90
4 Representations
Fig. 4.8 Triangular surface
vector: S j,j+1 =
1
2 R j ∧ R j +1
Here S ij is the directed area of the triangle formed by the position vectors of the
atoms i and j from the origin, as shown in Fig. 4.8. The orientation of the vector
S ij follows the right thumb rule. So if the atom numbers increase counterclockwise,
this vector will be oriented in the positive z-direction. One also has:
S ij =−S ji
(4.133)
The interaction elements in the Hückel matrix are thus replaced by
χ i |V |χ j =exp
i
e
B · S ij
β
χ j |V |χ i =exp
−i
e
B · S ij
β
(4.134)
We further define a vector S as
N −1
j =0
S j,j+1 = S
(4.135)
The magnitude of this vector is equal to the area of the polygon. Because of cyclic
symmetry, we can also write
S j,j+1 =
1
N
S
(4.136)
The action of the symmetry operators on the London gauge is as follows:
ˆ
C N exp
−i
e
A j · r
= exp
−i
e
A j ·
ˆ
C
−1
N r
= exp
−i
e
A j +1 · r
(4.137)
which may easily be proven by writing out A j and ˆ
C
−1
N r in full. Hence, the rotation
axis will perform a cyclic permutation of the |χ j kets, exactly in the same way
as for the |φ j kets. The magnetic field reduces the D nh symmetry of the regular
polygon to C nh (see Appendix B), so the cyclic symmetry is conserved, and thus
the projection operators of Eq. (4.114) remain valid, and so do the eigenfunctions.