= Excited-state variational constraint
Unrestricted minimization of a <Rayleigh quotient> seeks the ground state. To bound the $k$th excited <energy> from above, restrict normalized trial <wavefunctions> to the <orthogonal complement> of the exact lower-energy eigenspaces. In an exact <orthonormal eigenbasis>, all remaining expansion coefficients multiply energies at least $E_k$, so their weighted average is at least $E_k$. A known <symmetry> sector can supply some of these exclusions automatically. <Orthogonality> only to inaccurate approximate lower states does not automatically retain the same rigorous excited-state upper bound.
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