The three porbitals collectively are assigned to a T 1 symmetry species. (This
conclusion could have been reached by comparison ofto the irreducible
representations.) Since the character of the T 1 symmetry species is 3, this
indicates that the (hydrogen-like) porbitals are triply degenerate when a Td
symmetry is imposed on them.13.8 Using Symmetry in Integrals
The assignment of symmetry species to wavefunctions has some useful conse-
quences for evaluating integrals involving wavefunctions. Consider the integral
having the form
* 1 2 d (13.7)
where 1 and 2 represent the symmetry species of each wavefunction of a
system that has a certain symmetry. This integral, if it is nonzero, is simply
some numerical value. That value does not change if one operates on it with
any symmetry operation, so one can say that the character of the symmetry
operation on a number is 1. (Consider that v(3) 1 3 3.) Therefore, all
numerical values can be assigned to the totally symmetric irreducible repre-
sentation of any point group. Consider the opposite argument. If that integral
is to have a nonzero numerical value, then the irreducible representation of
the combination * 1 2 has to have totally symmetric symmetry. This im-
plies that
(1)* 2 A 1 (13.8)where A 1 in this point group happens to be the totally symmetric irreducible
representation, and the symbol is used to imply the proper multiplication
of appropriate characters of each representation. (The complex conjugate in
equation 13.8 is rarely invoked, because most characters are real numbers.) If
the product of the two irreducible representations is anything other than A 1
(or whatever the totally symmetric representation is),then the integral must be
exactly zero.This is a powerful tool to determine whether an integral must be
zero. (In fact, most combinations of wavefunctions of arbitrary symmetry
species are exactly zero from symmetry considerations. This idea drastically
simplifies the mathematical considerations of molecular wavefunctions if the
wavefunctions have symmetry elements.)
Although equation 13.8 is somewhat general, it does not cover all cases. For
example, in symmetry species that have Eor Tlabels, multiplication of the ir-
reducible representations yields a reducible representation that must be re-
duced using the great orthogonality theorem. In such cases, the integral is
identically zero unlessthe reducible representation can be broken down into ir-
reducible representations, one of which must be A 1 (or whatever the totally
symmetric representation is). Such a reducible representation is said to “con-
tain”A 1. Mathematically, this is written as
(1)* 2 A 1 (13.9)where the symbolmeans “contains.” This is the most applicable statement
of the requirement that an integral as in equation 13.7 might be nonzero. If
the product of the two irreducible representations does not contain A 1 , then
the integral must be zero.
13.8 Using Symmetry in Integrals 441