FROST: certified Hartree–Fock limits of helium and beryllium to hundreds of significant figures

We report rigorous upper bounds on the closed-shell Hartree–Fock (HF) energies of helium and beryllium whose first 503 and 501 significant figures are correct, EHF(He) = −2.861 679 995 612 238 878 775 543 740 022 563 429 412 879 776 752 447 053 906 864 . . . Eh, EHF(Be) = −14.573 023 168 316 399 582 411 447 359 870 014 453 877 721 357 . . . Eh, against previous records of 66 and 43. The method, which we call FROST (Fock radial orbital shooting with Taylor series), uses no basis set. When every occupied orbital is an s orbital, the HF equations reduce to coupled radial differential equations for the orbitals and for one potential per pair of orbitals. We integrate them with Taylor series of order up to 873 in arbitrary-precision arithmetic, sizing every step from the local radius of convergence. The orbital energies and normalisations are fixed by nested secant shooting to a hard wall, which is the only truncation and whose error decays as e−2κR. For beryllium each orbital has its own wall, and the canonical equations are solved without enforcing exact orthogonality, which costs energy far below the wall error. The orbitals are piecewise polynomials with dyadic rational coefficients. An independent program therefore evaluates the exact determinant energy in integer arithmetic, and by the variational principle the result is an upper bound free of rounding error. The orbitals are as accurate as the energies. Their nuclear cusp is exact, and for helium the density at the nucleus and ⟨rk⟩ are obtained to about 500 significant figures (previously 32 to 65), with a pointwise error given in closed form by Whittaker functions. On a laptop, the 503-digit helium calculation takes 7.4 minutes and beryllium to 500 digits 4.0 hours. The same code gives the HF energies of H− through Ne8+ to 100 significant figures in about 15 seconds each. We locate the nearest singularities of the helium HF orbital in the complex plane at r∗ = −1.099 ± 0.577 i. This explains why single-exponent Laguerre-type bases converge sub-geometrically and predicts the optimised exponent of the previous record to within 3%. Combined with our FLAKE energy, the correlation energy of helium follows to 58 significant figures.