11institutetext: Space Research Institute, Austrian Academy of Sciences,
Schmiedlstrasse 6, A-8042 Graz, Austria
11email<EMAIL_ADDRESS>22institutetext: Centre for Exoplanet
Science, University of St Andrews, North Haugh, St Andrews, KY169SS, UK
33institutetext: SUPA, School of Physics & Astronomy, University of St
Andrews, North Haugh, St Andrews, KY169SS, UK 44institutetext: Institute for
Astronomy, KU Leuven, Celestijnenlaan 200D, 3001 Leuven, Belgium
55institutetext: TU Graz, Fakultät für Mathematik, Physik und Geodäsie,
Petersgasse 16, 8010 Graz, Austria 66institutetext: Department of Chemistry
and Molecular Biology, University of Gothenburg, Kemigården 4, 412 96
Gothenburg, Sweden

# Infrared spectra of TiO2 clusters for hot Jupiter atmospheres

J.P. Sindel 11223344 Ch. Helling 1155 D. Gobrecht 4466 K.L. Chubb 22 L. Decin
44

(Received ——–; accepted ———)

###### Abstract

Context. Clouds seem unavoidable in cool and dense environments, and hence,
are necessary to explain observations of exoplanet atmospheres, most recently
of WASP 96b with JWST. Understanding the formation of cloud condensation
nuclei in non-terrestrial environments is therefore crucial to develop
accurate models to interpret present and future observations.

Aims. The goal of the paper is to support observations with infrared spectra
for (TiO2)N clusters in order to study cloud formation in exoplanet
atmospheres.

Methods. Vibrational frequencies are derived from quantum-chemical
calculations for 123 (TiO2)-clusters and their isomers, and line-broadening
mechanisms are evaluated. Cluster spectra are calculated for several
atmospheric levels for two example exoplanet atmospheres (WASP 121b-like and
WASP 96b-like) to identify possible spectral fingerprints for cloud formation.

Results.  Rotational motion of and transitions in the clusters cause
significant line broadening, so that individual vibrational lines are
broadened beyond the spectral resolution of the medium resolution mode of the
JWST mid-infrared instrument MIRI at R = 3000. However, each individual
cluster isomer exhibits a “fingerprint” IR spectrum. In particular, larger
(TiO2)-clusters have distinctly different spectra from smaller clusters.
Morning and evening terminator for the same planet can exhibit different total
absorbances due to different cluster sizes being more abundant.

Conclusions. The largest (TiO2)-clusters are not necessarily the most abundant
(TiO2)-clusters in the high-altitude regions of ultra-hot Jupiters, and the
different cluster isomers will contribute to the local absorbance. Planets
with a considerable day-night asymmetry will be most suitable to search for
(TiO2)-cluster isomers in order to improve cloud formation modelling.

###### Key Words.:

molecular data - spectral lines - exoplanets - stuff

##  1 Introduction

The presence of clouds has been inferred for many exoplanet atmospheres, as an
explanation for suppressed absorption features and overall flat spectra
(Charbonneau et al., 2002; Pont et al., 2008; Kreidberg et al., 2014). For hot
Jupiters, cloud models predict that cloud particles are primarily made of a
mix of silicates and other metal oxides (Helling et al., 2008; Mollière et
al., 2017). Silicate spectra exhibit a vibrational Si-O stretching feature at
$10\mu m$ (Hackwell et al., 1970). With the launch of JWST in December 2021
and its near infrared observation capabilities in NIRCam, NIRSpec, and MIRI,
the near infrared wavelength range ($\lambda=0.6-5\mu m$ for NIRCam and
NIRSpec and $\lambda=5-28\mu m$ for MIRI) became accessible for observations
at medium spectral resolutions. Miles et al. (2022) provided the first direct
detection of absorption around the $10\mu m$ band and therefore of silicate-
based clouds in a planetary mass companion, observing VHS 1256b with JWSTs
NIRSpec and MIRI MRS instruments. Observations released for the JWST initial
science release target WASP 96b showed that clouds were needed to explain its
spectral features. Additionally models for future JWST targets, such as WASP
121b, or WASP 43b, exist and predict their cloud coverage or lack thereof
(Helling et al., 2021). In order to quantify the impact of clouds on the
chemistry and elemental abundances in exoplanet atmospheres, their formation
processes need to be understood. Cloud particles condense from supersaturated
gas on so-called cloud condensation nuclei (CCN). On gaseous exoplanets, these
CCN need to be formed from the gas phase through a process called nucleation.
This process is the first step of cloud formation and starts as soon as the
densities of the nucleating species are high, and the temperatures low enough.
During the nucleation process differently sized small clusters or different
isomers of the same sized cluster of the CCN-forming species co-exist and it
is unknown if only one or all participate in the chemical path to a thermally
stable CCN which then triggers the formation of a cloud particle. These
clusters may have distinct spectral fingerprints, different to both that of
the monomer and the bulk. Finding such spectral fingerprints may put the cloud
formation modelling for extrasolar planets on much firmer grounds. The aim of
this paper is therefore to provide harmonic vibrational spectra of these small
clusters for the nucleating species Titanium dioxide (TiO2) to enable
discoveries similar to the unexpected SO2 detection in WASP-39b with JWST
(Rustamkulov et al., 2023). These spectra arise from excited vibrational
states of the clusters and their isomers that have energies that correspond to
wavelengths in the infra-red range that can be probed by JWSTs MIRI.
Rotational transitions have energies in the millimetre wavelength ranges, but
they also happen within vibrational state transitions, broadening the
corresponding vibrational bands.

