Biomedical Engineering Reference
In-Depth Information
but Ψ
2
and Ψ
4
have to be written as a sum of two determinants, for example
−
1
2
1
s
(
r
1
) α (
s
1
)
2
s
(
r
1
) β (
s
1
)
1
s
(
r
1
) β (
s
1
)
2
s
(
r
1
) α (
s
1
)
Ψ
2
=
1
s
(
r
2
) α (
s
2
)
2
s
(
r
2
) β (
s
2
)
1
s
(
r
2
) β (
s
2
)
2
s
(
r
2
) α (
s
2
)
14.8 Slater-Condon-Shortley Rules
Slater determinants are compact summaries of all possible permutations of electrons and
spin orbitals but the way I have written themdown is unwieldy, andmany authors adopt sim-
plified notations. Suppose for example we have a many-electron system whose electronic
configuration can be written
(ψ
A
)
2
(ψ
B
)
2
. . .(ψ
M
)
2
This is chemical shorthand for 2
M
spinorbitals ψ
A
αψ
A
βψ
B
α
ψ
B
β...ψ
M
αψ
M
β occupied
by 2
M
electrons. One convention is to write A for ψ
A
α
A
and
A
for ψ
A
β
A
with the Slater
determinant represented
AABB
...
MM
From time to time we need to know the expectation values of sums of certain one-electron
operators and certain two-electron operators. Suppose that there are
n
electrons; these are
indistinguishable and so any sum of operators must include all of them on an equal footing.
Expectation values are typically the electronic contribution to the molecular electric dipole
moment
=
D
Ψ
n
Ψ dτ
e
−
r
i
i
=
1
and the electron repulsion in a polyatomic system with
n
electrons is
Ψ
n
−
1
Ψ dτ
1
dτ
2
n
e
2
4πε
0
1
r
ij
i
=
1
j
=
i
+
1
Ψ has to be a linear combination of Slater determinants
D
1
,
D
2
, ... so we need a systematic
set of rules for working out such expectation values between single Slater determinants
that I will call
D
1
and
D
2
which are, in the simplified notation
D
1
= |
UVW
...
Z
|
D
2
= |
U
V
W
...
Z
|
The first step is to rearrange one of the determinants to make as many of the spinorbitals
equal as possible. This may introduce a sign change. The algebra is easier if we assume
that the individual orbitals are normalized and orthogonal. Consider the overlap integral
D
1
D
2
dτ
where the integration is over the space and spin coordinates of all the electrons. Each determ-
inant expands into
n
! terms, and the product has (
n
!)
2
terms. On integration, orthonormality
of the individual spinorbitals means that there will be just
n
! nonzero terms each equal to
