A text I am reading has used the variational principle not only to find the ground state of a system, but also to find some higher order states. (Specifically, it was used to derive the Roothaan equations, which are ultimately related to the LCAO method of orbital calculations.) I don't see how this could be valid. For finding the ground state energy of a system, it is obvious why minimizing the expected value of the Hamiltonian gets the best approximation to the ground state. But in what sense does the variational principle converge to the right result? I thought at first that it might minimize the variance of Hamiltonian in the trial wavefunction, but I do not believe that this is the case. So, does anyone know exactly in what precise sense the variational principle finds the "best" solutions to the eigenvalue problem?