Rayleigh-Ritz variational principle for ensembles of fractionally occupied states

The Rayleigh-Ritz minimization principle is generalized to ensembles of unequally weighted states. Given the M lowest eigenvalues E₁{≤}E₂{≤}...{≤}EM of a Hamiltonian H, and given M real numbers w₁{≥}w₂{≥}...{≥}wM>0, an upper bound for the weighted sum w₁{E}₁$ +${w}₂E₂+...+wM{E}M$ is established. Particular cases are the ground-state Rayleigh-Ritz principle (M=1) and the variational principle for equiensembles (${w}₁$=${w}₂$=...=${w}M$). Applications of the generalized principle are discussed.