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](C)/partial derivative lambda and partial derivative[H](G)/partial derivative lambda is established at the leading order of the saddle-point approximation.