136
Chapter 10 Pauling “3-Electron Bonds” and “Increased-Valence” Theory for N2O4
1
2
2
2
1
2
cov alent
1
2
ionic
{( –
)
(
)
} / 2
C x C
C x C
1
2
*
*
2
**
**
2
covalent
ionic
cov alent
ionic
C {xy(
) y
/ 2 }
(27)
The coefficients 1
C and 2
C are chosen so that the energy of this linear
combination is a minimum, a necessary condition for which is that
/
/
0
1
2
E C
E C
, where
2
2
2
2
1
11
2
22
1 2 12
1
2
(C
2
) /
E
H
C H
C C H
C C
. The
integral
12
1
2
ˆ
(MO)
(MO)d
H
H
may be shown to be equivalent to
2
1
3
4
12
2
4
3
d
d
)
2
(
ψ
)
1
(
ψ
/
)
2
(
ψ
)
1
(
ψ
r
e
, which is greater than zero. For a
finite N-N internuclear separation (r(NN)) with H 22 > H 11 , it is easy to deduce that
1
2
C
C
and that 1 0
C when 2 0
C . By substituting Eqs. (11), (20), (22) and
(24) into CI
, we obtain Eqn. (27), which indicates that cov alent
for “increasedvalence” structure (4) is the primary contributor to the lowest-energy linear combination of 1 (MO)
with 2 (MO)
.
When r(NN) = ,
1 (MO)
and
2 (MO)
are degenerate and therefore
1
2
1
2
2
C
C
. The parameters λ and μ are also equal for this distance, and therefore x = 1 and y = 0. The Ψ(CI) of Eqn. (27) then reduces to cov alent
, i.e. this C.I.
wave-function for 2 4
N O generates
2
NO radicals as dissociation products. The
lowest-energy molecular orbital configuration, 1 (MO)
of Eqn. (11), generates
both
2
NO radicals and
2
NO
and
2
NO
ions as dissociation products, and therefore is unsatisfactory at large internuclear separations; cf. 2
H of Section 3-3.
There are four other S = 0 spin excited configurations that may be linearly
combined with 1 (MO)
and 2 (MO)
, but these are of less importance for the
ground-state, i.e. the primary components of the “best” (lowest-energy) linear
combination of the six S = 0 spin configurations are 1 (MO)
and 2 (MO)
with
1
2
C
C
when r(NN) is finite.
In Sections 11-7, 11-9, 13-2, 13-8, 18-2 and 20-6, we shall discuss aspects of
the bonding for some other molecular systems that, (as does N 2 O 4 ), involve at
least one 6-electron 4-centre bonding unit . For each of these systems, we shall use
the “increased-valence” structure whose wave-function is the covalent
of the C.I.
wave-function of the Eqn. (26) type for the 6-electron 4-centre bonding unit.
10-4 Conclusions
In Sections 7-1 and 10-1, we have shown how dimerization of Pauling “3-electron
bond” structures for NO 2 leads to two equivalent types of valence-bond represent-
Chapter 10 Pauling “3-Electron Bonds” and “Increased-Valence” Theory for N2O4
1
2
2
2
1
2
cov alent
1
2
ionic
{( –
)
(
)
} / 2
C x C
C x C
1
2
*
*
2
**
**
2
covalent
ionic
cov alent
ionic
C {xy(
) y
/ 2 }
(27)
The coefficients 1
C and 2
C are chosen so that the energy of this linear
combination is a minimum, a necessary condition for which is that
/
/
0
1
2
E C
E C
, where
2
2
2
2
1
11
2
22
1 2 12
1
2
(C
2
) /
E
H
C H
C C H
C C
. The
integral
12
1
2
ˆ
(MO)
(MO)d
H
H
may be shown to be equivalent to
2
1
3
4
12
2
4
3
d
d
)
2
(
ψ
)
1
(
ψ
/
)
2
(
ψ
)
1
(
ψ
r
e
, which is greater than zero. For a
finite N-N internuclear separation (r(NN)) with H 22 > H 11 , it is easy to deduce that
1
2
C
C
and that 1 0
C when 2 0
C . By substituting Eqs. (11), (20), (22) and
(24) into CI
, we obtain Eqn. (27), which indicates that cov alent
for “increasedvalence” structure (4) is the primary contributor to the lowest-energy linear combination of 1 (MO)
with 2 (MO)
.
When r(NN) = ,
1 (MO)
and
2 (MO)
are degenerate and therefore
1
2
1
2
2
C
C
. The parameters λ and μ are also equal for this distance, and therefore x = 1 and y = 0. The Ψ(CI) of Eqn. (27) then reduces to cov alent
, i.e. this C.I.
wave-function for 2 4
N O generates
2
NO radicals as dissociation products. The
lowest-energy molecular orbital configuration, 1 (MO)
of Eqn. (11), generates
both
2
NO radicals and
2
NO
and
2
NO
ions as dissociation products, and therefore is unsatisfactory at large internuclear separations; cf. 2
H of Section 3-3.
There are four other S = 0 spin excited configurations that may be linearly
combined with 1 (MO)
and 2 (MO)
, but these are of less importance for the
ground-state, i.e. the primary components of the “best” (lowest-energy) linear
combination of the six S = 0 spin configurations are 1 (MO)
and 2 (MO)
with
1
2
C
C
when r(NN) is finite.
In Sections 11-7, 11-9, 13-2, 13-8, 18-2 and 20-6, we shall discuss aspects of
the bonding for some other molecular systems that, (as does N 2 O 4 ), involve at
least one 6-electron 4-centre bonding unit . For each of these systems, we shall use
the “increased-valence” structure whose wave-function is the covalent
of the C.I.
wave-function of the Eqn. (26) type for the 6-electron 4-centre bonding unit.
10-4 Conclusions
In Sections 7-1 and 10-1, we have shown how dimerization of Pauling “3-electron
bond” structures for NO 2 leads to two equivalent types of valence-bond represent-