Scheme for the energy-band calculation under the external magnetic field

In the framework of the current-density functional theory (CDFT), we propose three kinds of approximation schemes for the exchange-correlation energy functional on the basis of the coupling-constant expression. These should be called the CDFT version of the local-density approximation, average-density approximation and weighted-density approximation. The validity of these approximations is estimated by using the sum rules which have recently been derived