Environmental Engineering Reference
In-Depth Information
energy encompasses our ignorance of this problem, and is presumably
relatively small. In the
local
approximation, which is adopted to convert
the density-functional theory into a practical method, this energy is
written
ε
xc
[
n
(
r
)]
n
(
r
)
d
r
,
E
xc
{
n
(
r
)
}≈
(1
.
2
.
8)
and the effective potential is therefore
v
eff
(
r
)=
e
2
n
(
r
)
d
r
+
v
ext
(
r
)+
v
xc
[
n
(
r
)]
,
(1
.
2
.
9)
|
r
−
r
|
where
v
xc
[
n
(
r
)] =
d
[
nε
xc
(
n
)]
/dn
≡
µ
xc
[
n
(
r
)]
(1
.
2
.
10)
is the local approximation to the exchange-correlation contribution to
the chemical potential of the electron gas. Useful estimates of this quan-
tity have been obtained from calculations for a homogeneous electron gas
of density
n
(
r
) by Hedin and Lundqvist (1971), von Barth and Hedin
(1972), and Gunnarsson and Lundqvist (1976), and these are frequently
used in calculations on both atoms and solids.
In order to determine the atomic structure, the Schrodinger equa-
tion (1.2.5) must be solved by the Hartree self-consistent procedure, in
which, through a process of iteration, the potential (1.2.9) generates
wavefunctions which, via (1.2.6), reproduce itself. Since this potential is
spherically symmetric in atoms, the single-particle wavefunctions may
be written as the product of a radial function, a spherical harmonic and
a spin function
ψ
nlm
l
m
s
(
r
σ
)=
i
l
R
nl
(
r
)
Y
lm
l
(
r
)
χ
m
s
,
(1
.
2
.
11)
where
r
is a unit vector in the direction of
r
, the spin quantum number
m
s
can take the values
2
, and the phase factor
i
l
is included for later
convenience. The radial component satisfies the equation
1
±
+
v
eff
(
r
)+
l
(
l
+1)
h
2
2
mr
2
ε
[
rR
nl
(
r
)] = 0
.
h
2
2
m
d
2
[
rR
nl
(
r
)]
dr
2
−
−
(1
.
2
.
12)
Some radial wavefunctions for rare earth atoms are shown in Fig. 1.1.
The 4
f
electrons are well embedded within the atom, and shielded by
the 5
s
and 5
p
states from the surroundings. The 5
d
and 6
s
electrons
form the conduction bands in the metals. The incomplete screening of
the increasing nuclear charge along the rare earth series causes the lan-
thanide contraction of the wavefunctions, which is reflected in the ionic
and atomic radii in the solid state. In particular, as illustrated in Fig.
1.1, the 4
f
wavefunction contracts significantly between Ce, which has
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