  We combine recent ISO observations of the vibrational ground state lines of
H2 towards Photon-Dominated Regions (PDRs) with observations of vibrationally
excited states made with ground-based telescopes in order to constrain the
formation rate of H2 on grain surfaces under the physical conditions in the
layers responsible for H2 emission. We use steady state PDR models in order to
examine the sensitivity of different H2 line ratios to the H2 formation rate
Rf. We show that the ratio of the 0-0 S(3) to the 1-0 S(1) line increases with
Rf but that one requires independent estimates of the radiation field incident
upon the PDR and the density in order to infer Rf from the H2 line data. We
confirm the earlier result of Habart et al. (2003) that the H2 formation rate
in regions of moderate excitation such as Oph W, S140 and IC 63 is a factor of
5 times larger than the standard rate inferred from UV observations of diffuse
clouds. On the other hand, towards regions of higher radiation field such as
the Orion Bar and NGC 2023, we derive H2 formation rates consistent with the
standard value. We find also a correlation between the H2 1-0 S(1) line and PAH
emission suggesting that Rf scales with the PAH abundance. With the aim of
explaining these results, we consider some empirical models of the H2 formation
process. Here we consider both formation on big (a~0.1 microns) and small (a~10
Angstroms) grains by either direct recombination from the gas phase or
recombination of physisorbed H atoms with atoms in a chemisorbed site. We
conclude that indirect chemisorption is most promising in PDRs. Moreover small
grains which dominate the total grain surface and spend most of their time at
relatively low temperatures may be the most promising surface for forming H2 in
PDRs.
