Chemistry Reference
In-Depth Information
a
1
+
b
2
, which is precisely the meaning of the three terms on the right-hand
side of Eq. (
6.147
). This expression may be transformed in two steps to the term
which is required in the derivation. One first changes the substrate of the fibre from
Γ
σ
(e)
to
Γ
(e)
. This associates the edges with
b
1
objects, and combination with
a
1
+
b
1
+
b
1
+
b
2
will thus yield
b
1
+
a
1
+
a
2
,or:
Γ
(e)
×
Γ
T
=
Γ
(e)
+
Γ
σ
(e)
+
Γ
(e)
(6.148)
Finally, multiply this result by
Γ
:
Γ
(e)
×
Γ
R
=
Γ
⊥
(e)
+
Γ
(e)
+
Γ
σ
(e)
(6.149)
We now combine Eqs. (
6.146
) and (
6.149
), and find:
Trivalent:
Γ
e
P
×
Γ
R
−
Γ
σ
v
P
×
Γ
R
=
Γ
σ
e
P
=
Γ
Fries
(6.150)
This is precisely the representation of the Fries bonds. We can thus compare the
Fries and Clar structures in a general leapfrog, and find from Eq. (
6.144
):
Γ
Clar
−
Γ
Fries
=
Γ
T
+
Γ
R
(6.151)
The Clar structure thus has six extra bonding orbitals as compared with the Fries
structure. When both bonding schemes are correlated, as illustrated in Fig.
6.11
,
this sextet must correlate with the anti-bonding half of the Fries structure. It will
thus be placed on top of the Clar band, and actually be nearly non-bonding, forming
six low-lying virtual orbitals, which explains the electron deficiency of the leapfrog
fullerenes. Moreover, as the derivation shows, they transform exactly as rotations
and translations.
6.11 Problems
6.1 A three-electron wavefunction in an octahedron is given by:
(t
1
u
xα)(t
1
u
yα)(t
1
u
zα)
=
Ψ
(6.152)
The vertical bars denote a Slater determinant. Determine the symmetry of this
function, starting from parent two-electron coupled states, to which the third
electron is coupled. Make use of the coupling coefficients in Appendix
F
.
6.2 Write the Jahn-Teller matrix for a threefold degenerate
T
1
u
level in an icosahe-
dral molecule. How many reduced matrix elements are needed?
6.3 Do you expect octahedral
e
g
orbitals to show a magnetic dipole moment?
6.4 Binaphthyl consists of two linked naphthalene molecules. The dihedral angle
between the two naphthyl planes is around 70
◦
, and can be stabilized by bulky
substituents on the naphthyl units, as indicated below for the case of 2
,
2
-di-
biphenylphosphine-1
,
1
-binaphthyl. A circular dichroism signal is detected in
