Cyclic Conjugated Polyenes

 

The ‘Circle’ Method; Benzene and Aromaticity

 

 

See Murrell, Kettle and Tedder “Valence Theory” page 268,

Carroll ‘Perspectives on Structure and Mechanism in Organic Chemistry’ pgs. 205-208

 

 

It is easy to write down the secular equations and secular determinant for benzene.

 

We number the carbon atoms as below:

 

 

Consider atom No. 1.  It is next to atoms 6 and 2; so the corresponding secular equation is:

 

xc1 + c2 + 0´c3 + 0´c4 + 0´c5+ c6 = 0

or

xc1 + c2 + 0 + 0 + 0 + c6 = 0

 

·      Each secular equation involves just 3 non-zero terms. 

·      All the six secular equations have the same structure, and they can be obtained from each other by permuting the indices (i.e. by just adding one to the indices of the preceding equation. 

·      The equations can be solved analytically making use of trigonometric functions. 

·      These trigonometric solutions may in turn, rigorously, be represented by a simple geometric construction- this is the ‘Circle method’

 

 

The Circle method