ExoMol line lists XXVIII: The rovibronic spectrum of AlH

A new line list for AlH is produced. The WYLLoT line list spans two electronic states $X\,{}^1\Sigma^+$ and $A\,{}^1\Pi$. A diabatic model is used to model the shallow potential energy curve of the $A\,{}^1\Pi$ state, which has a strong pre-dissociative character with only two bound vibrational states. Both potential energy curves are empirical and were obtained by fitting to experimentally derived energies of the $X\,{}^1\Sigma^+$ and $A\,{}^1\Pi$ electronic states using the diatomic nuclear motion codes Level and Duo. High temperature line lists plus partition functions and lifetimes for three isotopologues $^{27}$AlH, $^{27}$AlD and $^{26}$AlH were generated using ab initio dipole moments. The line lists cover both the $X$--$X$ and $A$--$X$ systems and are made available in electronic form at the CDS and ExoMol databases.


INTRODUCTION
Aluminium is one of the commoner interstellar metallic elements, with a cosmic abundance of Al/H = 3 × 10 −6 but AlH has only been rather sparingly observed. AlH was detected in the photospheres of χ Cygni, a Mira-variable S-star, by Herbig (1956) and much more recently around Mira-variable o Ceti by Kaminski et al. (2016). AlH was also detected in sunspots through lines in its A 1 Π -X 1 Σ + electronic band, which lies in the blue region mospheres of extrasolar planets and cool stars. Rajpurohit et al. (2013) analysed BT-Settl synthetic spectra (Allard 2014) for M-dwarf stars and suggested that the CaOH band at 5570 A, and AlH and NaH hydrides in the blue part of the spectra constituted the main species still missing in the models. An ExoMol line list for NaH was subsequently computed by Rivlin et al. (2015); here we construct the corresponding line lists for isotopologues 27 AlH, 27 AlD and 26 AlH of aluminium hydride.
We previously provided line lists for isotopologues of AlO (Patrascu et al. 2015); this work follows closely on the methodology developed for treating this open shell system (Patrascu et al. 2014). Here we consider transitions within the X 1 Σ + -X 1 Σ + and electronic A 1 Π -X 1 Σ + bands. The A 1 Π potential energy curve (PEC) is very shallow with a strong pre-dissociative character and can accommodate only two vibrational states (Holst & Hulthén 1934) with a small barrier before the dissociation. Here we apply the Duo diatomic code (Yurchenko et al. 2016) to solve the nuclear motion problem for the X 1 Σ + and A 1 Π coupled electronic states of AlH and to generate a line list for the X-X and A-X bands using empirical PECs and high level ab initio (transition) dipole moment curves (DMC). The centrifugal correction due to the Born-Oppenheimer breakdown effect is also considered along with an empirical electronic angular momentum coupling between X 1 Σ + and A 1 Π. The empirical PECs were obtained by fitting the corresponding analytical representations to the experimental energies of AlH derived from the measured line positions available in the literature using the MARVEL (measured active rotation-vibration energy levels) methodology (Furtenbacher et al. 2007). Special measures were taken to ensure that the unbound and quasi-bound states are not included in the line lists. Lifetimes and partition functions are also provided as part of the line lists supplementary material, which are available from the CDS and ExoMol databases. Comparisons with experimental spectra and lifetimes are presented.
The paper is structured as follows. Section 2 describes the methods used and includes a discussion of previous laboratory data, Section 3 presents our results and Section 4 offers some conclusions.

METHOD
Rotation-vibration resolved lists for the ground X 1 Σ + and A 1 Π excited electronic states of AlH were obtained by direct solution of the nuclear-motion Schrödinger equation using c 2018 RAS, MNRAS 000, 1-23 the Duo program (Yurchenko et al. 2016) in conjunction with empirical PECs and ab initio (transition) DMCs. In principle the calculations could be performed using ab initio PECs and coupling curves (Tennyson et al. 2016a); however, in practice this does not give accurate enough transition frequencies or wavefunctions so the PECs was actually characterised by fitting to observed spectroscopic data. Conversely, experience (Tennyson 2014) suggests that retaining ab initio diagonal and transition dipole moment curves gives the best predicted transition intensities; this approach is adopted here.
The work of Halfen & Ziurys (2004 is hyperfine-resolved but hyperfine splittings are not present in the other studies and are not considered in this work. The works of White et al. (1993) and Deutsch et al. (1987) are important as the source of high v numbers (up to v = 5 and v = 8, respectively) in the X 1 Σ + state.
Running MARVEL on this network of 917 validated transitions gave 331 empirical energy levels, 283 in the X 1 Σ + state and 48 in the A 1 Π state of 27 AlH . For the X 1 Σ + state, J spanned the range 0 to 40 and v went from 0 to 8. For the A 1 Π state, J spanned the range 0 to 29 and v only included 0 and 1. Note that the A 1 Π state state is very shallow and supports, at most, only these two vibrational states. This issue is discussed further below.

