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, $N_c(Z)$, as a function of the nuclear charge $Z>0$. In particular, this new method allows us to establish the explicit lower bound $E(N,Z) \geq -0.8710\,N^{1/3}Z^2$ for the ground-state energy $E(N,Z)$ of the atom/ion with $N \in \mathbb{N}$ electrons and nuclear charge $Z \in (0,N]$, thereby considerably improving the best known constant $A=(3/2)^{1/3} \approx 1.1447$ in the estimate $E(N,Z) \geq -A\,N^{1/3}Z^2$ for $0 < Z \leq N$ and coming much closer to the best possible constant given by the Thomas-Fermi asymptotics $\lim_{N \to \infty} |E(N,N)|N^{-7/3}= -E_{\text{TF}}(1)\approx 0.7687$ 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 $2 \leq N \leq 86$ 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.