# Group Theory Solutions: #8**
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**8.* (1994 1 1) Shown below are the one s, three p, and
five d atomic orbitals with their phases indicated by shading. Pay
special attention to the axes, which are **__not__ all the
same!
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**A. Classify these
nine atomic orbitals by their symmetry under inversion, that is, by their
wavefunctions' change in sign when the symmetry operation i is
performed. Which of these atomic orbitals are
even (do not change sign) and which are odd (do change sign) under
inversion?**
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The s orbital and the 5 d orbitals are even under inversion. The 3 p
orbitals are odd under inversion.
**B. Consider a p**_{y}
atomic orbital on one nuclear center A and one of the five d atomic
orbitals on another nuclear center B. Let the nuclear centers A and B be
placed on the x axis with A on the left and B on the right. As A and B
approach, molecular orbitals can be formed from the two atomic orbitals if
these two atomic orbitals have the same symmetry with respect tot he
internuclear axis. With which atomic d orbital on B is there a nonzero
(noncancelling) overlap with the atomic p_{y} orbital on
A?
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All d orbitals except the d_{xy}
have the wrong spatial orientation for interaction with the p_{y}
orbital along an internuclear axis defined as the x axis.
C.
Draw pictures of the bonding and anitbonding molecular orbitals that
result from this linear combination of atomic orbitals.
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**D. Classify the resulting molecular orbital
as to its symmetry with respect to the internuclear axis.
If
we look along the internuclear axis of either the bonding or the
antibonding MO, we notice a nodal plane containing x:
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