We present ISO observations of several H2 pure-rotational lines (from S(0) to S(5)) towards a sample of 16 molecular clouds distributed along the central ~ 500 pc of the Galaxy. We also present C18O and 13CO J=1->0 and J=2->1 observations of these sources made with the IRAM-30m telescope. With the CO data we derive H2 densities of 10e(3.5-4.0) cm-3 and H2 column densities of a few 10e22 cm-2. We have corrected the H2 data for ~ 30 magnitudes of visual extinction using a self-consistent method. In every source, we find that the H2 emission exhibits a large temperature gradient. The S(0) and S(1) lines trace temperatures (T) of ~150 K while the S(4) and S(5) lines indicate temperatures of ~ 600K. The warm H2 column density is typically ~1-2 x 10e22 cm-2, and is predominantly gas with T=150 K. This is the first direct estimate of the total column density of the warm molecular gas in the Galactic center region. These warm H2 column densities represent a fraction of ~ 30 % of the gas traced by the CO isotopes emission. The cooling by H2 in the warm component is comparable to that by CO. Comparing our H2 and CO data with available ammonia NH3 observations from literature one obtains relatively high NH3 abundances of a few 10e(-7) in both the warm and the cold gas. A single shock or Photo-Dissociation Region (PDR) cannot explain all the observed H2 lines. Alternatives for the heating mechanisms are discussed.
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