134
Chapter 10 Pauling “3-Electron Bonds” and “Increased-Valence” Theory for N2O4
Because the identity
*
ab ab ab
ab
a
b
for the Pauling “3-electron bond”
configuration (Section 3-6) arises when ab
and
*
ab
are normalized excluding
atomic orbital overlapping integrals, the covalent
can be transformed further to
give Eqn. (14) and then Eqn. (15).
covalent
1
2
*
*
cov alent
1 L 2 3 R 4
1 L 2 3 R 4
(
h h
h h
) / 2
(14)
2
6
2
5
4
3
μ
1
/
μ
μ
(15)
In Eqn.(15), the 3
6
–
are the (S = 0 spin) bond-eigenfunctions or valencebond structure functions for the six electrons that occupy the 1
, 2
h , 3
h and 4
atomic orbitals of the Lewis structures (3)-(6) of Section 7-1.
10-3 “Increased-Valence” Theory and Configuration
Interaction for N 2 O 4
Although it is not required for the “increased-valence” theory of the subsequent
chapters, it is appropriate here to discuss aspects of configuration interaction (C.I.)
theory for N 2 O 4 . In particular, we shall demonstrate that the covalent
of Eqn. (12),
which contributes equally with
ionic
to the lowest-energy molecular orbital
configuration 1 (MO)
of Eqn. (8), is the primary component of the lower-energy
C.I. wave-function obtained by linearly combining 1 (MO)
with the 2 (MO)
of Eqn. (16). This result is similar to that which pertains for the ground-state of
2
H (Section 3-3).
2
1
1
2
2 4
4
(MO)
(16)
Because the molecular orbital 3
of Eqn. (8) is N-O antibonding (cf. Eqn. (75)), it is the highest-energy occupied orbital of 1 (MO)
. When two electrons are
excited from 3
into the vacant molecular orbital 4
of Eqn. (7-6), (which is
both N-O and N-N antibonding), the lowest-energy doubly-excited configuration
2 (MO)
of Eqn. (16) is obtained.
It is now helpful to express the molecular orbital 1
of Eqn. (16) in terms of
the molecular orbitals 1
and 3
of Eqs. (3) and (4), in which the latter orbitals
are defined in terms of the parameter μ. With 1
defined in terms of λ according
to Eqn. (7-3), we obtain Eqn. (17).
Chapter 10 Pauling “3-Electron Bonds” and “Increased-Valence” Theory for N2O4
Because the identity
*
ab ab ab
ab
a
b
for the Pauling “3-electron bond”
configuration (Section 3-6) arises when ab
and
*
ab
are normalized excluding
atomic orbital overlapping integrals, the covalent
can be transformed further to
give Eqn. (14) and then Eqn. (15).
covalent
1
2
*
*
cov alent
1 L 2 3 R 4
1 L 2 3 R 4
(
h h
h h
) / 2
(14)
2
6
2
5
4
3
μ
1
/
μ
μ
(15)
In Eqn.(15), the 3
6
–
are the (S = 0 spin) bond-eigenfunctions or valencebond structure functions for the six electrons that occupy the 1
, 2
h , 3
h and 4
atomic orbitals of the Lewis structures (3)-(6) of Section 7-1.
10-3 “Increased-Valence” Theory and Configuration
Interaction for N 2 O 4
Although it is not required for the “increased-valence” theory of the subsequent
chapters, it is appropriate here to discuss aspects of configuration interaction (C.I.)
theory for N 2 O 4 . In particular, we shall demonstrate that the covalent
of Eqn. (12),
which contributes equally with
ionic
to the lowest-energy molecular orbital
configuration 1 (MO)
of Eqn. (8), is the primary component of the lower-energy
C.I. wave-function obtained by linearly combining 1 (MO)
with the 2 (MO)
of Eqn. (16). This result is similar to that which pertains for the ground-state of
2
H (Section 3-3).
2
1
1
2
2 4
4
(MO)
(16)
Because the molecular orbital 3
of Eqn. (8) is N-O antibonding (cf. Eqn. (75)), it is the highest-energy occupied orbital of 1 (MO)
. When two electrons are
excited from 3
into the vacant molecular orbital 4
of Eqn. (7-6), (which is
both N-O and N-N antibonding), the lowest-energy doubly-excited configuration
2 (MO)
of Eqn. (16) is obtained.
It is now helpful to express the molecular orbital 1
of Eqn. (16) in terms of
the molecular orbitals 1
and 3
of Eqs. (3) and (4), in which the latter orbitals
are defined in terms of the parameter μ. With 1
defined in terms of λ according
to Eqn. (7-3), we obtain Eqn. (17).