Analytical Correlation in the H${}_2$ Molecule from the Independent Atom Ansatz

The independent atom ansatz of density functional theory yields an accurate analytical expression for dynamic correlation energy in the H${}_2$ molecule: $E_c=0.5(1-\sqrt{2})(ab\mathrm{\mid }ba)$ for the atom-additive self-consistent density $\rho =\mathrm{\mid }a{\mathrm{\mid }}^2+\mathrm{\mid }b{\mathrm{\mid }}^2$. Combined with exact atomic self-exchange, it recovers more than 99.5 % of nearly exact SCAN exchange-correlation energy at R > 0.5 $\mathring{A}$, differing by less than 0.12 eV. The total energy functional correctly dissociates the H-H bond and yields absolute errors of 0.002 $\mathring{A}$, 0.19 eV, and 13 cm${}^{-1}$ relative to experiment at the tight binding computational cost. The chemical bond formation is attributed to the asymptotic Heitler-London resonance of quasi-orthogonal atomic states ($-(ab\mathrm{\mid }ba)$) with no contributions from kinetic energy or charge accumulation in the bond.