Bound on the Excess Charge of Generalized Thomas-Fermi-Weizsäcker Functionals
Rafael D. Benguria, Heinz Siedentop
TL;DR
The paper analyzes generalized Thomas-Fermi-Weizsäcker functionals with exponent $p\in(3/2,2)$ and establishes a uniform-in-$Z$ bound on the excess charge $Q$ for $p$ in this range, improving prior estimates and clarifying the binding behavior as $p$ approaches the critical value $p=3/2$. It combines variational and potential-theoretic methods, Nam-type energy bounds, and Sommerfeld comparison principles to derive a sharp upper bound $Q\le 0.5211\,Z$ for $p\ge 6/5$, and proves a uniform bound $0\le Q\le B(p) \;A^{(3p-4)/(4p-6)}\gamma^{-1/(4p-6)}$ in the atomic case with a detailed minimax formulation for $B(p)$. At the critical exponent $p=\frac{3}{2}$, the analysis reveals a threshold coupling $\gamma_c=4\sqrt{\pi}$: for $\gamma\ge\gamma_c$ the excess charge vanishes ($Q=0$), while for $0\le\gamma<\gamma_c$ one obtains a linear-in-$Z$ bound $Q\le\frac{\gamma_c-\gamma}{\gamma}\,Z$, indicating a phase-transition-like change in binding. Overall, the work sharpens quantitative bounds on excess charge for generalized density-functionals and clarifies the competition between kinetic and Weizsäcker terms in determining electron binding.
Abstract
We bound the number of electrons $Q$ that an atom can bind in excess of neutrality for density functionals generalizing the classical Thomas-Fermi-Weizsäcker functional: instead of the classical power $5/3$ more general powers $p$ are considered. For $3/2<p<2$ we prove the excess charge conjecture, i.e., that $Q$ is uniformly bounded in the atomic number $Z$. The case $p=3/2$ is critical: the behavior changes from a uniform bound in $Z$ to a linear bound at the critical coupling $4\sqrtπ$ of the nonlinear term. We also improve the linear bound for all $p\geq6/5$.