180
6 Magnetism and Conduction
with g and u the bonding and anti-bonding combinations of the local orbitals a and
b. The energy of the two doublets is
E(D 1 ) ==hhg| ˆ
H|hhg=
1
2
hh(a + b)| ˆ
H|hh(a + b)
=
1
2
hha| ˆ
H|hha+2hha| ˆ
H|hhb++hhb| ˆ
H|hhb
(6.3)
E(D 2 ) ==hhu| ˆ
H|hhu=
1
2
hh(a − b)| ˆ
H|hh(a − b)
=
1
2
hha| ˆ
H|hha−2hha| ˆ
H|hhb++hhb| ˆ
H|hhb
(6.4)
Combining the energy difference of the two doublets
∆E 12 = E(D 1 ) − E(D 2 ) = 2hha| ˆ
H|hhb
(6.5)
with the definition given in Eq. 6.1a, the hopping parameter can be calculated by
t
+
ab =
1
2
∆E 12
(6.6)
In practice, an effective hopping parameter can be obtained from accurate ab initio
energies for the two doublets. For non-centrosymmetric systems, the calculation is
slightly more involved. The two doublets are now defined as
D 1 = c 1 |hha|+c 2 |hhb|
D 2 = c 2 |hha|−c 1 |hhb|
(6.7)
and the energy difference is
∆E 12 = (c
2
1 − c
2
2 )(H aa − H bb ) + 4c 1 c 2 H ab
(6.8)
with H ij ==hhi| ˆ
H|hhj. This leads to the following expression for t
+
ab
t
+
ab =
∆E 12 − (c 2
1 − c 2
2 )(H aa − H bb )
4c 1 c 2
(6.9)
To determine t, the energy difference is no longer sufficient and information is
required from the wave function. The magnetic orbitals have to be expressed in
orthogonal atomic-like orbitals and the wave functions projected on the model space
{|hha|, |hhb|}. After orthonormalization, a numerical 2 × 2 effective Hamiltonian
can be constructed
ˆ
H eff |hha| hhb
hha| H aa t
+
ab
hhb|
t
+
ab
H bb