  The uniform electron gas placed between two reservoirs is used as a model
system for molecular junctions under an external bias. The energetics of the
electron gas are calculated by generalizing the Thomas-Fermi-Dirac (TFD) model
to nonequilibrium cases. We show that when the bias voltage is not zero, the
first Hohenberg-Kohn (HK) theorem breaks down, and energies of the electron gas
can be determined by the total electron density together with the density of
nonequilibrium electrons, supporting the dual-mean-field (DMF) theory recently
proposed by us [J. Chem. Phys. 139, (2013) 191103]. The generalization of TFD
functionals to DMF ones is also discussed.
