NLTE for APOGEE: Simultaneous multi-element NLTE radiative transfer

Context. Relaxing the assumption of local thermodynamic equilibrium (LTE) in modelling stellar spectra is a necessary step to determine chemical abundances to better than about 10% in late-type stars. Aims. We describe our multi-element (Na, Mg, K, and Ca) non-LTE (NLTE) calculations, which can be applied to the APOGEE survey. Methods. The new version of TLUSTY allows for the calculation of restricted NLTE in cool stars using pre-calculated opacity tables. We demonstrate that TLUSTY gives consistent results with MULTI, a well-tested code for NLTE in cool stars. We used TLUSTY to perform LTE and a series of NLTE calculations that simultaneously used all combinations of one, two, three and four of the elements in NLTE. Results. We take into account that departures from LTE in one element can affect others through changes in the opacities of Na, Mg, K, and Ca. We find that atomic Mg, which provides strong UV opacity and exhibits significant departures from LTE in the low-energy states, can affect the NLTE populations of Ca, leading to abundance corrections as large as 0.07 dex. The differences in the derived abundances between the single-element and the multi-element cases can exceed those between the single-element NLTE determinations and an LTE analysis. We therefore caution that this is not always a second-order effect. Based on detailed tests for three stars with reliable atmospheric parameters (Arcturus, Procyon, and the Sun), we conclude that our NLTE calculations provide abundance corrections that can in the optical amount to 0.1, 0.2, and 0.7 dex for Ca, Na and K, but LTE is a good approximation for Mg. In the H-band, NLTE corrections are much smaller and always lower than 0.1 dex. The derived NLTE abundances in the optical and in the IR are consistent. In all three stars, NLTE line profiles fit the observations better than the LTE counterparts for all four elements. Conclusions. The atomic elements in ionisation stages where over-ionisation is an important NLTE mechanism are likely affected by departures from LTE in Mg. Particular care must be taken with the collisions that are adopted for high-lying levels when NLTE profiles of lines in the H-band are calculated. The derived NLTE corrections in the optical and in the H-band differ, but the derived NLTE abundances are consistent between the two spectral regions.