  The polytropic equation of state of an atomic hydrogen gas is examined for
primordial halos with baryonic masses of M_h~10^7-10^9 Mo. For roughly
isothermal collapse around 10^4 K, we find that line trapping of Lyman alpha
(HI and HeII) photons causes the polytropic exponent to stiffen to values
significantly above unity. Under the assumptions of zero H2 abundance and very
modest pollution by metals (<10^-4 Solar), fragmentation is likely to be
inhibited for such an equation of state. We argue on purely thermodynamic
grounds that a single black hole of ~0.02-0.003M_h can form at the center of a
halo for z=10-20 when the free-fall time is less than the time needed for a
resonantly scattered Lyman alpha photon to escape from the halo. The absence of
H2 follows naturally from the high, 10^4 K, temperatures that are attained when
Lyman alpha photons are trapped in the dense and massive halos that we
consider. An H2 dissociating UV background is needed if positive feedback
effects on H2 formation from X-rays occur. The black hole to baryon mass
fraction is suggestively close to what is required for these intermediate mass
black holes, of mass M_BH~10^4-10^6 Mo, to act as seeds for forming the
supermassive black holes of mass ~0.001M_spheroid found in galaxies today.
