# Overtone Transition $2\nu_{1}$ of HCO+ and HOC+: Origin, Radiative Lifetime,
Collisional Quenching

Miguel Jiménez-Redondo [a]+ Liliia Uvarova  [b]+ Petr Dohnal  [b] Miroslava
Kassayová  [b] Paola Caselli  [a] Pavol Jusko∗[a]

###### Abstract

## Introduction

The formyl cation, HCO+, is an abundant molecule in the interstellar medium,
being first detected already more than 50 years ago, interestingly under the
name “X-ogen”1, prior to its laboratory identification2. Although the
existence of its higher energy isomer isoformyl cation, HOC+, was clear, its
microwave spectrum was only measured in a dc glow discharge a decade later, 3
and its interstellar detection quickly followed. The study by Woods _et al._ 4
focused on searching for HOC+ in 14 interstellar sources, successfully
detected HOC+ towards Sgr B2, and was even able to estimate the ratio of
[HCO+]/[HOC+] $\approx$ 375\. HCO+/ HOC+, together with HCN/ HNC, are the two
most simple linear closed shell isomeric systems, playing a crucial role in
astrophysics, as well as in fundamental molecular physics, i. e., properties
of molecular ions.

Figure 1: Energy level diagram for HCO+ and HOC+ ions. The vibrational energy
levels are calculated from HCO+ 5 and HOC+ 6 spectroscopic constants, the zero
point energies are taken from ref. 7. The red arrows represent the induced
photon excitation, the magenta arrows radiative and collisional deexcitation.
The gray, green arrows represent the LIR reaction pathways to HCO2+ and ArH+,
respectively. The HCO+ does not proton transfer to ^15N2, unless sufficiently
excited. See text for details.

Of the two isomers, HCO+ is energetically more favourable compared to HOC+,
with the ground states separated by 1.74 eV (from proton affinities on O and
C, respectively8), and the isomerisation barrier from $\ce{HOC+}\to\ce{HCO+}$
is 1.52 eV 9. The energy diagram of levels within both isomers can be seen in
Fig. 1.

The Woods _et al._ 2 study was the first to perform ground rotational
spectroscopy on HCO+ in a discharge. IR spectroscopy, first determining
$\nu_{1}$ 10, then also reporting rotationally resolved P, R branches therein
11, the bending mode $\nu_{2}$ 12, 13 and C–O stretch $\nu_{3}$ 14, quickly
followed. The radiative lifetime of HCO+ in the $\nu_{1}$ state 15 and in the
bending mode $\nu_{2}$ up to 8, has been studied using an ICR trap16. Newer
experimental studies, after 2000, focused on higher excited vibrational
states, e. g., the determination of $\nu_{2}$, $x_{22}$, $g_{22}$, and
$B$(030) 17, and the acquisition of the pure rotational spectrum of HCO+ up to
1.2 THz 18. More recently, the $\nu_{1}$ band has been re-inspected in high
resolution with the help of a frequency comb19, and finally, Neese _et al._ 5
reported hot band spectra of 8 different vibrational modes of HCO+. Works
focused on the higher energy isomer, HOC+, are much more rare. The first
rotational spectrum 3 has only been followed by the infrared detection of the
$\nu_{1}$ band of HOC+ 20, the hot band transition $\nu_{1}$ in $\nu_{2}=1$
21, and another hot band study starting in the $\nu_{2}$ state by Amano and
Maeda 22. Both HCO+ and HOC+ were subject to numerous theoretical
spectroscopic studies 23, 24, 6, 25, 26, 27, 7 with an emphasis on the
prediction of the spectral properties of the lowest vibrational bands. While
the calculated vibrational band origins are consistent across the studies, at
least for the low lying states, the predicted lifetimes of the $\nu_{1}$ state
of HCO+ differ by up to an order of magnitude.

The abundance of HOC+, i. e., the [HCO+]/[HOC+] ratio, has been observed in
dense molecular clouds 28, in diffuse clouds 29, towards photodissociation
regions (PDRs)30, 31, and towards the Galactic Center32. It has also been used
as a diagnostic tool, e. g., starburst energy feedback seen through HCO+ and
HOC+ emission in NGC 253 33. Although the HCO+/ HOC+ column density ratio has
been found anywhere between 1000 and 9000 toward dark clouds, it has been
observed to be significantly smaller (50-400) toward PDRs and diffuse clouds,
suggesting an efficient HOC+ formation in these warmer environments.

In this work, we present a study of the first overtone vibrational transition
of the C–H and O–H stretching mode of HCO+ and HOC+. The measurements have
been performed in a cryogenically cooled 22 pole ion trap using a laser
induced reaction (LIR) action spectroscopy scheme. The setup allows us to
determine the spectroscopic constants of the excited vibrational state for
each ion as well as the corresponding radiative lifetimes and quenching rate
coefficients with selected neutral partners.

## Results and Discussion

### Overtone Spectroscopy of HCO+/HOC+

The vibrational energy levels of both isomers are denoted by their vibrational
quantum numbers, $\nu_{1}\nu_{2}^{l}\nu_{3}$, corresponding to the number of
vibrational quanta in a given vibrational mode (following the notation by
Neese _et al._ 5). As we probe the C–H/O–H stretching mode only, we further
abbreviate this notation such that $2\,\nu_{1}$ has the same meaning as
$20^{0}0$.

The laser induced reaction (LIR) technique 34 relies on the different
reactivities of the upper and the lower state involved in the transition. The
HCO+ spectra were measured by using the endothermic reaction with CO2

$\ce{HCO+}\ +\ce{CO2}\xrightarrow[]{}\ce{HCO2+}+\ce{CO},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \Delta E=0.55\leavevmode\nobreak\
\mathrm{eV},$ (1)

and, similarly, using Ar for HOC+

$\ce{HOC+}\ +\ce{Ar}\xrightarrow[]{}\ce{ArH+}+\ce{CO},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \Delta E=0.59\leavevmode\nobreak\
\mathrm{eV},$ (2)

where the endothermicities were calculated based on the proton affinities of
the reactants taken from NIST 35. These processes are best recognized by
following the reaction paths in the energy level diagram Fig. 1. At room
temperature and below, reactions (1) and (2) basically do not proceed due to
the ions being almost exclusively in their ground vibrational state. The
excitation of HCO+ or HOC+ to the $2\,\nu_{1}$ state and the “activation” of
the reactions result then in an improved production of $\ce{HCO2+}$ or
$\ce{ArH+}$ ions, respectively.