Section 2 describes how the harmonic vibrational spectra are extracted from
quantum-chemical calculations performed in Sindel et al. (2022) and the
broadening mechanisms considered. Section 3 compares the spectra for different
clusters and isomers and presents the resulting total absorbance caused by
(TiO2)N clusters for two exoplanet atmosphere models. In Section 4, we discuss
our results and their limits. Section 5 summarises the paper and gives and
outlook on possible future work.

##  2 Methods

###  2.1 Extraction of harmonic IR frequencies

This work investigates the infrared spectra of small (TiO2)N clusters. The
harmonic vibrational spectra discussed in this work are based on the results
of density functional theory (DFT) frequency calculations performed with the
B3LYP functional (Becke, 1993), the cc-pVTZ basis-set (Wilson et al., 1996)
and gd3bj empirical dispersion (Grimme et al., 2011). The cluster structure
and energy calculations were presented in Sindel et al. (2022), for (TiO2)N
clusters of sizes $N$ = 1-15. For all clusters with sizes $N>1$, multiple
isomers are considered. (see Table 1). For each cluster size $N$, there is one
most energetically favourable cluster configuration, the so-called global
minimum candidate (GM). The GM is given the suffix -A, e.g. TI6O12-A denotes
the GM cluster of size 6. The second lowest-energy isomer, i.e. the
energetically next favourable cluster at T = 0K is given the suffix -B, the
third -C, etc..

Table 1: Number of isomers for each cluster size $N$ $N$ | # of isomers
---|---
1 | 1
2 | 3
3 | 7
4 | 4
5 | 4
6 | 6
7 | 7
8 | 10
9 | 7
10 | 12
11 | 14
12 | 13
13 | 14
14 | 10
15 | 11
Total | 123

The frequencies and their corresponding infrared (IR) intensities are given in
units of [cm-1] and $\frac{\rm km}{\rm mol}$, respectively. We note that some
of the low energy / high wavelength modes ($\lambda\gtrsim 77\mu m$) are not
pure vibrational modes, but can arise from hindered rotation, where certain
rotations of the molecule are inhibited due to its geometry. In this work, we
treat all modes as purely harmonic vibrational modes. We convert the IR
intensities to molar absorption coefficients according to Spanget-Larsen
(2015). The unknown quantity for this transformation is the line-width for
each active IR mode. To quantify the line-width several broadening mechanisms
are evaluated for the physical parameterspaces of exoplanetary atmospheres:

####  2.1.1 Thermal broadening

Thermal broadening of a line is caused by the multi-directional movement of
individual line-sources within a gas. The broadening in frequency
($\Delta\nu$) by thermal broadening is dependent on the temperature ($T$), the
frequency ($\nu$) and the mass of the individual line-sources, i.e. the
cluster, ($m$) through:

$\frac{\Delta\nu}{\nu}=\frac{1}{c}\sqrt{\frac{2k_{B}T}{m}}$ (1)

where $k_{B}$ is the Boltzmann constant and $c$ is the speed of light. As a
consequence, large and more massive clusters show smaller thermal line
broadening than small clusters. Here, we assume that each active frequency
mode of a cluster is broadened equally.

