  Molecular hydrogen is quite underabundant in damped Lyman-alpha systems at
high redshift, when compared to the interstellar medium near the Sun. This has
been interpreted as implying that the gas in damped Lyman-alpha systems is
warm. like the nearby neutral intercloud medium, rather than cool, as in the
clouds which give rise to most H I absorption in the Milky Way. Other lines of
evidence suggest that the gas in damped Lyman-alpha systems -- in whole or part
-- is actually cool; spectroscopy of neutral and ionized carbon, discussed
here, shows that the damped Lyman-alpha systems observed at lower redshift z
$<$ 2.3 are largely cool, while those seen at z $>$ 2.8 are warm (though not
devoid of H2). To interpret the observations of carbon and hydrogen we
constructed detailed numerical models of H2 formation under the conditions of
two-phase thermal equilibrium, like those which account for conditions near the
Sun, but with varying metallicity, dust-gas ratio, $etc$. We find that the low
metallicity of damped Lyman-alpha systems is enough to suppress H2 formation by
many orders of magnitude even in cool diffuse clouds, as long as the ambient
optical/uv radiation field is not too small. For very low metallicity and under
the most diffuse conditions, H2 formation will be dominated by slow gas-phase
processes not involving grains, and a minimum molecular fraction in the range
$10^{-8}-10^{-7}$ is expected.
