ExoMol line lists -- LIV: Empirical line lists for AlH and AlD and experimental emission spectroscopy of AlD in $A$ $^1\Pi$ ($v=0, 1, 2$)

New ExoMol line lists AloHa for AlH and AlD are presented improving the previous line lists WYLLoT (Yurchenko et al., MNRAS 479, 1401 (2018)). The revision is motivated by the recent experimental measurements and astrophysical findings involving the highly excited rotational states of AlH in its $A\,^{1}\Pi-{X}\,^{1}\Sigma^{+}$ system. A new high-resolution emission spectrum of ten bands from the ${A}\,^{1}\Pi-{X}\,^{1}\Sigma^{+}$ system of AlD, in the region $17300 - 32000$ cm$^{-1}$ was recorded with a Fourier transform spectrometer, which probes the predissociative $A\,^1\Pi$ $v=2$ state. The AlD new line positions are combined with all available experimental data on AlH and AlD to construct a comprehensive set of empirical rovibronic energies of AlH and AlD covering the $X\,^1\Sigma^+$ and $A\,^1\Pi$ electronic states using the MARVEL approach. We then refine the spectroscopic model WYLLoT to our experimentally derived energies using the nuclear-motion code Duo and use this fit to produce improved line lists for $^{27}$AlH, $^{27}$AlD and $^{26}$AlH with a better coverage of the rotationally excited states of $A\,^1\Pi$ in the predissociative energy region. The lifetimes of the predissociative states are estimated and are included in the line list using the new ExoMol data structure, alongside the temperature-dependent continuum contribution to the photo-absorption spectra of AlH. The new line lists are shown to reproduce the experimental spectra of both AlH and AlD well, and to describe the AlH absorption in the recently reported Proxima Cen spectrum, including the strong predissociative line broadening. The line lists are included into the ExoMol database www.exomol.com.


INTRODUCTION
Aluminium hydride (AlH) has been been observed in the Mira-variable o Ceti (Kaminski et al. 2016), in the photospheres of χ Cygni, a Mira-variable S-star (Herbig 1956) as well as in the spectrum of Proxima Cen (Pavlenko et al. 2022).
The WYLLoT line lists (also known as AlHambra on the ExoMol website) were based on empirical potential energy curves (PECs), Born-Oppenheimer breakdown (BOB) curves, electronic angular momentum curves (EAMC) and ab initio (transition) dipole moment curves that made up the WYLLoT spectroscopic model.The PECs, EAMCs and BOBs curves were obtained by fitting to experimental data on AlH and AlD collected by Yurchenko et al. (2018b), who also provide a detailed review of the literature on AlH spectroscopy up to 2018.The AlH and AlD curves were fitted separately.
Very recently, the AlH WYLLoT line list was used to identify AlH lines in the spectra of cool star Proxima Centauri (M6 V) by Pavlenko et al. (2022).This study showed the limitations of WYLLoT for description of the high J predissociative states of AlH (J > 8) in the A 1 Π − X 1 Σ + (v ′ = 1) system as well as the associated transitions in the A 1 Π-X 1 Σ + (v ′ = 0, 1).In particular, the lines J ′ > 9, v ′ = 1, A 1 Π-X 1 Σ + , which appeared increasingly shifted, were also increasingly broadened through the predissociation of A 1 Π in the spectrum of Proxima Centauri thus indicating that an additional mechanism to describe the predisssociation in AlH is required in addition to the radiative, Doppler and collisional effects, included in WYLLoT spectra simulations.The limitations in the accuracy of the line positions of these lines were attributed to the limitations of the underlying experimental data used in WYLLoT, while the limitations of the WYLLoT line shapes are due to the absence of the predissociative effects in the model.Significantly, Pavlenko et al. (2022) were unable to establish the abundance of AlH in Proxima Centauri using standard bound-bound transitions as they were all saturated, and it was only by using the heavily-broadened predissociative transitions was it possible to retrieve abundances.Up until now ExoMol line lists have lacked any information on line broadening due to predissociation; this has necessitated development of a new data model (Tennyson et al. 2023) allowing inclusion of predissociation into the ExoMol data base.This paper presents our first calculations of lifetime broadening due to predissociaiton.It should be noted that state-resolved photo-dissociation cross sections of AlH were recently computed ab initio by Qin et al. (2021) using an ab initio icMRCI+Q model.
Apart from the rotational excitations in the A 1 Π − X 1 Σ + system of AlH, this work also aims to improve the description of the vibrational excitations in the A 1 Π state.To this end, here we present a new high-resolution emission study of ten bands of the AlD in the A 1 Π − X 1 Σ + system recorded with a Fourier transform spectrometer with the (2-1), (2-2) bands reported for the first time.The previous (lower resolution) conventional studies of these bands go back to Holst & Hulthén (1934) and Nilsson (1948), which were not included into the WYLLoT study due to their limited quality.As a result of this exclusion, the v ′ = 2 quasi-bound state of AlD was not predicted by the WYLLoT model at all.Being quasibound and predissociative, the A 1 Π (v = 2) vibronic level is especially important for modelling the A 1 Π state as it samples higher energies of the AlH potential energy curve (PEC), closer to the potential barrier.
Here we use the extended experimental data of 27 AlH and 27 AlD to improve the spectroscopic model WYLLoT for AlH and AlD and to produce new high-temperature line lists for 27 AlH, 26 AlH and 27 AlD which accurately represent the vibrational states of the shallow A 1 Π state: v = 0, 1 for AlH and v = 0, 1, 2 for AlD.Special attention is paid to the treatment of the predissociative states of AlD and AlH and the reproduction of the experimental predissociative spectra and lifetimes.We also compute a pure continuum contribution to the photo-absorption spectrum of AlH and AlD, which is in-cluded into the line list data following the recently proposed extension of the ExoMol data format (Tennyson et al. 2023).
This work illustrates the importance of experimental data for characterising complex potential energy curves, especially those with low dissociation limits or barriers, where extrapolations of the model can lead to inadequate or incorrect results.
About 50 standard Ne line positions (Palmer & Engleman 1983) were used for the frequency axis calibration.A linear calibration function of 0.999999697x + 0.002987348 was used.The absolute accuracy of the calibration (U cal. ) was estimated as 0.0020 cm −1 , and is limited by the accuracy of the Ne line measurements (Palmer & Engleman 1983).Molecular line positions were determined by fitting Voigt profiles to each measured contour using commercial Bruker software Opus TM (Bruker Optik GmbH 2005).Line position uncertainties (U fitt. ) were evaluated using an empirical relation similar to that given by Brault (1987) where f = 1 is used for the Voigt profile, FWHM is the full-width at half-maximum of the line, N is the true number of statistically independent points in a line width (taking into account the zero filling factor commonly used to interpolate FT spectra), and SNR is the signal-to-noise ratio.The profile fitting uncertainty was significantly smaller than 0.0020 cm −1 for the lines with a typical FWHM ca.0.08 cm −1 and a SNR ≥ 50.
Almost 500 rovibronic frequencies of AlD, A 1 Π − X 1 Σ + system bands were measured, see Tables A1 and A2 in Appendix.The total uncertainty in the measured line positions (U), calculated from the U = U 2 cal.+ U 2 fitt.relation, is about 0.0020 cm −1 for most strong and isolated lines.However, accuracy is lower for the a few weakest and/or blended ones.

