Environmental Engineering Reference
In-Depth Information
From mun-tin orbitals located on the lattice sites of a solid, with
one atom per unit cell, we now construct a wavefunction which is con-
tinuous and differentiable, and manifestly satisfies the Bloch condition
(1.3.1):
ψ
j
(
k
,ε,
r
)=
lm
lm
R
a
j
k
e
i
k
·
R
χ
lm
(
ε,
r
−
R
)
.
(1
.
3
.
6)
If we approximate the atomic polyhedra by spheres, and implicitly as-
sume that they fill space, the condition that (1.3.6) is a solution of the
Schrodinger equation is easily seen to be that the sum of the tails orig-
inating from terms of the form (
S/
)
l
+1
, from the surrounding
|
r
−
R
|
atoms, cancels the 'extra' contribution
a
j
k
lm
Y
lm
(
r
)
p
l
(
ε
)(
r/S
)
l
,
lm
in the atomic sphere at the origin. To satisfy this condition, we expand
the tails of the mun-tin orbitals centred at
R
about the origin, in the
form
e
i
k
·
R
S
|
r
−
R
|
l
+1
i
l
Y
lm
(
r
−
R
)
R
=
0
(1
.
3
.
7)
=
l
m
2(2
l
+1)
r
S
l
i
l
Y
l
m
(
r
)
−
1
k
S
l
m
,lm
,
where the expansion coecients, known as the
canonical structure con-
stants
,are
l
m
,lm
=
R
=
0
e
i
k
·
R
S
S
l
m
,lm
(
R
)
,
(1
.
3
.
8)
with
S
l
m
,lm
(
R
)=
g
l
m
,lm
√
4
π
(
i
)
λ
Y
λµ
(
R
)(
R/S
)
−λ−
1
,
−
where
1)
m
+1
2
(2
l
+ 1)(2
l
+1)
2
λ
+1
(
λ
+
µ
)!(
λ − µ
)!
(
l
+
m
)!(
l
− m
)!(
l
+
m
)!(
l − m
)!
g
l
m
,lm
≡
(
−
and
l
+
l
;
µ
m
.
λ
≡
≡
m
−
From (1.3.3) and (1.3.7), the required
tail-cancellation
occurs if
l
m
,lm
]
a
j
k
k
[
P
l
(
ε
)
δ
l
l
δ
m
m
−S
lm
=0
,
(1
.
3
.
9)
lm
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