One-electron integrals & the Boys function

With a Gaussian basis the one-electron integrals — overlap, kinetic energy, and nuclear attraction — all follow from the Gaussian product theorem, which collapses each two-centre product to a single Gaussian.

Overlap and kinetic energy are then ordinary Gaussian integrals with closed forms. Nuclear attraction carries a 1/r Coulomb factor, and a Gaussian basis evaluates it through one special function — the Boys function F₀(x). It is how Coulomb's law lives inside a Gaussian world, and it returns for the two-electron integrals too.