Ne present ISO observations of several H-2 pure-rotational lines (from S(0)
to S(5)) ton ards a sample of 16 molecular clouds distributed along the ce
ntral similar to 500 pc of the Galaxy. We also present (CO)-O-18 and (CO)-C
-13 J = 1 --> 0 and J = 2 --> 1 observations of these sources made with the
IRAM-30 m telescope. With the CO data we derive H-2 densities of 10(3.5-4.
0) cm(-3) and H-2 column densities of a few 10(22) cm(-2). We have correcte
d the H-2 data, for similar to 30 magnitudes of visual extinction using a s
elf-consistent method. In every source, we find that the Hz emission exhibi
ts a large temperature gradient. The S(0) and S(1) lines trace temperatures
(T) of similar to 150 K while the S(4) and S(5) lines indicate temperature
s of similar to 600 K. The warm H-2 column density is typically similar to
1-2 10(22) cm(-2). and is predominantly gas with T = 150 K. This is the fir
st direct estimate of the total column density of the warm molecular gas in
the Galactic center region. These warm H-2 column densities represent a fr
action of similar to 30% of the gas traced by the CO isotopes emission. The
cooling by H-2 in the warm component is comparable to that by CO. Comparin
g our H-2 and CO data with available ammonia (NH3) observations from litera
ture one obtains relatively high NH3 abundances of a few 10(-7) in both the
warm and the cold gas. A single shock or Photo-Dissociation Region (PDR) c
annot explain all the observed H-2 lines. Alternatives for the heating mech
anisms are discussed.