  The multiplicity of metal-free (Population III) stars may influence their
feedback efficiency within their host dark matter halos, affecting subsequent
metal enrichment and the transition to galaxy formation. Radiative feedback
from massive stars can trigger nearby star formation in dense self-shielded
clouds. In model radiation self-shielding, the H$_2$ column density must be
accurately computed. In this study, we compare two local approximations based
on the density gradient and Jeans length with a direct integration of column
density along rays. After the primary massive star forms, we find that no
secondary stars form for both the direct integration and density gradient
approaches. The approximate method reduces the computation time by a factor of
2. The Jeans length approximation overestimates the H$_2$ column density by a
factor of 10, leading to five numerically enhanced self-shielded, star-forming
clumps. We conclude that the density gradient approximation is sufficiently
accurate for larger volume galaxy simulations, although one must still caution
that the approximation cannot fully reproduce the result of direct integration.
