Linus Pauling explained the importance of orbital overlap in the molecular bond angles observed through experimentation; it is the basis for orbital hybridization.
[2] A quantitative measure of the overlap of two atomic orbitals ΨA and ΨB on atoms A and B is their overlap integral, defined as where the integration extends over all space.
The star on the first orbital wavefunction indicates the function's complex conjugate, which in general may be complex-valued.
The overlap matrix is a square matrix, used in quantum chemistry to describe the inter-relationship of a set of basis vectors of a quantum system, such as an atomic orbital basis set used in molecular electronic structure calculations.
In general, each overlap matrix element is defined as an overlap integral: where In particular, if the set is normalized (though not necessarily orthogonal) then the diagonal elements will be identically 1 and the magnitude of the off-diagonal elements less than or equal to one with equality if and only if there is linear dependence in the basis set as per the Cauchy–Schwarz inequality.