View source for Density functional theory/Hohenberg–Kohn theorems

==Hohenberg–Kohn theorems==

1. If two systems of electrons, one trapped in a potential <math> v_1(\vec r)</math> and the other in <math> v_2(\vec r)</math>, have the same ground-state density <math> n(\vec r)</math> then necessarily <math> v_1(\vec r)-v_2(\vec r) = const</math>.

Corollary: the ground state density uniquely determines the potential and thus all properties of the system, including the many-body wave function. In particular, the "HK" functional, defined as <math> F[n]=T[n]+U[n] </math> is a universal functional of the density (not depending explicitly on the external potential).

2. For any positive integer <math> N </math> and potential <math> v(\vec r)</math> it exists a density functional <math>F[n]</math> such that <math> E_{(v,N)}[n] = F[n]+\int{v(\vec r)n(\vec r)d^3r} </math> obtains its minimal value at the ground-state density of <math> N </math> electrons in the potential <math> v(\vec r)</math>. The minimal value of <math> E_{(v,N)}[n] </math> is then the ground state energy of this system.

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