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Chemistry LibreTexts

17: Determining whether an Integral can be Non-zero

As we continue with this course, we will discover that there are many times when we would like to know whether a particular integral is necessarily zero, or whether there is a chance that it may be non-zero. We can often use group theory to differentiate these two cases.

You will have already used symmetry properties of functions to determine whether or not a one-dimensional integral is zero. For example, cos(x) is an ‘even’ function (symmetric with respect to reflection through the origin), and it follows from this that

\[\int^{\infty}_{-\infty} \cos(x) dx = 0\]

In general integral between these limits for any other even function will be also be zero.

In the general case we may have an integral of more than one dimension. The key to determining whether a general integral is necessarily zero lies in the fact that because an integral is just a number, it must be invariant to any symmetry operation. For example, bonding in a diatomic (see next section) depends on the presence of a non-zero overlap between atomic orbitals on adjacent atoms, which may be quantified by an overlap integral. You would not expect the bonding in a molecule to change if you rotated the molecule through some angle \(\theta\), so the integral must be invariant to rotation, and indeed to any other symmetry operation.

In group theoretical terms, for an integral to be non-zero, the integrand must transform as the totally symmetric irreducible representation in the appropriate point group. In practice, the integrand may not transform as a single irreducible representation, but it must include the totally symmetric irreducible representation. These ideas should become more clear in the next section.


It should be noted that even when the irreducible representations spanned by the integrand do include the totally symmetric irreducible representation, it is still possible for the integral to be zero. All group theory allows us to do is identify integrals that are necessarily zero based on the symmetry (or lack thereof) of the integrand.