Sum rules for the exchange-correlation energy functional of the extended constrained-search theory: Application to checking the validity of the vorticity expansion approximation of the current-density functional theory

We present four kinds of sum rules for the exchange-correlation energy functional of the extended constrained-search theory. They are applicable even to the conventional density functional theory. As an application of these sum rules, we utilize them to check the validity of the vorticity expansion approximation (VEA) of the current-density functional theory (CDFT). The VEA formula fulfils three of them, though the local density approximation formula of the CDFT fulfills only one. The validity of the VEA formula is thus confirmed successfully from the viewpoint of the sum rules.