The Thomas - Fermi - von Weizsacker Theory of Atoms and Molecules *

We place the Thomas-Fermi-von Weizsacker model of atoms on a firm mathematical footing. We prove existence and uniqueness of solutions of the Thomas-Fermi-von Weizsacker equation as well as the fact that they minimize the Thomas-Fermi-von Weizsacker energy functional. Moreover, we prove the existence of binding for two very dissimilar atoms in the frame of this model. Introduction The Thomas-Fermi theory of atoms [1] (TF), attractive because of its simplicity, is not satisfactory because it yields an electron density with incorrect behavior very close and very far from the nucleus. Moreover, it does not allow for the existence of molecules. In order to correct this, von Weizsacker [2] suggested the addition of an inhomogeneity correction p (1) to the kinetic energy density. Here cw = /z /(32πm), where m is the mass of the electron. This correction has also been obtained as the first order correction to the TF kinetic energy in a semi-classical approximation to the Hartree-Fock theory [3]. The Thomas-Fermi-von Weizsacker (henceforth TFW) energy functional for nuclei of charges z > 0 (which need not be integral) located at Rt, i = 1, ... ,/c is defined by j V(x)p(x)dx + ±Sίp(x)p(y)\x -y\-dxdy, (2) in units in which /z(8m)"(3/π) = 1 and \e = 1. Here ρ(x) ^ 0 is the electron * Research supported by U. S. National Science Foundation under Grants MCS78-20455 (R. B.), PHY-7825390 A 01 (H. B. and E. L.), and Army Research Grant DAH 29-78-6-0127 (H. B.) 168 R. Benguria, H. Brezis, E. H. Lieb