Chemistry Reference
In-Depth Information
an extra term is added to Eq. (1.10), known as the exchange term. The Schrodinger
equation thus becomes:
−
R
j
)
¯h
2
2
m
e
∇
N
ion
H
1e
1e
2
i
1e
|
=
+
−
|
i
(
r
)
V
e
−
ion
(
r
i
i
(
r
)
i
=
j
1
e
2
1e
j
(
r
)
1e
j
(
r
)
N
e
d
r
|
1e
+
|
i
(
r
)
|
r
−
r
|
j
=
1
j
(
r
)
1e
1e
N
e
d
r
|
j
(
r
)
i
(
r
)
e
2
1e
−
|
.
(1.12)
|
r
−
r
|
j
=
1
Note that the exchange term is of the form
V
(
r
r
)
(
r
)d
r
,
instead of the
V
(
r
)
(
r
) type. Equation (1.12), known as the Hartree-Fock equation, is intractable
except for the free-electron gas case. Hence the interest in sticking to the concep-
tually simple free-electron case as the basis for solving the more realistic case of
electrons in periodic potentials. The question is how far can this approximation be
driven. Landau's approach, known as the Fermi liquid theory, establishes that the
electron-electron interactions do not appear to invalidate the one-electron picture,
even when such interactions are strong, provided that the levels involved are located
within
k
B
T
of
E
F
. For metals, electrons are distributed close to
E
F
according to the
Fermi function
f
(
E
):
1
f
(
E
)
=
e
(
E
−
E
F
)
/
k
B
T
.
(1.13)
1
+
A plot of
f
(
E
) is shown in Fig. 1.27.
E
F
is usually defined as the chemical potential
in the
T
0 limit.
However, if we consider strong electronic correlations it looks doubtful that
the simplified one-electron approximation can be applied at all. To circumvent
this conceptual problem Landau proposed the bright idea of substituting electrons
by something closely related, quasi-electrons (in general particles will be substi-
tuted by quasi-particles). In practice, this theoretical idea, so common in particle
physics, transforms the energy expression of a free electron,
E
→
2
k
2
=
/
2
m
e
into
2
m
e
, where
m
e
represents the effective mass, which is a function of
k
:
m
e
(
k
). In other words, the non-interacting electron case is maintained but the
energy scale is
renormalized
. The main goal is to preserve the free-electron model
because we can handle it analytically. This approach will be limited to the case
where correlations can be treated as perturbations.
So far we have studied the general many-electron problem of
N
e
electrons and
N
ion
fixed or very slow ions. Let us now apply the above-developed formalism to
molecules and solids.
2
k
2
E
=
/
