Pair-Correlation Effect and Virial Theorem in the Self-Consistent Density-Functional Theory

The self-consistent derivation of density-functional theory by cluster expansion gives each electron half of the interaction energy. The pair correlation effect maximizing the probability of particle separation recovers a missing energy that increases with the total number of electrons. It provides the first evidence that the virial theorem holds true for atomic and molecular systems. The $3d$ finite element method calculation of water molecule is also presented.