  We show that the diagonal elements of the second-order reduced density matrix
(RDM2) can be chosen as basic variables for describing the superconducting
state, instead of the off-diagonal elements of the RDM2 that are usually
adopted as basic variables in the density functional scheme. The diagonal
elements of the RDM2 are called pair-density (PD), which is explicitly related
to the fluctuation of the particle number of the system. In this paper, we
argue that the fluctuation of the particle number can become an indication of
the superconducting state, and that the density functional scheme in which the
PD is chosen as a basic variable would be a promising first-principles theory
for superconductors.
