The Hellmann–Feynman theorem and its relation with the universal density functional

The integrated Hellmann–Feynman theorem provides an expression for the energy as a functional of the density. However, this result is completely defined only for the densities corresponding to the eigen functions of the Schrödinger equation. Since this density functional is obtained by integrating the Hellmann–Feynman theorem, it is not possible to obtain bounds to the true energy by using the variation method together an approximate density. This point is discussed through some elementary examples. The relation between the integrated Hellmann–Feynman theorem and the Hohenberg and Kohn theorem, is also stressed in this Letter.