  We show in this paper why molecular millimeter absorption line searches in
DLAs have been unsuccessful. We use CO emission line maps of local galaxies to
derive the H2 column density distribution function f(N_H2) at z=0. We show that
it forms a natural extension to f(N_HI): the H2 distribution exceeds f(N_HI) at
N_H ~ 10^22 cm^-2 and exhibits a power law drop-off with slope ~ -2.5.
Approximately 97% of the H2 mass density rho_H2 is in systems above N_H2=10^21
cm^-2. We derive a value rho_H2 = 1.1 x 10^7 h_70 M_sun Mpc^-3, which is ~25%
the mass density of atomic hydrogen. Yet, the redshift number density of H2
above this N_H2 limit is only ~3 x 10^-4, a factor 150 lower than that for HI
in DLAs at z=0. Furthermore, we show that the median impact parameter between a
N_H2>10^21 cm^-2 absorber and the centre of the galaxy hosting the H2 gas is
only 2.5 kpc. Based on arguments related to the Schmidt law, we argue that H2
gas above this column density limit is associated with a large fraction of the
integral star formation rate density. Even allowing for an increased molecular
mass density at higher redshifts, the derived cross-sections indicate that it
is very unlikely to identify the bulk of the molecular gas in present quasar
absorption lines samples. We discuss the prospects for identifying this
molecular mass in future surveys.