Examples of obtained absorption lines are shown in Fig. 2. The measured
overtone transitions from the ground vibrational state to $2\,\nu_{1}$ state
for both HCO+ and HOC+ ions are summarized in Tab. 1. The upper and lower
levels can be described by a Hamiltonian 5

$H=T_{\nu}+B_{\nu}(J+1)-D_{\nu}\left[J(J+1)\right]^{2}+\\\
H_{\nu}\left[J(J+1)\right]^{3},$ (3)

where $\nu$ stands for the vibrational quantum numbers of a given level,
$T_{\nu}$ is the vibrational term energy, $J$ is the rotational quantum
number, $B_{\nu}$ denotes the rotational constant, and $D_{\nu}$ and $H_{\nu}$
are the centrifugal distortion constants. The vibrational band origin for the
$20^{0}0\leftarrow 00^{0}0$ transition is
$\omega_{2\nu_{1}}=T_{2\nu_{1}}-T_{0\nu_{1}}$.

The measured transition wavenumbers were fitted using Hamiltonian (3), where
the spectroscopic constants for the lower state were fixed at values
experimentally determined by Neese _et al._ 5 for HCO+ and by Amano and Maeda
22 for HOC+. The fitted spectroscopic constants are summarized in Tab. 2 and
compared to the previous theoretical and experimental studies in Tab. 3. Note
that two values, marked [N], acquired using cavity ring-down spectroscopy
(CRDS), were excluded from the fit due to presence of other (unidentified)
absorption features in immediate vicinity of the HCO+ transitions (see the
dataset for more details).

The derived HCO+ spectroscopic constants are in a very good agreement with the
hot band spectroscopy study by Neese _et al._ 5 and also with recent quantum
mechanical calculations 7. Although, in the HOC+ case, the older theoretical
studies 6, 26, 27 predicted the band origin $\omega_{2\nu_{1}}$ up to 30
$\text{cm}^{-1}$ away from the measured value, the newer spectroscopic
constants estimated by Koput7, 36 are in good agreement with the experiment
(the predicted band origin $\omega_{2\nu_{1}}$ is within two $B_{2\nu_{1}}$
from the experimental one).

Figure 2: P(3) and R(3) lines of HCO+ and HOC+, respectively, measured by LIR
technique in the 22 pole ion trap.  Table 1: Measured transition wavenumbers
for the $20^{0}0\leftarrow 00^{0}0$ vibrational bands of HCO+ and HOC+.
Statistical error reported in parenthesis. The absolute accuracy of the
wavemeter is better than $0.002\;\text{cm}^{-1}$.

HCO+ |  HOC+
---|---
P(1) | 6075.7093(1) | R(1) | 6366.0286(5)
P(2) | 6072.6892(1) | R(2) | 6368.8965(5)
P(3) | 6069.6258(1) | R(3) | 6371.7242(2)
P(4) | 6066.5171(1) | R(4) | 6374.5134(3)
P(5)[c,N] | 6063.3620(9) | R(8) | 6385.2699(2)
P(6)[c,N] | 6060.1689(2) | R(9) | 6387.8592(2)
P(7)[c] | 6056.9250(2) | R(10) | 6390.4088(3)
P(8)[c] | 6053.6385(3) | R(11) | 6392.9174(3)

All frequencies in $\text{cm}^{-1}$. Fitted temperature (i. e. line width)
varies between $130-210\;\text{K}$. [c] value obtained from the CRDS setup.
[N] not included in the fit (see text).

Table 2: Spectroscopic constants, radiative lifetimes $\tau_{\mathrm{rad}}$,
and quenching rate $k_{q\ce{}}$ for the $20^{0}0$ vibrational state of HCO+
and HOC+ ions derived in this work. The vibrational transition moment for the
$20^{0}0\leftarrow 00^{0}0$ transition and the reaction rate coefficients for
the vibrational quenching of the $20^{0}0$ state in collision with different
gases were determined only for HCO+ ions. Statistical error is reported in
parenthesis.

|  HCO+ |  HOC+
---|---|---
$\omega_{2\nu_{1}}$ | 6078.68411(19) | 6360.17630(26)
$B_{2\nu_{1}}$ | 1.465215(25) | 1.472957(12)
$D_{2\nu_{1}}$ |  3.56(44)$\cdot 10^{-6}$ |  4.03(11)$\cdot 10^{-6}$
$H_{2\nu_{1}}$ | $-$ |  3.3(2.2)$\cdot 10^{-10}$
$\langle 20^{0}0|\mu|00^{0}0\rangle$ |  0.0084(25)[a] |  $\mu_{\ce{HCO+}}/\,0.41(11)$[b]
$\tau_{\mathrm{rad}}$ (ms) | 1.53(34) | 1.22(34)
$k_{q\ce{He}}$[c] | 5.6(1.0) | $-$
$k_{q\ce{H2}}$[c] | 580(110) | $-$
$k_{q\ce{N2}}$[c] | 640(40) | $-$

Spectroscopic constants in $\text{cm}^{-1}$. [a] lower estimate from the CRDS
measurement in Debye. [b] derived from relative ion signal intensities in the
22 pole trap, see text, $R_{\mu}$, Eq. (16). [c] in
$10^{-12}\;\text{cm}^{3}\,\text{s}^{-1}$.

Table 3: Vibrational band origins and rotational constants of HCO+ and HOC+
obtained in previous theoretical and experimental studies.

