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March  2010, 28(1): 41-65. doi: 10.3934/dcds.2010.28.41

A convexified energy functional for the Fermi-Amaldi correction

Gisèle Ruiz Goldstein 1, , Jerome A. Goldstein 1, and  Naima Naheed 2,

1. 

Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, United States, United States
2. 

Department of Mathematics and Computer Science, Benedict College, Columbia, SC 29204, United States

Received  October 2009 Revised  January 2010 Published  April 2010

Consider the Thomas-Fermi energy functional $E$ for a spin polarized atom or molecule with $N_{1} $ [resp. $N_{2}$] spin up [resp. spin down] electrons and total positive molecular charge Z. Incorporating the Fermi-Amaldi correction as Benilan, Goldstein and Goldstein did, $E$ is not convex. By replacing $E$ by a well-motivated convex minorant $ \mathcal{E}$ ,we prove that $ \mathcal{E} $ has a unique minimizing density $( \rho _{1},\rho _{2}) \ $ when $N_{1}+N_{2}\leq Z+1\ $and $N_{2}\ $is close to $N_{1}.$
Keywords: $L^{1} $constrained minimization, ground state electron density, Fermi-Amaldi correction, convex minorant, spin polarized system, degree theory, Thomas-Fermi theory.
Mathematics Subject Classification: Primary: 35J47, 35J91, 49S05; Secondary: 81Q99, 81V55, 92E1.
Citation: Gisèle Ruiz Goldstein, Jerome A. Goldstein, Naima Naheed. A convexified energy functional for the Fermi-Amaldi correction. Discrete and Continuous Dynamical Systems, 2010, 28 (1) : 41-65. doi: 10.3934/dcds.2010.28.41
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