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.

The paper addresses rigorous lower bounds for nonrelativistic atomic and ionic ground-state energies $E(N,Z)$ with a fixed nucleus. It develops a recursive construction that propagates an initial lower bound using an upper bound $N_c(Z)<f(Z)$ on the maximum number of bound electrons. The construction applies to fermionic and bosonic/full-space Hamiltonians, with statistics entering through the noninteracting hydrogenic energy sum $\sigma_N$. The central estimate is $\sqrt{|E(N,Z_1)|}\leq\sqrt{|E(N,Z_0)|}+\sqrt{\sigma_N}(Z_1-Z_0)$ for $Z_1\geq Z_0$. It follows from the virial theorem, variational estimates and concavity in nuclear charge. Combining it with $E(N,Z)=E(N-1,Z)$ beyond the binding limit gives piecewise-linear upper bounds on $\sqrt{|E|}$. For spin-$1/2$ fermions, existing ionization bounds yield $E(N,Z)\geq-0.8710N^{1/3}Z^2$ for $0<Z\leq N$, improving the stated previous constant $1.1447$. The neutral-atom Thomas–Fermi asymptotic coefficient is approximately $0.7687$. Establishing the uniform bound combines analytical estimates with computer verification for $N\leq3600$. Comparisons for neutral atoms through $N=86$ show that the recursive bounds generally remain weaker than existing numerically obtained lower bounds, exceeding their magnitudes by over $20\%$ at $N=86$. Accuracy depends on the initial estimate and binding-number bound; improved comparisons assuming $N_c(Z)<Z+1$ and a numerical critical-charge estimate are explicitly conditional. The paper also establishes equality of kinetic, nuclear-attraction and interaction expectations across degenerate ground states wherever $E(N,Z)$ is differentiable in $Z$.