Variational principles which are functionals of electron density

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Abstract
The Ritz variational principle, ordinarily written as a functional of the ground‐state wavefunction, is rewritten in such a way that only one‐electron functions are included as variables. Three different such variational formulas are derived explicitly based on the formula given by Hohen Kohn. They constitute variational expressions of the integrated Hellmann–Feynman theorem, integral Hellmann–Feynman theorem, and virial theorem. These variational formulas are exact within the Born–Oppenheimer approximation, and when applied, they give the same results as does the conventional Ritz variational principle. However, they still have the defect that they require knowledge of the correct density function associated with a suitably chosen reference potential. Some applications are given for one‐electron systems. A new formula for the kinetic energy plus electron–electron repulsion energy is obtained from the integrated Hellmann–Feynman theorem.