####  2.1.2 Collisional broadening

In a dense gas, usually spontaneous emissions can be induced, i.e. triggered
prematurely, through collisions with other particles. This shortens the
average lifetime of the state and therefore increases the uncertainty in
energy and therefore frequency. The relative broadening at a frequency is
given by

$\frac{\Delta\nu}{\nu}=\frac{2\nu_{col}}{\nu}$ (2)

where $\nu_{col}$ is the frequency of collisions within the gas, which is
given by the root mean square (RMS) velocity of the gas particles ($v_{rms}$)
and the mean free path ($\lambda$):

$\nu_{col}=\frac{v_{rms}}{\lambda}$ (3)

with the RMS velocity of a gas being:

$v_{rms}=\sqrt{\frac{3RT}{M}},$ (4)

with the ideal gas constant $R$, the gas temperature $T$ and the mean
molecular weight of the gas $M$. The mean free path is given by:

$\lambda=\frac{RT}{\sqrt{2}\pi\sigma^{2}N_{A}p},$ (5)

with the gas pressure $p$, the effective cross-section of the cluster $\sigma$
and the Avogadro constant $N_{A}$.

####  2.1.3 Rotational broadening

A molecule with $n$ atoms has $3n-6$ vibrational modes. The rotational motion
of the molecule can affect the vibrational motion and vice versa. This means
that the vibrational energy levels are slightly dependent on the rotational
energy levels. As a result, when rotational transitions occur, they can
slightly shift the vibrational energy levels, which then leads to additional
broadening of the vibrational spectral lines. The population of rotational
states J follows a Boltzmann distribution. In order to quantify the broadening
mechanism induced by these rotational states, the maximum rotational quantum
number $J_{max}$ is used. It is defined by:

$J_{max}=\left(\frac{k_{B}T}{2hcB}\right)^{\frac{1}{2}}-\frac{1}{2}.$ (6)

$B$ is the rotational constant of the molecule and defines the spacing of
rotational energy levels. This maximum rotational quantum number specifies the
most likely occupied rotational energy state (Hollas, 2002). Since the
broadening effect is symmetric, the frequency space spanned by the rotational
states from $J=0$ to $J_{max}$ is taken as the FWHM for our broadening
approximation. The broadening induced by this process is therefore given as:

$\frac{\Delta\nu}{\nu}=\frac{2BJ_{max}}{\nu}$ (7)

####  2.1.4 Natural broadening

Natural broadening occurs due to the Heisenberg uncertainty principle. An
uncertainty in the life-time of an excited state ($\Delta t$) leads to an
uncertainty in its energy ($\Delta E$) and therefore in in its frequency
($\Delta\nu$).

$\Delta\nu\approx\frac{\Delta E}{h}\approx\frac{1}{2\pi\Delta t}$ (8)

Natural broadening effects are usually small when compared to other broadening
effects (Fig. 1) so they are not considered in this work.

###  2.2 Number densities of (TiO2)N clusters

In order to calculate not only the relative molar absorbances of the
individual (TiO2)N clusters at different pressure and temperature points, but
the absolute absorbance of each species at certain pressure levels in the
atmosphere, number densities for as many species as possible need to be
included, because they all contribute differently to the overall absorption.
In order to achieve that, we input the thermochemical cluster-data into the
equilibrium chemistry code GGChem Woitke et al. (2018). The molecular
equilibrium constants are fit using the approach by Stock et al. (2018). The
Gibbs free energies of formation ($\Delta G^{\circ}_{f}$) are calculated by

$\Delta G^{\circ}_{f}(({\rm TiO}_{2})_{N})=\Delta G_{DFT}^{\circ}({\rm
TiO})_{2})_{N}-N\Delta G_{Janaf}^{\circ}({\rm Ti})-2N\Delta
G_{Janaf}^{\circ}({\rm O}).$ (9)

Values for $\Delta G_{DFT}^{\circ}({\rm TiO}_{2})_{N}$ are taken from Sindel
et al. (2022), while $\Delta G^{\circ}_{Janaf}$ for Ti and O are taken from
the JANAF-NIST thermochemical tables (Malcolm W. Chase, 1998). The resulting
fit for the $k_{p}$ values was then done through Equations (13) and (17) in
Woitke et al. (2018) which combine into:

$\Delta
G^{\circ}_{f}(T)=-RT\left(\frac{a_{0}}{T}+a_{1}\ln{T}+b_{0}+b_{1}T+b_{2}T^{2}\right)$
(10)

with the fit-coefficients $a_{0}$, $a_{1}$, $b_{0}$, $b_{1}$, and $b_{2}$.
These were then implemented into GGChem for the 123 considered (TiO2)N cluster
isomers (Tab. 1), enabling the calculation of their number densities from
equilibrium chemistry at any pressure temperature point with $T>100$K.