MARVEL PROCEDURES
All currently available experimental transition frequencies (both extracted from literature and as part of this work) for AlH and AlD were analysed using the "Measured Active Rotational-Vibrational Energy Levels" (MARVEL) algorithm (Furtenbacher et al. 2007;Császár et al. 2007;Furtenbacher & Császár 2012;Tóbiás et al. 2019).This algorithm consists of inverting a set of transition frequencies with their respective uncertainties into a consistent set of energy levels with the uncertainties propagated from all relevant transitions.
The AlH and AlD experimental data extracted is primarily concentrated around the first two singlet states The work description of the sources is divided between AlH and AlD.
The complete set of experimentally derived energy term values for the AlH represented in the rotational decomposition, can be seen in Figure 3.The quantitative description of MARVEL-derived AlD term values can be found in Table 4.
The MARVEL input, transtion files, and output, energy files, for both AlH and AlD are given in the supporting material.

SPECTROSCOPIC MODEL AND REFINEMENT
We use the variational diatomic nuclear-motion code Duo (Yurchenko et al. 2016) to solve the coupled system of Schrödinger equations for a set of curves defining the spectroscopic model of the X 1 Σ + and A 1 Π system of AlH and AlD.We used the Sinc DVR method for the vibrational degree of freedom on a grid of 1601 points ranging from 0.5 to 13.5 Å.
The AlH/AlD PEC in its the A 1 Π state has a shallow minimum with a small barrier to the dissociation, which can hold only two bound vibrational states in AlH (v = 0, 1) and three (v = 0, 1, 2) in AlD as illustrated in Fig. 4. Furthermore, the highest vibrational states (v = 1 and v = 2, respectively) exhibit strong predissociative characters.
Following Yurchenko et al. (2018b), we use a diabatic representation to model the shallow A 1 Π PECs of AlH and AlD, with two diabatic PECs V1(r) and V2(r) coupled with a term W (r) via a 2 × 2 diabatic matrix The functions V1(r), V2(r) and W (r) are illustrated in Fig. 5 in the case of AlD.The diabatic PEC V1(r) is modelled with an EMO (Extended Morse Oscillator) function (Le Roy et al.

