Chemistry Reference
In-Depth Information
We shall denote these as
|
Ψ
k
. One has:
exp
2
πi
jk
N
N
−
1
1
√
N
|
Ψ
k
=
|
χ
j
(4.138)
j
=
0
The corresponding eigenvalues can be worked out in the same way as in the absence
of the field (see Eq. (
4.119
)):
E
k
=
Ψ
k
|
H
|
Ψ
k
exp
2
πi
k(
N
−
1
j
)
1
N
−
j
+
=
χ
j
|
H
|
χ
j
N
j,j
=
0
β
exp
S
j,j
−
1
exp
2
πik
S
j,j
+
1
N
−
1
1
N
2
πik
N
+
i
e
i
e
=
α
+
−
B
·
+
N
+
B
·
j
=
0
2
β
cos
2
π
N
k
S
e
=
α
+
+
B
·
(4.139)
The result is of the same form as in the absence of the field, except for a shift of the
quantum number
k
under the influence of the magnetic field. The magnitude of this
shift, e
/
S
, is equal to the magnetic flux through the area of the ring, multiplied
by the constant e
/
B
·
. As a result of this shift, the energy levels that were originally
degenerate now display a Zeeman splitting. For
B
S
>
0, the Zeeman contribution
adds to
k
in the energy expression. This implies that the points on the
k
axis in
Fig.
4.7
are displaced to the right. The roots with
k
·
=
1
,
2 thus increase in energy,
while their counterparts,
k
=−
1
,
−
2, become lower in energy. Likewise, the root at
the bottom (
k
=
0) increases in energy, while the root at the top,
k
=
3, decreases,
but the changes in these extremal points are only of second order.
This simple model is at the basis of a whole corpus of electromagnetic studies
of conjugated polyenes, involving, inter alia, the calculation of magnetic suscep-
tibilities, current densities, ring currents, and chemical shifts in nuclear magnetic
resonance (NMR). From the point of view of symmetry, it is to be noted that the
magnetic field has removed all degeneracies. The time-reversal symmetry is indeed
no longer valid. However, if one reverses the momenta,
k
→−
k
, and at the same
time switches the magnetic field,
B
B
, the energies are still invariant. This op-
eration is no longer an invariance operation of one measurement though, but rather
a comparison between two separate experiments with opposite fields.
→−
Polyhedral Hückel Systems of Equivalent Atoms
The polygonal system of the annulenes can be extended to polyhedral systems of
equivalent atoms. Atoms are equivalent if the symmetry group of the molecule—or
