ExoMol line lists -- XLVII. Rovibronic molecular line list of the calcium monohydroxide radical (CaOH)

Any future detection of the calcium monohydroxide radical (CaOH) in stellar and exoplanetary atmospheres will rely on accurate molecular opacity data. Here, we present the first comprehensive molecular line list of CaOH covering the \A--\X\ rotation-vibration-electronic and \X--\X\ rotation-vibration bands. The newly computed OYT6 line list contains over 24.2 billion transitions between 3.2 million energy levels with rotational excitation up to $J=175.5$. It is applicable to temperatures up to $T=3000$~K and covers the 0\,--\,35\,000~cm$^{-1}$ range (wavelengths $\lambda>0.29$~$\mu$m) for rotational, rotation-vibration and the \A--\X\ electronic transition. The strong band around 16\,000~cm$^{-1}$ ($\lambda = 0.63$~$\mu$m) is likely to be of interest in future astronomical observations, particularly in hot rocky exoplanets where temperatures can become extremely high. The OYT6 line list has been generated using empirically-refined \X\ and \A\ state potential energy surfaces, high-level \textit{ab initio} transition dipole moment surfaces and a rigorous treatment of both Renner-Teller and spin-orbit coupling effects, which are necessary for correctly modelling the CaOH spectrum. Post-processing of the CaOH line list has been performed so as to tailor it to high-resolution applications, i.e.\ by replacing calculated energy levels with more accurate empirically-derived values (where available), hence improving the accuracy of the predicted line positions in certain regions. The OYT6 line list is available from the ExoMol database at http://www.exomol.com and the CDS astronomical database.


INTRODUCTION
The calcium monohydroxide radical ( 40 Ca 16 O 1 H) has been cited as a missing opacity source in the study of M-dwarf photospheres using the BT-Settl model (Rajpurohit et al. 2013). The atmospheres of hot rocky super-Earth exoplanets are also expected to contain spectroscopic signatures of CaOH (Bernath 2009;) but a lack of molecular line list data in the necessary spectral regions inhibits detections. Notably, it is the rotation-vibration-electronic (rovibronic) spectrum composed of the strongÃ 2 Π-X 2 Σ + band system that is of interest. This has motivated the ExoMol database (Tennyson et al. , 2016Tennyson & Yurchenko 2012) to work on the production of a high-accuracy line list for CaOH and this paper is the final stage in that process.
In previous work, we performed an extensive literature search and extracted all meaningful rovibronic spectroscopic transitions (around 3200) of CaOH (Wang et al. 2020) and processed them using the MARVEL (measured active rotation vibration energy levels) algorithm Császár et al. 2007;Furtenbacher & Császár 2012;Tóbiás et al. 2019). This procedure produced a consistent dataset of empirical-quality energy levels, each possessing a measurement uncertainty and unique quantum number labelling. The advantages of having such a dataset are twofold: Firstly, the theoretical spectroscopic model of the molecule, primarily the molecular potential energy surface (PES), can be fine-tuned in calculations so as to improve the accuracy of the predicted line positions and to a lesser extent the line intensities. Secondly, when post-processing the final line list the computed energy levels can be substituted with the more accurate MARVEL values (where available). Doing so can dramatically improve the accuracy of certain line positions, making the final list list suitable in certain windows for studying exoplanet atmospheres at high spectral resolution (Snellen 2014;Birkby 2018). Currently, the adaptation of molecular line lists in the ExoMol database for high-resolution applications is being actively pursued, for example, see Bowesman et al. (2021).
A full-dimensional ab initio spectroscopic model of theÃ 2 Π-X 2 Σ + band system of CaOH has also been reported by the authors (Owens et al. 2021a). High-level ab initio theory was used to construct new potential energy and transition dipole moment surfaces, and both Renner-Teller and spin-orbit coupling effects, which are essential for correctly reproducing the CaOH spectrum, were accounted for in the calculations. Whilst the groundX 2 Σ + potential energy surface (PES) was empirically refined to the MARVEL dataset of energy levels, vastly improving its accuracy, theÃ 2 Π state PES was not rigorously refined as further development of the computer program EVEREST (Mitrushchenkov 2012) was necessary. This has now been completed for the present work, ultimately leading to considerably more accurateÃ 2 Π state energy levels andÃ-X transition wavenumbers (up to several orders-of-magnitude). Doing so has paved the way for the production of a comprehensive, high-temperature line list of CaOH which we report below.