2006) as given by
where Ae is a dissociation asymptote, Ae − Ve is the dissociation energy, re is an equilibrium distance of the diabatic PEC, and ξp is the Šurkus variable given by: V2(r) in Eq. ( 1) is modelled by a repulsive curve playing a role of a dummy state (called here 1 1 Π) and represented by: with the asymptote A A e fixed to the dissociation asymptote of the A 1 Π state, Ae = 25500 cm −1 .
For the coupling function W (r), an inverted EMO PEC with an asymptote of W (r) → 0 at r → ∞ was used where W0 is the height of the coupling at r0, see Fig. 5.The adiabatic PEC of A 1 Π is then given by the lower eigenvalue of the diabatic matrix A in Eq. (1) as The upper diabatic component is disregarded in the rest of the calculations.The expansion parameters defining the diabatic curves were obtained in the fit to the MARVEL energies of AlH/AlD and are given in the supplementary material (see also below).
We used the EMO function to represent the PEC of the X 1 Σ + state with the corresponding expansion parameters taken from and constrained to the values of Yurchenko et al. (2018b).A Born-Oppenheimer Breakdown (BOB) correction curve was added and modelled using the following function: where z is taken as a damped-coordinate given by see also Prajapat et al. (2017) and Yurchenko et al. (2018a).
Here r ref is a reference position equal to re by default and β2 and β4 are damping factors.In order to model the deviation of PEC of AlD from the AlH PEC, a diabatic correction term ∆V (r) was added to the adiabatic PEC V A 1 Π (r), which was modelled with the same form as in Eq. ( 7).
A Λ-doubling empirical curve q(r) was also included in the fit modelled using Eq. ( 7) with a single expansion term where ξp as in Eq. (3).
The AlD PEC has an extra vibrational state, v = 2 (see Fig. 4), which samples a larger range of the PEC than that of AlH.We, therefore, decided to process the AlD curves first by fitting to the experimentally derived (MARVEL) energies, and then refine the AlD spectroscopic model for AlH by fitting to the corresponding MARVEL energies (see above).
Because of the limited amount of experimental data and high complexity of the diabatic model, the fit is highly degenerate.As a work around we applied a rather subjective criterion of physically sensible shapes of the diabatic curves.
During this user-guided fit, attention was paid to the predissociative lifetimes of the A 1 Π v = 2 states of AlD and v = 1 states of AlH, which had to be consistent with the experimental data: predissociative line shapes as in Pavlenko et al. (2022) as well as lifetimes (see discussion below).
The final spectroscopic model of AlD consists of 14 varying parameters reproducing the AlD 423 MARVEL energies with the root-mean-square (rms) error of 0.06 cm −1 .The corresponding curves are illustrated in Figs. 4 and 6.The residuals are shown in Fig. 7.
In the AlH fit, the A 1 Π PEC was constrained to that of AlD.In order to allow for variation in the shapes of the corresponding curves, an 'adiabatic' potential correction term was added to the model using Eq. ( 7).We also introduced a BOB term for A 1 Π of AlH and varied the parameter q0 of the Λ-doubling curve q(x).The X 1 Σ + PEC parameters were still constrained to the values from Yurchenko et al. (2018b), but we refitted the X 1 Σ + BOB term to improve the quality of the model.The AlH spectroscopic model consists of 8 parameters reproducing 346 MARVEL energies (see above) with an rms error of 0.08 cm −1 .The dipole moment and transition dipole moment curves were taken from Yurchenko et al. (2018b).
All curves or parameters defining the AlH and AlD spectroscopic models are given as part of the supplementary material to the paper in the form of Duo input files.

Lifetimes and predissociation line broadening
As part of the ExoMol States files, the lifetimes of species are usually included (see Table 6).In most cases of negligible predissociation effects, the radiative lifetime (of a state i) is computed via where Aij are the Einstein A coefficients for all states j lower than i.According to the recent changes to the ExoMol format (Tennyson et al. 2023), predissociative lifetimes τ prediss are to be included into the line list with the radiative lifetimes, if non-negligible, which is the case for many A 1 Π rovibronic states of AlH and AlD.Here we used the LEVEL program (Le Roy 2017) to estimate lifetimes for the predissociative A 1 Π states of AlH (v = 1) and AlD (v = 2) with the our new PECs.LEVEL uses the uniform semiclassical procedure of Connor & Smith (1981) to compute the widths γ (cm −1 ) of the predissociative states, which we converted to lifetimes via where c is the speed of light in cm s −1 .These are shown in Table 5; our lifetimes show reasonable agreement with the laboratory values obtained by Baltayan & Nedelec (1979) using in a hollow cathode discharge by dye laser excitation as well as the astrophysical estimates of (Pavlenko et al. 2022) from analysis of Proxima Cen predissociative spectrum of AlH.
The LEVEL predissociative values τ prediss are then added to the radiative lifetime to give total lifetime in the States file: The lifetimes can be then used to evaluate the line broadening of the the predissociated lines by inverting Eq. ( 10) for the HWHM γ prediss and apply alone side the collisional value of γ col : This feature is now implemented in the spectrum simulator ExoCross (Yurchenko et al. 2018c;Zhang et al. 2023).

