Direct collapse to supermassive black hole seeds: the critical
  conditions for suppression of $\rm H_2$ cooling
  Observations of high-redshift quasars imply the presence of supermassive
black holes already at z~ 7.5. An appealing and promising pathway to their
formation is the direct collapse scenario of a primordial gas in atomic-cooling
haloes at z ~ 10 - 20, when the $\rm H_2$ formation is inhibited by a strong
background radiation field, whose intensity exceeds a critical value, $J_{\rm
crit}$. To estimate $J_{\rm crit}$, typically, studies have assumed idealized
spectra, with a fixed ratio of $\rm H_{2}$ photo-dissociation rate $k_{\rm
H_2}$ to the $\rm H^-$ photo-detachment rate $k_{\rm H^-}$. This assumption,
however, could be too narrow in scope as the nature of the background radiation
field is not known precisely. In this work we argue that the critical condition
for suppressing the $\rm H_2$ cooling in the collapsing gas could be described
in a more general way by a combination of $k_{\rm H_2}$ and $k_{\rm H^-}$
parameters. By performing a series of cosmological zoom-in simulations with an
encompassing set of $k_{\rm H_2}$ and $k_{\rm H^-}$, we examine the gas flow by
following evolution of basic parameters of the accretion flow. We test under
what conditions the gas evolution is dominated by $\rm H_{2}$ and/or atomic
cooling. We confirm the existence of a critical curve in the $k_{\rm
H_2}-k_{\rm H^-}$ plane, and provide an analytical fit to it. This curve
depends on the conditions in the direct collapse, and reveals domains where the
atomic cooling dominates over the molecular cooling. Furthermore, we have
considered the effect of $\rm H_{2}$ self-shielding on the critical curve, by
adopting three methods for the effective column density approximation in $\rm
H_{2}$. We find that the estimate of the characteristic length-scale for
shielding can be improved by using $\lambda_{\rm Jeans25}$, which is 0.25 times
that of the local Jeans length.
