Chemistry Reference
In-Depth Information
thus enabling interaction or mixing between states or orbitals. Equations (1.51a)
and (1.51b) are no longer solutions of the perturbed one-electron Hamiltonian and
new Bloch functions have to be built. The most simple way consists in expressing
the new wave functions as linear combinations of
1e
(
k
1e
(
k
|
φ
,
x
)
and
|
φ
−
2
k
F
,
x
)
.
Such linear combinations are shown in Eqs. (1.52a) and (1.52b):
1e
1e
(
k
,
x
)
e
−
i
δ
|
φ
1e
(
k
−
|
φ
mix
(
k
,
x
)
∝|
φ
+
ϕ
2
k
F
,
x
)
,
(1.52a)
1e
e
i
δ
|
φ
1e
(
k
1e
(
k
|
φ
mix
(
k
−
2
k
F
,
x
)
∝−
ϕ
,
x
)
+|
φ
−
2
k
F
,
x
)
,
(1.52b)
where
δ
stands for an arbitrary phase. For
ϕ
=
0we recuperate the
H
per
=
0 situation
1e
1e
(
k
1e
developed above and therefore
|
φ
mix
(
k
,
x
)
=|
φ
,
x
)
and
|
φ
mix
(
k
−
2
k
F
,
x
)
=
1e
(
k
|
φ
−
2
k
F
,
x
)
. The electronic densities for both
k
and
k
−
2
k
F
states are respec-
1e
1e
1e
1e
tively given by
φ
mix
(
k
,
x
)
|
φ
mix
(
k
,
x
)
and
φ
mix
(
k
−
2
k
F
,
x
)
|
φ
mix
(
k
−
2
k
F
,
x
)
:
1e
1e
1e
(
k
1e
(
k
φ
mix
(
k
,
x
)
|
φ
mix
(
k
,
x
)
∝
φ
,
x
)
|
φ
,
x
)
2
1e
(
k
1e
(
k
+
ϕ
φ
−
2
k
F
,
x
)
|
φ
−
2
k
F
,
x
)
e
−
i
δ
φ
1e
(
k
1e
(
k
+
ϕ
,
x
)
|
φ
−
2
k
F
,
x
)
e
i
δ
φ
1e
(
k
1e
(
k
+
ϕ
−
2
k
F
,
x
)
|
φ
,
x
)
.
(1.53)
1e
(
k
1e
(
k
1e
(
k
1e
(
k
1e
(
k
The terms
φ
,
x
)
|
φ
,
x
)
,
φ
−
2
k
F
,
x
)
|
φ
−
2
k
F
,
x
)
,
φ
,
x
)
|
1e
(
k
1e
(
k
1e
(
k
φ
−
2
k
F
,
x
)
and
φ
−
2
k
F
,
x
)
|
φ
,
x
)
are given by the expressions:
N
ion
N
ion
1
N
ion
1e
(
k
1e
(
k
e
i
k
(
n
−
m
)
a
at
m
at
n
φ
,
x
)
|
φ
,
x
)
=
ψ
|
ψ
,
(1.54a)
m
=
1
n
=
1
N
ion
N
ion
1
N
ion
1e
(
k
1e
(
k
e
i(
k
−
2
k
F
)(
n
−
m
)
a
at
m
at
n
φ
−
2
k
F
,
x
)
|
φ
−
2
k
F
,
x
)
=
ψ
|
ψ
,
m
=
1
n
=
1
(1.54b)
N
ion
N
ion
1
N
ion
e
i[(
k
−
2
k
F
)
n
−
km
]
a
1e
(
k
1e
(
k
at
m
at
n
φ
,
x
)
|
φ
−
2
k
F
,
x
)
=
ψ
|
ψ
,
m
=
1
n
=
1
(1.54c)
N
ion
N
ion
1
N
ion
1e
(
k
1e
(
k
e
i[
kn
−
(
k
−
2
k
F
)
m
]
a
at
m
at
n
φ
−
2
k
F
,
x
)
|
φ
,
x
)
=
ψ
|
ψ
.
m
=
1
n
=
1
(1.54d)