LINE LISTS
Using the new empirical spectroscopic models of AlH and AlD, line lists AloHa for the X 1 Σ + , A 1 Π system were computed with Duo.In intensity calculations, we distinguish bound-to-bound and bound-to-free transitions and compute two line lists, bound-bound and continuum (bound-free).The transitions to the quasi-bound states, especially important in the A 1 Π-X 1 Σ + band, are included in the bound-bound line list.In order to improve the resolution of the continuum spectrum, we use a significantly larger calculation box, with the bond length ranging from 0.5 to 60 Å.Since Duo is a pure bound state variational method, it produces both (quasi-)bound and continuum eigenfunctions ψ λ (r) as part of the same variational calculations.All eigenfunctions are ortho-normal, including the continuum ones, and all satisfy the boundary conditions that they vanish exactly, together with their first derivatives, at both edges of the box.
In order to identify continuum states and then separate them from the (quasi-)bound states, we check if they have non-zero density across a region ∆r adjacent to the outer border rmax against some threshold value ϵmax as given by (see Yurchenko et al. (2023)): where the value of ϵmax must be tuned to the specific case.For the box size L of 59.5 Å, the integration region was chosen as 40 Å. Figure 8 shows examples of reduced radial densities for the bound state A 1 Π, v = 0, J = 9 and two quasi-bound states A 1 Π, v = 1, J = 9 and J = 12 together with an integration box used.The corresponding values of integrated densities ϵ are 0, 2.4 × 10 −4 and 4.8 × 10 −2 , respectively and of the average densities ϵ/L of 0, 3.7 × 10 −6 Å −1 and 7.8 × 10 −4 /Å −1 .Here we adopted the threshold value of ϵ = 0.46, which was tuned to allow the 1930), to be included in the AlH line list.
According to the new ExoMol data structure (Tennyson et al. 2023), bound and quasi-bound states and the corresponding Einstein A coefficients (X-X, X-A) are stored in the bound ExoMol line lists, while continuum A 1 Π 'states' and the corresponding bound-free transitions to/from the bound X 1 Σ + states form temperature-dependent photoabsorption cross sections, see also Pezzella et al. (2022).

(Quasi-)bound line lists of AlH/AlD
A bound ExoMol line list consists of a States file, Transition file and Partition function file computed using bound and quasi-bound wavefunctions.The AlH/AlD line lists cover the wavenumber range up to 30 000 cm −1 (< 0.3333 µm), J = 0 . . .60 of the X 1 Σ + state, Jmax = 25 (A 1 Π) of AlH A 1 Π and Jmax = 36 (A 1 Π) of AlD.The vibrational excitations  of the X 1 Σ + are limited to v = 22 for both AlH and AlD, which is just below the AlH dissociation limit, while for the A 1 Π state these are vmax = 1 (AlH) and vmax = 2 (AlD).A States file (see an extract in Table 6) consists of state IDs, energy term values (cm −1 ), total degeneracies, quantum numbers, energy uncertainties (cm −1 ) and lifetimes (s −1 ).The calculated energies are replaced with the MARVEL values where available.The uncertainties are taken as the MAR-VEL uncertainties for the substituted values.Otherwise, we use the following empirical and rather conservative expression as an estimate for uncertainties of the calculated energies: unc.= 0.01 v + 0.008 v 2 + 0.0002 J(J + 1), X 0.05 v + 0.008 v 2 + 0.002 J(J + 1), A The Transition files (see an extract in Table 7) consists of the IDs of the upper f and lower i states and Einstein A f i coefficients.The latter are the calculated values, i.e. not modified using the MARVEL energies and are given as reference only.We always recommend using energies from the State file for any practical purposes.
The partition function of AlH has been recomputed with the new line list but is very close to the one computed using the WYLLoT line list.This is unsurprising as the main contribution to the partition function is from the ground electronic state, and we, therefore, do not expect any significant changes from the current model of AlH.As before the partition function agrees well with the ones derived by Sauval & Tatum (1984) and Barklem & Collet (2016).
As a part of the AloHa data set, a set of bound-free temperature-dependent photo-absorption cross sections of AlH and AlD are provided.The cross sections are generated on a wavenumber grid of 0.01 cm −1 ranging from 0 to 30 000 cm −1 for a set of 50 temperatures, 100 K, 200 K, . . ., 5000 K.The AlH cross sections should be considered as addons for the spectra produced using the bound-bound line lists of AlH, see Tennyson et al. (2023).
A line list and photo-absorption data for the minor isotopologue 26 AlH was also produced using the 27 AlH spectroscopic model and the same calculation parameters but for a different mass of Al. 26 Al is radioactive with a half-life of about 710 000 years and has been clearly detected in the Milky Way (Diehl et al. 2006).We are not aware of any experimental data on the spectroscopy of 26 AlH.