###  2.3 Atmospheric pressure - temperature profiles

We use 1D pressure-temperature ($p_{\rm gas}$, $T_{\rm gas}$)-profiles that
were extracted from Baeyens et al. (2021) and extended to pressures of
$10^{-15}$ bar. We focus on two specific cases: A WASP-121b like planet
($T_{*}=6500$K, $T_{\rm eff}=2200$K, log(g) = 3) and a WASP-96b like planet
($T_{*}=5650$K, $T_{\rm eff}=1200$K, log(g) = 3). WASP 96b is a puffy hot
Jupiter, that is cold enough to produce cloud coverage everywhere but the
substellar point, and the existence of clouds has been predicted by cloud
formation models (Samra et al., 2023). WASP 121b is an ultra-hot Jupiter with
extremely high day/nightside temperature differences, where clouds are only
expected to form on the nightside (Helling et al., 2021). For both of the
planets a ($p_{\rm gas}$, $T_{\rm gas}$)-profile for the morning and the
evening terminator are used.

###  2.4 Absorbance of clusters

For all of the ($p_{\rm gas}$, $T_{\rm gas}$)-profiles, we start from a gas of
solar composition (Asplund et al., 2009) and run GGChem for gas-phase species
only at every point along the ($p_{\rm gas}$, $T_{\rm gas}$)-curve. This
provides the number densities for all isomers. At each pressure and
temperature point, all broadening mechanisms are evaluated for all vibrational
lines of all clusters and the strongest broadening mechanism is chosen. The
broadening through this method is used as the input broadening width $w$ to
convert the lines IR intensity into an absorption profile $\sigma$ with units
of [cm2/molecule]. All absorption profiles are mutliplied by the number
density [$\frac{\rm molecule}{\rm cm^{3}}$] of their respective cluster,
giving their total absorbance $A_{tot}$ in [$\frac{1}{\rm cm}$]:

$A_{tot,cluster}=n_{\rm cluster}\sigma_{\rm cluster}$ (11)

##  3 Results

###  3.1 Broadening mechanisms

The impact of the broadening mechanisms at different pressure levels at the
morning side terminator of a WASP-96b like planet is investigated as an
example (Figure 1).

Figure 1: Comparison of the considered broadening mechanisms along a ($p_{\rm
gas}$, $T_{\rm gas}$)-profile. All broadening mechanisms are evaluated for
their potential minimum and maximum contribution. Thermal broadening is
weakest for large clusters ($N=15$, ’Thermal, min’), and strongest for small
clusters ($N=1$, ’Thermal, max’). Collisional broadening is weakest for small
clusters at high frequencies ($N=1$, $\lambda=10\mu m$, ’Collisional, min’)
and strongest for large clusters at low frequencies ($N=15$, $\lambda=100\mu
m$, ’Collisional, max’). Rotational broadening is weakest for large clusters
($N=15$, ’Rotational, min’) and strongest for small clusters ($N=1$,
’Rotational, max’). Natural broadening is weakest for long state life-times at
high frequency ($\tau=1\mu s$, $\lambda=10\mu m$, ’Natural, min’) and
strongest for short state life-times at low frequency ($\tau=1ns$,
$\lambda=100\mu m$, ’Natural, max’). The T(p)-profile is one of the
WASP-96b-like profiles from Sec. 2.3.

It is of interest whether the individual vibrational modes could be resolved
by JWSTs MIRI instrument at its working spectral resolution of $R\sim 3000$.
Any mechanisms that broaden the lines further than that will make them
resolvable at this resolution.

The impact of collisional broadening depends strongly on the pressure. In our
test case atmosphere it is only relevant at high pressures low in the
atmosphere. For the atmosphere used in this test case, the upper layers are
largely isothermal. Therefore broadening mechanisms that depend predominantly
on temperature such as thermal and rotational broadening are constant
throughout most of the atmosphere and only change when the temperature changes
too. For natural broadening, the smallest broadening effects are characterised
by a long state life-time and a small emission wavelength (Eq. 8). $\tau=1\mu
s$ and $\lambda=10\mu m$ are chosen representatively. The maximum broadening
scenario corresponds to a short state life-time ($\tau=1ns$) and a large
emission wavelength ($\lambda=100\mu m$). In both cases, the relative line
broadening is far below the rotational broadening. Therefore, it is justified
to neglect natural broadening. The most impactful broadening mechanism
throughout most of the atmosphere is rotational broadening, dominating
everywhere but deep in the atmosphere.

