Density matrices and density functionals in strong magnetic fields

The equation of motion for the first-order density matrix (1DM) is constructed for interacting electrons moving under the influence of given external scalar and vector potentials. The 1DM is coupled there to the 2DM by means of the electron-electron interaction. This equation is then employed to obtain the differential virial equation for interacting electrons moving in a magnetic field of arbitrary strength. Suitable integration leads back to the virial theorem derived recently by Erhard and Gross. The exchange-correlation scalar potential of the current-density functional theory of Vignale and Rasolt is derived in two forms, in terms of 1DMs and 2DMs and their noninteracting-system counterparts, involving also (in a linear way) the vector potentials: external and exchange-correlation (xc) ones in the first form, and the xc one in the second form. An equation is obtained also for determining the corresponding xc vector potential in terms of the same DMs and the external vector potential. Approximate exchange-only scalar and vector potentials are proposed in terms of noninteracting 1DM. Finally the Hartree-Fock 1DM for atoms and molecules in magnetic fields is shown to satisfy the same equation of motion as the fully interacting 1DM.