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SOLUTIONSMANUAL TO ACCOMPANY ATKINS' PHYSICAL CHEMISTRY 313
�e secular determinant and hamiltonian matrix for octatetraene are
RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR
α − E β 0 0 0 0 0 0
β α − E β 0 0 0 0 0
0 β α − E β 0 0 0 0
0 0 β α − E β 0 0 0
0 0 0 β α − E β 0 0
0 0 0 0 β α − E β 0
0 0 0 0 0 β α − E β
0 0 0 0 0 0 β α − E
RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR
H = α1+β
⎛
⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜
⎝
0 1 0 0 0 0 0 0
1 0 1 0 0 0 0 0
0 1 0 1 0 0 0 0
0 0 1 0 1 0 0 0
0 0 0 1 0 1 0 0
0 0 0 0 1 0 1 0
0 0 0 0 0 1 0 1
0 0 0 0 0 0 1 0
⎞
⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟
⎠
�is is diagonalized by the matrix
⎛
⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜
⎝
0.161 0.303 0.408 0.464 0.464 0.408 0.303 0.161
0.303 0.464 0.408 0.161 −0.161 −0.408 −0.464 −0.303
0.408 0.408 0 −0.408 −0.408 0 0.408 0.408
0.464 −0.161 −0.408 −0.303 0.303 0.408 −0.161 −0.464
0.464 −0.161 −0.408 0.303 0.303 −0.408 −0.161 0.464
0.408 −0.408 0 0.408 −0.408 0 0.408 −0.408
0.303 −0.464 0.408 −0.161 −0.161 0.408 −0.464 0.303
0.161 −0.303 0.408 −0.464 0.464 −0.408 0.303 −0.161
⎞
⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟
⎠
�e diagonal elements are ±1.88, ±1.53, ±1.00, and ±0.347, implying that the
energies are α ± 1.88 β, α ± 1.53 β, α ± 1.00 β, and α ± 0.347 β.
Each columnof the diagonalizationmatrices gives the coe�cients of the atomic
orbitals for the corresponding molecular orbitals. For each molecule, the �rst
column gives the coe�cients for the lowest energy π molecular orbital. All of
these coe�cients have positive signs, therefore the lowest energy π molecular
orbital has no nodes and the lowest energy π molecular orbital is delocalized
over all carbon atoms in the chain. On going to successive columns to the right
the energy of the molecular orbital increases and the number of sign changes
within a column increases. �ese sign changes correspond to nodes in the
molecular molecular orbitals.
P9E.6 �e bonding and antibonding e�ects of theHOMOand LUMOhave to be con-
sidered to predict the changes in bond strength that accompany the transition.
�e ground state electron con�guration of butadiene is 1π2 2π2, therefore the
HOMO of butadiene is the 2π molecular orbital. Looking at Fig. 9E.2 on page
376, this molecular orbital is bonding between carbon atoms C1 and C2 and
likewise between C3 and C4, but antibonding between C2 and C3.�e LUMO
of butadiene is the 3πmolecular orbital, which is antibonding between carbon
atoms C1 and C2 and likewise between C3 and C4, but bonding between C2
and C3. �erefore promotion of an electron from the HOMO to the LUMO
weakens the bonding between C1 and C2 and likewise between C3 and C4, but
strengthens the bonding between C2 and C3.
�e HOMO of benzene is the e1g orbital and the LUMO is the e2u orbital. One
of the e1g orbitals, the one on the right in Fig. 9E.4 on page 377, is bonding
between two pairs of neighbouring carbon atoms, and nonbonding between all