###  3.2 Raw spectra of (TiO2)N clusters

Every (TiO2)N cluster contains $X=3N$ atoms and shows $3X-6$ vibrational
modes. Therefore, the overall opacity is expected to increase with the cluster
size $N$. Wavelength dependent cross-sections $\sigma_{\rm cluster}(\lambda)$
[$\frac{\rm cm^{2}}{\rm molecule}$] are computed for 123 isomers of sizes
$N=1-15$ of the (TiO2)N clusters. The broadening for each line at a
temperature of $T=1000{\rm K}$ and a pressure of $p=1$ bar is calculated and
the lines are broadened according to Spanget-Larsen (2015): Each line profile
is given by a Lorentzian with a FWHM of $w$ and a maximum of $\epsilon_{\rm
max}=27.648\frac{I_{\rm IR}}{w}$. The width $w$ is computed by comparing the
broadening mechanisms discussed in Sec. 2.1 and using the width produced by
the dominant one. A comparison is made to show trends between different sizes
in Figure 2.

Figure 2: Wavelength dependent cross-sections $\sigma_{\rm cluster}(\lambda)$
for the GM clusters of sizes $N=1,2,10,15$.

.

In the whole spectral range of $\lambda=5-100\mu m$ cross-sections of larger
clusters tend to be larger than ones of smaller clusters. Especially in the
wavelength-range observed by the JWST/MIRI instrument of $5-28\mu m$, larger
clusters are stronger absorbers than smaller clusters, while the TiO2 monomer
has no significant absorption cross-section over most of this range. Cluster
growth (i.e. nucleation) will therefore lead to an increase of absorption in
the $\sim 4-25\mu m$ wavelength region, with the largest clusters dominating
the absorption.

Generally, not only the GM cluster will be present, but energetically less
favourable isomers will also be present in the atmosphere. Since their
geometries are different, their vibrational spectra also differ. A comparison
is made of cross-sections for the 3 isomers of size $N=2$ (Fig. 3) and the 4
lowest-energy isomers of size $N=15$ (Fig. 4).

Figure 3: Wavelength dependent cross-sections $\sigma_{\rm cluster}(\lambda)$
for all 3 isomers of the (TiO2)2 cluster.

.

Figure 4: Wavelength dependent cross-sections $\sigma_{\rm cluster}(\lambda)$
for the 4 energetically most favorable isomers of the (TiO2)15 cluster.

.

For the molecular cross-sections of a small cluster, (TiO2)2, there are
differences between the isomers, especially for wavelengths towards the tail
at $100\mu m$. Within the MIRI wavelength range ($5-28\mu m$) they agree in
key features, such as the peak just short of $\sim 10\mu m$ and the double
peak at $\sim 15\mu m$ for the 2 most favourable isomers. For the large
cluster (TiO2)15, the forest of lines at shorter wavelengths, covering most of
the MIRI wavelength range can only be disentangled at high spectral
resolutions. At longer wavelengths differences in line positions become more
clear. It is important to note that all 4 energetically favorable isomers
share a strong line feature at $\sim 10\mu m$.

###  3.3 Cluster number densities in chemical equilibrium

In order to calculate the total absorbance of a species the cross-sections for
each isomer of each cluster need to be multiplied by its respective number
density in the atmosphere. The chemical equilibrium code GGChem is used to
compute the number densities of all Ti-bearing molecules and clusters from
solar metallicity in a grid of pressures from $p=10^{-15}-700$ bar and
temperatures of $T=300-3000$K (Fig. 5).

Figure 5: Most abundant Ti-containing species in chemical equilibrium at solar
metallicity as a function of temperature and pressure. Letters after clusters
indicate energetic isomer ordering within isomers of the same size, i.e. -A
indicates the GM isomer, while -B would indicate the second most energetically
favourable isomer. Overplotted are temperature pressure profiles for morning
(in black) and evening (in white) terminators of a WASP 96b-like planet and a
WASP 121b-like extrasolar planet.

