Why was the existence of molecular hydrogen ($H_2$) critical for the initial cooling pathways of Population III clouds?

$H_2$ molecules enabled necessary cooling pathways, unlike atomic hydrogen alone.

While the early universe was dominated by atomic hydrogen, atomic hydrogen gas is inefficient at radiating away the heat generated during gravitational compression, making cooling insufficient for collapse at the required densities. The formation of molecular hydrogen ($H_2$) molecules became a critical prerequisite because these molecules possess rotational and vibrational energy states that allow them to couple with photons and radiate thermal energy away much more effectively than atomic hydrogen alone. Although this metal-free cooling mechanism was less efficient than the metal-line cooling seen in later eras, the presence of $H_2$ was essential to establish the runaway cooling that allowed the gas to condense into protostars rather than remaining hot and diffuse.