The helium ground-state energy to 60 significant figures on a laptop
Abstract
We report a variational upper bound on the nonrelativistic, infinite-nuclear-mass ground-state energy of helium whose first 60 significant figures are converged, E0 = −2.903 724 377 034 119 598 311 159 245 194 404 446 696 925 311 117 290 339 042 34 Eh, 16 more than the best previous record. The calculation runs on a laptop and the method, which we call FLAKE (Fock-log anisotropic Kronecker eigensolver), combines four ingredients. (i) A change of variables making Schwartz’s Hylleraas-type basis with logarithms an exact tensor product, so the Hamiltonian is a sum of 27 Kronecker products of small matrices. (ii) Exact, arbitrary-precision orthonormalisation of the small radial and angular Gram factors removes all cancellation from the large computation. (iii) Fixed-point products at about 280 bits are split without error into ordinary double-precision matrix multiplications. (iv) The variational space is anisotropic: it carries extra radial degree and a tower of (ln s)p functions only at low angular order, as dictated by channelresolved convergence rates. Every energy quoted is certified by an independent arbitrary-precision evaluation of the Rayleigh quotient of the explicit trial function. The remaining error is estimated from geometric convergence laws that predicted subsequent energy increments to within 2%.
Related
- Asymptotics of Ground States for Helium-Like and Bosonic Atoms in a Two-Cluster Region
- Low-energy positron scattering from metastable helium
- Energy-dependent trapping of backscattered helium during magnetron sputtering growth: A molecular dynamics study
- Third Order Upper Bound for the Ground State Energy of the Dilute Bose Gas.
- Reverse Quantum Annealing for Ground-State Energy Calculation in Molecule
BibTeX
@misc{baskerville2026helium,
title = {{The helium ground-state energy to 60 significant figures on a laptop}},
author = {Adam Baskerville},
howpublished = {ChemRxiv},
year = {2026},
month = oct,
doi = {10.26434/chemrxiv.15010182/v1},
url = {https://chemrxiv.org/doi/full/10.26434/chemrxiv.15010182/v1},
note = {CC BY},
}
RIS
TY - UNPB
TI - The helium ground-state energy to 60 significant figures on a laptop
AU - Adam Baskerville
PY - 2026
DA - 2026/10/09
PB - ChemRxiv
DO - 10.26434/chemrxiv.15010182/v1
UR - https://chemrxiv.org/doi/full/10.26434/chemrxiv.15010182/v1
AB - We report a variational upper bound on the nonrelativistic, infinite-nuclear-mass ground-state energy of helium whose first 60 significant figures are converged, E0 = −2.903 724 377 034 119 598 311 159 245 194 404 446 696 925 311 117 290 339 042 34 Eh, 16 more than the best previous record. The calculation runs on a laptop and the method, which we call FLAKE (Fock-log anisotropic Kronecker eigensolver), combines four ingredients. (i) A change of variables making Schwartz’s Hylleraas-type basis with logarithms an exact tensor product, so the Hamiltonian is a sum of 27 Kronecker products of small matrices. (ii) Exact, arbitrary-precision orthonormalisation of the small radial and angular Gram factors removes all cancellation from the large computation. (iii) Fixed-point products at about 280 bits are split without error into ordinary double-precision matrix multiplications. (iv) The variational space is anisotropic: it carries extra radial degree and a tower of (ln s)p functions only at low angular order, as dictated by channelresolved convergence rates. Every energy quoted is certified by an independent arbitrary-precision evaluation of the Rayleigh quotient of the explicit trial function. The remaining error is estimated from geometric convergence laws that predicted subsequent energy increments to within 2%.
ER -