Temperature-dependent photo-absorption cross sections of AlH/AlD
Using energies and Einstein coefficients from the bound (X 1 Σ + ) and continuum (A 1 Π) solutions, a set of temperature-dependent cross sections of AlH and AlD are computed, using a wavenumber grid of 0.01 cm −1 and a temperature grid ranging from 100 K to 5000 K in steps of 100 K.Here we use the procedure established in Pezzella et al. (2021), where all discrete transition intensities to the continuum states are re-distributed in their vicinity to form continuum photo-absorption cross sections using a Gaussian line profile where αG is the Gaussian half-width-at-half-maximum (HWHM).For the size box of ∼ 60 Å, the distance between the 'continuum' lines does not exceed 26 cm −1 , which adopt as the values of αG.
Figure 9 shows the continuum (bound-unbound) spectrum of AlH at T = 1000 K generated using the Gaussian profile smoothing with HWHM of 26 cm −1 .As an illustration, the original separation between the 'unbound' discrete absorption lines before the smoothing applied can be seen in the same spectrum generated using HWHM = 2 cm −1 .
When computing the total cross sections of a molecule using the extended ExoMol format (Tennyson et al. 2023), we first compute cross sections for a given temperature and pressure using the (quasi-)bound line list and then add them to the photo-absorption cross section for the temperature in question.The pressure dependence of the continuum transitions is ignored.
Figure 10 (left) shows total (bound+continuum) cross sections of AlH for four temperatures and zero pressure computed using the procedure described above.In the same figure, where the 1000 K spectrum is also compared to the ab initio cross sections by Qin et al. (2021).Despitte a generally good agreement between the continuum contributions, our semi-empirical model provides more accurate data for highresolution applications.
In Figure 11, we compare absorption spectra of AlH and AlD simulated using the WYLLoT and AloHa line lists at T = 1000 K.The main differences are (i) the continuum contributions in the spectra and (ii) the v ′ = 2 bands in the spectrum of AlD, missing in the WYLLoT simulations.

Simulations of spectra of AlH and AlD
As illustrations, here we simulate emission spectra of AlH and AlD to compare to the experimental spectra from Szajna et al. ( 2023) and the current work, respectively.All spectra were generated using our open-access Fortran code Ex-oCross (Yurchenko et al. 2018c) 1 .Figure 12 shows a general overview of the AlH emission spectra generated using a Gaussian line profile with the HWHM of 0.08 cm −1 and the rotational temperature of 750 K, where the vibrational temperature of T vib = 4500 K was assumed as in Yurchenko et al. (2018b).
Figure 13 provide a similar illustration for AlD, where the simulations of the regions containing the (1-1), (0-0) and (1-0) bands are shown.The appearance of an extra line in the right display is due to the predissociative effects and discussed below.
For the sake of completeness, we reproduce a comparison of the IR of AlH (X 1 Σ + -X 1 Σ + ) with the emission measurements by White et al. (1993), see Yurchenko et al. (2018b).2023) and simulated AloHa A 1 Π − X 1 Σ + spectrum of AlH in the region of the (1-1), (0-0) and (1-0) bands.For the theoretical spectrum, a rotational temperature of T = 750 K and vibrational temperature T = 4500 K were used; and a Gaussian line profile with FWHM of 0.08 cm −1 was assumed The current line list preserves the high quality of the original ExoMol line list WYLLoT.Pavlenko et al. (2022) recently studied the absorption of AlH in the spectrum of Proxima Cen from the HARPS ESO public data archive (Mayor et al. 2003), recorded over the spectral range from 3780 to 6810 Å with a resolving power R ∼ 115000.This study has demonstrated the importance of the accurate description of line lists of AlH.In particular, the previous AlH line list WYLLoT was shown to deteriorate the higher J spectral lines of AlH in the v = 0 and v = 1.It also could not describe the predissociative broadening effects in this band for J > 19 in v ′ = 0 and J > 8 in v ′ = 1.In Figs. 15 and 16 we simulate the high resolution AlH spectrum in model of stellar atmosphere appropriate for Proxima Cen in the spectral region covering (1-0), (0-0) and (1-1) bands of A 1 Π -X 1 Σ + system.For details of the calculations please consult Pavlenko et al. (2022).To underline the prominent presence of AlH molecular lines in the spectrum one of the two synthetic spectra (the red one) includes only molecular lines.Fig. 15 shows (0-0) band and Fig. 15 presents bands which upper level is v ′ = 1 -the first two panels show the (1-0) band and the next two panels show (1-1) band.The actual list of lines of AlH shows very good consistency with the observed spectrum both in line position and in profiles of diffusive lines.Some differences in line shapes and in depths of broad atomic lines and of diffusive molecular lines may be ascribed to the uncertainty in the continuum tracing of the observed spectrum before its normalization.Simulations show a much better description of the AlH spectrum in Proxima Cen, including the predissociative broadening effects.Even the heavily predissociated lines Q(13), Q(14), Q(15) of v ′ = 1 and Q(23) and Q(24) of v ′ = 0 can be clearly recognised.The presence of Q(25) (See the bottom panel of Fig. 15) is less evident in the observed spectrum.The approach used to model the predissociation line broadening is described in details in Section 4.1.

