86
4 Representations
The symmetry of an N -atom ring is D Nh , but in practice the cyclic group C N is
sufficient to solve the eigenvalue problem. Atoms are numbered from 0 to N − 1.
The cyclic projection operator, ˆ
P k , is given by
ˆ
P k =
1
N
N −1
j =0
exp
2πi
jk
N
ˆ
C
j
N
(4.114)
Projectors are characterized by an integer k in a periodic interval. We may choose
the range ]−N/2, +N/2] as the standard interval. The total number of integers in
this interval is N . Keeping in mind the active view, where the rotation axis will
rotate all the orbitals one step further in a counterclockwise way, we now act with
the projection operator on the starting orbital, |φ 0 :
ˆ
P k |φ 0 =
1
N
N −1
j =0
exp
2πi
jk
N
|φ j
(4.115)
The result is an unnormalized SALC, which we denote as |Φ k . Neglecting overlap
between adjacent atoms, we obtain the normalized SALC as
|Φ k =
√
N ˆ
P k |φ 0
(4.116)
The transformation properties of this SALC under the rotation axis are characterized
as
ˆ
C N |Φ k =
1
√
N
N −1
j =0
exp
2πi
jk
N
|φ j +1
=
1
√
N
N −1
j =0
exp
2πi
(j − 1)k
N
|φ j
= exp
−2πi
k
N
|Φ k
(4.117)
Applying this symmetry element N times is identical to the unit operation and raises
the exponential factor in this expression to the N th power:
exp
−2πi
k
N
N
= exp(−2πik) = 1
(4.118)
Each integer value of k in the periodic interval ]−N/2, +N/2] thus characterizes a
different SALC. The corresponding energy eigenvalues are also easily extracted:
E k ==Φ k |H|Φ k
=
1
N
N −1
j,j ′ =0
exp
2πi
k(−j + j ′ )
N
φ j |H|φ j ′
4 Representations
The symmetry of an N -atom ring is D Nh , but in practice the cyclic group C N is
sufficient to solve the eigenvalue problem. Atoms are numbered from 0 to N − 1.
The cyclic projection operator, ˆ
P k , is given by
ˆ
P k =
1
N
N −1
j =0
exp
2πi
jk
N
ˆ
C
j
N
(4.114)
Projectors are characterized by an integer k in a periodic interval. We may choose
the range ]−N/2, +N/2] as the standard interval. The total number of integers in
this interval is N . Keeping in mind the active view, where the rotation axis will
rotate all the orbitals one step further in a counterclockwise way, we now act with
the projection operator on the starting orbital, |φ 0 :
ˆ
P k |φ 0 =
1
N
N −1
j =0
exp
2πi
jk
N
|φ j
(4.115)
The result is an unnormalized SALC, which we denote as |Φ k . Neglecting overlap
between adjacent atoms, we obtain the normalized SALC as
|Φ k =
√
N ˆ
P k |φ 0
(4.116)
The transformation properties of this SALC under the rotation axis are characterized
as
ˆ
C N |Φ k =
1
√
N
N −1
j =0
exp
2πi
jk
N
|φ j +1
=
1
√
N
N −1
j =0
exp
2πi
(j − 1)k
N
|φ j
= exp
−2πi
k
N
|Φ k
(4.117)
Applying this symmetry element N times is identical to the unit operation and raises
the exponential factor in this expression to the N th power:
exp
−2πi
k
N
N
= exp(−2πik) = 1
(4.118)
Each integer value of k in the periodic interval ]−N/2, +N/2] thus characterizes a
different SALC. The corresponding energy eigenvalues are also easily extracted:
E k ==Φ k |H|Φ k
=
1
N
N −1
j,j ′ =0
exp
2πi
k(−j + j ′ )
N
φ j |H|φ j ′