44
Chapter 3 Wave-Functions and Valence-Bond Structures for 1-Electron Bonds, …
spin (cf. Eqn. (17)), i.e.
1
1
1
1
*
ab
ab
a b
. Therefore for the molecular
orbital configuration
2
* 1
ab
ab
( ) ( )
, we can write
ii
2
1
1
2
1
1
1
1
1
1
ab
1
*
ab
2
ab
b
a
b
a
b
a
b
a
b
a
ψ
ψ
ψ
k
k
(18)
to demonstrate that resonance between the valence-bond structures A: ·B and
A· :B (with atomic orbital configurations
2
1
(a) (b) and
1
2
(a) (b) ) is involved, just
as it is for the Pauling “3-electron bond” structure A···B .
Because the configuration
1
1
1
ab
( ) (a) (b)
involves only one bonding electron
(namely, the electron that occupies ab
), and two non-bonding electrons, in 1960,
Green and Linnett
9 modified the symbolism for the “3-electron bond” valencebond structure and wrote it as
A · B (or ·A · B· ), which indicates clearly that the
Pauling “3-electron bond” involves only one A-B bonding electron. Further,
because the a and b electrons of this structure are non-bonding electrons with
parallel spins, the spin of the bonding ab
electron must be opposite to those of
the a and b electrons. (This follows because only one of the ab
electrons of
ab
2
* 1
ab
( ) ( )
can have the same spin as that of
ab
*
.) Therefore, the electron spins
of
A · B may be those of either
, according to whether
the antibonding
ab
*
electron has a 1/ 2
or
z
1/ 2 ( )
s spin quantum number. In
Figure 3-2, the orbitals and electron spin assignments are displayed for these latter
valence-bond structures.
The above considerations show that the Green and Linnett structure
A · B ,
with one bonding and two non-bonding electrons, provides a “better and clearer
diagramatic representation of the electron distribution”
10 than does the Pauling
structure A···B . Therefore we shall use the
A · B representation in the
remainder of this book. However, (perhaps misleadingly), we shall continue to
refer to this valence-bond structure as a Pauling “3-electron bond” structure.
3-7 Slater Determinants and the Pauling “3-Electron Bond”
Although it is not needed for a reading of much of this book, it is useful here to
elaborate further the discussion of the Pauli exclusion principle in order to formaii Our concern here is with orbital occupancies, and therefore this equivalence is appropriate
whether or not atomic orbital overlap integrals are included in the normalization constants for
the molecular orbitals. See Section 3-10 for a discussion on the inclusion of atomic orbital
overlap integrals on the energies of Pauling “3-electron bonds”.
Chapter 3 Wave-Functions and Valence-Bond Structures for 1-Electron Bonds, …
spin (cf. Eqn. (17)), i.e.
1
1
1
1
*
ab
ab
a b
. Therefore for the molecular
orbital configuration
2
* 1
ab
ab
( ) ( )
, we can write
ii
2
1
1
2
1
1
1
1
1
1
ab
1
*
ab
2
ab
b
a
b
a
b
a
b
a
b
a
ψ
ψ
ψ
k
k
(18)
to demonstrate that resonance between the valence-bond structures A: ·B and
A· :B (with atomic orbital configurations
2
1
(a) (b) and
1
2
(a) (b) ) is involved, just
as it is for the Pauling “3-electron bond” structure A···B .
Because the configuration
1
1
1
ab
( ) (a) (b)
involves only one bonding electron
(namely, the electron that occupies ab
), and two non-bonding electrons, in 1960,
Green and Linnett
9 modified the symbolism for the “3-electron bond” valencebond structure and wrote it as
A · B (or ·A · B· ), which indicates clearly that the
Pauling “3-electron bond” involves only one A-B bonding electron. Further,
because the a and b electrons of this structure are non-bonding electrons with
parallel spins, the spin of the bonding ab
electron must be opposite to those of
the a and b electrons. (This follows because only one of the ab
electrons of
ab
2
* 1
ab
( ) ( )
can have the same spin as that of
ab
*
.) Therefore, the electron spins
of
A · B may be those of either
, according to whether
the antibonding
ab
*
electron has a 1/ 2
or
z
1/ 2 ( )
s spin quantum number. In
Figure 3-2, the orbitals and electron spin assignments are displayed for these latter
valence-bond structures.
The above considerations show that the Green and Linnett structure
A · B ,
with one bonding and two non-bonding electrons, provides a “better and clearer
diagramatic representation of the electron distribution”
10 than does the Pauling
structure A···B . Therefore we shall use the
A · B representation in the
remainder of this book. However, (perhaps misleadingly), we shall continue to
refer to this valence-bond structure as a Pauling “3-electron bond” structure.
3-7 Slater Determinants and the Pauling “3-Electron Bond”
Although it is not needed for a reading of much of this book, it is useful here to
elaborate further the discussion of the Pauli exclusion principle in order to formaii Our concern here is with orbital occupancies, and therefore this equivalence is appropriate
whether or not atomic orbital overlap integrals are included in the normalization constants for
the molecular orbitals. See Section 3-10 for a discussion on the inclusion of atomic orbital
overlap integrals on the energies of Pauling “3-electron bonds”.