  The direct collapse black hole (DCBH) model attempts to explain the observed
number density of supermassive black holes in the early Universe by positing
that they grew from seed black holes with masses of $10^{4}$-$10^{5} \: {\rm
M_{\odot}}$ that formed by the quasi-isothermal collapse of gas in metal-free
protogalaxies cooled by atomic hydrogen emission. For this model to work,
H$_{2}$ formation must be suppressed in at least some of these systems by a
strong extragalactic radiation field. The predicted number density of DCBH
seeds is highly sensitive to the minimum value of the ultraviolet (UV) flux
required to suppress H$_{2}$ formation, $J_{\rm crit}$. In this paper, we
examine how the value of $J_{\rm crit}$ varies as we vary the strength of a
hypothetical high-redshift X-ray background. We confirm earlier findings that
when the X-ray flux $J_{\rm X}$ is large, the critical UV flux scales as
$J_{\rm crit} \propto J_{\rm X}^{1/2}$. We also carefully explore possible
sources of uncertainty arising from how the X-rays are modelled. We use a
reaction-based reduction technique to analyze the chemistry of H$_{2}$ in the
X-ray illuminated gas and identify a critical subset of 35 chemical reactions
that must be included in our chemical model in order to predict $J_{\rm crit}$
accurately. We further show that $J_{\rm crit}$ is insensitive to the details
of how secondary ionization or He$^{+}$ recombination are modelled, but does
depend strongly on the assumptions made regarding the column density of the
collapsing gas.
