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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.

A trial function is a chosen candidate used to approximate a solution or evaluate a variational principle. For an eigenvalue upper bound from a Rayleigh quotient, it must be nonzero, have finite numerator and denominator, and lie in the appropriate variational domain. In a Dirichlet problem it satisfies the fixed endpoint values. Natural boundary conditions need not always be imposed on a trial in the quadratic-form domain, but a candidate violating an essential boundary condition does not give the stated bound.