Rigorous Lower Bounds for Atomic and Ionic Ground-State Energies
Abstract
We establish a new iterative method for computing rigorous lower bounds for the ground-state energies of all atoms and ions. The accuracy of the resulting bounds depends on two ingredients: the quality of the initial energy estimate and the quality of the used upper bound for the maximum number of bound electrons, Nc(Z), as a function of the nuclear charge Z>0. In particular, this new method allows us to establish the explicit lower bound E(N,Z) ≥ −0.8710 N1/3Z2 for the ground-state energy E(N,Z) of the atom/ion with N ∈ ℕ electrons and nuclear charge Z ∈ (0,N], thereby considerably improving the best known constant A=(3/2)1/3 ≈ 1.1447 in the estimate E(N,Z) ≥ −A N1/3Z2 for 0 < Z ≤ N and coming much closer to the best possible constant given by the Thomas-Fermi asymptotics limN → ∞ |E(N,N)|N−7/3= −ETF(1)≈ 0.7687 derived in Adv. Math., 23(1) by Lieb and Simon. To demonstrate the potential and the limits of our method, we also compute explicit lower bounds for the ground-state energies E(N,N) of all atoms with 2 ≤ N ≤ 86 and we compare them with the bounds obtained in J. Chem. Phys., 62(8) by Hertel, Lieb, Thirring and the bounds obtained by Seco in his dissertation. Finally, we also provide a uniqueness result for the expectation values of the kinetic, potential and interaction energies on the ground-state space.
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BibTeX
@misc{hofacker2026rigorous,
title = {{Rigorous Lower Bounds for Atomic and Ionic Ground-State Energies}},
author = {Michael Hofacker},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34480},
eprint = {2610.34480},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34480},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Rigorous Lower Bounds for Atomic and Ionic Ground-State Energies
AU - Michael Hofacker
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.34480
UR - https://arxiv.org/abs/2610.34480
AB - We establish a new iterative method for computing rigorous lower bounds for the ground-state energies of all atoms and ions. The accuracy of the resulting bounds depends on two ingredients: the quality of the initial energy estimate and the quality of the used upper bound for the maximum number of bound electrons, N_c(Z), as a function of the nuclear charge Z>0. In particular, this new method allows us to establish the explicit lower bound E(N,Z) ≥ −0.8710 N^1/3Z^2 for the ground-state energy E(N,Z) of the atom/ion with N ∈ ℕ electrons and nuclear charge Z ∈ (0,N], thereby considerably improving the best known constant A=(3/2)^1/3 ≈ 1.1447 in the estimate E(N,Z) ≥ −A N^1/3Z^2 for 0 < Z ≤ N and coming much closer to the best possible constant given by the Thomas-Fermi asymptotics lim_N → ∞ |E(N,N)|N^−7/3= −E_TF(1)≈ 0.7687 derived in Adv. Math., 23(1) by Lieb and Simon. To demonstrate the potential and the limits of our method, we also compute explicit lower bounds for the ground-state energies E(N,N) of all atoms with 2 ≤ N ≤ 86 and we compare them with the bounds obtained in J. Chem. Phys., 62(8) by Hertel, Lieb, Thirring and the bounds obtained by Seco in his dissertation. Finally, we also provide a uniqueness result for the expectation values of the kinetic, potential and interaction energies on the ground-state space.
ER -