Density functional variational principle ($E_0={\min }_n\{F[n]+\int vn\}$)

The constrained-search functional $F[n]={\min }_{\Psi \to n}⟨\Psi |T+W|\Psi ⟩$ is universal for a fixed interaction. Minimization of $F[n]+\int vn$ over admissible densities with the correct electron number gives the ground-state energy. Each trial density represents a constrained set of antisymmetric states, so minimizing over densities and their representing states recovers the ordinary variational principle. The admissible domain and possible degeneracy must be handled explicitly.