Potential energy and dipole moment curves
There are a number of previous studies of the A 1 Π and X 1 Σ + curves of AlH, see Brown & Wasylishen (2013), Seck et al. (2014), and references therein. The ground electronic PEC has a nice Morse-like structure. Experimental data on the X state cover vibrational excitations up to v = 8; therefore we decided to obtain the X-state PEC fully empirically by fitting it to the experimental frequencies from Deutsch et al. (1987); White et al. (1993); Ito et al. (1994).
The X-state PEC was represented using the Extended Morse Oscillator (EMO) potential (Lee et al. 1999) given by where (A e − V e ) is the dissociation energy, V e is the minimum of the PEC, which for the X 1 Σ + state was set to zero, N is the expansion order parameter, r e is the equilibrium and ξ p is theSurkus variable (Šurkus et al. 1984) given by with p as a parameter. Use of the EMO has two advantages. First, it guarantees a correct dissociation limit and second, allows extra flexibility in the degree of the polynomial around a reference position r ref , which was defined as the equilibrium internuclear separation (r e ) in this case. Figure 1 shows our empirical PEC of AlH in its X state.
This closed shell X 1 Σ + ground state was fitted to an EMO using Level (Le Roy 2017).
To allow for rotational Born-Oppenheimer breakdown (BOB) effects (Le Roy 2007) which become important for J > 20, the vibrational kinetic energy operator was extended by where the unitless BOB functions g BOB are represented by the polynomial where ξ p as theSurkus variable and p, A k and A ∞ are adjustable parameters.
Given the shallow nature of the A curve which also appears to undergo an avoided crossing we decided to perform our own calculations using a high level of electronic structure theory. Ab initio PECs and Dipole Moment Curves (DMCs) were computed using the MOLPRO electronic structure package (Werner et al. 2012) at the multi-reference configuration interaction (MRCI) level using an aug-cc-pV5Z Gaussian basis set. Calculations were performed at 120 bond lengths over the range of r = 2 to 8 a 0 . Figure 1 shows the ab initio PEC of A 1 Π, which only supports two bound vibrational states. It also shows a maximum at about 4.5 a 0 which is probably associated with an avoided crossing. Figure 2 shows our ab initio DMCs, which agree well with the ab initio dipole moment values from Bauschlicher & Langhoff (1988), although, as discussed below, the magnitude of our transition dipole is slightly smaller. Our calculations give a permanent dipole moment of 0.158 D (absolute value) at r = 1.646Å at equilibrium, which is slightly higher than that by Bauschlicher & Langhoff (1988), 0.12 D. This is significantly less than the absolute value of 0.186 used by CDMS (Müller et al. 2005), which is taken from an old calculation by Meyer & Rosmus (1975). We also note that the X dipole also changes sign close to equilibrium. We return to these issues below. Matos et al. (1988) in their ab initio work showed a strong variation of the dipole and obtain a value of 0.3 D for µ 0 (i.e. a vibrational averaged in the ground vibrational state), while our value is 0.248 D.
No experimental values exist.
The A 1 Π -X 1 Σ + transition dipole moment of AlH also undergoes a change in behaviour in the region around 4.5 a 0 .
In order to represent the complex shape of the shallow A 1 Π potential energy curve (see Fig. 1), we used a diabatic-like scheme, where the effect of the avoided crossing is described by a 2 × 2 matrix: Here V 1 (r) is given by the EMO potential function in Eq. (1), while V 2 (r) is represented by a simple repulsive form V 2 (r) = w 6 r 6 . The coupling W (r) is given by where r cr is a crossing point. The two eigenvalues of B are given by where the lowest root V low (r) corresponds to the A 1 Π adiabatic PEC.
Initially, the expansion parameters representing this form were obtained by fitting to the ab initio A-state PEC shown in Fig. 1 and then refined by fitting to the MARVEL energies.
Currently Duo does not support quasi-bound or continuum solutions, see Yurchenko et al. The BOB-correction in the form given in Eq. (5) was used for the A-state as well. In these fits the X-state parameters were fixed to the values obtained using Level. The BOB-curves are shown in Fig. 3.
In order to account for the Λ-doubling effect, we also used an empirical electronic angular momentum (EAM) coupling between the A 1 Π and X 1 Σ + states, which was represented by which is nothing else than Eq. (5) truncated after the leading term. The final value of A EAM 0 is 0.1475 cm −1 and the EAM curve is shown in Fig. 3.
The final fit gave an observed minus calculated root-mean-square (rms) error of 0.025  cm −1 , when compared to our MARVEL energy levels for the X 1 Σ + state. The MARVEL energy levels of the A 1 Π state are reproduced with an rms error of 0.59 cm −1 .
Baltayan & Nedelec (1979) reported AlH dissociation energies measured using a hollow cathode discharge by dye laser excitation, 3.16 ±0.01 eV and 0.24 ±0.01 eV for the X 1 Σ + and A 1 Π states, respectively. The dissociation energy (D e ) of our empirical PEC of the X 1 Σ + state is 3.644 eV which overestimates the experimental value. This should not be a problem for our line list since the contributions from the highly excited vibrational states of X 1 Σ + is practically zero at such energies. For the A 1 Π PEC we obtained D e = 0.209 eV (ab initio) and 0.210 eV (refined PEC), which compare well to the experimental value by Baltayan & Nedelec (1979). It should be noted that Bauschlicher & Langhoff (1988) also reported ab initio FCI dissociation energies which coincide with the experimental values by Baltayan & Nedelec (1979).
Since the current version of Duo does not account for the isotopic-effect explicitly and thus is not capable of treating a mass-independent model as, for example, in Level, we had to create independent models for different isotopologues. Therefore the same fitting procedure was repeated for AlD, where the model curves were fitted to the AlD experimentally derived energies (MARVEL). Fortunately, the experimental data set for AlD is almost as large as that for AlH.
Final parameters for all the curves representing our two spectroscopic models for AlH and AlD (PECs, DMCs, and other empirical curves) and used in Duo are given in the supplementary material in the form of the Duo input. The program Duo is freely available via the www.exomol.com web site. The actual curves can be extracted from the Duo outputs, which are also provided.

