304
Chapter 23 Some Comparisons of Types of Wave-Funcions
IV . In Table 23-3, neither Ψ(VBHL) nor Ψ(VΒΒΟ) has a low energy, and we
may conclude that the standard valence-bond resonance does not give a suitable
representation for the electronic structures of these systems.
23-4 Linnett Non-Paired Spatial Orbital
Valence-bond structure (10)
(10)
Y · A · Β
is the general non-paired spatial orbital (NPSO) structure for 4-electron 3-centre
bonding
1-6,12,13 as occurs in the valence-bond structures
and
1
1
2
2
( )
( )
H · H · H
for 3
O and 3
H
. In structure (10), two electrons occupy y and b
atomic orbitals, and two electrons occupy the two-centre bond orbitals
L
y a
k
and R b a
k
with k > 0.
Because the four electrons occupy different spatial orbitals, we may construct
two singlet wave-functions, both with spin quantum numbers
Z
0
S S
. The
details are described in Section 15-2 and it is possible to form linear combinations
of these wave-functions
1-6,12
. For illustrative purposes here, the special linear
combination that generates Eqn. (7),
L R
L R
(NPSO) y
b
y
b
(7)
in which electrons that occupy spatially adjacent orbitals have opposite spins, is a
satisfactory wave-function and this is the NPSO wave-function that we shall
examine. In terms of the functions I to IV, we may express it
1-6 as
(NPSO) = k I - k
2
II + 2 III
(8)
thereby showing that it includes the “long-bond” function II , and excludes the
unimportant function IV . The opposite pertains for the Ψ(VBBO) of Section
23-3, and therefore we can understand why the Ψ(NPSO) of Table 23-3 generate
much lower energies than do the Ψ(VBBO).
Chapter 23 Some Comparisons of Types of Wave-Funcions
IV . In Table 23-3, neither Ψ(VBHL) nor Ψ(VΒΒΟ) has a low energy, and we
may conclude that the standard valence-bond resonance does not give a suitable
representation for the electronic structures of these systems.
23-4 Linnett Non-Paired Spatial Orbital
Valence-bond structure (10)
(10)
Y · A · Β
is the general non-paired spatial orbital (NPSO) structure for 4-electron 3-centre
bonding
1-6,12,13 as occurs in the valence-bond structures
and
1
1
2
2
( )
( )
H · H · H
for 3
O and 3
H
. In structure (10), two electrons occupy y and b
atomic orbitals, and two electrons occupy the two-centre bond orbitals
L
y a
k
and R b a
k
with k > 0.
Because the four electrons occupy different spatial orbitals, we may construct
two singlet wave-functions, both with spin quantum numbers
Z
0
S S
. The
details are described in Section 15-2 and it is possible to form linear combinations
of these wave-functions
1-6,12
. For illustrative purposes here, the special linear
combination that generates Eqn. (7),
L R
L R
(NPSO) y
b
y
b
(7)
in which electrons that occupy spatially adjacent orbitals have opposite spins, is a
satisfactory wave-function and this is the NPSO wave-function that we shall
examine. In terms of the functions I to IV, we may express it
1-6 as
(NPSO) = k I - k
2
II + 2 III
(8)
thereby showing that it includes the “long-bond” function II , and excludes the
unimportant function IV . The opposite pertains for the Ψ(VBBO) of Section
23-3, and therefore we can understand why the Ψ(NPSO) of Table 23-3 generate
much lower energies than do the Ψ(VBBO).