On the WASP 96b-like planet (solid lines) the cluster composition in the upper
atmosphere is dominated by the largest cluster available in the calculations
(TiO2)15 for both terminators. It is reasonable to assume that larger clusters
would be more favourable if included in the calculations. In the lower
atmosphere between $\sim 1-100$bar, the largest cluster is no longer the most
favoured Ti-bearing compound , but the second-largest cluster, (TiO2)14 is
more abundant. At these pressure levels, a limit to the stability of larger
clusters has been reached, making the smaller cluster more favourable. For the
WASP 121b-like planet (dotted lines), the terminators probe different
conditions leading to different atmospheric compositions. At the top of the
atmosphere at the morning terminator, the TiO2 monomer is dominating, as any
larger clusters are less favourable at these temperatures and pressures. This
changes when moving downwards in the atmosphere, increasing the pressure and
therefore also the densities of the atmosphere. At pressures between $\sim
10^{-11}-10^{-6}$ bar, some growth of clusters is favoured, again leaving the
second largest cluster (TiO2)14 as the most abundant one. As pressures further
increase, i.e. deeper in the atmosphere, larger clusters, here represented by
(TiO2)15, become the most favourable species ($p_{\rm gas}\approx
10^{-6}-10^{-1}$ bar), after which the temperature increases beyond the
stability limit and follows a similar path to the terminators of the WASP
96b-like planet. The hotter evening terminator starts within the TiO-dominated
regime, where temperatures are high enough for the smaller molecule to be
favoured over the TiO2 monomer. This dominance continues down to $p\sim
10^{-5}$ bar, interrupted by a small section ($\sim 10^{-13}-10{{}^{-11}}$
bar) where TiO2 is most abundant. Larger clusters ($N\geq 15$) are never
favoured along this terminator, only the (TiO2)14 cluster is most abundant
between $p\approx 10^{-5}-10^{-2}$ bar. Deep in the atmosphere, where
temperatures are higher, the titanium content is dominated by first the more
thermally stable TiO and at higher pressures the atomic Ti. A slice through
this graph at $p=10^{-8}$ bar showcases the prevalence of different clusters
at different temperatures (Fig. 6).

Figure 6: Concentration of Ti-bearing gas-phase species across a temperature
range of $300-3000$K. Letters after cluster names indicate different isomers,
with -A being the energetically most favourable (GM) isomer. Only species that
reach a concentration of $\frac{n_{mol}}{n_{tot}}>10^{-12}$ are shown.

For temperatures $T<600$K the GM candidate of the largest cluster, (TiO2)15,
is the only relevant Ti-bearing species. As the temperature rises, the second
largest cluster (TiO2)14 rises in abundance, indicating a comparatively
enhanced preference at certain temperatures and pressures, as it becomes the
most abundant species at $T\approx 1100$K. In the region between $\sim
900-1300$K low abundances of many intermediate sized GM clusters and their
respective metastable isomers can be seen. Especially the size $N=12$ cluster
(solid gold) and the $N=2$ cluster (solid red) play a role. The relative
energies between isomers of the same size vary with temperature, so that
different isomers may become the most abundant of their size at different
temperature points. For $N=15$, the energy difference of isomers A and B at
300K is $\Delta(\Delta G_{f}^{\circ}(N=15,T=300{\rm K},{\rm
A-B})))=-92.9\frac{\rm kJ}{\rm mol}$. At 1500K, this difference decreases to
$\Delta(\Delta G_{f}^{\circ}(N=15,T=1500{\rm K},{\rm A-B})))=-37.8\frac{\rm
kJ}{\rm mol}$. At 3000K, isomer B is more energetically favourable, as the
difference is $\Delta(\Delta G_{f}^{\circ}(N=15,T=3000{\rm K},{\rm
A-B})))=+31.5\frac{\rm kJ}{\rm mol}$. The first isomer of $N=15$ that does not
reach the cut-off concentration of $\frac{n_{mol}}{n_{tot}}>10^{-12}$ is
TI15O30-D. The relative energy to the most stable $N=15$ isomer is always
$\Delta(\Delta G_{f}^{\circ})>-92.9\frac{\rm kJ}{\rm mol}$ across the entire
temperature range 300-3000K. The relative cut-off energy is therefore at
around $90\frac{\rm kJ}{\rm mol}$ for $N=15$.

As the temperature increases beyond the stability for clusters, the abundance
of the TiO2 monomer rises. The difference in peak abundance between TiO2 and
(TiO2)15 is about an order of magnitude, indicating that a similar amount of
Ti and O is used to produce these concentrations.

###  3.4 Spectra at different atmospheric pressure levels

To investigate the change of the absorption profiles of (TiO2)N clusters at
different pressure levels in the atmosphere, the spectra are computed at even
logarithmic steps from $p=10^{-15}-10^{2}$ bar. At each pressure level, GGChem
is run to obtain the number densities of all considered species, i.e. the 576
atoms and molecules included in GGChem as well as the 123 clusters from this
work. As the molecular cross-sections span no more than 5 orders of magnitude,
contributions from clusters with abundances more than 5 magnitudes lower than
the most abundant cluster are neglected to save computational cost for the
absorbance calculation. For each of the remaining species, the absorbance is
calculated by multiplying the molecular cross-section with its number density.
All absorbances are then added up to the total absorbance of (TiO2)N clusters
at the given pressure level.

Figure 7: Total absorbance from (TiO2)N clusters at different pressure levels
of a WASP 96b-like exoplanet atmosphere as a function of the wavelength. Top:
Morning terminator. Bottom: Evening terminator.

