REPRESENTATION THEORY
1
2
3
(^4) x
y
Figure 29.4 A molecule consisting of four atoms of iodine and one of
manganese.
29.11.1 Bonding in molecules
We have just seen that whether chemical bonding can take place in a molecule
is strongly dependent upon whether the wavefunctions of the two atoms forming
a bond transform according to the same irrep. Thus it is sometimes useful to be
able to find a wavefunction that does transform according to a particular irrep
of a group of transformations. This can be done if the characters of the irrep are
known and a sensible starting point can be guessed. We state without proof that
starting from anyn-dimensional basis vector Ψ≡(Ψ 1 Ψ 2 ···Ψn)T,where{Ψi}
is a set of wavefunctions, the new vector Ψ(λ)≡(Ψ 1 (λ)Ψ( 2 λ)···Ψ(nλ))Tgenerated
by
Ψ
(λ)
i =
∑
X
χ(λ)
∗
(X)XΨi (29.24)
will transform according to theλth irrep. If the randomly chosen Ψ happens not
to contain any component that transforms in the desired way then the Ψ(λ)so
generated is found to be a zero vector and it is necessary to select a new starting
vector. An illustration of the use of this ‘projection operator’ is given in the next
example.
Consider a molecule made up of four iodine atoms lying at the corners of a square in the
xy-plane, with a manganese atom at its centre, as shown in figure 29.4. Investigate whether
the molecular orbital given by the superposition ofp-state (angular momentuml=1)
atomic orbitals
Ψ 1 =Ψy(r−R 1 )+Ψx(r−R 2 )−Ψy(r−R 3 )−Ψx(r−R 4 )
can bond to thed-state atomic orbitals of the manganese atom described by either(i)φ 1 =
(3z^2 −r^2 )f(r)or(ii)φ 2 =(x^2 −y^2 )f(r),wheref(r)is a function ofrand so is unchanged by
any of the symmetry operations of the molecule.Such linear combinations of atomic orbitals
are known as ring orbitals.
We have eight basis functions, the atomic orbitals Ψx(N)andΨy(N), whereN=1, 2 , 3 , 4
and indicates the position of an iodine atom. Since the wavefunctions are those ofp-states
they have the formsxf(r)oryf(r) and lie in the directions of thex-andy-axes shown in
the figure. Sinceris not changed by any of the symmetry operations,f(r) can be treated as
a constant. The symmetry group of the system is 4mm, whose character table is table 29.4.