  Hydroxyl (OH) is expected to be abundant in diffuse interstellar molecular
gas as it forms along with $H_2$ under similar conditions and within a similar
extinction range. We have analyzed absorption measurements of OH at 1665 MHz
and 1667 MHz toward 44 extragalactic continuum sources, together with the J=1-0
transitions of $^{12}$CO, $^{13}$CO , and C$^{18}$O, and the J=2-1 of
$^{12}$CO. The excitation temperature of OH were found to follow a modified
log-normal distribution, $ f(T{\rm_{ex}}) \propto \frac{1}{ \sqrt{2\pi}\sigma }
\rm{exp}\left[-\frac{[ln(\textit{T}_{ex})-ln(3.4\ K)]^2}{2\sigma^2}\right] $,
the peak of which is close to the temperature of the Galactic emission
background (CMB+synchron). In fact, 90% of the OH has excitation temperature
within 2 K of the Galactic background at the same location, providing a
plausible explanation for the apparent difficulty to map this abundant molecule
in emission. The opacities of OH were found to be small and peak around 0.01.
For gas at intermediate extinctions (A$\rm_V$ $\sim$ 0.05-2 mag), the detection
rate of OH with detection limit $N(\mathrm{OH})\simeq 10^{12}$ cm$^{-2}$ is
approximately independent of $A\rm_V$. We conclude that OH is abundant in the
diffuse molecular gas and OH absorption is a good tracer of `dark molecular gas
(DMG)'. The measured fraction of DMG depends on assumed detection threshold of
the CO data set. The next generation of highly sensitive low frequency radio
telescopes, FAST and SKA, will make feasible the systematic inventory of
diffuse molecular gas, through decomposing in velocity the molecular (e.g. OH
and CH) absorption profiles toward background continuum sources with numbers
exceeding what is currently available by orders of magnitude.
