Intuition for the overlap integral

Question

I want to ask a question about the overlap integral.

I was studying about the MO diagram of $\ce{H_2^+}$ and we managed to derive an expression which had the following term:

$$\langle \mathrm{1s}_\ce{A}(r_\ce{A}) | \mathrm{1s}_\ce{B}(r_\ce{B}) \rangle$$

and our lecturer later explained that this is known as the overlap integral.

The overlap integral is the magnitude to which the atoms overlap when combining AOs to produce MOs.

The trouble is, I later wanted to find a proof for this idea in order to confirm my understanding of this idea, but could not find anything that could explain this phenomenon from principle.

I did come across this question, but this only explained its main definition and not actually any derivation from basic principle.

How can the overlap integral be explained from basic principle?

Answer 1

Let $\Phi, \Psi$ be normalized wave functions.

Then $0 \le|\langle\Phi\vert\Psi\rangle| \le 1$, with $\langle\Phi\vert\Phi\rangle = 1$ and $\langle\Phi\vert\Psi\rangle = 0$ for orthogonal $\Phi$ and $\Psi$.

Since we can argue that any wave function completely contains itself and for two orthogonal wave functions none of them contains a part of the other, we can interpret the scalar product $\langle\Phi\vert\Psi\rangle$ as a measure for overlap.