Computational Chemistry

Sincebis negative, thep-electronic energy of the propenyl cation is calculated
to be below that of ethene: providing an extra, formally emptyporbital for the
electron pair causes the energy to drop. Actually, resonance energy is usually
presented as a positive quantity, e.g. “100 kJ mol"^1 ”. We can interpret this as
100 kJ mol"^1 below a reference system. To avoid a negative quantity in SHM
calculations like these, we can use |b| instead ofb.
The propenyl radical, Fig.4.16. The totalpelectronic energy by the SHM is

Epðprop:radicalÞ¼ 2 ðaþ 1 : 414 bÞþa¼ 3 aþ 2 : 828 b

For the reference energy we use one ethene molecule and one nonbonding
pelectron (like the electron in a methyl radical):

EpðreferenceÞ¼ð 2 aþ 2 bÞþa¼ 3 aþ 2 b

The stabilization energy is then

E(stab, radical)¼Epðprop:radicalÞ"EpðreferenceÞ

¼ð 3 aþ 2 : 828 bÞ"ð 3 aþ 2 bÞ¼ 0 : 828 b

The propenyl anion. An analogous calculation (cf. Fig.4.16, with four electrons
for the anion) gives

E(stab anion)¼Epðprop:anionÞ"EpðreferenceÞ

¼ð 4 aþ 2 : 828 bÞ"ð 4 aþ 2 bÞ¼ 0 : 828 b

Thus the SHM predicts that all three propenyl species will be lower in energy
than if thepelectrons were localized in the formal double bond and (for the radical
and anion) in oneporbital. Because this lower energy is associated with the ability
of the electrons to spread or be delocalized over the wholepsystem, what we have
calledE(stab) is often denoted as the delocalization energy, and designatedED.
Note thatER(resonance energy, orED, delocalization energy) is always some
multiple ofb(or is zero). Since electron delocalization can be indicated by the
familiar resonance symbolism the H€uckel delocalization energy is often equated
with resonance energy, and designatedER. The accord between calculated delocal-
ization and the ability to draw resonance structures is not perfect, as indicated by the
next example.
Cyclobutadiene(Fig.4.17). The totalpelectronic energy is

EpðcyclobutadieneÞ¼ 2 ðaþ 2 b)+2a¼ 4 aþ 4 b

Using two ethene molecules as our reference system:

EpðreferenceÞ¼ 2 aþ 2 b

140 4 Introduction to Quantum Mechanics in Computational Chemistry