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### Cyclic Hydrocarbons

The secular equations, in the Huckel approximation for a cyclic hydrocarbon are
 (72)
where each orbital is expanded in the  atomic orbitals , and cyclic symmetry implies cir=ci,r+N. We show by substitution that two solutions are possible
 (73)
Substituting the first into the secular equation yields
 (74)
Using , and cancelling  gives
 (75)
The argument similarly follows for . Note that each solution obeys the periodic boundary condition. Thus each energy level appears to be degenerate. However for i=0. the lowest orbital only has  solution, and  only has the  solution if N is even. Thus

for N=4, the energy levels are

for N=6, the energy levels are

There is another theoretical model for a hydrocarbon, called the mobius hydrocarbon, for which the periodic condition is cir=-ci,r+N. For this problem the solutions are

 (76)
with energy levels given by
 (77)
for N=4, the energy levels are
for N=6, the energy levels are

The interesting observation is that the regular N=4 hydrocarbon gives the TS orbital energy level diagram for the disrotatory ring opening of cyclbutene, and the mobius N=4 hydrocarbon gives the TS orbital energy level diagram for the conrotatory ring opening.

Next: The Symmetry of Molecular Vibrations Up: Contents - Previous: Symmetry and the Woodward-Hoffmann Rules
Nicholas Handy

1998-09-23