  In recent years, several experimental groups have reported measurements of
the current-voltage (I-V) characteristics of individual or small numbers of
molecules. Our purpose in this chapter is to provide an intuitive explanation
for the observed I-V characteristics using simple models to illustrate the
basic physics. In contrast to the MOSFET, whose I-V is largely dominated by
classical electrostatics, the I-V characteristics of molecules is determined by
a more interesting interplay between nineteenth century physics
(electrostatics) and twentieth century physics (quantum transport) and it is
important to do justice to both aspects.
  We start with a qualitative discussion of the main factors affecting the I-V
characteristics of molecular conductors, using a simple toy model to illustrate
their role. Including the effects of: (1) Shift in the energy level due to
charging effects and (2) broadening of the energy levels due to the coupling to
the two contacts we obtain realistic I-V plots, even though the toy model
assumes that conduction takes place independently through individual molecular
levels. In general, however, the full non-equilibrium Green's function (NEGF)
formalism is needed. Here, we describe the NEGF formalism as a generalized
version of the one-level model. This formalism provides a convenient framework
for describing quantum transport and can be used in conjunction with ab initio
or semi-empirical Hamiltonians. A simple semi-empirical model for a gold wire
serves to illustrate the full NEGF formalism. This example is particularly
instructive because it shows the lowest possible "Resistance of a 'Molecule'"
per channel.
