Valence Bond Theory and the Chemical Bond - Methods of Molecular Quantum Mechanics - page 149

g Þ¼ N fjj ab jjjj ab jjg 1

Y ð HL ; S

a ð r 1 Þ b ð r 2 Þþ b ð r 1 Þ a ð r 2 Þ
2 þ 2S 2

1

p

p

¼ ½ að s 1 Þbð s 2 Þbð s 1 Það s 2 Þ

S ¼ M S ¼ 0 ð 9 : 28 Þ

giving the HL energy in the Born-Oppenheimer approximation:

H jYð HL g Þ¼hYð HL g g

1 1 1

E ð HL

; S ; S Þj ; S Þi

* +

ab þ ba
2 þ 2S 2

1
r 12 þ

1
R

ab þ ba
2 þ 2S 2

h 1 þ h 2 þ

¼ p p

þ ð a 2

j b 2

h aa þ h bb þ S ð h ba þ h ab Þ
1 þ S 2

Þþð ab j ab Þ
1 þ S 2

1
R

¼ þ

ð 9 : 29 Þ

It is seen that theHL two-electron component of themolecular energy is
much simpler than itsMO counterpart and now has the correct behaviour
as R !¥ .

In the hydrogenic approximation (c 0 ¼ 1), the HL interaction energy
for 1 S g H 2 is

1 g Þ¼ E ð HL 1 g

Þ 2E H ¼ D E cb þD E exch-ov

1 g Þ 9

D E ð HL ; S ; S ð S : 30 Þ

D E cb is the same as the MO expression (9.21), while

where

g Þ¼ S ð ab Sa 2

j V B Þþð ba Sb 2

þ ð ab j ab Þ S 2

ð a 2 j b 2 j V A Þ Þ

D E exch-ov

1

ð S

1 þ S 2 1 þ S 2

ð 9 : 31 Þ

Both components of the interaction energy now vanish as R !¥ , there-

fore describing correctly the dissociation of the ground-state H 2 molecule
into neutral ground-state H atoms (top part of Figure 9.2, HL curve).
For the excited stateof theH 2 molecule, wehave the tripletwavefunction

<
:

k ab k

S ¼ 1

; M S ¼ 1

1

p ½k a b kþk ab k

S ¼ 1

; M S ¼ 0

3 u Þ¼

Yð HL ; S ð 9 : 32 Þ

k a b k S ¼ 1 ; M S ¼ 1