  The Landauer expression for computing current-voltage characteristics in
nanoscale devices is efficient and widely applicable but not suited to
transient phenomena and time dependent currents because it assumes that the
charge carrier population attains a time independent dynamic equilibrium as
soon as the external voltage is turned on. In this article, we construct a very
general expression for a time dependent current in an
electrode-molecule-electrode arrangement. Utilizing a model Hamiltonian, we
propagate the Schrodinger wave function equation to numerically compute the
time dependent population in the individual sub-systems. The current in each
electrode (defined in terms of the rate of change of the corresponding
population) has two components, one due to the charges originating from the
same electrode and the other due to the charges initially residing at the other
electrode. We derive an analytical expression for the first component and
illustrate that it agrees reasonably with the numerical counterpart at early
times. The structural form reveals that the initial occupancy can be factored
out of the time dependent segment of the expression. We take this cue to
construct a Landauer style formula and demonstrate that the current obtained
from this simplified formula overlaps with our most general numerical current
only after the charge flow settles into a steady state. Thus, we illustrate the
emergence of Landauer transport from a true first-principles quantum dynamics
calculation without any prior assumptions. Subsequently, we investigate the
ingredients in our model that regulate the onset time scale of this Landauer
regime. We compare the performance of our general current expression with the
Landauer current for time dependent electronic coupling. Finally, we comment on
the applicability of the Landauer formulas to compute hot-electron current
arising upon plasmon decoherence.
