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 ∼ 6 by assuming that they grow from seeds with masses M > 104 M⊙ that form by the direct collapse of metal-free gas in atomic cooling haloes 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 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 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 analyse the chemistry of H2 in these conditions, allowing us to identify the key chemical reactions that are responsible for determining the value of J crit. We construct a reduced network of 26 reactions that allows us to determine J 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 3 into previous determinations of J crit.