$\omega_{2\nu_{1}}$ | $B_{2\nu_{1}}$ | Method/ Ref.
---|---|---
HCO+
6078.68411(19) | 1.465215(25) | E [present]
6078.6839[a] | 1.4651993(13) |  D 5
6079.8 | 1.46512 |  T 7
$\ce{HOC+}$
6360.17630(26) | 1.472957(12) | E [present]
6362.3 | 1.4736 |  T 36
6391.5[a] |  1.4464[a] |  T 6
6377.9 |  1.4707[a] |  T 25, 26
6390.5 |  |  T 27

Spectroscopic constants in units of $\text{cm}^{-1}$. Method: T – theory, E –
experiment, D – derived from experimental data (fundamental + hot-band
spectra). [a] Values calculated from published spectroscopic constants.

### Radiative Lifetime of the $2\,\nu_{1}$ State

Measuring the overtone transitions while varying the number density of the LIR
reactant allowed us to determine the radiative lifetime of the $2\,\nu_{1}$
state of HCO+ and HOC+. In the steady state, with low laser power, the number
of HCO2+ ions produced in the laser induced reaction (1) is proportional to
the number of trapped HCO+ ions (predominantly in their ground vibrational
state)

$N_{\ce{HCO2+}}=\frac{k_{1}[\ce{CO2}]r_{1}N_{\ce{HCO+}}}{k_{1}[\ce{CO2}]+1/\tau_{\mathrm{rad}}+k_{q\ce{M}}[\ce{M}]}t,$
(4)

where $t$ is the irradiation time, $r_{1}$ corresponds to the rate of photon
excitation to the $2\,\nu_{1}$ state, $k_{1}$ to the reaction rate coefficient
of vibrationally excited HCO+ with CO2, $\tau_{\mathrm{rad}}$ is the time
constant of the radiative deexciation of the $2\,\nu_{1}$ state to lower lying
vibrational levels, and $k_{q\ce{M}}$ is the collisional reaction rate
coefficient for the quenching of the $2\,\nu_{1}$ state in collision with
particle $M$. It is inherently assumed that $k_{1}$ is close to the Langevin
collisional rate coefficient and every collision of the $2\,\nu_{1}$ state of
HCO+ with CO2 leads to HCO2+, so the vibrational quenching induced by
collisions with CO2 can be neglected.

The radiative lifetime, $\tau_{\mathrm{rad}}$, is then obtained by fitting
equation (4) to the dependence of the measured ratio of secondary (HCO2+ or
ArH+) to primary (HCO+ or HOC+) ions as a function of the number density of
the probing gas (CO2 or Ar) as shown in Fig. 3 for HCO+ ions. The resulting
radiative lifetime was $\tau_{\mathrm{rad}}=1.53\pm 0.34$ ms for
HCO+($2\,\nu_{1}$) and $\tau_{\mathrm{rad}}=1.22\pm 0.34$ ms for
HOC+($2\,\nu_{1}$) ions. The measurements were performed using the P(3) and
R(3) lines for HCO+ and HOC+, respectively, and the corresponding $k_{1}$ were
calculated using polarisabilities from refs 37, 38. Given that typical
vibrational selection rules strongly favor transitions with vibrational
quantum number changing by one, the measured radiative lifetimes probably
pertain to the transition $20^{0}0\rightarrow 10^{0}0$ (note that the
$10^{0}0$ level is endothermic in reaction with CO2 (for HCO+) as well as Ar
(for HOC+), i. e., it is not probed in the selected LIR schemes).

Figure 3: Determination of the $\tau_{rad}$ for HCO+ from the increase of the
LIR signal as a function of CO2 number density in the trap using Eq. (4).

To our best knowledge, there are no experimental or theoretical data on the
radiative lifetimes of the $2\,\nu_{1}$ states of either HCO+ or HOC+. An
approximate comparison with present data can be done applying the $1/\nu$
scaling law 16, i. e., that the lifetime of the $2\,\nu_{1}$ state should be
about half of the $1\,\nu_{1}$ state. The predicted lifetimes for the
$1\,\nu_{1}$ state of HCO+ range from 0.4 ms 27 to 4.46 ms 23 with the
majority of the studies closer to the upper value (e. g. 3.9 ms24 or 3.7 ms
6). Keim _et al._ 15 in their DLASFIB (direct laser absorption in a fast ion
beam) experiment 15 measured an absolute vibrational band intensity that
corresponds to a lifetime of $5.9\pm 0.9$ ms for the $1\,\nu_{1}$ state of
HCO+. Considering the aforementioned $1/\nu$ scaling law, the value of
$\tau_{\mathrm{rad}}$ for the $2\,\nu_{1}$ state of HCO+ measured in the
present experiment is thus within experimental error, comparable to that
predicted for the $1\,\nu_{1}$ state by theoretical calculations 23, 24, 6,
and lower than the value that could be extrapolated from the experiment by
Keim _et al._ 15.

For HOC+ ions, only theoretical data on vibrational lifetimes are available.
Kraemer and Špirko 27 predicted the lifetime of the $1\,\nu_{1}$ state of HOC+
to be 0.4 ms, very close to the value they calculated for the same transition
in HCO+. Slightly higher values were reported by Rogers and Hillman 23 (0.6
ms), Botschwina 24 (0.7 ms) and Martin _et al._ 6 (0.7 ms). These results are
substantially lower than the value obtained in present experiment with the
lifetime of the $2\,\nu_{1}$ state of HOC+ re-scaled by the $1/\nu$ law. As
this scaling law is valid for the harmonic approximation, it is difficult to
distinguish whether the apparently longer than predicted radiative lifetime is
due to anharmonic contributions to the wavefunctions or due to coupling to the
other vibrational levels. Note that the scaling law is an approximation only,
e. g., a deviation from the $1/\nu$ scaling by a factor of two was observed
for the higher bending modes of DCO+ in an ion storage ring study 39.