Breaking-off of predissociation lines of AlH and AlD
Figure 17 shows the experimental spectrum of the (2,2) band of AlD from this work and our attempt to model it using the new AloHa line list.Only the J ′ ≤ 4 lines appear in the experiment while the theory predicts lines with higher J.In fact, higher J (J ≤ 11) predissociative lines were observed experimentally by Nilsson (1948).The effect of "breaking-off" of the predissociative lines in different experimental setups was studied by Bengtsson & Rydberg (1930) and discussed by Herzberg (1939) and was attributed to the non-local thermal equilibrium (non-LTE) effects present in some low pressure conditions.In LTE, the number of predissociating molecules is compensated by new molecules formed by inverse predissociation.This effect can be nicely demonstrated in the comparison of the experimental FT spectrum of AlH from (Szajna et al. 2023) with our that of Proxima Cen as shown in Fig. 19.This figure reproduces our simulation of the Proxima Cen from the bottom display of Fig. 15 and the experimental spectrum is converted to air for a better comparison.It is evident how the emission lines from the FT spectrum break off for J ′ > 8 in comparison to the spectrum of Proxima Cen.It should be noted that this is not due to the lower temperature conditions of the FT spectrum.Indeed, if we assumed the LTE, the population of the corresponding states with J ≥ 8 is comparable to those visible in the spectrum at T = 750 K, indicating that the breaking-off of J ≥ 8 in the experiment is due to non-LTE effects.
The effect can be also seen in right display of Fig. 13, where extra Q(J) lines (J > 7) of AlH appear compared to the experimental spectrum.

Collisional line-broadening parameters
Collisional line-broadening parameters of AlH for the X 1 Σ + state with different partners (H2, He, N2, and AlH) have been computed using the MCRB approach (Antony et al. 2006).This is a semi-classical approach where internal degrees of freedom of the radiator and the perturber are treated quantum-mechanically and their relative translational motion is described classically.Line broadening can be said to appear as a consequence of monochromatic wave-train interruption when the radiating molecule is interacting with a perturber during a collision.The magnitude of this effect for completed collisions is described with a scattering matrix, which in this approach is expanded up to the second order in perturbation theory (Hartmann et al. 2008).The model interaction potential between the radiator (AlH) and a perturber is constructed from short-ranged and long-ranged parts.The former, repulsive part is obtained from atom-atom Lennard-Jones contributions (Svehla 1962), while the latter is composed from electrostatic interactions and uses molecular multipole moments from NIST (Johnson 2022).Trajectories are computed within the rigid rotor approximation using equilibrium geometries of the X 1 Σ + state and a different isotropic potential to drive them (Loukhovitski & Sharipov 2021).Vibrational dependence of broadening parameters have also been modelled, assuming that only the changes in longranged van-der-Waals interactions with vibrational state significantly change scattering cross-sections.Diagonal rovibrational matrix elements of the X 1 Σ + electric dipole moment ⟨vJ |µ 2 (r)| vJ⟩ and isotropic polarizability ⟨vJ | αiso(r) | vJ⟩ curves required for this part were computed using Duo's ro-vibrational wavefunctions |vJ⟩.The polarizability curve αiso(r) was computed ab initio with MOLPRO (Werner et al. 2020) using the CCSD(T)/aug-cc-pVQZ level of theory as second order derivatives of the X 1 Σ + energy with respect to the electric field.It is shown in Fig. 18.The dipole moment curve of X 1 Σ + was taken from Yurchenko et al. (2018b).
It is worth mentioning that this semi-classical approach works best when the interaction potential is fitted to improve agreement with experimental broadening coefficients.Without this adjustment, theoretical values usually overestimate experimental ones (Ma et al. 2013).However, to the best of our knowledge, no experimental measurements of AlH broadening by any molecules are available, so our broadening coefficients are presented without any adjustments.
The new broadening parameters of AlH are included into the ExoMol database using the ExoMol diet format (Barton et al. 2017), which is based on the representation of the temperature-and pressure-dependence of the half-width-athalf-maximum γ (cm −1 /atm) by a single-power law: where T ref is the HITRAN reference temperature of 296 K and P ref is the reference pressure of 1 atm.The J-dependence is best parameterised by the standard HITRAN | m | dependence , where m = J lower + 1 for the R-branch and m = −J lower for the P-branch.We have therefore introduced a new ExoMol diet type m0.An example of the the diet file for AlH broadened by H2 is given in Table 8.Our MCRB calculations predict a mildly sloping dependence on m (and therefore J).The m0 type is implemented and now available in ExoCross.

