Tight-binding method in a nutshell
Consider first, for the sake of simplicity, a solid built up from atoms of the same type.
A tight binding variational wavefunction is given by
ψ(r) =
inlm
ainlmφnlm(r − Ri) , (1)
where φnlm(r) are the atomic orbitals, the index i refers to atoms of the lattice, Ri is
the position vector of the i-th nucleus. The values of the coefficients ainlm are to be
found by using the variational method. In the following, we consider a simple crystal
lattice with one atom per unit cell. We may thus restrict our attention to functions
satisfying the Bloch theorem,
ψ(r + Rq) = eikRq
ψ(r) ∀ Rq
of the crystal lattice. By inserting the expression on the r. h. s. of Eq. (1) we obtain
inlm
ainlmφnlm[r − (Ri − Rq)] = eikRq
inlm
ainlmφnlm(r − Ri) . (2)
By comparing the expressions on the left and the right hand side of the above equation
(the coefficients associated with the wave function φnlm(r) “at the origin") we find
that
aqnlm = a0nlmeikRq
. (3)
The expansion of Eq. (1) then becomes
ψ(r) =
nlm
a0nlm
q
eikRq
φnlm(r − Rq) =
=
nlm
anlmϕk,nlm(r) , (4)
where anlm = a0nlm
√
N,
ϕk,nlm =
1
√
N q
eikRq
φnlm(r − Rq) , (5)
and N is the number of unit cells in the Born Kármán region. The factor 1/
√
N is
introduced because of normalization: if the set of the atomic orbitals is orthonormal
then the wavefunctions defined by Eq. (5) are normalized. For any k, the number
of basis wavefunctions ϕk,nlm, and also the corresponding number of equations to be
solved, is the same as the number of atomic orbitals considered!
It remains to find the equations determining the coefficients anlm(k) (the argument
has been added, the coefficients are k-dependent) of Eq. (4). The strategy is the same
as in the studies of the electronic structure of simple molecules; the variational condition
leads to the following set of equations:
nlm
anlm(k)[ ϕk,n′l′m′ |h|ϕk,nlm − ǫ(k) ϕk,n′l′m′ |ϕk,nlm ] = 0 . (6)
The matrix elements in the above equation are then expressed in terms of those
between the atomic orbitals: φn′l′m(r − Ri)|h|φnlm(r − Rj) and
φn′l′m(r − Ri)|φnlm(r − Rj) .
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Frequently it is assumed that the only relevant matrix elements of h are
(a) the “on-site matrix elements”, φnlm(r − Ri)|h|φnlm(r − Ri) = ǫnlm, and
(b) the “hopping matrix elements” corresponding to “nearest neighbours”
φn′l′m′ (r − Ri)|h|φnlm(r − Rj) = tij,n′l′m′,nlm (i, j nearest neighbours).
The values of the on-site matrix elements and the hopping-matrix elements may be
estimated according to various semiempirical rules. The overlap matrix
φn′l′m′ (r−Ri)|φnlm(r−Rj) is frequently approximated by the unit matrix δijδn′l′m′;nlm.
For crystals with several atoms per unit cell, we obtain an analogue of Eq. (6),
where the sum runs, in addition to n,l,m, over the atoms.
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