Chemistry Reference
In-Depth Information
<
:
e
2
a
0
¼
me
4
1
E
h
¼
4
p
«
0
2
h
2
ð
4
p
«
0
Þ
4
10
31
10
19
kg C
4
C
4
m
2
J
2
s
2
:
ð
:
Þ
9
109 382
1
602 176
¼
2
2
10
10
10
34
ð
1
:
112 650
Þ
ð
1
:
054 571
Þ
10
18
J
¼
4
:
359 744
:
ð
1
:
79
Þ
1.5 THE ELECTRON DISTRIBUTION IN MOLECULES
The one-electron spatial function P(r) describing the distribution of the
electrons (the electron density) in the doubly occupied MO
f
(r):
x
A
ð
Þ
1
r
Þþ
l
x
B
ð
r
fð
r
Þ¼x
A
ð
r
Þ
c
A
þx
B
ð
r
Þ
c
B
¼
p
ð
1
:
80
Þ
2
þ
l
þ
2
l
S
where
l
¼
c
B
=
c
A
denotes here the polarity parameter of the bond orbital
and S
¼hx
A
jx
B
i
the overlap integral, is simply given by:
Þ¼r
a
Þþr
b
Þf
ð
2
P
ð
r
ð
r
ð
r
Þ¼
2
fð
r
r
Þ¼
2
jfð
r
Þj
ð
1
:
81
Þ
the factor 2 comes from the equal contribution of electrons with either
spin (
a¼
spin-down).
The electron density can be further analysed in terms of elementary
contributions from the AOs, giving the so-called population analysis,
11
which shows how the electrons are distributed between the different
atomic orbitals in the molecule. We obtain from Equation (1.81):
spin-up,
b¼
q
AB
x
A
ð
r
Þx
B
ð
r
Þ
q
BA
x
B
ð
r
Þx
A
ð
r
Þ
q
A
x
A
2
q
B
x
B
2
P
ð
r
Þ¼
ð
r
Þþ
ð
r
Þþ
þ
ð
1
:
82
Þ
S
S
x
A
ð
r
Þx
B
ð
r
Þ
and
x
B
ð
r
Þ
x
A
ð
r
Þ
S
where
x
A
ð
and
x
B
ð
r
Þ
r
Þ
are atomic densities,
are
S
overlap densities, all normalized to 1, while the coefficients:
2
2
2
l
q
A
¼
S
;
q
B
¼
ð
1
:
83
Þ
2
2
1
þ
l
þ
2
l
1
þ
l
þ
2
l
S
11
The extension to N-electron LCAO-MO wave functions is due to Mulliken (1955).
