Chemistry Reference
In-Depth Information
v
(
r
)
Position,
r
V
(
r
)
Figure 3.15
The valence wavefunctions in a solid are expected to oscillate rapidly
near each nucleus, because of the strong Coulomb interaction between
the nucleus and the electron. How then can the valence wavefunction be
approximated by a small number of plane waves when we use the NFE
model?
where the coefficients
a
i
k
are chosen to ensure the valence basis functions
φ
v
(
r
)
are orthogonal to the core states within each unit cell, that is,
φ
c
i
(
d
3
r
k
v
r
)φ
(
r
)
=
0
(3.41)
This can be achieved by choosing the
a
i
k
values such that
d
3
r
φ
c
i
(
e
i
k
·
r
a
i
k
=−
r
)
(3.42)
The orthogonalised plane waves of eq. (3.40) are, therefore, constructed
to have the necessary rapid oscillations in the core regions while having a
slow plane-wave-like variation in the remainder of the structure. It is then
possible to calculate the band structure using a relatively small number of
OPW basis functions.
We do this by using linear combinations of orthogonalised plane waves
to solve the full crystal Hamiltonian for the valence states
H
E
k
n
k
m
v
k
m
v
m
α
φ
(
r
)
=
m
α
φ
(
r
)
(3.43)
mn
mn
k
m
v
where the
φ
(
r
)
are a finite set of OPWs and theHamiltonian
H
is given by
2
2
m
∇
=−
2
H
+
V
(
r
)
(3.44)
where
V
(
r
)
is the full crystal potential.
