The propenyl (allyl) system has two, three or fourpelectrons, depending on
whether we are considering the cation, radical or anion (Fig.4.19b; cf. Fig.4.16).
The cation might be expected to be resistant to oxidation, which requires removing
an electron from a low-lyingporbital (c 1 ) and to be moderately readily reduced, as
this involves adding an electron to the nonbondingporbitalc 2 , a process that
should not be strongly favorable or unfavorable. The radical should be easier to
oxidize than the cation, for this requires removing an electron from a nonbonding,
rather than a lower-lying bonding, orbital, and the ease of reduction of the radical
should be roughly comparable to that of the cation, as both can accommodate an
electron in a nonbonding orbital. The anion should be oxidized with an ease
comparable to that of the radical (removal of an electron from the nonbonding
c 2 ), but be harder to reduce (addition of an electron to the antibondingc 3 ).
The cyclobutadiene system (Fig.4.19c; cf. Fig.4.17) can be envisaged with,
amongst others, two (the dication), four (the neutral molecule) and sixp(the
dianion) electrons. The predictions one might make for these the behavior of
these three species toward redox reactions are comparable to those just outlined
for the propenyl cation, radical and anion, respectively (note the analogous occu-
pancy of bonding, nonbonding and antibonding orbitals). The neutral cyclobuta-
diene molecule is, however, predicted by the SHM to have an unusual electronic
arrangement for a diene: in filling theporbitals, from the lowest-energy one up, one
puts electrons of the same spin into the degeneratec 2 andc 3 in accordance with
Hund’s rule of maximum multiplicity. Thus the SHM predicts that cyclobutadiene
will be a diradical, with two unpaired electrons of like spin. Actually, more
advanced calculations [ 39 ] indicate, and experiment confirms, that cyclobutadiene
is a singlet molecule with two single and two double C/C bonds. A square
cyclobutadiene diradical with four bond order 1.5 C/C bonds would distort to a
rectangular, closed-shell (i.e. no unpaired electrons) molecule with two single and
two double bonds (Fig.4.20). This could have been predicted by augmenting the
SHM result with a knowledge of the phenomenon known as the Jahn–Teller effect
[ 40 ]: cyclic systems (and certain others) with an odd number of electrons in
degenerate (equal-energy) MOs will distort to remove the degeneracy.
What general pattern of molecular orbitals emerges from the SHM? Acyclicp
systems (ethene, the propenyl system, 1,3-butadiene, etc.), have MOs distributed
singly and evenly on each side of the nonbonding level; the odd-AO systems also
have one nonbonding MO (Fig.4.21). Cyclicpsystems (the cyclopropenyl system,
cyclobutadiene, the cyclopentadienyl system, benzene, etc.) have a lowest MO and
pairs of degenerate MOs, ending with one highest or a pair of highest MOs,
depending on whether the number of MOs is even or odd. The total number of
MOs is always equal to the number of basis functions, which in the SHM is, for
organic polyenes, the number ofporbitals (Fig.4.21). The pattern for cyclic
systems can be predicted qualitatively simply by sketching the polygon, with one
vertex down, inside a circle (Fig.4.22). If the circle is of radius 2|b| the energies can
even be calculated by trigonometry [ 41 ]. It follows from this pattern that cyclic
species (not necessarily neutral) with 2, 6, 10, ...pelectrons have filledpMOs and
might be expected to show particular stability, analogously to the filled AOs of the
136 4 Introduction to Quantum Mechanics in Computational Chemistry