In the tight-binding approach [15],
the wavefunction is expanded in terms of a set of
localized states 
in each atomic layer j
The coefficients 
can be thought of as forming a block-structured vector 
with vector elements 
. The Hamiltonian then
becomes a block-structured matrix 
, of which the diagonal blocks
are hermitian and
describe interactions within a plane and the off-diagonal blocks 
are not necessarily hermitian
(
)
and describe the coupling between planes. If only nearest-neighbor interactions are
included, 
is block-tridiagonal [16,17]
(
only for j = i-1, i, i+1).
Because the tight-binding representation is intrinsically discrete, we need to
modify somewhat our concepts of probability and current density. A total probability
density 
is associated with each atomic plane i, and is equal to
. The current density represents the flux between
adjacent planes; we will write the flux between planes i and i+1 as
[18].
Applying (2) to the tight-binding Hamiltonian 
and
assuming only nearest-neighbor interactions, we get
This can be written as a discrete continuity equation,
if 
is identified as
Here we see the machinery of Green's identity operating in a discrete space.
A variation of the tight-binding scheme is the ``Wannier Orbital Model''
[19], which draws upon the discrete form of the Wannier-Slater theory.
Interactions with remote neighbors are included to fit the 
dispersion, and
typically M=1, so that only one band is modeled. Inserting such a Hamiltonian
into (2) leads to many terms, which cannot be associated with
a particular position. Thus, the notion
of a local current density disappears, due to the direct interactions between remote
sites. Instead, we may define an antisymmetric current matrix with elements
where 
is the current flowing out of site i into site j.
The nonlocal continuity equation is then