Lifetimes
The lifetimes of AlH in the A 1 Π state were measured by Baltayan & Nedelec (1979)   This transition DMC is put forward to produce the AlH line lists.

Line list generation
Line lists for AlH and AlD were generated using the program Duo.
In order to reduce the numerical noise in the intensity calculations of high overtones characterized by small transition probabilities in the spectra of the X 1 Σ + state (see recent recommendations by Medvedev et al. (2016)) the DMCs are represented analytically. We use the following expansion (Prajapat et al. 2017;Yurchenko et al. 2018b): where z is the damped-coordinate given by: Here r ref is a reference position equal to r e by default and β 2 and β 4 are damping factors. The expansion parameters are given in the supplementary material. As an additional measure to reduce numerical noise in the overtone intensities, a dipole moment cutoff of 10 −7 D was applied to the vibrational dipole moments: all transitions for which the vibrational dipole moments are smaller than 10 −7 D were ignored.
The A 1 Π -X 1 Σ + (bound) spectrum only contains transitions to/from the upper states v ′ = 0 and v ′ = 1, i.e. no overtones, and thus should not suffer from the numerical noise issue as much as the X 1 Σ + -X 1 Σ + band. Therefore the A 1 Π-X 1 Σ + transition dipole moment was given directly in the (scaled) ab initio grid representation of 120 points. The latter points are interpolated by Duo onto the sinc DVR grid using the cubic splines method (see Yurchenko et al. (2016) for details).
Only bound vibrational and rotational states were retained which meant for 27 AlH considering J 82 for the X 1 Σ + state and J 25 for the A 1 Π state. For 27 AlD the range of J was increased to J max = 108 and 35, respectively. Duo input files used to generate the line lists are included as part of the supplementary data. This procedure was then simply repeated for 26 AlH by changing the mass of Al and nuclear statistics factor from 12 to 22.
All A 1 Π empirical energies in the .states file of 27 AlH were replaced by the MARVEL values, or by values generated using PGOPHER from the constants by Szajna et al. (2015) if the MARVEL energies were not available; for 27 AlD we used the experimentally derived term values by Szajna et al. (2015).

