  We propose a new method for simulating electron dynamics in open quantum
systems out of equilibrium, using a finite atomistic model. The proposed method
is motivated by the intuitive and practical nature of the driven Liouville
von-Neumann equation approach of S\'anchez et al. [J. Chem. Phys., 124, 214708
(2006)]. A key ingredient of our approach is a transformation of the
Hamiltonian matrix from an atomistic to a state representation of the molecular
junction. This allows us to uniquely define the bias voltage across the system
while maintaining a proper thermal electronic distribution within the finite
lead models. Furthermore, it allows us to investigate complex molecular
junctions, including non-linear setups and multi-lead configurations. A
heuristic derivation of our working equation leads to explicit expressions for
the damping and driving terms, which serve as appropriate electron sources and
sinks that effectively "open" the finite model system. Although the method does
not forbid it, in practice we find neither violation of Pauli's exclusion
principles nor deviation from density matrix positivity throughout our
numerical simulations of various tight-binding model systems. We believe that
the new approach offers a practical and physically sound route for performing
atomistic time-dependent transport calculations in realistic molecular
junctions.
