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 ${E}_{1}$\ensuremath{\le}${E}_{2}$\ensuremath{\le}...\ensuremath{\le}${E}_{M}$ of a Hamiltonian H, and given M real numbers ${w}_{1}$\ensuremath{\ge}${w}_{2}$\ensuremath{\ge}...\ensuremath{\ge}${w}_{M}$>0, an upper bound for the weighted sum ${w}_{1}$${E}_{1}$ +${w}_{2}$${E}_{2}$+...+${w}_{M}$${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.