15-1
“Increased-Valence” Wave-Functions for 4-Electron 3-Centre
207
Inspection of Eqn. (6) shows that a Heitler-London type wave-function has
been used to construct the wave-function for the fractional Y-A bond of structure
(1). If it is preferred to use bonding molecular orbitals to describe the wavefunctions for all bonds, then we may construct the S = 0 spin wave-function of
Eqn. (7)
(7)
for the four electrons of structure (1), in which ya y a
and
0
. The properties of Eqs. (6) and (7) are different, and on most occasions, we shall be
implying the usage of Eqn. (6). Some further considerations of Eqn. (7), and elaborations of it (via the use of the orbitals = y + ka and ’ = y + k’a instead of
doubly-occupied ya ), will be provided in Chapters 20 and 23.
“Increased-valence” structure (1) is the fundamental “increased-valence”
structure. Together with “increased-valence” structure (4),
Y—A · B
Y · A—B
(1)
(4)
A · B—C · D
A · B C · D
(5)
(6)
it is appropriate whenever four electrons can participate in 3-centre bonding. But
as we have discussed in Chapter 13, it is possible to construct longer “increasedvalence” structures that are appropriate for different types of N-centre bonding
units, with N ≥ 4. Here, we shall construct a Slater-determinantal wave-function
for the 6-electron 4-centre “increased-valence” structure (5) (Section 12-3), which
is obtained by spin-pairing the odd-electrons of the Pauling “3-electron bond”
structures of structure (6). For the latter structures, the odd electrons occupy the
antibonding
ab
*
and
cd
*
molecular orbitals, and the bonding molecular orbitals
ab
and cd
are doubly occupied. When S = 0 spin-pairing of the odd-electrons
of structure (6) occurs, the two singly-occupied antibonding orbitals will also
pertain for the wave-function for structure (5). Consequently, the S = 0 spin wavefunction for (5) is given by Eqn. (8).
*
*
*
*
ab ab ab cd cd cd
ab ab cd ab cd cd
(A·B — C·D) |
| |
|
(8)
Orbital-occupancy diagrams that correspond to the Slater determinants of Eqs.
(5) and (8) are displayed in Figs. 11-2 and 11-3.
“Increased-Valence” Wave-Functions for 4-Electron 3-Centre
207
Inspection of Eqn. (6) shows that a Heitler-London type wave-function has
been used to construct the wave-function for the fractional Y-A bond of structure
(1). If it is preferred to use bonding molecular orbitals to describe the wavefunctions for all bonds, then we may construct the S = 0 spin wave-function of
Eqn. (7)
(7)
for the four electrons of structure (1), in which ya y a
and
0
. The properties of Eqs. (6) and (7) are different, and on most occasions, we shall be
implying the usage of Eqn. (6). Some further considerations of Eqn. (7), and elaborations of it (via the use of the orbitals = y + ka and ’ = y + k’a instead of
doubly-occupied ya ), will be provided in Chapters 20 and 23.
“Increased-valence” structure (1) is the fundamental “increased-valence”
structure. Together with “increased-valence” structure (4),
Y—A · B
Y · A—B
(1)
(4)
A · B—C · D
A · B C · D
(5)
(6)
it is appropriate whenever four electrons can participate in 3-centre bonding. But
as we have discussed in Chapter 13, it is possible to construct longer “increasedvalence” structures that are appropriate for different types of N-centre bonding
units, with N ≥ 4. Here, we shall construct a Slater-determinantal wave-function
for the 6-electron 4-centre “increased-valence” structure (5) (Section 12-3), which
is obtained by spin-pairing the odd-electrons of the Pauling “3-electron bond”
structures of structure (6). For the latter structures, the odd electrons occupy the
antibonding
ab
*
and
cd
*
molecular orbitals, and the bonding molecular orbitals
ab
and cd
are doubly occupied. When S = 0 spin-pairing of the odd-electrons
of structure (6) occurs, the two singly-occupied antibonding orbitals will also
pertain for the wave-function for structure (5). Consequently, the S = 0 spin wavefunction for (5) is given by Eqn. (8).
*
*
*
*
ab ab ab cd cd cd
ab ab cd ab cd cd
(A·B — C·D) |
| |
|
(8)
Orbital-occupancy diagrams that correspond to the Slater determinants of Eqs.
(5) and (8) are displayed in Figs. 11-2 and 11-3.