H₂⁺ 1s-1s overlap S(R) = (R²/3 + R + 1) e^{-R}: S(0)=1, S(1)=7 e^{-1}/3, S(∞)=0

Sympy-exact witness of the canonical LCAO overlap integral S(R) for two 1s hydrogen-like orbitals at internuclear separation R (Bohr units). Closed form: S(R) = e^{−R} · (R²/3 + R + 1). Three canonical anchors: S(0) = 1 (same orbital,…