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The Magellanic Stream (MS), a tail of diffuse gas formed from tidal and ram pressure interactions between the Small and Large Magellanic Clouds (SMC and LMC) and the Halo of the Milky Way, is primarily composed of neutral atomic hydrogen (HI). The deficiency of dust and the diffuse nature of the present gas make molecular formation rare and difficult, but if present, could lead to regions potentially suitable for star formation, thereby allowing us to probe conditions of star formation similar to those at high redshifts. We search for 
$\text{HCO}^{+}$, HCN, HNC, and C
$_2$H using the highest sensitivity observations of molecular absorption data from the Atacama Large Millimeter Array (ALMA) to trace these regions, comparing with HI archival data from the Galactic Arecibo L-Band Feed Array (GALFA) HI Survey and the Galactic All Sky Survey (GASS) to compare these environments in the MS to the HI column density threshold for molecular formation in the Milky Way. We also compare the line of sight locations with confirmed locations of stars, molecular hydrogen, and OI detections, though at higher sensitivities than the observations presented here.
We find no detections to a 3
$\sigma$ significance, despite four sightlines having column densities surpassing the threshold for molecular formation in the diffuse regions of the Milky Way. Here we present our calculations for the upper limits of the column densities of each of these molecular absorption lines, ranging from 
$3 \times 10^{10}$ to 
$1 \times 10^{13}$ cm
$^{-2}$. The non-detection of 
$\text{HCO}^{+}$ suggests that at least one of the following is true: (i) 
$X_{\text{HCO}^{+}{}, \mathrm{MS}}$ is significantly lower than the Milky Way value; (ii) that the widespread diffuse molecular gas observed by Rybarczyk (2022b, ApJ, 928, 79) in the Milky Way’s diffuse interstellar medium (ISM) does not have a direct analogue in the MS; (iii) the HI-to-
$\text{H}_{2}$ transition occurs in the MS at a higher surface density in the MS than in the LMC or SMC; or (iv) molecular gas exists in the MS, but only in small, dense clumps.
We present observations of the four 
$^2 \Pi _{3/2}\,J=3/2$ ground-rotational state transitions of the hydroxyl molecule (OH) along 107 lines of sight both in and out of the Galactic plane: 92 sets of observations from the Arecibo telescope and 15 sets of observations from the Australia Telescope Compact Array (ATCA). Our Arecibo observations included off-source pointings, allowing us to measure excitation temperature (
$T_{\rm ex}$) and optical depth, while our ATCA observations give optical depth only. We perform Gaussian decomposition using the Automated Molecular Excitation Bayesian line-fitting Algorithm ‘Amoeba’ (Petzler, Dawson, & Wardle 2021, ApJ, 923, 261) fitting all four transitions simultaneously with shared centroid velocity and width. We identify 109 features across 38 sightlines (including 58 detections along 27 sightlines with excitation temperature measurements). While the main lines at 1665 and 1667 MHz tend to have similar excitation temperatures (median 
$|\Delta T_{\rm ex}({\rm main})|=0.6\,$K, 84% show 
$|\Delta T_{\rm ex}({\rm main})|<2\,$K), large differences in the 1612 and 1720 MHz satellite line excitation temperatures show that the gas is generally not in LTE. For a selection of sightlines, we compare our OH features to associated (on-sky and in velocity) Hi cold gas components (CNM) identified by Nguyen et al. (2019, ApJ, 880, 141) and find no strong correlations. We speculate that this may indicate an effective decoupling of the molecular gas from the CNM once it accumulates.
