FROST II: certified Hartree–Fock limits of the closed-shell atoms from neon to xenon

We extend FROST (Fock radial orbital shooting with Taylor series) from atoms whose occupied orbitals are all s orbitals to closed-shell atoms with p and d shells, and report rigorous upper bounds on the Hartree–Fock (HF) energies of the closed-shell atoms from neon to xenon. Their leading digits are correct to 504 significant figures for neon, 103 for magnesium, 105 for argon and 103–106 for the heavier atoms, against previous records of 30 and 25. Three changes to the method make this possible. First, exchange with multipoles k ≥ 1 gives every canonical orbital the exponential tail of the outermost one, so all orbitals must share one wall; this also makes them exactly orthogonal. Second, over that common wall the inner shells are extremely stiff: the growing mode of the krypton 1s orbital rises by 10^1701. The equations are therefore solved by multiple shooting, with Taylor-series segments short enough that no mode grows by more than 10g, coupled by continuity conditions and solved by Newton’s method with a sparse LU factorization, so that the working precision grows with the target accuracy rather than with the stiffness. Third, a homotopy in the electron–electron coupling, from hydrogenic orbitals, supplies a starting point that converges to the ground state. The exact-rational certificate carries over to any angular momentum. Writing each radial function as P = rl+1f with f piecewise polynomial, the triangle rule k ≤ la + lb cancels the factor r−k−1 of every multipole, so all overlap, one-electron and Slater integrals are exact rationals. All 25 published digits of the quadruple-precision energies of Cinal are reproduced, as are his positions of the exchange-induced nodes of inner orbitals, which we locate to about 100 significant figures.