Hellmann-Feynman theorem in a mixed ensemble

The Hellmann-Feynman theorem in a pure ensemble is extended to a quasidegenerate mixed ensemble (canonical) as an ensemble average value. The theorem is also extended to a grand canonical ensemble wherein average particle number is conserved. Connections with the thermodynamic observables are made. An approximate relation between {partial_derivative}[H]{sub C}/{partial_derivative}{lambda} and {partial_derivative}[H]{sub G}/{partial_derivative}{lambda} is established at the leading order of the saddle-point approximation.