The First Hohenberg-Kohn Theorem Proof of Existence

1 The First Hohenberg-Kohn Theorem Proof of Existence 

Therefore F and TU respectively, are different, and we can use as trial wave function for H We must then have by virtue of the variational principle 

Interchanging the unprimed with the primed quantities and repeating the above steps of equations (4-1) to (4-3) we arrive at the corresponding equation 

After adding equations (4-3) and (4-4), this leaves us with the clear contradiction 

This concludes the proof that there cannot be two different Vext that yield the same ground state electron density, or, in other words, that the ground state density uniquely specifies the external potential Vext. Using again the terminology of Section 1.2 we can simply add p0 as the property which contains the information about N, ZA, RA and summarize this as 

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