Partition function
Partition functions were generated for each isotopologue by explicit summation of the energy levels. Comparison for 27 AlH with the recent results of Barklem & Collet (2016) and with the partition function generated using parameters from Sauval & Tatum (1984) shows excellent agreement for temperatures below 5000 K (see Fig. 5) once allowance is made for the fact that ExoMol adopts the HITRAN convention (Gamache et al. 2017) which includes the full nuclear spin degeneracy factor in the partition function (12 in case of 27 AlH, 18 in case of  Figure 5. Temperature dependence of the partition function of AlH computed using our line lists and compared to those by Sauval & Tatum (1984) and Barklem & Collet (2016).
It should be noted that AlH is unlikely to be important at temperatures above 5000 K.
Partition functions, Q(T ), on a 1 K grid up to 5000 K are given for each isotopologue in the supplementary material. For ease of use we also provide fits in the form proposed by Vidler & Tennyson (2000): with the values given in Table 4.

Spectra
In the following, we present different spectra of AlH computed using the new line lists and utilizing the program ExoCross (Yurchenko et al. 2018a). Figure 6 gives an overview of the AlH line list in the form of absorption cross sections for a range of temperatures from 300 to 3000 K.
Our line lists can be used to generate spectra for a variety of conditions. First we compare with available laboratory spectra. Figure 7 compares an emission infrared spectrum of AlH recorded by White et al. (1993) with that generated using our line list assuming a temperature of 1700 K. Although the experimental spectrum does not provide the absolute scale for the intensities, there is good agreement for the relative intensities of the hot bands in this region between the experiment and our predictions. Our R-branch appears to be slightly stronger relative to the P-branch than the observations of White et al. (1993), but given the variable baseline and presence of self-absorption in the observed spectrum this may not be significant. Figure 8 shows a comparison with the emission 0-0 and 1-1 bands of the A 1 Π -X 1 Σ + system of AlH and AlD by Szajna et al. (2015). The observed spectrum was produced from an electric discharge in an aluminium hollow-cathode lamp. Our spectrum has been synthesized assuming a vibrational temperature of 4500 K and rotational temperature of 900 K.
Comparisons with the figure suggest that the experiments had an even lower effective rotational temperature and a higher effective vibrational temperature. Inspection of Fig. 8   suggests that our 1-1 band is blue-shifted relative to the experiment by about 7 cm −1 . Our actual numerical agrement is much better (within experimental uncertainly), which suggest some problems with the original figure from this paper. Figure 9 illustrate a good agreement of the theoretical emission spectrum of the A 1 Π -X 1 Σ + band (1, 0) at T = 600 K of AlD with the experiment by Szajna et al. (2017a). Rice et al. (1992) reported experimentally determined ratios of Einstein coefficients for a number of vibrational bands of A 1 Π -X 1 Σ + , which we use to assess our transition probabilities in Table 5. Our ratios are found to be in excellent agreement with experiment.
Finally, Figure 10 gives a comparison with the long-wavelength, rotational spectrum WYLLoT Figure 8. Comparison between the 0-0 and 1-1 bands of the A 1 Π -X 1 Σ + emission spectrum of AlH and AlD measured by Szajna et al. (2015) (upper) and computed using our line list (lower) assuming a temperature of 4500 K (vibrational) and 900 K (rotational). The intensities of the AlD theoretical spectrum, given in blue and annotated in the experimental spectrum, are scaled by a factor 0.5. Reprinted from Szajna et al. (2015), Copyright (2015), with permission from Elsevier.
taken from the CDMS database (Müller et al. 2005). The agreement between the line positions is excellent, although we recommend using the highly-accurate CDMS frequency directly for long-wavelength studies of cool sources. However, there is approximately a factor of two discrepancy in the predicted line intensities. This difference is almost exactly in line with the square of the ratio of our dipole to that of Meyer & Rosmus (1975) used by CDMS. Our vibrationally averaged value for the ground state of AlH is 0.248 D, which is significantly different from the permanent µ e value 0.158 D. We would recommend that CDMS adopts our value in future and note that using this value will approximately double the upper limit for AlH in IRC+10216 determined by Cernicharo et al. (2010).

CONCLUSIONS
Line lists for three AlH isotopologue species were computed using a mixture of empirical and