306
Chapter 23 Some Comparisons of Types of Wave-Funcions
meters. However, it is possible to improve these wave-functions by introducing
additional variational parameters. We shall now discuss how this can be done.
For the molecular orbital description of H 2 , we have shown in Section 3-3 that
the bonding molecular orbital configuration
2
( 1s)
could be improved through
configuration interaction by linearly combining it with the antibonding configuretion
2
( *1s)
. Through configuration interaction, we may also improve the
(M )
1
of Section 23-2. By constructing the configuration interaction (CI)
wave-function of Eqn.(11)
1 1
2
2
3
3
4
4
(CI)
(MO)
(MO)
(MO)
(MO)
C
C
C
C
(11)
in which
1
2
2
2
2 3
3
3
1 1 3
3
4
1
3 2
2
3 1 2 2
(MO)
,
(MO)
,
(MO) (
) / 2
(12)
and the coefficients 1
C to 4
C are chosen so that the energy of Ψ(CΙ) is a minimum, we obtain an (over-parametrized) wave-function which is equivalent to the
Ψ(best). Without some transformation, this Ψ(CI) has the disadvantage that it does
not correspond to one or two simple valence-bond structures.
For Ψ(VBBO) and Ψ(NPSO), we may use a different bond-parameter for each
bond orbital
6b,13 . Thus, instead of using L y a
k
for both Y-A bonding electrons of the valence-bond structure (2), we may use L y a
k
for one electron
and L y
a
k
for the other electron. Similarly, for the two A-B bonding electrons of structure (1), we may use the orbitals R b a
k
and R b
a
k
instead of R b a
k
for both electrons. The Ψ(VBBO) wave-function may now
be expressed as
(VBBO) (
)
4
2
k k
kk
III
IV
(13)
which still omits the “long-bond” wavefunction II .
In Table 23-4, we report the energies for some two-parameter wave functions
13 .
They show that Ψ(VBBO) remains a high-energy function.
Table 23-4: Energies (in eV) of two-parameter VBBO, NPSO and IVBO wave-functions relative to Ψ(best).
2
XeF
3 3
C H
2
NO
2
HCO
VBBO
8.08
1.20
2.98
3.37
NPSO
0.060
0.058
0.225
0.266
IVBO
0.190
0.003
0.00001
0.005
Chapter 23 Some Comparisons of Types of Wave-Funcions
meters. However, it is possible to improve these wave-functions by introducing
additional variational parameters. We shall now discuss how this can be done.
For the molecular orbital description of H 2 , we have shown in Section 3-3 that
the bonding molecular orbital configuration
2
( 1s)
could be improved through
configuration interaction by linearly combining it with the antibonding configuretion
2
( *1s)
. Through configuration interaction, we may also improve the
(M )
1
of Section 23-2. By constructing the configuration interaction (CI)
wave-function of Eqn.(11)
1 1
2
2
3
3
4
4
(CI)
(MO)
(MO)
(MO)
(MO)
C
C
C
C
(11)
in which
1
2
2
2
2 3
3
3
1 1 3
3
4
1
3 2
2
3 1 2 2
(MO)
,
(MO)
,
(MO) (
) / 2
(12)
and the coefficients 1
C to 4
C are chosen so that the energy of Ψ(CΙ) is a minimum, we obtain an (over-parametrized) wave-function which is equivalent to the
Ψ(best). Without some transformation, this Ψ(CI) has the disadvantage that it does
not correspond to one or two simple valence-bond structures.
For Ψ(VBBO) and Ψ(NPSO), we may use a different bond-parameter for each
bond orbital
6b,13 . Thus, instead of using L y a
k
for both Y-A bonding electrons of the valence-bond structure (2), we may use L y a
k
for one electron
and L y
a
k
for the other electron. Similarly, for the two A-B bonding electrons of structure (1), we may use the orbitals R b a
k
and R b
a
k
instead of R b a
k
for both electrons. The Ψ(VBBO) wave-function may now
be expressed as
(VBBO) (
)
4
2
k k
kk
III
IV
(13)
which still omits the “long-bond” wavefunction II .
In Table 23-4, we report the energies for some two-parameter wave functions
13 .
They show that Ψ(VBBO) remains a high-energy function.
Table 23-4: Energies (in eV) of two-parameter VBBO, NPSO and IVBO wave-functions relative to Ψ(best).
2
XeF
3 3
C H
2
NO
2
HCO
VBBO
8.08
1.20
2.98
3.37
NPSO
0.060
0.058
0.225
0.266
IVBO
0.190
0.003
0.00001
0.005