Biomedical Engineering Reference
In-Depth Information
As we noted earlier, this expression contains too much information and we ask instead
about the probability that any given element will be occupied by one electron with the other
electrons anywhere. This is often called the
one-electron density function
n
ρ
1
(
x
1
)
=
Ψ
∗
(
x
1
,
x
2
, ...,
x
n
) Ψ (
x
1
,
x
2
, ...,
x
n
)dτ
2
d
s
2
... dτ
n
d
s
n
(19.1)
The
x
1
on the left refers to 'point 1' at which the density is to be evaluated rather than the
coordinates of electron 1, the indistinguishability of the electrons being accounted for by
the factor
n
.
If we want to know the probability of finding an electron of either spin in the spatial
element dτ
1
then we integrate over d
s
1
to give the charge density discussed in Chapter 16
and measured by crystallographers:
n
Ψ
∗
(
x
1
,
x
2
, ...,
x
n
) Ψ (
x
1
,
x
2
, ...,
x
n
)d
s
1
dτ
2
d
s
2
... dτ
n
d
s
n
P
1
(
r
1
)
=
(19.2)
It is written either
P
or
P
1
; I have used
P
in earlier chapters. It also proves desirable to
introduce probabilities for finding different configurations of any number of particles. Thus
the
two-electron
(or
pair
)
density function
1)
Ψ
∗
(
x
1
,
x
2
, ...,
x
n
) Ψ (
x
1
,
x
2
, ...,
x
n
)dτ
3
d
s
3
... dτ
n
d
s
n
ρ
2
(
x
1
,
x
2
)
=
n
(
n
−
(19.3)
determines the probability of two electrons being found simultaneously at points
x
1
and
x
2
, spins included, whilst
1)
Ψ
∗
(
x
1
,
x
2
, ...,
x
n
) Ψ (
x
1
,
x
2
, ...,
x
n
)d
s
1
d
s
2
dτ
3
d
s
3
... dτ
n
d
s
n
(19.4)
P
2
(
x
1
,
x
2
)
=
n
(
n
−
determines the probability of finding them at points
r
1
and
r
2
in ordinary space, irrespective
of spin.
Many common one-electron properties depend only on
P
1
and since the Schrödinger
equation only contains pair interactions we need not consider distributions higher than the
pair functions. For a state of definite spin, as distinct from a mixed state, the one-electron
density function has the form
P
1
(
r
1
) β
2
(
s
1
)
ρ
1
(
x
1
)
=
P
1
(
r
1
) α
2
(
s
1
)
+
(19.5)
where the
P
's are spatial functions. There are no cross terms involving both α and β.
In orbital models, the
P
's are just sums over the squares of the occupied orbitals. The
two-electron density function is also given by
P
α
2
(
r
1
,
r
2
) α
2
(
s
1
) β
2
(
s
2
)
ρ
2
(
x
1
,
x
2
)
=
P
αα
2
(
r
1
,
r
2
) α
2
(
s
1
) α
2
(
s
2
)
+
(19.6)
P
βα
2
P
β
2
(
r
1
,
r
2
) β
2
(
s
1
) β
2
(
s
2
)
+
(
r
1
,
r
2
) β
2
(
s
1
) α
2
(
s
2
)
+
