Sum rules and bounds on the exchange and correlation energy functional of current-density functional theory

The exact expression for the exchange-correlation energy functional Exc@r ,jp# of the current-density functional theory ~CDFT! is derived by means of the coupling-constant integration technique. It contains the coupling-constant-averaged pair correlation function, which is a functional of the electron density r(r) and paramagnetic current density jp(r). On the basis of this expression, the local density approximation and its modifications for Exc@r ,jp# are proposed within the CDFT. In addition, we present sum rules and bounds on Exc@r ,jp# by considering the behaviors of the basic variables and Exc@r ,jp# under the various types of the nonuniform coordinate scaling of electrons. They are useful in estimating the validity of the approximate forms of Exc@r ,jp# proposed.