It is worth mentioning that the spectroscopic model and associatedÃ-X line list produced in this work could greatly assist the design of laser cooling schemes in ultracold molecule research and precision tests of fundamental physics. These fields have been focusing on the alkaline earth monohydroxide radicals, notably CaOH (Kozyryev et al. 2019;Augustovičová & Bohn 2019), because of the favourable rovibronic energy level structure. Already, direct laser cooling of CaOH to temperatures near 1 millikelvin has been performed in a one-dimensional magneto-optical trap (Baum et al. 2020) and robust experimental schemes to extend cooling of CaOH into the microkelvin regime have been suggested (Baum et al. 2021) and successfully implemented (Vilas et al. 2022). The calculation of Einstein A coefficients between the different molecular states in CaOH would allow branching ratios and decay routes to be analysed, and such work has previously been carried out for other molecules using ExoMol line lists (Owens et al. 2017).

Refinement of theÃ 2 Π potential energy surface
In CaOH, the Renner-Teller effect (Renner 1934) causes theÃ 2 Π state to split into two Renner surfaces A and A at bent molecular geometries. Any correct description of the spectrum of CaOH must account for the Renner-Teller effect (Li & Coxon 1996, 1995Coxon et al. 1994;Jarman & Bernath 1992;Coxon et al. 1991;Hilborn et al. 1983). In our spectroscopic model, the A and A surfaces are each represented by their own analytic function, essentially an eighth-order polynomial in terms of linear expansion variables of the bond lengths and bond angle, see Owens et al. (2021a) for a detailed description. The A and A potential functions share the same parameter values for stretching only terms but possess different parameter values for terms involving the bending mode.
Despite numerous studies of theÃ-X band system of CaOH (Bernath & Brazier 1985;Coxon et al. 1991;Coxon et al. 1992Coxon et al. , 1994Li & Coxon 1995;Dick et al. 2006), only four vibrational levels of theÃ 2 Π state have been characterised experimentally, namely the ground vibrational state (0, 0, 0), the first excited Ca-O stretching ν1 state (1, 0, 0), and the first and second excited bending ν2 state (0, 1 1 , 0) and (0, 2 0 , 0). Here, vibrational states are labelled with normal mode notation (v1, v L 2 , v3), where v1 and v3 correspond to the symmetric and asymmetric stretching modes, respectively, and v2 labels the bending mode. The quantum number L is related to the absolute value of the vibrational angular momentum quantum number l associated with the ν2 bending mode, L = |l|. It is the ν2 bending mode which exhibits Renner-Teller splitting. No studies of the O-H stretching ν3 mode have been carried out. As a result, we do not allow potential parameters related to the ν3 mode to vary in the refinement and they are instead fixed to the original ab initio values determined in Owens et al. (2021a). Computed energy levels and transitions involving the ν3 mode will therefore not be as accurate as those involving only the ν1 and ν2 modes in the final line list. This should not be overly problematic, however, as rovibronic transitions from the groundX 2 Σ + electronic state to the ground, ν1 and ν2 vibrational levels in theÃ 2 Π state are the strongest and these levels have all been characterised by experiment and utilised in the refinement.
TheÃ 2 Π state PESs were refined using the EVEREST code and the new implementation is discussed in Sec. 2.2. A total of 10 potential parameters of the A and A surfaces were simultaneously varied in the refinement along with the zeroth-order parameter for the spin-orbit coupling surface, which was fine-tuned to ensure accurate spin-orbit splitting in the energy level structure of CaOH. 308 term values up to J = 15.5, where J is the total angular momentum quantum number, were used in the refinement and were reproduced with a root-mean-square (rms) error of 0.318 cm −1 . A weighting scheme based on the measurement uncertainty and number of transitions involving the energy level was utilised. This information was obtained from the CaOH MARVEL analysis (Wang et al. 2020), essentially giving more importance in the refinement to well characterised term values with smaller uncertainties.
