4.3.5 The Simple Huckel Method – Applications€
Applications of the SHM are discussed in great detail in several books [ 21 ]; here we
will deal only with those applications which are needed to appreciate the utility of
the method and to smooth the way for the discussion of certain topics (like bond
orders and atomic charges) in later chapters. We will discuss: the nodal properties
of the MOs; stability as indicated by energy levels and aromaticity (the 4nþ2 rule);
resonance energies; and bond orders and atomic charges.
4.3.5.1 The Nodal Properties of the MOs
A node of an MO is a plane at which, as we proceed along the sequence of basis
functions, the sign of the wavefunction changes (Figs.4.15–4.17). For a given
molecule, the number of nodes in theporbitals increases with the energy. In the
two-orbital system (Fig.4.15),c 1 has zero nodes andc 2 has one node. In the three-
orbital system (Fig.4.16),c 1 ,c 2 andc 3 have zero, one and two nodes, respec-
tively. In the cyclic four-orbital system (Fig.4.17),c 1 has zero nodes,c 2 andc 3 ,
which are degenerate (of the same energy) each have one node (one nodal plane),
andc 4 has two nodes. In a given molecule, the energy of the MOs increases with
the number of nodes. The nodal properties of the SHMporbitals form the basis of
one of the simplest ways of understanding the predictions of the Woodward–Hoff-
mann orbital symmetry rules [ 38 ]. For example, the thermal conrotatary and
disrotatary ring closure/opening of polyenes can be rationalized very simply in
terms of the symmetry of the highest occupiedpMO of the open-chain species.
That the highestpMO should dominate the course of this kind of reaction is
indicated by more detailed considerations (including extended H€uckel calculations)
[ 38 ]. Figure4.18shows the situation for the ring closure of a 1,3-butadiene to a
cyclobutene. The phase (+ or") of thepHOMO (c 2 ) at the end carbons (the atoms
that bond) is opposite on each face, because this orbital has one node in the middle
of the C 4 chain. You can see this by sketching the MO as the four AOs contributing
to it, or even – remembering the node – drawing just the end AOs. For the electrons
inc 2 to bond, the end groups must rotate in the same sense (conrotation) to bring
orbital lobes of the same phase together. Remember that plus and minus phase has
nothing to do with electric charge, but is a consequence of the wave nature of
electrons (Section 4.2.6): two electron waves can reinforce one another and form a
bonding pair if they are “vibrating in phase”; an out-of-phase interaction represents
an antibonding situation. Rotation in opposite senses (disrotation) would bring
opposite-phase lobes together, an antibonding situation. The mechanism of the
reverse reaction is simply the forward mechanism in reverse, so the fact that the
thermodynamically favored process is the ring-opening of a cyclobutene simply
means that the cyclobutene shown would open to the butadiene shown on heating.
Photochemical processes can also be accommodated by the Woodward–Hoffmann
orbital symmetry rules if we realize that absorption of a photon creates an electron-
ically excited molecule in which the previous lowest unoccupied MO (LUMO) is
4.3 The Application of the Schr€odinger Equation to Chemistry by H€uckel 133