CONCLUSIONS
Improved line lists for AlH and AlD (X 1 Σ + , A 1 Π) are presented.They now provide a better description of the high J predissociation effects in the A 1 Π state and a proper description of predissociative line broadening via the inclusion of the predissociative lifetimes into the ExoMol States file.The AlD line list now contains the v ′ = 2 A 1 Π predissociative band, which was not present in WYLLoT.As part of the AloHa line list, we also provide temperature-dependent photo-absorption cross sections of AlH/AlD.These data are complimentary and should be added to the temperature-and pressure-dependent cross sections produced from the boundbound line lists.The AlH AloHa line lists are freely available at www.exomol.com.
The new AlH/AlD line lists can be used for some modelling and analysis of non-LTE spectral effects, at least as far as the radiative rates are concerned.As it is typical for diatomics, the hot vibronic bands of AlH are well separated (see Fig. 11), which helps estimate the vibrational temperatures (populations) of the corresponding (lower) states and thus to assess the presence and magnitude of non-LTE effects, see, e.g.Wright et al. (2022Wright et al. ( , 2023)).However, a full non-LTE treatment would also require the other contributions to the statistical population balance, including collisional rates and reaction rates (van der Tak et al. 2007), which will need further work.
It should be noted, that there are experimental data on R( 5) R( 6) R( 7) R( 8) R( 9) R( 10) R( 11)  6) R( 7) R( 8) R( 9) R(10) 0-0 P( 23 Normalized Flux P(5) P( 6) P( 7) P(8) P( 9) P( 10) In AlH the lifetime broadening is due to tunneling in the A 1 Π state.More commonly predissociation is caused by tun-neling.Work reporting extension of Duo to allow for predissociation due to curve crossing will be reported elsewhere and line lists for molecules such as OH, for which this mechanism is important, will presented in this journal in due course.The energy of the spin-rotation interaction is defined as ESR = C ⊥ C, where C is from Eq. (B4) and C ⊥ is nuclear magnetic coupling constant.It has been shown that C ⊥ ( 1 H) is too small (on the scale of 10 kHz Gee & Wasylishen (2001)) to perturb the spectra and hence is ignored in our calculations.From the above definitions we can derive the following relationship: where Y ′ and C ′ represent the final state energy, Y ′′ and C ′′ represent the initial state energy in a hyperfine transition and ν0 is the "unperturbed" rotational transition frequency if there were no quadrupole and spin-rotation interactions.
After solving for ν0 in all hyperfine transitions, the "true" ν0 is found by the method of weighted averages: where ν0,i is individually derived rotational transition frequency using Eq.(B5) and wi = 1 σ 2 0 with σ0 as the propagated standard deviation for each transition.A standard Gaussian error propagation was used to obtain the value of σ0, with the overall expression being the following: where σorig is the originally reported uncertainty in hyperfine measurement, σeQq is the uncertainty in eQq and σC ⊥ is the uncertainty in the C ⊥ ( 27 Al).The values for eQq and C ⊥ used in our calculations and their respective uncertainties can be in Table B.

APPENDIX C: ASTROPHYSICAL MEASUREMENTS OF ALH
As part of this work, by using an output from the updated Duo model for AlH, a number of new ro-vibronic transitions in the Proxima Cen spectrum from the (1-1) A 1 Π − X 1 Σ + band of AlH have been identified, following the same procedure as described previously by Pavlenko et al. (2022).The remeasured line positions can be seen in Table C1. Figure C1 shows the Proxima Cen spectrum, with comparison to the calculated AlH spectrum and nearby atomic lines.20) Q( 21) Q( 22) Q( 23) Q( 24) P( 10) P( 11) P( 12) P( 13) P( 14) P( 15) P( 16) P( 17

Figure 1 .
Figure1.High-resolution FT-VIS emission spectrum of the AlD, A 1 Π − X 1 Σ + (1-0) band recorded with SNR of about 200 : 1 and FWHM of lines about 0.08 cm −1 .Bands of the 1 − v ′′ progression are observed up to the J ′ max = 19 lines due to the predissociation in the A 1 Π (v = 1) level.