### Collisional Quenching of HCO+($2\,\nu_{1}$)

In order to determine the rate coefficients for the quenching of the
$2\,\nu_{1}$ vibrational state of HCO+ in collisions with different species,
the number density of the probing gas (CO2) was kept constant and the number
density of the collisional partner was varied over several orders of
magnitude. The measured dependence of the ratio of secondary HCO2+ to primary
HCO+ ions as a function of the quenching partner number density at a
temperature of 125 K is shown in Fig. 4. The quenching reaction rate
coefficients $k_{q\ce{\mathrm{M}}}$ for the particular species M were obtained
by fitting the data in Fig. 4 using equation (4) where $\tau_{\mathrm{rad}}$
was kept constant (at the value obtained in present study, $1.53\pm 0.34$ ms).
The resulting fitted reaction rates, summarized in Tab. 2, show that the
quenching of the $2\,\nu_{1}$ vibrational state of HCO+ by molecular gases H2,
and N2, is two orders of magnitude more efficient than quenching by atomic He.
It is important to note that the number density of the LIR reactant (CO2) has
to be chosen so that the LIR signal is high enough but at the same time not in
saturation mode, i. e., somewhere in the first third of the curve shown in
Fig. 3. In our quenching experiment, this value was ca. $3\cdot
10^{11}\;\text{cm}^{-3}$. The sensitivity to the changes of CO2 number density
(change by $20\,\%$) is shown on two different traces for He quenching. The
method seems to be reliable, as implied by Eq. (4), as long as the CO2 number
density is not too large.

When the quenching partner is a molecule, its rovibrational structure makes
vibrational energy transfer (V–V) possible 40, in addition to vibration-to-
translation (V–T). This V–V transfer is a resonant process, and HCO+ possesses
three normal modes with vibrational frequencies of
$\nu_{1}=3089\leavevmode\nobreak\ \text{cm}^{-1}$,
$\nu_{2}=831\leavevmode\nobreak\ \text{cm}^{-1}$, and
$\nu_{3}=2183\leavevmode\nobreak\ \text{cm}^{-1}$ 7, thus, there are many
possible vibrational states where HCO+ can end up after a collision with a
quenching gas (see Figure 1). Therefore, it can be expected that the measured
quenching coefficients for collisions with molecular gases will have higher
values than those that are typically obtained for vibrationally excited
diatomic ions 41, 40. In fact, the measuered quenching coefficients
$k_{q\ce{H2}}=5.8\pm 1.1\cdot 10^{-10}\leavevmode\nobreak\
\text{cm}^{3}\,\text{s}^{-1}$ and $k_{q\ce{N2}}=6.4\pm 0.4\cdot
10^{-10}\leavevmode\nobreak\ \text{cm}^{3}\,\text{s}^{-1}$ are close to the
corresponding Langevin collisional rates of $1.5\cdot
10^{-9}\leavevmode\nobreak\ \text{cm}^{3}\,\text{s}^{-1}$ and $8.2\cdot
10^{-10}\leavevmode\nobreak\ \text{cm}^{3}\,\text{s}^{-1}$, respectively.

In an ion trap study by Schlemmer _et al._ 42, the collisional quenching
relaxation of antisymetric C–H stretch mode $\nu_{3}$ of C2H2+ with H2 has
been estimated using a fit of a kinetic model to the number of reaction
product as a function of H2 number density as $1.3\cdot
10^{-9}\;\text{cm}^{3}\,\text{s}^{-1}$. Their approach could be described as
more sophisticated in comparison with the one presented in this work, as the
applied LIR scheme C2H2+(ν_3=1) + H2 →C2H3+ + H was not a simple proton
transfer, i. e., the assumption of the Langevin behaviour did not hold and the
reaction rate of the excited state had to be investigated concurrently. Their
simulation also showed, that the quenching rate with H2 was a “substantial
fraction ($0.81-0.87$) of the Langevin rate”.

Atomic helium seems to be surprisingly efficient at quenching the $2\,\nu_{1}$
state of HCO+ with $k_{q\ce{He}}=5.6\pm 1.0\cdot 10^{-12}\leavevmode\nobreak\
\text{cm}^{3}\,\text{s}^{-1}$. The weak helium-ion interaction potential often
results in small ($<10^{-15}\leavevmode\nobreak\ \text{cm}^{3}\,\text{s}^{-1}$
at 300 K) deexcitation rates as was observed, e. g., for N2+ and NO+ 40, 43.
Even smaller reaction rate coefficients were predicted for low temperature
collisional quenching of diatomic anions (CN-, C2-) by helium 44, 45.
Wisthaler _et al._ 46 reported a rate coefficient of $1.3\cdot
10^{-13}\leavevmode\nobreak\ \text{cm}^{3}\,\text{s}^{-1}$ for the quenching
of vibrationally excited HCN+ and DCN+ ions in collision with helium at a
collisional energy corresponding to a temperature of 1500 K, noting that the
quenching reaction rate coefficient fell down below
$10^{-14}\leavevmode\nobreak\ \text{cm}^{3}\,\text{s}^{-1}$ at lower energies.

The HCO+–He potential energy surface has a minimum of approximately 300
$\text{cm}^{-1}$ below the dissociation limit 47. For comparison, the N2+–He
system has minimum of only 140 $\text{cm}^{-1}$ 48 (with a corresponding
vibrational quenching reaction rate coefficient of $1\cdot
10^{-15}\leavevmode\nobreak\ \text{cm}^{3}\,\text{s}^{-1}$) and the O2++Ar
system has a lowest dissociation energy of 765 $\text{cm}^{-1}$ 49 and the
reported quenching coefficient for the $\nu=1$ state of O2+ was $1\cdot
10^{-12}\leavevmode\nobreak\ \text{cm}^{3}\,\text{s}^{-1}$ 41. We note that it
is very hard to predict the magnitude of the quenching coefficient based only
on the interaction potentials and the polarisability of the neutral without a
full quantum mechanical calculation 45.

Figure 4: Determination of the quenching rate $k_{q}$ for HCO+ with He, H2,
and N2 from the decrease of the LIR signal as a function of the number density
of the quenching partner in the trap using Eq. (4). The two different data
sets for He, acquired with $>20\,\%$ difference in the CO2 LIR reactant number
density, demonstrate the robustness of the technique.

