6.11 Problems
159
a 1 + b 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 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 ǫ :
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:
Ψ =
(t 1u xα)(t 1u yα)(t 1u zα)
(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 1u level in an icosahedral 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 ′ -dibiphenylphosphine-1, 1 ′ -binaphthyl. A circular dichroism signal is detected in
159
a 1 + b 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 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 ǫ :
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:
Ψ =
(t 1u xα)(t 1u yα)(t 1u zα)
(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 1u level in an icosahedral 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 ′ -dibiphenylphosphine-1, 1 ′ -binaphthyl. A circular dichroism signal is detected in