Product representations

29.10 PRODUCT REPRESENTATIONS
give a large selection of character tables; our aim is to demonstrate and justify
the use of those found in the literature specifically dedicated to crystal physics or
molecular chemistry.
Variations in notation are not restricted to the naming of groups and their
irreps, but extend to the symbols used to identify a typical element, and hence
all members, of a conjugacy class in a group. In physics these are usually of the
types nz , n̄z or mx . The first of these denotes a rotation of 2π/n about the z-axis,
and the second the same thing followed by parity inversion (all vectors r go to
−r), whilst the third indicates a mirror reflection in a plane, in this case the plane
x = 0.
Typical chemistry symbols for classes are NCn , NCn2 , NCnx , NSn , σv , σ xy . Here
the first symbol N, where it appears, shows that there are N elements in the
class (a useful feature). The subscript n has the same meaning as in the physics
notation, but σ rather than m is used for a mirror reflection, subscripts v, d or h or
superscripts xy, xz or yz denoting the various orientations of the relevant mirror
planes. Symmetries involving parity inversions are denoted by S; thus Sn is the
chemistry analogue of n̄. None of what is said in this and the previous paragraph
should be taken as definitive, but merely as a warning of common variations in
nomenclature and as an initial guide to corresponding entities. Before using any
set of group character tables, the reader should ensure that he or she understands
the precise notation being employed.
29.10 Product representations
In quantum mechanical investigations we are often faced with the calculation of
what are called matrix elements. These normally take the form of integrals over all
space of the product of two or more functions whose analytic forms depend on the
microscopic properties (usually angular momentum and its components) of the
electrons or nuclei involved. For ‘bonding’ calculations involving ‘overlap integrals’
there are usually two functions involved, whilst for transition probabilities a third
function, giving the spatial variation of the interaction Hamiltonian, also appears
under the integral sign.
If the environment of the microscopic system under investigation has some
symmetry properties, then sometimes these can be used to establish, without
detailed evaluation, that the multiple integral must have zero value. We now
express the essential content of these ideas in group theoretical language.
Suppose we are given an integral of the form
J = Ψφ dτ or J = Ψξφ dτ
to be evaluated over all space in a situation in which the physical system is
invariant under a particular group G of symmetry operations. For the integral to
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REPRESENTATION THEORY
be non-zero the integrand must be invariant under each of these operations. In
group theoretical language, the integrand must transform as the identity, the onedimensional representation A1 of G; more accurately, some non-vanishing part of
the integrand must do so.
An alternative way of saying this is that if under the symmetry operations
of G the integrand transforms according to a representation D and D does not
contain A1 amongst its irreps then the integral J is necessarily zero. It should be
noted that the converse is not true; J may be zero even if A1 is present, since the
integral, whilst showing the required invariance, may still have the value zero.
It is evident that we need to establish how to find the irreps that go to make
up a representation of a double or triple product when we already know the
irreps according to which the factors in the product transform. The method is
established by the following theorem.
Theorem. For each element of a group the character in a product representation is
the product of the corresponding characters in the separate representations.
Proof. Suppose that {ui } and {vj } are two sets of basis functions, that transform
under the operations of a group G according to representations D(λ) and D(µ)
respectively. Denote by u and v the corresponding basis vectors and let X be an
element of the group. Then the functions generated from ui and vj by the action
of X are calculated as follows, using (29.1) and (29.4):
T T D(λ) (X)
ul ,
ui = Xui = D(λ) (X) u = D(λ) (X) ii ui +
i
il
l=i
T T D(µ) (X)
vj = Xvj = D(µ) (X) v = D(µ) (X) jj vj +
j
m=j
jm
vm .
Here [D(X)]ij is just a single element of the matrix D(X) and [D(X)]kk = [DT (X)]kk
is simply a diagonal element from the matrix – the repeated subscript does not
indicate summation. Now, if we take as basis functions for a product representation Dprod (X) the products wk = ui vj (where the nλ nµ various possible pairs of
values i, j are labelled by k), we have also that
wk = Xwk = Xui vj = (Xui )(Xvj )
= D(λ) (X) ii D(µ) (X) jj ui vj + terms not involving the product ui vj .
This is to be compared with
T T Dprod (X)
wn ,
wk = Xwk = Dprod (X) w = Dprod (X) kk wk +
k
n=k
kn
where Dprod (X) is the product representation matrix for element X of the group.
The comparison shows that
prod
D (X) kk = D(λ) (X) ii D(µ) (X) jj .
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