Variational and diagrammatic evaluations of the ground-state energy of quantum liquids

The Feynman-Hellmann theorem shows that the exact ground-state energy of a quantum system may be obtained by integrating the potential energy with respect to a coupling constant. For approximation schemes in which the energy is actually calculated from an energy functional, we show that direct and coupling-constant integration evaluations of the energy functional are identical, provided that the energy functional is optimized.