{\prime}}$
$\leftarrow$ $J"_{Ka"Kc"}$ = $8_{08}$ $\leftarrow$ $7_{07}$ (blue trace and
blue arrow) of H2CCCH+, recorded using the double resonance spectroscopic
scheme. For this measurement, the IR laser frequency (red arrow) was kept
fixed on resonance with the $\nu_{1}$ $7_{07}$ $\leftarrow$ $8_{08}$
rovibrational transition at 3230.208 cm-1, while the mm-wave frequency was
stepped in an up-and-down manner in 10 kHz steps.

###  3.4 Spectroscopic fits

The spectroscopic constants for the $\nu_{1}$ and $\nu_{3}+\nu_{5}$ bands were
obtained by fitting the assigned lines using a Watson’s S-reduced Hamiltonian
in the _I r_ representation as implemented in Western’s PGOPHER program [26].
In these fits, the values of the ground state rotational and centrifugal
distortion constants were kept fixed to those determined from pure rotational
transitions due to their higher accuracy [21]. As expected, the ground state
parameters of H2CCCH+ are very similar to those of its deprotonated
counterpart, H2CCC [33], or those of ketene, H2CCO [34, 28]. Such molecules
are distinguished by their large $A$ rotational and $D_{K}$ centrifugal
parameters. As the centrifugal distortion constants in the vibrationally
excited states could not be well determined, they were also fixed to their
ground state values, see Table 1. For the $\nu_{1}$ band, 54 assigned lines
were used in the fit, 49 of which are from the $K_{a}=1\leftarrow 1$ subband.
Although many transitions with $K_{a}=0\leftarrow 0$ could be identified in
the $\nu_{1}$ state via combination differences (see the complete line list in
the supplementary material), only five were used in the final fit due to
strong perturbations (shifts and splittings). For the $\nu_{3}+\nu_{5}$ band,
65 lines were assigned (see supplementary material), of which 62 lines within
the $K_{a}=0\leftarrow 0$ and $K_{a}=1\leftarrow 1$ bands were used in the
final fit. It is worth noting that to obtain a satisfactory fit of the 62
lines in the $\nu_{3}+\nu_{5}$ band, the $D_{JK}$ constant had to be released
and its derived value (18.2(1) MHz) deviates significantly to that of the
ground state (0.4388 MHz). Thus, the value of $D_{JK}$ presented here should
be taken with caution as this is likely an artefact of the fit most probably
related to perturbations in the $K_{a}=0\leftarrow 0$ subband of
$\nu_{3}+\nu_{5}$, as exemplified above for the transitions with upper
$J_{KaKc}=6_{06}$ quantum numbers. In an alternative fitting procedure, we
kept the $D_{JK}$ constant fixed to the ground state value and only included
transitions within the $K_{a}=0\leftarrow 0$ subband with J $<$ 5\. In this
fit, a reasonable rms error of the obs-calc values of 0.0019 cm-1 was obtained
with the $K_{a}=0\leftarrow 0$ lines exhibiting the largest deviations on the
order of 0.006 cm-1, in agreement with the observed perturbations.
Nevertheless, the fitted parameters of the first fit are reproduced in Table 1
as this allowed the rotational constants and band origin to be derived with
smaller uncertainties. In the future, the observation of transitions with
$K_{a}\geq 2$ (e.g. experiments at higher temperatures) will certainly improve
the quality of the fit and permit $D_{JK}$ to be better determined within
$\nu_{3}+\nu_{5}$.

Overall, the fits for both states presented in Table 1 are quite reasonable
with rms errors of the obs-calc values of 0.0004 cm-1 and 0.0003 cm-1 for
$\nu_{1}$ and $\nu_{3}+\nu_{5}$, respectively. Please note that the fit
results may be somewhat optimistic, as many parameters were fixed. For
instance, the uncertainties in the values of $A_{1}$ and $A_{3+5}$ reflect
only that of the fit and do not take into account the large uncertainty of
$A_{0}$ itself, which is 247 MHz [21]. Future investigations of a b-type
perpendicular band of H2CCCH+, e.g. the antisymmetric CH2 stretch at around
3093 cm-1 [27], may help to constrain $A_{0}$ and $D_{K}$ better. A similar
strategy has been followed for ketene, H2CCO [34], 50 years ago.

