Tetrahedral s-orbital symmetry combinations

Four equivalent $s$ atomic orbitals at the vertices of a regular tetrahedron carry the tetrahedral symmetry representation $A_1\oplus T_2$. The normalized equal sum is $A_1$; three coordinate-sign combinations span $T_2$. Every normalized coefficient vector has entries $\pm 1/2$ in a suitable orthonormal real basis. These functions simplify a Hamiltonian invariant under the tetrahedral operations, but equivalent symmetry copies on other sites may still mix.