Szabo/Ostlund exercise on spin orbital orthonormality

I have a question about exercise 2.1 in Szabo and Ostlund's *Modern Quantum Chemistry*, which concerns the inner product between different spin orbitals: $\require{begingroup}\begingroup\newcommand{\br}{\mathbf{r}}\newcommand{\bx}{\mathbf{x}}$



> 
> **Exercise 2.1.** Given a set of $K$ orthonormal spatial functions, $\{\psi\_i^\alpha(\br)\}$, and another set of $K$ orthonormal functions, $\{\psi\_j^\beta(\br)\}$, such that the first set is not orthogonal to the second set, i.e.,
> 
> 
> $$\int d\br \, \psi\_i^{\alpha\*}(\br)\psi\_j^\beta(\br) = S\_{ij}$$
> 
> 
> where $\mathbf{S}$ is an overlap matrix, show that the set $\{\chi\_i\}$ of $2K$ spin orbitals, formed by multiplying $\psi\_i^\alpha(\br)$ by the $\alpha$ spin function and $\psi\_j^\beta(\br)$ by the $\beta$ spin function, i.e.,
> 
> 
> \begin{align}
> \chi\_{2i-1}(\bx) &= \psi\_i^\alpha(\br)\alpha(\omega) \\
> \chi\_{2i}(\bx) &= \psi\_j^\beta(\br)\beta(\omega)
> \end{align}
> 
> 
> with $i = 1, 2, \ldots, k$ is an orthonormal set.
> 
> 
> 


I feel quite confused about this question. 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.


The result can only be orthogonal if both ket and bra vectors are in the $i$-indexed set. If so, why does the exercise mention a $j$-indexed set in the beginning?





> 
> 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 "alpha" set can be multiplied only by the $\alpha$ spin function while spatial orbitals from the second "beta" set can be multiplied only by the $\beta$ spin function. Thus, there are only three types of integrals you need to consider:
$$
\langle \psi\_i^{\alpha} \alpha | \psi\_j^{\alpha } \alpha \rangle \, ,
\quad
\langle \psi\_i^{\alpha} \alpha | \psi\_j^{\beta } \beta \rangle \, ,
\quad
\langle \psi\_i^{\beta} \beta | \psi\_j^{\beta } \beta \rangle \, .
$$


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