Chemistry Reference
In-Depth Information
Also, Pauli's exclusion principle is satisfied by (6.7):
c
1
ð
x
1
Þ c
1
ð
x
1
Þ c
N
ð
x
1
Þ
c
1
ð
x
2
Þ c
1
ð
x
2
Þ c
N
ð
x
2
Þ
c
1
ð
x
N
Þ c
1
ð
x
N
Þc
N
ð
x
N
Þ
1
N
!
Yð
x
1
;
x
2
; ...;
x
N
Þ¼
p
¼
0
ð
6
:
11
Þ
since (6.11) now has two identical columns and the determinant vanishes.
For the N-electron system, the wavefunction
is antisymmetric if it is
left invariant by an even number of electron interchanges (permutations)
and it changes sign with an odd number of interchanges. The probability
density (as the Hamiltonian operator) is instead left unchanged after any
number of interchanges among the electrons.
2
A single Slater determinant is often sufficient as a first approximation
for closed-shell systems (S
¼
M
S
¼
0), but some components of the open-
shell systems, like the M
S
¼
0 components of the triplet states 1s2s(
3
S) of
He or 1
Y
u
Þ
of H
2
, require a linear combination of Slater deter-
minants to get a state of definite S.
A few atomic and molecular examples are given below.
3
s
g
1
s
ð
S
u
(i) Ground state of the He atom (N
¼
2)
1
S
Þ¼jj
1s 1
Yð
1s
2
s
jj
S
¼
M
S
¼
0
;
(ii) Ground state of the Be atom (N
¼
4)
1
S
Þ¼jj
1s 1
Yð
1s
2
2s
2
s2s2
s
jj
S
¼
M
S
¼
0
;
(iii) Ground state of the H
2
molecule (N
¼
2)
2
g
1
g
Þ¼k
1
Yð
1
s
;
S
s
g
1
s
k
S
¼
M
S
¼
0
g
where
a
þ
b
2
þ
2S
1
s
¼
p
a
¼
1s
A
;
b
¼
1s
B
ð
6
:
12
Þ
g
is the bondingmolecular orbital doubly occupied by the electrons.
2
Recall that we simply interchange spin-orbitals between the electrons kept in dictionary order.
