Chemistry Reference
In-Depth Information
(Abramowitz and Stegun, 1965) gives the cubic equation in x:
<
x
3
þ
ax
þ
b
¼
0
1
3
ð
1
27
ð
ð
2
:
143
Þ
p
2
2p
3
a
¼
3q
Þ;
b
¼
9pq
þ
27r
Þ
:
Since
H
is Hermitian, we must have:
b
2
4
þ
a
3
27
<
0
ð
2
:
144
Þ
with three different real roots, which are easily expressed in terms of the
trigonometric relations:
s
<
a
3
cos
3
x
1
¼
2
!
s
a
3
3
þ
120
x
2
¼
2
cos
ð
2
:
145
Þ
:
!
s
a
3
3
120
x
3
¼
2
cos
with:
r
cos
3
¼
b
2
a
3
27
ð
:
Þ
2
146
Using one-term SCF/STO values for the fluorine atom (Clementi and
Roetti, 1974):
«
s
a
s
¼
:
43 E
h
;
«
p
a
p
¼
:
53 E
h
;
«
h
¼ a
h
¼
:
1
0
0
5 E
h
ð
2
:
147
Þ
the roots of the cubic secular equation will depend in a parametric way
on the values given to
. Assuming:
19
jbj
5 kcal mol
1
jbj¼
0
:
114E
h
¼
71
:
ð
2
:
148
Þ
gives, in atomic units:
«
1
¼
0
:
3928 E
h
;
«
2
¼
1
:
4440 E
h
;
«
3
¼
0
:
6238 E
h
:
ð
2
:
149
Þ
19
This is roughly the value assumed by
jbj
in the bond-orbital approximation.
