  Under rapid cooling from molecular hydrogen, the accretion disks around
Population III (PopIII) stars are believed to fragment, resulting in multiple
accreting cores. In this paper, we build a theoretical framework for
calculating the optical depth of H$_2$ ro-vibrational line cooling based on the
vertical structure in these accretion disks. Applying this physically motivated
prescription for the optical depth, we find that cooling in the inner disk with
$r \lesssim 10 {\rm\ AU}$ is attenuated significantly as a result of high
surface density; $PdV$ heating becomes more efficient than cooling, which
prevents fragmentation in the inner disk. Despite this, cooling becomes
dynamically important in the outer disk, favoring fragmentation. We argue that
most of the resultant fragments are initially at the outer disk, and that any
surviving fragment has to migrate slower than the disk-scale photo-evaporation
process. Since type I migration is fast, migration slows down as a result of
gap-opening in the disk structure. Two possible processes for gap-opening are
studied: (1) through a massive, densely-cored ($\rho \gtrsim 10^{-8} {\rm\ g\
cm^{-3}}$) clump able to radiate away the excess gravitational potential
energy, and (2) through a fast-growing central star, with $\dot{M} \gtrsim 2
\times 10^{-3} \, M_\odot {\rm\ yr^{-1}}$, whose gravity dominates the
star-disk system and favors gap opening.