The results of theÃ 2 Π PES refinement are illustrated in Fig. 1, where we have plotted the residual errors ∆E(obs − calc) between the empirically-derived MARVEL and calculated energies up to J = 15.5. Generally speaking, the residual errors in each vibrational state exhibit similar behaviour with steadily increasing errors with increasing J (or energy as seen in Fig. 1) in both positive and negative directions. For example, in the ground vibrational state (yellow dots in Fig. 1) the residual errors for the e-parity levels become more negative with J (and energy), while the errors for the f -parity levels increase positively with J. There is also an outlier corresponding to the J = 1.5, f -parity energy level at 16032.5597 cm −1 and we suspect this to be an issue with the empirically-derived MARVEL value given the consistent behaviour of the residual errors for the other energy levels in the ground vibrational state.
Interestingly, the second excitedB 2 Σ + state of CaOH has a minimum around 18 000 cm −1 and couples to theÃ 2 Π state through spin-orbit coupling and linear vibronic coupling. As we have neglected theB 2 Σ + state and its associated couplings in our spectroscopic model, there will no doubt be an impact on the accuracy that could be achieved had we been able to incorporate them. Even adding a "dummy" coupling curve as a free fitted parameter can improve the accuracy of the spectroscopic model as is commonly seen for diatomics studied by the ExoMol project, for example, see CaH (Owens et al. 2022). TheÃ 2 Π state is expected to acquire a significant fraction ofB 2 Σ + character through the vibronic interaction (Coxon et al. 1994) and recently it was shown that the inclusion of theB-Ã couplings can improve the calculation of vibronic branching ratios in CaOH (Zhang et al. 2021), which are needed when designing efficient laser-cooling schemes. That said, the effect of theB 2 Σ + state on theÃ 2 Π state is to some extent absorbed into the model by empirically refining theÃ 2 Π PESs.

Extension of the EVEREST code
The EVEREST code (Mitrushchenkov 2012) was used in all rotation-vibration (rovibrational) and rovibronic line list calculations and for theÃ 2 Π PES refinement. EVEREST is capable of treating interacting electronic states with spin-dependent coupling and Renner-Teller effects in triatomic molecules. The program is general in its design and employs an exact kinetic energy operator. Extending the functionality of EVEREST to be able to perform PES refinement required several developments.
The PESs of the ground state VX and excited states V A , V A are represented using the Taylor-type expansion where the coordinates with the internal stretching coordinates r1 = rCaO, r2 = rOH, the interbond angle α = ∠(CaOH), and the equilibrium parameters r eq 1 , r eq 2 and αeq (see Owens et al. (2021a)). The refinement procedure is a least-squares fitting of the potential parameters f (S) ijk and structural parameters r eq i and αeq to the observed (MARVEL) energies of CaOH, realized through the minimization of the functional using Newton's steepest descent method. Here E where |n is the corresponding rovibronic EVEREST wavefunction. This method takes advantage of the linear form of the f Some of the A and A potential parameters of the two Renner-Taylor components corresponding to the linear geometry are shared between the two surfaces, namely the stretch-only terms f ij0 . The corresponding derivatives are given by the chain rule as follows, ∂En The derivatives wrt to the non-linear parameters r eq 1 , r eq 2 and αeq are obtained using finite differences, for example, ∂En ∂r eq 1 = En(r eq 1 + ∆r) − En(r eq 1 − ∆r) 2∆r , where ∆r is a small displacement, typically taken as 0.1% of the parameter value, 0.002Å or 0.2 • for the bond lengths or inter-bond angles, respectively. The implementation of the refinement procedure closely follows a previously developed Fortran wrapper for the triatomic nuclear motion program DVR3D reported in Polyansky et al. (2018), where the underlying ab initio PES was used to constrain the shape of the refined PES via a simultaneous fit to the original ab initio energies. The fitting weights were initially set to be inversely proportional to the experimental (MARVEL) uncertainties and then adjusted via Watson's robust weighting scheme (Watson 2003). The expectation values of the potential expansions are now a part of the EVEREST program. The least-squares fitting procedure was implemented as an external, generalized fitting wrapper written in Fortran 95.

