  The direct collapse model for the formation of massive seed black holes in
the early Universe attempts to explain the observed number density of
supermassive black holes (SMBHs) at $z \sim 6$ by assuming that they grow from
seeds with masses M > 10000 solar masses that form by the direct collapse of
metal-free gas in atomic cooling halos in which H2 cooling is suppressed by a
strong extragalactic radiation field. The viability of this model depends on
the strength of the radiation field required to suppress H2 cooling, $J_{\rm
crit}$: if this is too large, then too few seeds will form to explain the
observed number density of SMBHs. In order to determine $J_{\rm crit}$
reliably, we need to be able to accurately model the formation and destruction
of H2 in gas illuminated by an extremely strong radiation field. In this paper,
we use a reaction-based reduction technique to analyze the chemistry of H2 in
these conditions, allowing us to identify the key chemical reactions that are
responsible for determining the value of $J_{\rm crit}$. We construct a reduced
network of 26 reactions that allows us to determine $J_{\rm crit}$ accurately,
and compare it with previous treatments in the literature. We show that
previous studies have often omitted one or more important chemical reactions,
and that these omissions introduce an uncertainty of up to a factor of three
into previous determinations of $J_{\rm crit}$.