For the WASP 96b-like planet (Fig. 7) there are only minor differences between
morning and evening terminator, owing to the fact that both temperature-
pressure profile occupy similar regions in the phase-diagram (Fig. 5). At the
bottom of the atmosphere at 1 bar, there are many cluster-sizes and their
isomers contributing to the total spectrum, blending lines and causing the
lack of distinct peaks towards the longer-wavelength part of the spectrum.
Additionally the higher pressure causes more collisional broadening, further
blending lines. This changes when moving upwards in the atmosphere, where the
Ti content is dominated by the large cluster (TiO2)15, with its distinct lines
across the wavelength range. The most visible difference between the morning
and evening terminators are the small peaks at $\sim 10$ and $12\mu m$, that
exist for the evening terminator but not for the morning terminator. As the
evening terminator is hotter, (TiO2)15 becomes less favourable and some
contribution from the next smaller cluster is visible, as these peaks are
caused by absorption from (TiO2)14.

An overall trend is that the absorbance is higher at higher pressures, i.e.
lower in the atmosphere. This is due to the fact that number densities are
much higher at the bottom of the atmosphere.

Figure 8: Total absorbance from (TiO2)N clusters at different pressure levels
of a WASP 121b-like exoplanet atmosphere as a function of the wavelength. Top:
Morning terminator. Bottom: Evening terminator.

The spectra for the WASP 121b-like planet (Fig. 8) are more easily
distinguishable with respect to their terminators. At the evening terminator
the only contributing TiO2 species is the monomer for low pressures levels
($p=10^{-12},10^{-8}$ bar) and high pressure levels ($p=1$bar). The TiO
molecule is the dominant Ti-bearing species for most high and low pressures
($p<10^{-6}$ bar and $p>10^{-2}$ bar). Only at an intermediate pressure level
($p=10^{-4}$ bar) clusters may contribute to the spectrum. As the second
largest cluster (TiO2)14 is the most abundant here, the peaks at 10 and 12$\mu
m$ are well pronounced. It also showcases the strength of absorption of larger
clusters in comparison to the monomer, as the total absorbance with the
contribution of the clusters approaches, and sometimes exceeds, the absorbance
of the monomer at much higher pressures. For the morning terminator only the
upper atmosphere is dominated by the TiO2 monomer. Lower in the atmosphere the
spectra have contributions from many clusters and isomers, as both (TiO2)14
and (TiO2)15 and its isomers are highly abundant. At the bottom of the
atmosphere ($p=1$ bar), fewer isomers and clusters are present due to the
higher temperature, leading to less distinct lines, as well as the higher
collisional broadening causing the lines to blend into each other.

##  4 Discussion

The (TiO2)N clusters considered in this work extend to a maximum size of
$N=15$. As the largest cluster is the most abundant one over a large area of
the parameter space (Fig. 5), this has some implications for our results. When
the largest considered cluster is also the most abundant one, larger clusters,
which are more stable when extrapolating observed trends, are likely to be
more abundant. It is therefore important to note that the spectra at pressure
levels in the ($p_{\rm gas}$, $T_{\rm gas}$)-profiles where (TiO2)15 is the
most abundant species are likely to also contain contributions from even
larger clusters or the bulk-phase of TiO2. A general point can be made about
the differences of the spectra of different cluster sizes. The absorption
cross-sections of larger clusters exhibit a denser forest of stronger lines at
shorter wavelengths and more distinct lines at longer wavelengths (Fig. 2),
when compared to the absorption cross-sections of smaller clusters.

