In LCAO method for approximate solutions to Shrodinger equation of homonuclear biatomic molecules, it is defined an integral called "resonance integral" whose expression is:
$$
\beta = \langle\Psi_{a}|\Psi_{b}\rangle = \int \Psi_{a}^*(\vec{x}) \Psi_{b}(\vec{x})\, \mathrm{d}^3x
$$
Why it is called $\textit{resonance}$ integral?
Best Answer
I think this is an accident of history and actually not very interesting. The word resonance refers to the concept of Lewis resonance used in chemistry.
Note that the expression you give is the overlap integral not the resonance integral. The resonance integral is:
$$ H_{ab} = \int \Psi_{a}^* \hat{H} \Psi_{b}\, \mathrm{d}^3x $$
And it describes (in an arm waving sort of way) the interaction between the atomic orbitals on two different atoms. Typically the interaction will be written as a sum of all the interactions between all the different atomic orbitals:
$$ H = \int \left(\sum_i a_i^*\psi_i^*\right) \,\hat{H} \left(\sum_j a_j\psi_j\right)\, \mathrm{d}^3x = \sum_{ij} a_i^*a_j H_{ij} $$
So the overall bond is written as a sum of all possible individual bonds just as in Lewis resonance the overall structure is written as a sum of all possible structures, hence the name resonance.
I did say it wasn't very interesting ...