Page 65 - Mechanism and Theory in Organic Chemistry

The secular equation then takes the form of the N-by-N determinental Equation
A2.17. The asterisk in row i, column j is zero if atom i is not bonded to atom j,

and it is /3 if i is bonded to j. A further simplification of the algebraic manipula-
tions required is obtained by setting the zero of energy equal to a and the unit of
energy equal to /3. (That is, we measure energy relative to a in units of 8.) Then
we have Equation A2.18, where *,, is zero if i and j are not bonded and unity if
they are.

The Huckel orbitals have a number of properties that make them parti-
cularly convenient starting points for arguments of a qualitative or semiquantita-
tive nature about conjugated systems./ Huckel calculations are the only ones that
are practical to do without the aid of a computer, and then only when deter-
minants are of an order no larger than about four.g The Huckel method gives
rather poor energies and orbital functions but does reproduce faithfully sym-
metry properties of orbitals. Despite its many deficiencies, the method has been
successful in correlating a variety of experimental data and has pointed the way to
much interesting experimental chemistry.

As an example of the Huckel method we will examine the ally1 system. There are
three basis orbitals, numbered as shown in 1. Atoms 1 and 2 are bonded to each

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f M. J. S. Dewar, The Molecular Orbital Theory of Organic Chemistry, McGraw-Hill, New York, 1969.
9 Symmetry properties can sometimes be used to reduce the size of determinants.