3-3 Electron-Pair Bonds
39
Table 3-1: Simple wave-functions, dissociation energies and equilibrium internuclear distances
for the H2 ground-state. (Adapted from Table 5-12 of “Atoms and Molecules”, by M. Karplus
and R.N. Porter, (Benjamin, N.Y., 1970).)
Wave-function
Parameters
e (eV)
D
e
o
( )
R a
A
B
A
B
{1s (1) 1s (1)}{1s (2) 1s (2)}
1.0
2.695
1.61
"
1.197
3.488
1.38
A
B
B
A
1s (1)1s (2) 1s (1)1s (2)
1.0
3.156
1.64
"
1.166
3.782
1.41
A
B
B
A
(1)
(1) (2)
1s
2p
1.19
4.04
1.416
z
1s 2p
0.105
A
B
B
A
1s (1)1s (2) 1s (1)1s (2)
1
3.230
1.67
A
A
B
B
{1s (1)1s (2) 1s (1)1s (2)}
0.105
"
1.194
4.025
1.43
0.265
A
B
B
A
(1) (2)
(1) (2)
1.190
4.122
1.41
A
A
B
B
{1s (1)1s (2) 1s (1)1s (2)}
0.175
0.07
Experimental
4.74759
1.4006
contributing equally to the resonance. Alternatively we may replace the a and b
atomic orbitals of the Heitler-London wave-function with the semi-localized
orbitals a + κb and b + κ′a in which κ and κ′ are parameters. If the parameters that
arise in each of these three “improved” wave-functions are chosen so that the total
energy for each wave-function is minimized, it can be shown that the three wavefunctions are equivalent and will generate the same dissociation energy.
For H 2 , the three improved wave-functions of the previous paragraph are given
by Eqs. (7)-(9)
(MO, CI) {a(1) b(1)}{a(2) b(2)}
{a(1) – b(1)}{a(2) – b(2)}
K
(7)
(VB, resonance) a(1)b(2) b(1)a(2) {a(1)a(2) b(1)b(2)}
(8)
(VB, semi-localized) {a(1) b(1)}{b(2) a(2)} {b(1) a(1)}{a(2) b(2)}
(9)
with
A
a 1s
and
B
b 1s
. Equality is obtained when
2
(1
) / (1 – ) 2 / (1
)
K
K
.
In Table 3-1, we report the results of calculations of the dissociation energy for
H 2 , which use the above types of orbital wave-functions for 2
H ·Because for
Eqn. (8) is calculated to be << 1 (for example, 0.105 or 0.265), the primary
component for the 2
H ground-state wave-function must be the Heitler-London
covalent wave-function.
39
Table 3-1: Simple wave-functions, dissociation energies and equilibrium internuclear distances
for the H2 ground-state. (Adapted from Table 5-12 of “Atoms and Molecules”, by M. Karplus
and R.N. Porter, (Benjamin, N.Y., 1970).)
Wave-function
Parameters
e (eV)
D
e
o
( )
R a
A
B
A
B
{1s (1) 1s (1)}{1s (2) 1s (2)}
1.0
2.695
1.61
"
1.197
3.488
1.38
A
B
B
A
1s (1)1s (2) 1s (1)1s (2)
1.0
3.156
1.64
"
1.166
3.782
1.41
A
B
B
A
(1)
(1) (2)
1s
2p
1.19
4.04
1.416
z
1s 2p
0.105
A
B
B
A
1s (1)1s (2) 1s (1)1s (2)
1
3.230
1.67
A
A
B
B
{1s (1)1s (2) 1s (1)1s (2)}
0.105
"
1.194
4.025
1.43
0.265
A
B
B
A
(1) (2)
(1) (2)
1.190
4.122
1.41
A
A
B
B
{1s (1)1s (2) 1s (1)1s (2)}
0.175
0.07
Experimental
4.74759
1.4006
contributing equally to the resonance. Alternatively we may replace the a and b
atomic orbitals of the Heitler-London wave-function with the semi-localized
orbitals a + κb and b + κ′a in which κ and κ′ are parameters. If the parameters that
arise in each of these three “improved” wave-functions are chosen so that the total
energy for each wave-function is minimized, it can be shown that the three wavefunctions are equivalent and will generate the same dissociation energy.
For H 2 , the three improved wave-functions of the previous paragraph are given
by Eqs. (7)-(9)
(MO, CI) {a(1) b(1)}{a(2) b(2)}
{a(1) – b(1)}{a(2) – b(2)}
K
(7)
(VB, resonance) a(1)b(2) b(1)a(2) {a(1)a(2) b(1)b(2)}
(8)
(VB, semi-localized) {a(1) b(1)}{b(2) a(2)} {b(1) a(1)}{a(2) b(2)}
(9)
with
A
a 1s
and
B
b 1s
. Equality is obtained when
2
(1
) / (1 – ) 2 / (1
)
K
K
.
In Table 3-1, we report the results of calculations of the dissociation energy for
H 2 , which use the above types of orbital wave-functions for 2
H ·Because for
Eqn. (8) is calculated to be << 1 (for example, 0.105 or 0.265), the primary
component for the 2
H ground-state wave-function must be the Heitler-London
covalent wave-function.