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: $E_{c} = 0.5(1 - \sqrt{2})(ab|ba)$ for the atom-additive self-consistent density $\rho = |a|^{2} + |b|^{2}$. Combined with exact atomic self-exchange, it recovers more than 99.5 % of nearly exact SCAN exchange-correlation energy at R > 0.5 $\r{A}$, differing by less than 0.12 eV. The total energy functional correctly dissociates the H-H bond and yields absolute errors of 0.002 $\r{A}$, 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 ($- (ab|ba)$) with no contributions from kinetic energy or charge accumulation in the bond.