Rayleigh-Ritz variational principle for ensembles of fractionally occupied states

Abstract

The Rayleigh-Ritz minimization principle is generalized to ensembles of unequally weighted states. Given the M lowest eigenvalues $E_1$≤$E_2$≤...≤$E_M$ of a Hamiltonian H, and given M real numbers $w_1$≥$w_2$≥...≥$w_M$>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.