Chemistry Reference
In-Depth Information
that enables a realistic evaluation of more complex systems. A more elaborated
approach is called the extended Huckel method, which allows the determination of
H
ij
for
i
=
j
from the following expression:
+
75
S
ij
H
ii
H
jj
H
ij
=
1
.
.
(1.21)
2
Some relevant references concerning the application of the Huckel method are
Hoffmann, 1963; Cotton, 1971; Iung & Canadell, 1997.
Periodic solids
In principle we could apply the methodology described in the previous section for
very large molecules and solids, where
N
at
→∞
, but in this case the resolution of
the secular determinant given in Eq. (1.19) would be impossible. Some boundaries
have to be given in order to make the problemmanageable and the most elegant way
is by taking symmetry into account. An infinite solid can be quite simply described
if it exhibits real-space long-range periodicity (period negligible compared with the
crystal dimensions), remaining invariant under primitive or lattice translations
T
R
n
of vectors
R
n
and
a
1
,
a
2
and
a
3
are
non-coplanar basis vectors. The set of all
R
n
vectors leads to all equivalent points
in the lattice. In an infinite crystal with a real-space period
R
n
, the application of
T
R
n
would transform the crystal into itself, so that any physical property should be
invariant under such operation. The effective interaction potential
V
(
r
i
) must thus
be periodic:
=
n
1
a
1
+
n
2
a
2
+
n
3
a
3
, where
n
1
,
n
2
and
n
3
∈ Z
T
R
n
V
(
r
i
)
=
V
(
r
i
+
R
n
)
=
V
(
r
i
)
,
(1.22)
and as a consequence theHamilton and the lattice translation operator
T
R
n
commute:
[
H
1e
H
1e
T
R
n
−
T
R
n
H
1e
,
T
R
n
]
=
=
0
.
(1.23)
Hence, applying the
T
R
n
operator to the left of Eq. (1.8) we obtain:
T
R
n
H
1e
1e
1e
H
1e
T
R
n
1e
1e
|
=
T
R
n
E
|
=
|
=
ET
R
n
|
.
(1.24)
1e
In conclusion, the invariance under lattice translations means that if
|
is a
1e
wave function of the Hamilton operator,
T
R
n
|
is a solution as well. It can be
shown that if two operators commute they have a common orthonormal basis of
wave functions. In the case of the Hamilton and lattice translation operator this
common base is given by the Bloch functions
|
φ
(
k
,
r
)
:
e
i
kR
n
T
R
n
|
φ
(
k
,
r
)
=|
φ
(
k
,
r
+
R
n
)
=
|
φ
(
k
,
r
)
.
(1.25)
Bloch's theorem establishes that the eigenstates of the one-electron Hamiltonian
H
1e
, where
V
(
r
i
)
=
V
(
r
i
+
R
n
) for all
R
n
in a Bravais lattice, can be chosen to
