Szabo/Ostlund exercise on spin orbital orthonormality
Consider the spin-orbital integral
$$ \langle \psi_i^{\alpha} \alpha
(\omega) | \psi_j^{\beta } \alpha (\omega) \rangle  = \langle\psi_i^{\alpha}  |  \psi_j^{\beta } \rangle
$$
which is not necessarily orthogonal.

Yes, it is not necessarily orthogonal, but the thing is that you don't need to consider such integrals in the exercise. Read more carefully: spatial orbitals from the first &quot;alpha&quot; set can be multiplied only by the $\alpha$ spin function while spatial orbitals from the second &quot;beta&quot; set can be multiplied only by the $\beta$ spin function. Thus, there are only three types of integrals you need to consider:
$$
\langle \psi_i^{\alpha} \alpha | \psi_j^{\alpha } \alpha \rangle \, ,
\quad
\langle \psi_i^{\alpha} \alpha | \psi_j^{\beta } \beta \rangle \, ,
\quad
\langle \psi_i^{\beta} \beta | \psi_j^{\beta } \beta \rangle \, .
$$