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t Matrix MethodWe find the transmission probability T(E) used in the Landauer formula, Eq.(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
and has different forms on the left
lead (L), molecule (M), and right lead (R). The total
wavefunction is a sum of these three,
,
where
In the above, are forward/backward propagating Bloch waves in the mode, and is the j^{th} MO on the molecule.
We then consider our Hamiltonian in the tightbinding (or
Hückel) approximation and use Schroedinger's equation to
arrive at a system of linear equations in the unknown
quantities,
,
c_{j} and
.
We solve numerically for the reflection
and transmission coefficients for each rightward propagating
mode
at energy E in the left lead. The total
transmission is then given by

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