Figure 2 .
Figure 2. Experimentally derived reduced energy term values for the AlH in the rotational decomposition.

Figure 3 .
Figure 3. Experimentally derived reduced energy term values for the AlD in the rotational decomposition.

Figure 4 .
Figure 4. Refined potential energy curves of X 1 Σ + and A 1 Π of AlH and AlD and the corresponding (quasi-)bound vibrational energy term values.

Figure 7 .
Figure 7. Obs.-calc.residuals between the MARVEL and calculated energies of AlH and AlD using the corresponding refined models.

Figure 9 .
Figure9.Example of the photo-absorption continuum cross sections of AlH at T = 1000 K generated using a Gaussian line profile of HMWM = 26 cm −1 (red line), overlaid with the same spectrum but using HMWM = 2 cm −1 (grey line).

Figure 10 .Figure 11 .
Figure10.Temperature dependence of the AlH absorption spectrum using the Gaussian profile with HWHM = 1 cm −1 (left) and a comparison with the T = 1000 K cross-sections of AlH byQin et al. (2021)

Figure 12 .
Figure12.Comparison of the experimental FT-VIS bySzajna et al. (2023) and simulated AloHa A 1 Π − X 1 Σ + spectrum of AlH in the region of the (1-1), (0-0) and (1-0) bands.For the theoretical spectrum, a rotational temperature of T = 750 K and vibrational temperature T = 4500 K were used; and a Gaussian line profile with FWHM of 0.08 cm −1 was assumed

Figure 15 .
Figure15.Comparison with the observed spectrum of AlH of Proxima Cen analysed byPavlenko et al. (2022) shown by black line for the (0-0) band spectral range.Here and onward, we have adhered to the procedure of computation and identification of spectral features outlined byPavlenko et al. (2022).The blue line marks the synthetic spectrum including atomic and molecular species, the red line spectrum is calculated including AlH lines only.A version of this figure, with the atomic lines also indicated, is provided in the Appendix, see C1.

Figure 16 .
Figure 16.Comparison of the spectrum of AlH of Proxima Cen by Pavlenko et al. (2022) with the synthetic spectrum.Two upper panels show the spectral range of the 1-0 band and the two lower of the 1-1 band.The observed spectrum is shown by black line.The blue line marks the synthetic spectrum including atomic and molecular species, the red line spectrum is calculated including AlH lines only.

Figure 17 .Figure 18 .
Figure17.Comparison of the experimental FT-VIS and simulated ExoMol A 1 Π-X 1 Σ + (2-2) emission band of AlD.For the theoretical spectrum a rotational temperature of T = 750 K and a Gaussian line profile with HWHM of 0.08 cm −1 were assumed.

Figure 19 .
Figure 19.Comparison of the experimental FT-VIS (Szajna et al. 2023) of AlH (bottom) and the Proxima Cen spectrum (top), observation and simulation (T = 2900 K) in the region of A 1 Π-X 1 Σ + (1-1) band using the new ExoMol line list as in the bottom display of Fig. 15.The blue/red arrows indicate the AlH lines that are present/disappear in the experiment.

Figure C1 .
Figure C1.The same as in Fig.15, but with atomic lines indicates: Comparison with the observed spectrum of AlH of Proxima Cen byPavlenko et al. (2022) shown by black line for the (0-0) band spectral range.Here and onward, we have adhered to the procedure of computation and identification of spectral features outlined byPavlenko et al. (2022).The blue line marks the synthetic spectrum including atomic and molecular species, the red line spectrum is calculated including AlH lines only.

Table 1 .
List of transition lines used in MARVEL procedure for AlH grouped by source.

Table 2 .
Description of AlH energy levels derived from MARVEL.
Statev J Range Unc.Range cm −1 Avg. of Unc.cm −1 Range of energy levels cm −1

Table 3 .
List of transition lines used in MARVEL procedure for AlD grouped by source.

Table 4 .
Description of AlD energy levels derived from MARVEL.

Table 6 .
Extract from the states file of the line list for AlH .State energy term values in cm −1 , MARVEL or Calculated (Duo).

Table 7 .
Extract from the transitions file of the line list for AlH.

Table 8 .
ExoMole diet line broadening file for AlH with H 2 as perturber.

Table B1 .
Spectroscopic constant values and uncertainties that were used to derive pure rotational transition from hyperfine data.