Mathematical approach to N -centered ensemble DFT and associated methods for the study of open systems
Résumé
A fundamental aspect of the study of N-electronic systems is to obtain information on the states in which these systems have minimal energy. In practice a numerical search of such states is impossible to carry out, so that alternative approaches have been developed, the one around which this work revolves being to consider electronic systems through their electronic density rather than their state. A first implementation of this idea was proposed by Thomas and Fermi and was later revisited by Hohenberg and Kohn who set the basis for the method known today as Density Functional Theory (DFT) for which an in-depth mathematical study has been established by Lieb. Since then, the ideas leading to the construction of DFT have been adapted to the context of electronic systems with a fractional number of electrons (open systems), first through PPLB DFT and more recently through the definition of N-centered ensemble DFT. In both cases it is unclear whether the mathematical properties established for regular DFT can be expected to hold true despite a growing use by quantum chemistry community and physicists. This question is the main problematic of our work, in which we shall study the analogy between N-centered ensemble and regular DFT, from their construction to the methods that are derived from them. This will lead us to construct a Kohn-Sham scheme for N-centered ensemble DFT. We also investigate the links between this theory and optimal transport formulation of DFT-like functionals and finally, we present the Hubbard Dimer calculations framework for N-centered ensemble DFT.
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