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We show that the diagonal elements of the second-order reduced density matrix(RDM2) can be chosen as basic variables for describing the superconductingstate, instead of the off-diagonal elements of the RDM2 that are usuallyadopted as basic variables in the density functional scheme. The diagonalelements of the RDM2 are called pair-density (PD), which is explicitly relatedto the fluctuation of the particle number of the system. In this paper, weargue that the fluctuation of the particle number can become an indication ofthe superconducting state, and that the density functional scheme in which thePD is chosen as a basic variable would be a promising first-principles theoryfor superconductors.

Basic variables to be reproduced in the first-principles theory for superconductors: Fluctuation of the particle number