Additionally there are areas in the ($p_{\rm gas}$, $T_{\rm gas}$) parameter
space where not the largest, but the second largest cluster is the most
energetically favourable (green area in Fig. 5, $\sim 1000$K at $10^{-12}$ bar
and $\sim 1200-2000$ K at $10^{3}$ bar). Figure 6 shows that also smaller
cluster sizes have significant abundances. If this trend continues, not all
areas of the parameterspace that are dominated by (TiO2)15 ($\sim<1000$K at
all pressures) in this work are necessarily dominated by larger clusters if
taken into consideration. Another caveat of the spectra produced in this work
originates from the harmonic oscillator approximation to the vibrational modes
of the individual clusters. Guiu et al. (2021) have shown that anharmonicities
and non-uniform thermal effects in the vibrational modes have an impact on
line position and strength starting already at low temperatures. However,
treating all frequency calculations at a full anharmonic level is
prohibitively computationally expensive for the scope of this paper. General
trends of line intensity or line density increasing with cluster size are not
impacted by anharmonicities, enabling qualitative statements on the resulting
spectra.
This work shows, that it is possible to distinguish between the absorption
caused by the TiO2 monomer and any (TiO2)N clusters with $N>1$, as the
absorption cross-sections at wavelengths $\lambda\sim 12-25\mu m$ are up to 3
orders of magnitudes larger for clusters than they are for the monomer.
Detection of such an absorption feature with e.g. JWST/MIRI could indicate the
presence of TiO2 clusters and confirm their role in the formation of cloud
condensation nuclei for cloud formation. This work shows where in the
investigated atmospheres (TiO2)N clusters play a role as absorbing species.
Other absorbing species could play a larger role or, in the case of the
presence of clouds, the atmosphere can become optically thick above the
pressure levels where the clusters contribute to the total absorbance.
Silicate clouds have a feature around 10 $\mu m$ (Wakeford & Sing, 2015),
which is in the vicinity of the shared $\sim 9.5\mu m$ feature of the clusters
from this work. The clusters could therefore explain the existence of a 10
micron feature in the absence of clouds. We note that, beside (TiO2)N, N=1-15,
also other metal oxide clusters exhibit their most intense emissions between
9.5 and 12.5 $\mu$m including clusters of silicates (Zamirri et al., 2019) and
alumina (Gobrecht et al., 2022). This might represent an additional challenge
in identifying and discriminating the cluster species and sizes by IR
spectroscopy. The most interesting ($p_{\rm gas}$, $T_{\rm gas}$) points to
probe are the ones contained in the green area in Fig. 5, where (TiO2)14 is
the most abundant species, as cluster growth towards larger clusters and the
solid phase is inhibited in these regions, but medium sized clusters have
already formed and are contributing to absorption with denser and stronger
cross-sections. These pressures and temperatures can be probed in the upper
atmosphere of a WASP 121b-like exoplanet (black dashed line in Fig. 5).

##  5 Conclusions

This work has shown that (TiO2)N clusters have unique vibrational absorption
spectra. When considering broadening mechanisms of the individual vibrational
modes, the lines are broadened beyond JWST/MIRI spectral resolution throughout
most of the atmosphere of these exoplanet examples. There are clear
differences between the spectra of smaller and larger clusters, as increasing
size correlates with more lines overall, denser lines at shorter wavelengths
and an overall higher cross-section. Different cluster isomers of the same
size have different line positions and intensities, trends in cross-section
such as higher cross-sections towards shorter wavelengths. Different cluster
sizes dominate different areas of the pressure-temperature parameterspace
(Fig. 5), including isomers differing from the GM candidate dominating the
spectra at different points in the studied exoplanet atmospheres. For the WASP
96b-like atmosphere, there is a small difference between the morning and
evening terminator spectra due to a higher relative abundance of the (TiO2)14
cluster in the latter. For the WASP 121b-like planet, the upper atmosphere is
always dominated by the TiO2 monomer and the TiO molecule, giving a possible
clue as to how deep the atmosphere is probed. When it comes to detecting these
building blocks that are relevant for cloud formation on their way to become
CCN, the atmosphere of an ultra-hot Jupiter such as WASP 121b is better suited
for the search when compared to a WASP 96b-like planet.

These model spectra of TiO2 clusters can be implemented into radiative
transfer models of exoplanet atmospheres to gauge whether their detection with
modern instruments is possible. This potential detection or non-detection of
the specific fingerprints of the individual clusters could give an indication
towards the formation of the clusters and their role as cloud condensation
nuclei in these exoplanet atmospheres. When used with observational data
however, the caveats need to be addressed. The harmonic oscillator
approximation used in this work provides a useful and effective means of
analysing vibrational spectra. More complex descriptions including
anharmonicities, ro-vibrational coupling and non-uniform thermal effects can
improve the precision of line positions, but come at much higher computational
cost. The changes in line position induced by these effects are small enough
(Guiu et al. (2021) find line shifts of a few percent at wavelengths
$\lambda\sim 10\mu m$) to be blurred out at low spectral resolutions, e.g.
MIRIs LR mode. Treating these effects can make the data useful at MIRIs medium
spectral resolution. The templates this paper provides can be used to model
the absorption caused by these CCN-forming clusters at lower spectral
resolutions.

###### Acknowledgements.

J.P.S. acknowledges a St Leonard’s Global Doctoral Scholarship from the
University of St Andrews and funding from the Austrian Academy of Science.
Ch.H. and L.D. acknowledge funding from the European Union H2020-MSCA-ITN-2019
under Grant Agreement no. 860470 (CHAMELEON). D.G. acknowledges funding by the
project grant ”The Origin and Fate of Dust in Our Universe” from the Knut and
Alice Wallenberg Foundation (KAW 2020.0081).

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