### Transition Dipole Moments for $20^{0}0\leftarrow 00^{0}0$ Transition of
HCO+ and HOC+

In order to estimate the value of the integrated molar infrared intensity of
absorption $\gamma$ for the $20^{0}0\leftarrow 00^{0}0$ transition of HCO+ in
the Stationary Afterglow with Cavity Ring-Down Spectrometer (SA-CRDS)
apparatus, we have followed the approach used for H3+ ions by Shapko _et al._
50. Aided by a chemical kinetics model, the experimental conditions were set
to maximize the amount of HCO+. The measured absorption signal can then be
compared to the value of the electron number density determined at the same
time in the afterglow by microwave diagnostics. Using the P(4), P(7) and P(8)
transitions of HCO+, the obtained lower estimate for the molar infrared
intensity of absorption was $\gamma=18\pm 5\leavevmode\nobreak\
\text{cm}^{2}\cdot\text{mol}^{-1}$, corresponding to a vibrational transition
moment $\langle 20^{0}0|\mu|00^{0}0\rangle=0.0084\pm 0.0025$ D. This is lower
than the value of $\gamma=23.7\leavevmode\nobreak\
\text{cm}^{2}\cdot\text{mol}^{-1}$ predicted by Botschwina 24. The
measurements were performed at temperatures of 80 K and 170 K and the H2 and
CO number densities were varied in the range of $5\cdot 10^{13}-5\cdot
10^{15}\;\text{cm}^{-3}$ with helium buffer gas pressure in the range of 500
to 1000 Pa. The stated error arises mainly from the estimated uncertainty of
the electron number density determination.

Although it is inherently difficult to derive absolute intensities in an
action spectroscopy experiment in an ion trap, relative evaluation is
possible. We show how to correctly assess the relative intensities of two
different species in a LIR action spectra in Eqs. (15–19). For high number
densities of LIR probing gas and no quenching, equation (4) simplifies to

$\displaystyle N_{\ce{HCO2+}}/N_{\ce{HCO+}}$
$\displaystyle=\rho_{1}=r_{1}\left(\ce{HCO+}\right)t,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \text{for}\leavevmode\nobreak\ \ce{HCO+}$ (5)
$\displaystyle N_{\ce{ArH+}}/N_{\ce{HOC+}}$
$\displaystyle=\rho_{2}=r_{1}\left(\ce{HOC+}\right)t.\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \text{for}\leavevmode\nobreak\
\ce{HOC+}$

The measured values were $\rho_{1}=0.0099\pm 0.0013$ from the P(3) line of
HCO+ at 125 K, and $\rho_{2}=0.025\pm 0.005$ from the R(3) line of HOC+ at 115
K, at the same trapping time $t=1.7\;\text{s}$. In the latter case, the
chemical probing by ^15N2 has shown that only ca. 70 % of all ions with mass
29 $m/z$ were HOC+, i. e., $\rho_{2}$ has to be increased accordingly.
Assuming that Herman-Wallis factors are close to unity for both HCO+ and HOC+,
the resulting calculated ratios for the Einstein A coefficients (Eq. (16)) and
vibrational transition moment squared (Eq. (19)) are $R_{A}=0.48\pm 0.10$ and
$R_{\mu}=0.41\pm 0.11$. The stated uncertainty is only the statistical error.
Systematic uncertainties arise mainly from the completely unknown overlap of
the laser beam (different light sources/optics/path for both species) with the
ion cloud in the trap. Even thought HCO+/HOC+ have the same $m/z$, the
distribution of the ions inside the trap will most probably differ, since
neither effective potential $V^{*}$, nor the number of ions were the same, and
on top, in case of HOC+, also the presence of the dominant lighter C+ ion will
play a significant role. Therefore, the results of this relative intensity
determination technique are only illustrative, as we are unable to provide a
reliable confidence interval.

## Conclusions

We present the first spectra of the $2\,\nu_{1}$ overtone transition of HCO+
and HOC+. The availability of this data and cheap telecommunication laser
diodes opens up the possibility of very cheap monitoring of these two isomers
in any optically thin environment or in emission. Although the accuracy of the
theoretical prediction of the HOC+ band origin by Koput 36 is in the
$\text{cm}^{-1}$ range, we hope that the experimental spectroscopic constants
will be helpful for further improvement of the ab initio theoretical models.

We explore the versatility of the cryogenic ion trapping technology and
present a technique to improve the LIR action spectroscopy for very weak
signals into a background free action spectroscopic method, by actively
removing the background in the ion signal by selective $m/z$ ejection from the
trap. Furthermore, we extensively describe the method of determination of
radiative lifetimes, $\tau_{\mathrm{rad}}$, and quenching reaction rates,
$k_{q}$, and its caveats inside an ordinary 22 pole rf trap, easily applicable
on a plethora of molecular ions.

Although similar experiments have been done in the past, e. g.,
$\tau_{\mathrm{rad}}$ in an ICR 16, 51, or measurements of quenching reaction
rates for various, mainly diatomic, ions 41, 43, 52, 45, this data exist only
for a select few systems. Moreover, there are not many experiments focused on
higher lying vibrational levels. The collisional quenching rates are needed
for the treatment of the molecular emission lines in astronomy. We aspire to
stimulate further study of $\tau_{\mathrm{rad}}$, and quenching rates for more
ion molecular systems, as well as for the different transitions, e. g.,
different vibrational modes (bending, stretching, fundamentals, overtones
etc.).

Last but not least, our quenching technique only shows the removal of the
vibrational excitation in the molecular ion. We can not estimate what happened
with the energy, whether it redistributed towards translational energy or into
internal excitation of the quenching neutral molecule. These fundamental ion-
molecule processes are not only of interest for astrophysics, but also closely
related to action spectroscopy technology, i. e., LOS (leak-out spectroscopy)
scheme53, where only the kinetic energy release is monitored in a quenching
event. Consequently, a study of one system by both techniques could provide a
full picture of energy distribution in the de-exciting collision.

