H Solutions to Problems
251
4.9 Since all irreps are one-dimensional, the characters can only consist of a phase
factor:
D(C 5 ) = e
iλ I
(5)
The fifth power of the generator will yield the unit element, and hence,
e
5iλ = 1( 6 )
This is the Euler equation. Its solutions are the characters in the table of C 5 ,
as given in Appendix A.
4.10 The product of inversion with a ˆ
C 2 axis must yield a reflection plane, perpendicular to this axis. As an example, a product of type ˆ
ı · ˆ
C ′
2 must yield a
reflection plane of ˆ
σ d type, as this is perpendicular to the primed twofold axis.
For the one-dimensional irreps of D 6h , one thus should have
χ(ı)χ
C
′
2
= χ(σ d )
(7)
This is indeed verified to be the case.
4.11 The a ′′
2 distortion is antisymmetric with respect to 3 ˆ
C 2 , ˆ
σ h , and 2 ˆ
S 3 .A sa
result, when the mode is launched, all these symmetry elements will be destroyed, and the symmetry reduces to the subgroup C 3v . In general, the result
of a distortion will always be the maximal subgroup for which the distortion
is totally symmetric [14].
4.12 The group of this fullerene is D 6d . The 24 atoms separate into two orbits: a 12orbit containing the top and bottom hexagons and another 12-orbit containing
the crown of the 12 atoms, numbered from 7 to 18. In both cases the site group
is only C s , and hence both orbits will span the same irreps:
a
′ C s ↑ D 6d = A 1 + B 2 + E 1 + E 2 + E 3 + E 4 + E 5
Quite remarkably, the Hückel spectrum for this fullerene has a nonbonding
level of E 4 symmetry.
5.1 Let r i and r j denote the position vectors of electrons i and j . The electron
repulsion operator contains the distance between both electrons as |r i − r j |.The
matrix D(R) expresses the transformation of the Cartesian coordinates under a
rotation. This matrix will also rotate the coordinate differences:
ˆ
R
⎛
⎝
x i − x j
y i − y j
z i − z j
⎞
⎠ = D(R)
⎛
⎝
x i − x j
y i − y j
z i − z j
⎞
⎠
(8)
Exactly as in the derivation for Problem 1.2, the square of the distance between
the two electrons is then found to be invariant under any orthogonal transformation of the coordinates.
5.2 For the G irrep, it is noted from Sect. C.1 that a tetrahedral splitting field will
branch G into A + T . It thus acts as a splitting field to isolate the unique Ga
component. Symmetry adaptation to ˆ
C
z
2 will yield two totally symmetric components, one of which will be the Ga already obtained; the remaining one is
251
4.9 Since all irreps are one-dimensional, the characters can only consist of a phase
factor:
D(C 5 ) = e
iλ I
(5)
The fifth power of the generator will yield the unit element, and hence,
e
5iλ = 1( 6 )
This is the Euler equation. Its solutions are the characters in the table of C 5 ,
as given in Appendix A.
4.10 The product of inversion with a ˆ
C 2 axis must yield a reflection plane, perpendicular to this axis. As an example, a product of type ˆ
ı · ˆ
C ′
2 must yield a
reflection plane of ˆ
σ d type, as this is perpendicular to the primed twofold axis.
For the one-dimensional irreps of D 6h , one thus should have
χ(ı)χ
C
′
2
= χ(σ d )
(7)
This is indeed verified to be the case.
4.11 The a ′′
2 distortion is antisymmetric with respect to 3 ˆ
C 2 , ˆ
σ h , and 2 ˆ
S 3 .A sa
result, when the mode is launched, all these symmetry elements will be destroyed, and the symmetry reduces to the subgroup C 3v . In general, the result
of a distortion will always be the maximal subgroup for which the distortion
is totally symmetric [14].
4.12 The group of this fullerene is D 6d . The 24 atoms separate into two orbits: a 12orbit containing the top and bottom hexagons and another 12-orbit containing
the crown of the 12 atoms, numbered from 7 to 18. In both cases the site group
is only C s , and hence both orbits will span the same irreps:
a
′ C s ↑ D 6d = A 1 + B 2 + E 1 + E 2 + E 3 + E 4 + E 5
Quite remarkably, the Hückel spectrum for this fullerene has a nonbonding
level of E 4 symmetry.
5.1 Let r i and r j denote the position vectors of electrons i and j . The electron
repulsion operator contains the distance between both electrons as |r i − r j |.The
matrix D(R) expresses the transformation of the Cartesian coordinates under a
rotation. This matrix will also rotate the coordinate differences:
ˆ
R
⎛
⎝
x i − x j
y i − y j
z i − z j
⎞
⎠ = D(R)
⎛
⎝
x i − x j
y i − y j
z i − z j
⎞
⎠
(8)
Exactly as in the derivation for Problem 1.2, the square of the distance between
the two electrons is then found to be invariant under any orthogonal transformation of the coordinates.
5.2 For the G irrep, it is noted from Sect. C.1 that a tetrahedral splitting field will
branch G into A + T . It thus acts as a splitting field to isolate the unique Ga
component. Symmetry adaptation to ˆ
C
z
2 will yield two totally symmetric components, one of which will be the Ga already obtained; the remaining one is