2nd HK theorem

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The first Hohenberg-Kohn theorem guarantees that all observable ground state properties (including, in this case, the ground state total energy) are uniquely defined by the electron density, and that a functional exists to relate the two together. In this case, a functional that defines the ground state energy for a certain density distribution is guaranteed to exist, but the 1st theorem says nothing about applying it to other densities. The 2nd theorem likely comes from the definition of ground state energy – the very lowest energy state that a system can possibly occupy.

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