## Experimental Section

The measurements have been predominantly conducted in the CCIT 22 pole rf
cryogenic trap setup 54, with the exception of the P(5)–P(8) transitions of
the HCO+ ion acquired in the CRDS setup55. The experimental setups are
extensively described in the references provided, we will only focus on
particular improvements of the 22 pole trap setup needed for this experiment
(the CRDS setup has been used without any modifications).

#### 22 Pole Trap Setup

The 22 pole trap setup CCIT54 consists of a storage ion source, quadrupole
mass filter used to select the mass of the ion to be stored in the trap, the
22 pole trap mounted on variable temperature cryostat and a mass sensitive
detection system. Each data point in the experiment, i. e., ion production,
trapping, storage/ irradiation/ LIR reaction and detection, is acquired in a 3
s cycle time.

Figure 5: Experimental setup in a laser multi-pass configuration (HCO+
experiments – (a)), and an on axis molecular beam configuration (HOC+
experiments – (b)). Not to scale. For a detailed description of the experiment
see ref. 54

HCO+ was produced directly from CO and H2 precursor gases using electron
bombardment in the storage ion source56, where the produced ions have many
collisions with both CO and H2, assuring dominant production of the lowest
isomer. Subsequently, HCO+ was mass selected in a quadrupole filter and
injected into the trap, where a short intense He pulse was used to trap it and
provide collisional cooling into the ground state (see Fig. 5(a)).

In order to produce HOC+in sufficient quantities, the following ion-molecule
reaction57 inside the trap was used

$\displaystyle\ce{C+}+\ce{H2O}$
$\displaystyle\xrightarrow[]{}\ce{HOC+}+\ce{H},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \Delta
E=-2.60\leavevmode\nobreak\ \mathrm{eV},$ (6)
$\displaystyle\xrightarrow[]{}\ce{HCO+}+\ce{H},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \Delta
E=-4.34\leavevmode\nobreak\ \mathrm{eV}.$ (7)

The C+ ion was produced in the ion source from CH4 precursor gas using
electron bombardment, mass selected and injected into the trap, where a short
intense He pulse was used to trap and provide collisional cooling.
Subsequently, an intense water pulse from the side of the trap (on the axis,
ca. 200 ms long) was used to convert approx. 1/3 of the C+ ions into mass 29
$m/z$ (HCO+ and HOC+) (see Fig. 5(b)). The on axis water beam had to be used
to avoid the freezing of the water molecules at temperatures $<200$ K. The
ratio of the HOC+ to HCO+ ions was determined using proton transfer to ^15N2
as a chemical probing reaction. This reaction is exothermic for the HOC+
reactant only (see Fig. 1. Note that ^14N2H+ has the same mass as HOC+). This
method of HOC+ production and isomer fraction determination has been recently
described by Yang _et al._ 58. The gas pulses into the trap, as well as on the
axis, are delivered using a custom built piezo element actuated valves,
operated at the resonant frequency of a few kHz.

In both cases, following the ion preparation/ trapping, the ions are left
stored in the trap for a predetermined time (1.75 s), during which the IR
radiation can excite the ions in the ground state. The probing reactant gas
(CO2 in the case of HCO+ or Ar in the case of HOC+ spectroscopy) is flowing
directly into the trap freely, maintaining a constant number density therein
during the whole cycle time. Neither of the products of the LIR processes (1)
and (2), HCO2+ or ArH+, can undergo any significant further reaction inside
the trap and thus accumulate until being detected at the end of the cycle. In
the spectroscopy experiment, the wavelength of the laser radiation is scanned
by a small step after every single data point taken and the spectrum as seen
in Fig. 2 is acquired.

In case of the radiative lifetime experiment and quenching experiment (Figs.
3, 4), the light wavelength is kept constant (on resonance), while the number
density of the probing gases has to be varied using a variable leak valve. The
pressure inside the vacuum vessel has been recorded using a Bayard-Alpert ion
gauge (calibrated by an absolute pressure baratron gauge), the temperature
transpiration effect was taken into account to accurately determine the number
density in the trap (for details see ref.54).

We exclusively used continuous wave fiber pigtailed telecommunication grade
distributed feedback (DFB) laser diodes FLPD-1647-07-DFB-BTF (HCO+ P(1)–P(4)),
QDFBLD-1650-5 (HCO+ P(5)–P(6)), NX8570SD654Q-55 (HOC+ R(8)–R(11)), and
QDFBLD-1570-20 (HOC+ R(1)–R(4)). A small part of the light has been split
(fiber beam splitter, $1\,\%$) and fed into the WA-1650 (EXFO) wavemeter, with
an absolute accuracy better than 0.3 ppm at 1500 nm. The majority of the light
was guided into the trap using an adjustable focus collimator. Since the laser
diode power for the HCO+ experiment was only ca. 7 mW, we decided to use a
Pfund cell (two spherical mirrors $f=750$ mm with a central $\phi 1.5$ mm
hole) in a 3-pass configuration, effectively multiplying the power inside the
trap by a factor of $\approx 2.5$ (see Fig. 5(a)). The radiation power has
been regularly monitored with a Ge photodiode.

#### Mass Selective Ejection – Zero Background Spectroscopy

The ions coming from the source can easily be mass selected to contain only
the $m/z$ ion of interest. The ion (HCO+) is consequently injected into the
trap, where the stopping He pulse slows the ion down and removes its internal
energy. However, if some other gas, e. g., probing gas (CO2), is present in
the trap, some of the incoming ions will collide with it prior to
thermalisation (kinetic, internal degrees of freedom) and undergo a chemical
reaction (product HCO2+). As a consequence, it is impossible to fill the trap
containing a probing gas and not produce any of the ions which are detected in
the LIR scheme. This is usually of no concern for fundamental vibrational
spectroscopy, where the absorption rates are usually two orders of magnitude
stronger than for overtone vibrational transitions and the “injection” only
creates a few percent of the total ions detected in the LIR scheme.

