Hohenberg-Kohn and Two Other Density Theorems

Hohenberg-Kohn and Two Other Density Theorems.—Below a proof is given that a unique charge density exists for each external potential, following Hohenberg and Kohn.135 

Now let us suppose further that the electron densities associated with the wave functions Wx and W are the same. One can then write 

By addition these two inequalities yield E+Ex E+EX and one must conclude to escape this absurdity that two different external potentials cannot generate the same electron density. But E is uniquely determined by the external potential and hence one deduces that the ground-state energy is a unique functional of the electron density p(r). This result, known as the Hohenberg-Kohn theorem, was assumed in all the early work on the density description (see refs. 4 and 16). Of course, the problem of finding the energy functional remains to date it is a matter of judicious approximation for the problem under consideration. 

It is relevant in this context to refer to the earlier result of Wilson18 that, for a molecule with nuclear charges Za, and internuclear distances Rap, the 

A better known relation, also involving the density, is the theorem of Kato.11 For a spherically symmetric atom or ion, it relates the electron density at the nucleus to the derivative of p also taken at the nucleus through 

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