Symmetry of the hydrogen molecule wave function

Question

Due to their overlapping wave functions, electrons in an $H_2$ molecule must posses opposite spins. The nuclei (two protons) on the other hand are far enough apart for the Pauli exclusion principle to be irrelevant. From my understanding is the total wave function of a molecule given by the product of a spin, rotational and vibrational wave function.

However, why does the Pauli exclusion principle require the total wave function of the molecule to be antisymmetric? I am only familiar with the principle being applied to individual identical particles.

Answer 1

$H_2$ is a composite boson. Thus, the total wave function of the molecule must be symmetric, not anti-symmetric. In contrast, the two-electron wave function must be anti-symmetric, because these are two identical fermions -- see spin-statistics theorem.