EVEREST line list calculations
The final list list calculations in EVEREST used the same set up and parameters as the PES refinement. Specifically, valence bond length-bond angle coordinates with a discrete variable representation (DVR) basis composed of 100 Sinc-DVR functions on both the Ca-O bond in the 2.6-7.0 a0 interval and the O-H bond in the 1.1-6.0 a0 interval, and with 120 Legendre functions for the ∠(CaOH) bond angle. The J = 0 vibrational eigenfunctions were computed up to 10 000 cm −1 above the lowest vibronic state for 0 K 27, where K = |Λ + l| (Λ and l are the projections of the electronic and vibrational angular momenta along the linear axis) from a Hamiltonian with a dimension of 10 000. For K 1, the Renner-Teller effect was explicitly taken into account by solving the coupled A /A problem, see Mitrushchenkov (2012) for the full methodology. The full rovibronic Hamiltonian including spin-orbit coupling was built and diagonalized using these vibronic states for J up to 175.5, where J is the total angular momentum quantum number. Convergence of the computed rovibronic states was checked by running test calculations with K 50, increasing the Hamiltonian dimension, and using larger DVR grids for the coordinates. The final selected EVEREST calculation parameters produced sufficiently converged results and ensured that calculations were computationally tractable. All rovibrational transitions in theX 2 Σ + -X 2 Σ + band and rovibronic transitions in theÃ 2 Π-X 2 Σ + band were calculated to produce a line list containing 24,215,753,701 transitions between 3,187,522 states with rotational excitation up to J = 175.5. The CaOH line list covers the 0 -35 000 cm −1 range (wavelengths λ > 0.29 µm).

Line list format
The ExoMol data format has been discussed in detail ) and is used for all the line lists in the ExoMol database. The .trans file, see Table 1, contains the computed transitions with upper and lower state ID labels, Einstein A coefficients (in s −1 ) and transition wavenumbers (in cm −1 ). The .states file, see Table 2, contains the computed rovibronic energy levels (in cm −1 ), each labelled with a unique state ID counting number and quantum number labelling.
To adapt the CaOH line list for high-resolution applications we have replaced the computed EVEREST energy levels and their uncertainties with the empirical MARVEL values where available. This was done for a total of 1614 states up to J = 61.5 for the groundX 2 Σ + and excitedÃ 2 Π states. The final column in the .states file indicates whether the energy level is either calculated (Ca) or from MARVEL (Ma). All computed energy levels have an estimated uncertainty of 10 cm −1 , which is  Wang et al. (2020). State energy (in cm −1 ). g i : Total statistical weight, equal to gns(2J + 1).
a conservative estimate to ensure that users are aware of the difference in reliability between calculated and "MARVELised" lines, particularly for high-resolution applications. The penultimate column of the .states file is the EVEREST calculated energy for reference. The quantum numbers used to label the rovibronic states of CaOH are listed in Table 2 and discussed in detail in Wang et al. (2020).

Temperature-dependent partition functions
Intensity simulations require knowledge of the temperature-dependent partition function Q(T ), defined as where gi = gns(2Ji + 1) is the degeneracy of a state i with energy Ei and total angular momentum quantum number Ji, and the nuclear spin statistical weight gns = 2 for CaOH. Partition function values were computed on a 1 K grid in the 1 -4000 K range and can be downloaded from the ExoMol website at www.exomol.com along with the OYT6 line list. The convergence of Q(T ) as a function of J is shown in Fig. 2 for different temperatures. At T = 3000 K, the convergence of the partition function with J is slower and we recommend this as a soft upper limit for using the OYT6 line list at elevated temperatures. Application of the OYT6 line list above this temperature may result in a progressive loss of opacity. In Fig. 3 we have compared partition function values up to T = 4000 K computed using the OYT6 line list against the JANAF CaOH values (Irwin 1988). The JANAF values were calculated using a simple molecular model that employs a rigid rotor, simple harmonic oscillator approximation. As such, although the general behaviour between the OYT6 and JANAF values is in reasonable agreement, we expect the OYT6 values to be more accurate at least up to T = 3000 K. This is because the OYT6 values are computed by summing over all meaningful rotation-vibration energy levels in the lowest two electronic states of CaOH. Interestingly, the contribution fromÃ 2 Π state energy levels in the summation of Eq. (10) to the partition function is near-negligible. For example, the partition function at T = 3000 K calculated using only groundX 2 Σ + state energy levels in the summation of Eq. (10) showed a 0.08% difference to the full OYT6 value and this is the largest difference encountered.

