Universality in the Thomas-Fermi-von Weizsäcker model of atoms and molecules
The Thomas-Fermi-von Weizsäcker (TFW) theory is a nonlinear model for the ground state of an atom or a molecule given (in the atomic case) by the energy functional \[ {\mathcal E}(\psi)=A\int | \nabla \psi |^ 2dx+\frac{3}{5}\gamma \int | \psi |^{10/3}dx-z\int \frac{| \psi |^ 2}{| x|}dx+\frac{1}{2}\int \frac{| \psi (x)|^ 2| \psi (y)|^ 2}{| x-y|}dx dy \] on functions \(\psi\) in \({\mathbb{R}}^ 3\). The minimization problem for this model has been studied by \textit{R. Benguria}, \textit{H. Brezis} and \textit{E. H. Lieb} [Commun. Math. Phys. 79, 167-180 (1981; Zbl 0478.49035)]. In this paper we prove that as \(z\to \infty\) all physical quantities converge. The idea is to renormalize the infinite nuclear charge. In the atomic case above one of the results is that the unique absolute minimizer \(\psi_ z\) converges, i.e., \(\psi_ z(x)\to \psi_{\infty}(x)\), \(| x| \neq 0\). \(\psi^ 2_{\infty}\) is what we call the density for the universal infinite atom. The main step in the renormalization is to characterize the singularity of \(\psi_{\infty}\) at \(x=0\). This is done by showing that both \(\psi_ z\) and \(\psi_{\infty}\) solve a system of partial differential equations. Solutions to this TFW-system have either weak \((z<\infty)\) or strong \((z=\infty)\) singularities at \(x=0\), and the strong singularities have a unique Laurent expansion. Recently we have shown by a completely different method that the highest order term in this expansion also holds for the much more complicated Hartree-Fock model (without the exchange term).
scientific article; zbMATH DE number 4163407
scientific article