Abstract
Since the Schrödinger equation can be solved exactly only in very few special cases, approximation methods to determine the density ρ of the electric charge have been developed. This chapter discusses some basic mathematical aspects of these approximations. Here some types of variational problems emerge which have not been mentioned previously, namely minimization of functions defined on Banach spaces which are not reflexive. We begin with an overview of the semi-classical theories of density functionals (mainly Thomas-Fermi and variations). Then we proceed to explain and prove the basic facts of a theory mainly developed by Hohenberg, Kohn and Sham (uniqueness theorem, Hohenberg-Kohn variational principle and the Kohn-Sham equations).
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Blanchard, P., Brüning, E. (2015). Density Functional Theory of Atoms and Molecules. In: Mathematical Methods in Physics. Progress in Mathematical Physics, vol 69. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-14045-2_37
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DOI: https://doi.org/10.1007/978-3-319-14045-2_37
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