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# A convexified energy functional for the Fermi-Amaldi correction

• 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 $N_{1}.$
Mathematics Subject Classification: Primary: 35J47, 35J91, 49S05; Secondary: 81Q99, 81V55, 92E10.

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