Rigorous lower bounds for atomic and ionic ground-state energies

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 $E(N,Z)\ge −0.8710 N^{1/3} Z^2$ 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 $E(N,Z)\ge −AN^{1/3} Z^2$ for $0<Z\le N$ and coming much closer to the best possible constant given by the Thomas–Fermi asymptotics $\lim_{N\to\infty}|E(N, N )|N^{−7/3} = −E_{TF}(1) \approx 0.7687$ [22]. 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 $2\le N\le 86$ and we compare them with the bounds obtained in [11] and [29]. Finally, we also provide a uniqueness result for the expectation values of the kinetic, potential and interaction energies on the ground-state space.