
120 Nanosensors: Physical, Chemical, and Biological
a net attraction and chemical bonding. In the middle at the energy α, there
is a nonbonding orbital. The energy levels of these orbitals, E
j
, are obtained
easily on the Huckel level as
Annulene 1
/2 for even
: ( ) cos , , ,
(
E j
j
N
j
N N
= +
⎛
⎝
⎜
⎞
⎠
⎟
= ± …
±
α β
π
2
2
0
)) { ( ) } or 1 /2for odd± −N N
(2.52)
Polyene 1 2 : ( ) cos , , ,E j
j
N
j N= +
+
⎛
⎝
⎜
⎞
⎠
⎟
= …α β
π
2
1
(2.53)
By dening a new index k, these equations are rewritten in a different form
E k ka( ) cos( )= +α β2
(2.54)
k = 0 to (±π/a for annulene) or (or π/a for polyene). Here k = 2jπ/Na for annu-
lene and jπ/(N + 1)a for polyene; i