Fortunately, in this work, the products of the LIR reaction (HCO2+: $45\;m/z$,
ArH+: $41\;m/z$) are significantly heavier than the ions of interest
$29\;m/z$. Therefore, we can vary the effective potential $V^{*}$ in the trap,
by lowering the driving amplitude $V_{0}$ for few hundred of ms, to achieve
conditions where the higher mass ion is no longer efficiently trapped,
effectively cleaning the trap from masses higher than a set threshold of ca.
40 $m/z$ (see Gerlich 56 for details on trap technology). The effect of this
technique can be best observed on the left panel of Fig. 2, where the off
resonance background signal of HCO2+ is essentially 0 (note that the value is
not kept at 0 on purpose, since it is difficult to spot any instrument
malfunction while scanning without having any signal at all).

The same technique is applied for the HOC+ spectroscopy, where the few hundred
ms long ejection time additionally allows the excited HOC+/HCO+, produced in a
strongly exothermic reaction (6), to radiatively decay to its vibrational
ground state. Unfortunately, in this configuration, some of the water vapor
and precursor ion C+ is always present, effectively creating excited products
of the reaction (6) all the time. These can further react with the probing
neutral Ar, even off HOC+ resonance, and contaminate the trap with ArH+ during
the 1.75 s irradiation time. To provide some quantitative values, in case of
HOC+ experiment, the trap is operated at $f_{0}=19.5\,\text{MHz}$ and $V_{0}$
from $41$ V (everything trapped) to $26$ V (ejection of masses $>40\;m/z$).
The change in effective potential $V^{*}(0.8\,r_{0})$ for mass $29\;m/z$ is
then $5.2\to 2.1\,\text{meV}$ and respectively for mass $41\;m/z$ it is
$3.7\to 1.5\,\text{meV}$.

#### CRDS Setup

The SA-CRDS apparatus (Stationary Afterglow with Cavity Ring-Down
Spectrometer) was utilized for the measurement of 4 line positions in this
study. Originally developed for the determination of recombination rate
coefficents for collisions of electrons with molecular ions 59, 60, it was
also used for measurements of the line positions of overtone transitions for
several polyatomic ions 55, 61.

The main diagnostic technique is a continuous wave modification of cavity
ring-down spectroscopy based on a design by Romanini _et al._ 62. The HCO+
ions were produced in a pulsed microwave discharge (discharge power of
$\approx$ 17 W, period of 4.9 ms, 40% duty cycle) ignited in a He/Ar/H2/CO
mixture with reactant number densities of $8\cdot 10^{17}/2\cdot
10^{14}/4\cdot 10^{14}/4\cdot 10^{14}$ $\text{cm}^{-3}$. The discharge tube
made of fused silica was immersed in liquid nitrogen.

We employed a model of chemical kinetics and varied the reactant number
densities by almost an order of magnitude to maximize the number of HCO+ ions
in the discharge and afterglow plasma.

### Radiative Deexcitation/ Collisional Quenching Derivation

In the following, we derive Eq. (4), used for the determination of
$\tau_{\mathrm{rad}}$ and $k_{q}$, for the HOC+($2\,\nu_{1}$) case. The
technique can be applied on any analogous LIR system and transition
(fundamental, overtone etc.), e. g., for HOC+ ions with Ar probing gas, etc.
For simplicity, let’s consider only the most important processes involving
HCO+ ions in the 22 pole rf trap:

1\. Overtone excitation

$\ce{HCO+}(0\,\nu_{1})+h\nu\xrightarrow[]{r_{1}}\ce{HCO+}(2\,\nu_{1}),$ (8)

where $r_{1}$ is the rate for excitation from the vibrational ground state by
two vibrational quanta in the stretching mode $\nu_{1}$. It depends on the
corresponding transition line strength, the intensity of the laser and on the
overlap between the ion cloud in the trap and the laser beam.

2\. Deexcitation of the vibrationally excited HCO+ ions to lower states

$\ce{HCO+}(2\,\nu_{1})\xrightarrow[]{1/\tau}\ce{HCO+}(0\,\nu_{1},1\,\nu_{1})+h\nu,$
(9)

here $1/\tau=1/\tau_{\mathrm{rad}}+1/\tau_{\mathrm{q}}$, with
$\tau_{\mathrm{rad}}$ representing spontaneous emission, and
$\tau_{\mathrm{q}}$ deexcitation due to collisions with another particle. In
our particular case, it can be safely assumed that $\tau_{\mathrm{rad}}$ is
mainly given by the transition to the $1\,\nu_{1}$ state.

3\. Proton transfer in reaction with the probing gas

$\ce{HCO+}(2\,\nu_{1})+\ce{CO2}\xrightarrow[]{k_{1}}\ce{HCO2+}+\ce{CO}.$ (10)

The corresponding rate equations for HCO+($2\,\nu_{1}$) and HCO2+ can then be
written as

$\displaystyle\frac{\mathrm{d}N_{\ce{HCO+}(2\,\nu_{1})}}{\mathrm{d}t}$
$\displaystyle=-k_{1}N_{\ce{HCO+}(2\,\nu_{1})}[\ce{CO2}]-\frac{1}{\tau}N_{\ce{HCO+}(2\,\nu_{1})}+$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ r_{1}N_{\ce{HCO+}(0\,\nu_{1})},$ (11)
$\displaystyle\frac{\mathrm{d}N_{\ce{HCO2+}}}{\mathrm{d}t}$
$\displaystyle=k_{1}N_{\ce{HCO+}(2\,\nu_{1})}[\ce{CO2}],$ (12)

where $N_{\mathrm{i}}$ denotes number of ions of species $i$ in the trap.

