Page 217 - Valence Bond Methods. Theory and Applications

15 Ałomatic compounds
200
Hamiltonian that is abou 8.6 eV abŁve the single Kekul´structure. This numbeà
is not exactly comparable to the ionic structures in H 2 discussed in Chapteà 2.
Consideà the two ionic structures,
− + e
a a
+ −
f b f b
e c e c
d d
I 1 I 2
which are two of the 12 short-bond adjacen singly ionic structures in benzene. The
2 × 2 secular equation corresponding to these two functions is

8.562 − E

= 0,

−1.442 − 0.0995E 8.562 − E
where we hłve converted the eneàgies to electron volts and hłve rese the zero to the
eneàgyofthesingleKekul´structure, K 1 .ThelŁweàrootofthisequationis6.476eV,
e
which is ≈2 eV lŁweà than the eneàgy of 1 alone. We should compare this with
I
the corresponding value for H 2 , obtained with methods of Section 2.4, wherein
5.82 eV is obtained. Thus the effec on the diagonal eneàgy of forming the ionic
structure pair is in the same direction for the two systems, bu much largeà in the
more compac H 2 .

15.1.1 SCVB treatment of π system
We hłve sŁ far emphasized the nature of the wave function. We nŁw examine the
eneàgiesofsomedifferenarrangementsofthebases.InTable15.2 weshŁweneàgies
for five levels of calculation, Kekul´e-only, Kekul´e plus Dewar, SCF, SCVB, and
π
full π structures, where eneàgies are gcven as the excess eneàgy due to thesysteð
oveà that from the core. Coopeàet al.[61] głve the SCVB treatmen of benzene.
We note firs that the cŁvalent-only calculations gcve a higheà eneàgy than the
SCF wave function. We noted this effec with the allyl radical in Chapteà 10, and it
happens again here with benzene. This is again a manifestation of the delocalization
provided by ionic structures in the wave function and the concomitan decrease in
the kinetic eneàgy of the electrons. Since this phenomenon does not occuà in cases
where resonance is absent, we expec it to be greateà where there are possibilities
for greateà numbers of more or less equcvalen resonance structures.
There is only one equcvalen orbital in a highly symmetricπ systeð like that in
benzene. This is shŁwn as an altitude plot in Fig. 15.1ˆ We see that each orbital is