Simulated spectra
There is no meaningful intensity information in the literature on theÃ-X band system that we can compare to, making it difficult to evaluate the accuracy of our computed line intensities against measured spectra. Transition intensities computed using ab initio DMSs are comparable to, and in certain instances, more reliable than experiment (Yurchenko 2014;Tennyson 2014). As an estimate we would expect the CaOH line intensities to be within 5-10% of experimentally determined values. It is also worth noting that strong electronic transitions are less sensitive to the shape of the transition DMSs, given that vibronic intensities can be relatively well modelled using only a single value for the transition dipole times a Franck-Condon factor.
In Fig. 4, the temperature dependence of the CaOH spectrum is illustrated and we have simulated absolute absorption cross-sections at a resolution of 1 cm −1 using a Gaussian line profile with a half width at half maximum (HWHM) of 1 cm −1 . All spectral simulations were done using the ExoCross program (Yurchenko et al. 2018b). The strongest rovibronic features  of the OYT6 line list are around 16 000 cm −1 and even at high temperatures (T = 3000 K) these still dominate the CaOH spectrum. This region is significant for atmospheric studies of hot rocky super-Earths, a class of exoplanets that are in close proximity to their host star and exposed to extremely high temperatures, e.g. up to 4000 K.
Closer inspection of the main infrared part of the spectrum is shown in Fig. 5 at T = 1000 K. The fundamental wavenumbers of theX 2 Σ + state of CaOH occur at ν1 ≈ 609, ν2 ≈ 353 and ν3 ≈ 3792 cm −1 (Owens et al. 2021a). The spectrum exhibits a relatively slow flattening of band intensities with increasing wavenumber and similar behaviour was also observed in line list calculations of the alkali metal hydroxides KOH and NaOH Owens et al. (2021b), which both possess linear molecular structures.
In Fig. 6, absolute absorption line intensities (in units of cm/molecule) at T = 700 K have been plotted for the strongest regions of the OYT6 spectrum. Line intensities were simulated using the expression,   Figure 7. Comparison of the OYT6 stick spectrum at T = 300 K in the microwave region against all transition data from the Cologne Database for Molecular Spectroscopy (CDMS) (Müller et al. 2001;Müller et al. 2005).
where A f i is the Einstein A coefficient of a transition with wavenumber ν f i (in cm −1 ) between an initial state with energy Ei and a final state with rotational quantum number J f . Here, k is the Boltzmann constant, h is the Planck constant, c is the speed of light and T is the absolute temperature. The nuclear spin statistical weight gns = 2 for CaOH and Q(T ) is the temperature-dependent partition function. Interestingly, the structure of the bands using the refinedÃ 2 Π state PES are quantitatively different to those predicted using the original ab initio PES, see Fig. 5 in Owens et al. (2021a). This behaviour has been encountered before by the ExoMol project and highlights the fact that rovibronic spectra can be very sensitive to the quality of the underlying PESs with both line positions and intensities being affected.
In Fig. 7, a CaOH OYT6 stick spectrum at T = 300 K, computed using Eq. (11), is plotted against pure rotational transitions in the groundX 2 Σ + state extracted from the Cologne Database for Molecular Spectroscopy (CDMS) (Müller et al. 2001;Müller et al. 2005). The rotational band is reproduced well with the OYT6 spectrum possessing more structure and lines since it extends up to J = 175.5, compared to CDMS which contains 102 lines up to J = 50.5. The CDMS line intensities are noticeably stronger, however, and they were computed using a more simplified model based on a single dipole moment value of µ = 1.47 Debye (Steimle et al. 1992). Our value for the groundX 2 Σ + state dipole is µ = 1.09 Debye and this was computed using high-level ab initio theory (restricted coupled cluster with a large augmented correlation consistent basis set RCCSD(T)/aug-pwCVQZ-PP). This value is closer to other CaOH dipole values of 0.98 Debye from ab initio calculations (Bauschlicher et al. 1990) and a semiempirical value of 1.2 Debye (Mestdagh & Visticot 1991), and would explain the weaker OYT6 line intensities. These comparisons also give us confidence in the reliability of the DMSs used in intensity calculations for the OYT6 line list.