We have searched for molecular hydrogen in damped Lyman-α (DLA) and sub-DLA systems at high redshift (zabs> 1.8) using UVES at the VLT down to a detection limit of typically N(H2) = 2 ×10 14 cm −2. Out of the 33 systems in our sample, 8 have firm and 2 have tentative detections of associated H2 absorption lines. Considering that 3 detections were already known from past searches, molecular hydrogen is detected in 13 to 20 percent of the newly-surveyed systems. We report new detections of molecular hydrogen at zabs = 2.087 and 2.595 toward, respectively, Q1444+014 and Q0405−443, and also reanalyse the system at zabs = 3.025 toward Q0347−383. In all of the systems, we measure metallicities relative to Solar, [X/H] (with either X=Zn, or S, or Si), and depletion factors of Fe, [X/Fe], supposedly onto dust grains, and compare the characteristics of our sample with those of the global population of DLA systems (60 systems in total). We find that there is a correlation between metallicity and depletion factor in both our sample and also the global population of DLA systems. Although H2 molecules are detected in systems with [Zn/Fe] as smal
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