By utilising the steady-state approximation, we can express
$N_{\ce{HCO+}(2\,\nu_{1})}$ from equation (11) and insert it into equation
(12). After simple integration we obtain

$N_{\ce{HCO2+}}=\frac{k_{1}[\ce{CO2}]r_{1}N_{\ce{HCO+}(0\,\nu_{1})}}{k_{1}[\ce{CO2}]+1/\tau}t$
(13)

for the number of detected HCO2+ ions, where $t$ is trapping time. Noting that
$1/\tau=1/\tau_{\mathrm{rad}}+1/\tau_{\mathrm{q}}$ and that
$1/\tau_{\mathrm{q}}=k_{q\ce{\mathrm{M}}}[\mathrm{M}]$, equation (13) can be
rewritten as

$N_{\ce{HCO2+}}=\frac{k_{1}[\ce{CO2}]r_{1}N_{\ce{HCO+}(2\,\nu_{1})}}{k_{1}[\ce{CO2}]+1/\tau_{\mathrm{rad}}+k_{q\ce{\mathrm{M}}}[\mathrm{M}]}t,$
(14)

where $k_{q\ce{\mathrm{M}}}$ is the reaction rate coefficient for the
quenching of the $2\,\nu_{1}$ state of HCO+ in collisions with particles
$\mathrm{M}$. This relation can be used to determine the lifetime of
spontaneous emission by varying the LIR reactant number density, as well as to
evaluate the quenching rate by varying the number density of the quenching
particles $[\mathrm{M}]$ (provided $\tau_{\mathrm{rad}}$ is known), from the
measured LIR signal, $N_{\ce{HCO2+}}/N_{\ce{HCO+}(0\,\nu_{1})}$, as a function
of the respective number density.

### Relative Transition Intensities in Action Spectroscopy

In saturation conditions, i. e., conditions with high LIR reactant gas number
density such that neither $\tau_{\mathrm{rad}}$ nor quenching play a role, the
intensity of the LIR signal, defined by Eq. (14), reduces to Eq. (5), i. e.,
$\rho=r_{1}t$. The rate of excitation to the $2\,\nu_{1}$ vibrational state is
proportional to the laser power and to the Einstein coefficient for given
transition. Following the derivation in refs. 63, 50 and assuming a thermal
population of states, the Einstein coefficient $A_{JvJ^{\prime}v^{\prime}}$
for spontaneous emission between two rovibrational states of HCO+ is
proportional to the measured ratio $\rho$ of the numbers of secondary (HCO2+
or ArH+) and primary (HCO+ or HOC+) ions obtained at high probing gas number
densities

$A_{JvJ^{\prime}v^{\prime}}\sim\rho\frac{8\pi\nu^{2}cQ(T)}{g_{\mathrm{u}}}\exp{\frac{E}{k_{\mathrm{B}}T}}\sqrt{\frac{\pi}{4\ln{2}}}w/I,$
(15)

where $\nu$ is the transition wavenumber, $c$ is speed of light,
$k_{\mathrm{B}}$ is the Boltzmann constant, $Q(T)$ is the partition function
at temperature $T$, $g_{\mathrm{u}}$ denotes the statistical weight of the
upper state, $E$ is the energy of the lower state of the transition, $w$ is
the full width at half maximum (FWHM) of the Doppler broadened line, and $I$
is the number of photons passing through the trap per second. The ratio
between the Einstein coefficients of two rovibrational transitions is then

$R_{A}=\frac{A_{1}}{A_{2}}=\frac{\rho_{1}\nu_{1}^{2}w_{1}g_{\mathrm{l1}}g_{\mathrm{u2}}P_{2}I_{2}}{\rho_{2}\nu_{2}^{2}w_{2}g_{\mathrm{l2}}g_{\mathrm{u1}}P_{1}I_{1}},$
(16)

where the subscripts 1 and 2 distinguish between the transitions and $P$ is
the thermal population of the lower state with statistical weight
$g_{\mathrm{l}}$. The state to state transition dipole moment squared $\langle
Jv|\mu|J^{\prime}v^{\prime}\rangle^{2}$ is connected to the corresponding
vibrational transition moment $\langle v|\mu|v^{\prime}\rangle$ through the
relation 64

$\langle Jv|\mu|J^{\prime}v^{\prime}\rangle^{2}=\langle
v|\mu|v^{\prime}\rangle^{2}S^{\Delta J}_{J}F(m),$ (17)

where $S^{\Delta J}_{J}$ and $F(m)$ are Hönl-London and Herman-Wallis factor,
respectively. Given that 63

$A_{JvJ^{\prime}v^{\prime}}=\frac{64\pi^{4}}{3h}\nu^{3}\frac{g_{\mathrm{l}}}{g_{\mathrm{u}}}\langle
Jv|\mu|J^{\prime}v^{\prime}\rangle^{2}\times 10^{-36}\leavevmode\nobreak\
\mathrm{s^{-1}},$ (18)

with $h$ in cm2gs, $\nu$ in $\text{cm}^{-1}$ and $\langle
Jv|\mu|J^{\prime}v^{\prime}\rangle$ in Debye, it follows that

$R_{\mu}=\frac{\langle v|\mu|v^{\prime}\rangle^{2}_{\mathrm{HCO^{+}}}}{\langle
v|\mu|v^{\prime}\rangle^{2}_{\mathrm{HOC^{+}}}}=\frac{\rho_{1}w_{1}\nu_{2}P_{2}I_{2}S^{\Delta
J}_{J2}F_{2}(m)}{\rho_{2}w_{2}\nu_{1}P_{1}I_{1}S^{\Delta J}_{J1}F_{1}(m)}.$
(19)

## Acknowledgements

This work was supported by the Max Planck Society, the Czech Science
Foundation (Grant Nos. GACR 23-05439S and GACR 22-05935S), and the Charles
University (Grant Nos. GAUK 337821 and GAUK 332422). L.U. acknowledges support
by the COST Action CA21101. The authors gratefully acknowledge the work of the
electrical and mechanical workshops and engineering departments of the Max
Planck Institute for Extraterrestrial Physics.

## Conflict of Interest

The authors report there are no competing interests to declare.

## Additional Information

The data that support the findings of this study are openly available in
Zenodo at https://doi.org/10.5281/zenodo.10473481, reference number 10473481.

Keywords:

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