Table 1:  Spectroscopic parameters of H2CCCH+ (Watson’s S-reduced Hamiltonian)
obtained by fitting the rovibrational transitions given in the supplementary
material, using Western’s PGOPHER program [26]. The uncertainty in the last
digits of each parameter is reported in parentheses. The units are MHz, unless
otherwise noted.

Parameter | ground | $\nu_{1}$ | $\nu_{3}+\nu_{5}$
---|---|---|---
| exp a | ab initio b | this work | ab initio b | this work | ab initio b
$\nu$ / cm-1 |  |  | 0 |  | 3235 | .3338(2) | 3239 | .0 | 3184 | .9154(1) | 3201 | .8
$A$ | 281856 |  | 281911 | .9 | 274403 | (6) | 281933 | .2 | 267652 | (4) | 281534 | .8
$B$ | 9675. | 841 | 9580 | .2 | 9637 | .9(1) | 9554 | .3 | 9625 | .3(1) | 9501 | .2
$C$ | 9342. | 877 | 9251 | .9 | 9325 | .6(1) | 9227 | .4 | 9315 | .1(1) | 9175 | .9
$D_{J}\times 10{{}^{3}}$ | 2. | 942 | 3 |  | 2 | .942 c | - |  | 2 | .942 c | - |
$D_{JK}$ | 0. | 4388 | 0 | .479 | 0 | .4388 c | - |  | 18 | .2(1) | - |
$D_{K}$ | 20. | 678 | 20 | .862 | 20 | .678 c | - |  | 20 | .678 c | - |
$d_{1}\times 10{{}^{3}}$ | -0. | 121 | 0 |  | -0 | .121 c | - |  | -0 | .121 c | - |
$d_{2}\times 10{{}^{3}}$ | -0. | 051 | 0 |  | -0 | .051 c | - |  | -0 | .051 c | - |

a: ground state parameters fixed to those of Silva et al. [21]. $D_{K}$ and
$d_{2}$ are calculated.
b: values from Huang et al. [6]
c: Fixed to ground state value

##  4 Discussion and Outlook

The successful measurement of the rovibrational spectrum of the fundamental
$\nu_{1}$ band (C-H stretch) and the combination band $\nu_{3}+\nu_{5}$ (C-C
stretches) for the H2CCCH+ cation presented here illustrates the broad
applicability and remarkable sensitivity of the recently developed LOS method.
The rotational-vibrational double resonance method using LOS, recently
demonstrated in our group for the HC3O+ ion [25], enabled not only pure
rotational transitions for H2CCCH+ to be obtained but also the perturbed
$\nu_{1}$ band to be deciphered. A search in the cold interstellar cloud TMC-1
finally led to the detection of this ion in space [21]. A similar approach was
followed for another member of the C3H${}_{3}^{+}$ family, the singly
deuterated c-C3H2D+ cation [20], but unfortunately its low dipole moment and
low abundance inhibited detection in TMC-1, and only an upper limit for its
column density could be given [21]. In the future, we expect that LOS will
enable the characterization of more cations in the laboratory, potentially
leading to their detection in space by radio astronomy.

While these results are relevant to astronomers, interesting insights into the
molecular structure of H2CCCH+ upon vibrational excitation are obtained. In
particular, the value of the rotational constant $A$ (Table 1) decreases upon
excitation into the $\nu_{1}$ and $\nu_{3}+\nu_{5}$ states, by 2.7 % and 5 %,
respectively. This suggests significant structural changes when the terminal
C-H or C-C bonds are excited. The experimentally obtained vibration-rotation
interaction constants are about $\alpha^{A}_{1}=7486$ MHz and
$\alpha^{A}_{3+5}=14483$ MHz, respectively. As the backbone of H2CCCH+ lies
along the $a$-axis with only the methylene hydrogens pointing away from this
axis, the value of $A$ is intimately related to the value of the average
$\angle$HCH angle. Considering that $A$ is inversely proportional to the
moment of inertia component $I_{a}$, it becomes evident that the smaller
values of $A_{1}$ and $A_{3+5}$ are likely due to the C-H and C-C stretches
that move the methylene hydrogens further apart by increasing the $\angle$HCH
angle when vibrating. This effect is more pronounced in the $\nu_{3+5}$ band
as the C-C bonds are spatially closer to the CH2 group. Although the
predictions from Huang et al. [6] support the experimental assignments and
show overall good agreement with the determined values, the theoretical
calculations do not capture well the trend observed for the changes in the
values of $A$ upon vibrational excitation, as seen in Table 1. The calculated
predictions are $\alpha^{A}_{1}=-21.277$ MHz and $\alpha^{A}_{3+5}=377.099$
MHz. These discrepancies may be associated with perturbations in H2CCCH+,
which are not well described by standard computational methods. Thus, our
high-resolution study of H2CCCH+ also provides data which can contribute to
the improvement and benchmark of theoretical models.

