Orbital overlap
= Orbital overlap
{title2=$S$}
The <orbital overlap> of normalized real <atomic orbitals> is their <inner product> $S=\int\phi_A\phi_B$. It is a <dimensionless> measure of their common amplitude, not an <electron> count. For the <Gaussian function> toy, increasing $a$ from $1$ to $2$ reduces $S$ from $e^{-1}\approx0.367879$ to $e^{-4}\approx0.018316$. As $S$ tends to zero, both <LCAO> normalization denominators tend to $\sqrt2$.