Basis sets: why Gaussians win

To treat molecules we expand each orbital in a fixed set of simple functions — a basis. The physically natural choice, Slater functions e^{−ζr}, have the right shape (a cusp at the nucleus, a slow exponential tail) — but their integrals across several atomic centres have no closed form.

Gaussians e^{−αr²} have the wrong shape but one magic property: the product of two Gaussians on different atoms is a single Gaussian on a point between them. That collapse makes every multi-centre integral analytic. We stack a few Gaussians (STO-3G uses three per orbital) to imitate the Slater shape we gave up.