Applications of Symmetry to Integrals
One of the most useful applications of symmetry is to the evaluation of various integrals. It is possible with many integrals that are commonly encountered in chemistry to use symmetry arguments to tell whether an integral must necessarily be zero.
We will consider a general integral of the product of two functions f1 and f2:
I = ∫ f1f2 dτ
For example, if f1 and f2 are both atomic orbitals, then I is their overlap integral. If symmetry requires that this integral is zero, then we can immediately conclude that there is no overlap of these orbitals to give a molecular orbital in the molecule under consideration.
The fundamental point from which this argument is constructed is that the value of any integral is independent of the coordinate system used to evaluate it.(Consider that the area of any shape is independent of the orientation of that shape on a page.)
In other words, the value of an integral such as an overlap integral is independent of the orientation of the molecule (rotating or reflecting an entire molecule does not affect the degree of overlap of two atomic orbitals within the molecule). This implies that I must be invariant under any symmetry operation of the molecule.
The volume element dτ is invariant under any symmetry operation, so it is the integrand, the product f1f2, which must be unchanged by any symmetry operation of the molecular point group under consideration. (If the integrand did change sign under a symmetry operation, then the integral would be the sum of equal and opposite values, and would hence be equal to zero.)
Thus the integral may only be non-zero if the character of the integrand under every symmetry operation is 1, i.e. if the integrand belongs to the totally symmetric species (often A1.)
To decide whether or not the integrand does belong to the totally symmetric symmetry species, we must first assign the symmetry species of the individual functions f1 and f2 (by reference to the character table of the molecular point group). We then consult a table of direct products to see if the direct product of the functions’ symmetry species contains A1. In practice this requires the two functions to have at least one symmetry species in common, as A1 (or the equivalent totally symmetric species) is only found in the direct products of identical symmetry species.
We can thus conclude that only orbitals of the same symmetry can overlap with each other and give rise to a molecular orbital.
Although discussion on this page has centred on overlap integrals, symmetry arguments may be used on any integrand to demonstrate whether or not it must be zero. This approach provides the technique for determination of selection rules in infra-red and Raman spectroscopy (rules that say which transitions are permitted and which are not).
One very important point to note is that symmetry considerations can only tell us which integrals must be zero. However, integrals that symmetry permits to be non-zero can be zero for other reasons. For example, orbitals that may overlap on symmetry grounds will not do so if they are too far apart, or if they differ greatly in energy.