Dissociation limits in Density Functional Theory

In this paper we consider the {\it Density Functional Theory} (DFT) framework, where a functional of the form $$F_\eps(\rho)=\eps T(\rho)+bC(\rho)-U(\rho)$$ has to be minimized in the class of non-negative measures $\rho$ which have a prescribed total mass $m$ (the total electronic charge). The parameter $\eps$ is small and the terms $T$, $C$, $U$ respectively represent the kinetic energy, the electronic repulsive correlation, the potential interaction term between electrons and nuclei. Several expressions for the above terms have been considered in the literature and our framework is general enough to include most of them. It is known that in general, when the positive charge of the nuclei is small, the so-called {\it ionization phenomenon} may occur, consisting in the fact that the minimizers of $F_\eps$ can have a total mass lower than $m$; this physically means that some of the electrons may escape to infinity when the attraction of the nuclei is not strong enough. Our main goal, continuing the research we started in \cite{bbcd18}, is to study the asymptotic behavior of the minimizers of $F_\eps$ as $\eps\to0$. We show that the $\Gamma$-limit functional is defined on sums of Dirac masses and has an explicit expression that depends on the terms $T$, $C$, $U$ that the model takes into account. Some explicit examples illustrate how the electrons are distributed around the nuclei according to the model used.