Although H2CCCH+ has been well characterized in this work, many open questions
remain concerning its rovibrational spectroscopy. The above mentioned
perturbations affect both rovibrational bands, but particularly $\nu_{1}$
exhibits severe line shifts and splittings. Such severe perturbations have
also been observed for ketene [34, 35, 28], and traced back to the presence of
low-lying vibrational states, whose overtones and combination bands give rise
to Fermi and Coriolis interactions [35, 28]. The unusual effective value for
$D_{JK}$ determined for $\nu_{3}+\nu_{5}$ is certainly a consequence of this,
as well as the intensity pattern observed both for $\nu_{1}$ and
$\nu_{3}+\nu_{5}$. As H2CCCH+ contains a plane of symmetry with two equivalent
hydrogen atoms, H2CCCH+ has two nuclear spin states, ortho (levels with odd
values of $K_{a}$) and para (levels with even values of $K_{a}$) with an
expected ortho-para ratio of 3:1. This situation is very similar to, e.g.,
neutral H2CCC [5, 33] or the aforementioned ketene, H2CCO [34, 28]. From the
radio astronomical observations of H2CCCH+, in which the para 202-101 and
ortho 212-111 and 211-110 transitions were seen [21], a consistent ortho-para
ratio of 3.3($\pm$0.6):1 was derived. In the rovibrational spectra in Fig. 1,
however, the $K_{a}=0\leftarrow 0$ para-transitions exhibit somewhat lower
than expected intensities. To reproduce the observed IR spectra, an
’artificial’ ortho-para ratio of roughly 4:1 would be needed. We can exclude
that the unusual ratio is related to the formation processes of H2CCCH+ by
electron impact ionization of allene (C3H4) in our ion source, because
essentially the same spectrum was obtained using a HCCD/CH4 precursor mixture,
see Fig. S3 in the supplementary material. Also, we do not suspect that our
novel LOS method to be responsible for this peculiar observation, as
experiments with other ions containing two identical protons exhibited the
expected nuclear spin statistics, e.g. C2H${}_{2}^{+}$ [36] and c-C3H2D+ [20].
The intensities observed in our spectrum, however, depend on multiple factors
like the transition dipole moment, population distribution, trap time, and
more. So, to determine an accurate ortho-para population ratio of H2CCCH+ in
the ion trap all ortho (or para) species shall be leaked out from the trap and
the remaining para (or ortho) fraction shall be measured. This new method of
determining the nuclear-spin composition in a trap111Patent pending: DE 10
2021 127 556.3 (Universität zu Köln), 22.10.2021 is currently tested in our
laboratory for a number of simple cases. Results from these measurements will
be published separately. In addition, we plan to perform rovibrational
spectroscopy of the b-type antisymmetric CH2 stretch or one of the fundamental
a-type C-C stretches [37] of H2CCCH+. Measurements at higher trap temperature
to populate the $K_{a}=2$ states or the rovibrational measurement of the
perdeuterated version, D2CCCD+, are also in reach.

##  5 Acknowledgements

The authors want to dedicate this contribution to Prof. Attila Császár on the
occasion of his 60th birthday. The authors also want to thank Xinchuan Huang
for reading the manuscript prior to submission. This work has been supported
by an ERC advanced grant (MissIons: 101020583) and via Collaborative Research
Centre 1601 (project ID: 500700252, sub-project C4) funded by the Deutsche
Forschungsgemeinschaft (DFG) and DFG SCHL 341/15-1 (“Cologne Center for
Terahertz Spectroscopy”). W.G.D.P.S. thanks the Alexander von Humboldt
Foundation for support through a postdoctoral fellowship. J.L.D. acknowledges
the support from the MCINN project
PID2020-113084GB-I00/AEI/10.13039/501100011033 and the CSIC project ILINK+
LINKA20353.

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