Szabo/Ostlund exercise on spin orbital orthonormality

Question

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?

Answer 1

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: $$ \langle \psi_i^{\alpha} \alpha | \psi_j^{\alpha } \alpha \rangle \, , \quad \langle \psi_i^{\alpha} \alpha | \psi_j^{\beta } \beta \rangle \, , \quad \langle \psi_i^{\beta} \beta | \psi_j^{\beta } \beta \rangle \, . $$