1), by solving the Schroedinger equation
directly for the scattered wavefunction of the electron. The electron
is initially propagating in a Bloch wave in one of the modes of the
source lead. The molecule will reflect some of this wave back into
the various modes of the source lead. The molecule is represented by
a discrete set of molecular orbitals (MO's) through which the electron
can tunnel. Hence, some of the wave will be transmitted through the
molecule and into the modes of the drain lead. By finding the
scattered wavefunction it is then possible to determine how much was
transmitted, yielding T(E).
We start with Schroedinger's equation,
where H is the Hamiltonian for the
entire MW system consisting of the leads and the molecule.
is the wavefunction of the electron
propagating initially in the
mode of the left
lead with energy E. It is expressed in terms of the
transmission and reflection coefficients,
and has different forms on the left
lead (L), molecule (M), and right lead (R). The total
wavefunction is a sum of these three,
In the above, are forward/backward propagating Bloch waves in the mode, and is the jth MO on the molecule.
We then consider our Hamiltonian in the tight-binding (or
Hückel) approximation and use Schroedinger's equation to
arrive at a system of linear equations in the unknown
We solve numerically for the reflection
and transmission coefficients for each rightward propagating
at energy E in the left lead. The total
transmission is then given by
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