 Solution

For the tetrahedral symmetry classes $(E,8C_3,3C_2,6S_4,6{\sigma }_d)$, each set of four $s$ atomic orbitals hasThe counts are the fixed sites of each symmetry operation: a threefold rotation fixes one vertex of each tetrahedron, a twofold or improper fourfold operation fixes none, and a diagonal mirror fixes two. Character orthogonality reduces each set toThe dimensions $1+3=4$ provide a check. Thus the complete eight-orbital model contains two copies of each of $A_1$ and $T_2$, which can mix across the two species.