42
Chapter 3 Wave-Functions and Valence-Bond Structures for 1-Electron Bonds, …
replaced by –1/(
1
2
ab
2
1 – )
S
. For the general orthogonal orbitals ab a b
   k and
ab
*
*a – b
  k
, we obtain the identity of Eqn. (16).
ab
ab
*
*
ab
ab
(1) (2) –
(1) (2) –(1
*){a(1)b(2) – b(1)a(2)}
kk






(16)
With respect to orbital occupancy for a diatomic system, we may therefore
conclude that
one antibonding electron
two “non bonding” electrons
one bonding electron



(17)
provided that the two electrons have parallel spins
i and the molecular orbitals are
constructed from the same set of atomic orbitals. The non-bonding property of
Eqn.(17) arises because the a and b electrons have parallel spins. With respect to
energy, this configuration is net antibonding when the overlap integral is included
in the normalization constants for the molecular orbitals; this is because
ab
*
 is
more antibonding than ab
 is bonding relative to the component atomic orbitals
(see, for example, Eqs. (1) and (2)).
3-6 The Pauling “3-Electron Bond”
In 1931, Pauling
8 introduced the “3-electron bond” structure A···B as a way to
summarize resonance between the Lewis valence-bond structures A: ·B and
A· :B , i.e., Pauling wrote A···B  A: ·B  A· :B .
In Figure 3-2, the 1s atomic orbital occupations for A: ·B and A· :B are
displayed for the helium molecule ion, He 2
+
. Because each of these valence-bond
structures has only one unpaired electron, Pauling deduced that the length and
i If the two electrons have antiparallel spins, then the spatial wave-function of Eqn. (14) for
one bonding + one antibonding electron is equivalent to


( * –1) {a(1)b(2) b(1)a(2)} 2 * a(1)a(2) – b(1)b(2)


kk
k
k
.