A vanishing integral is a continuous sum of a function that evaluates to zero. Group theory allows us to identify a vanishing integral as a result of the symmetry of the system.
Consider a set of basis functions that transforms according to a representation of a point group . The integral of an element of the function is a scalar, which is invariant to any symmetry operation of the coordinate system, i.e.
Since there are number of symmetry operations of , we can write equations of eq82 and sum them to give
Let , where and belong to the sets of linearly independent functions and respectively, and where and are bases for the irreducible representations and respectively of . Substituting and eq77i in eq83,
Rewrite eq20 as and substitute it in eq84 to give
We conclude that:
Rule 1 |
If and transform according to two different non-equivalent irreducible representations and respectively, the integral of their product over all space is necessarily zero |
This implies that basis functions belonging to different non-equivalent irreducible representations are orthogonal. With respect to eq76 and eq77i, if transforms according to the totally symmetric representation of and transforms according to a different irreducible representation of , then must transform according to , which is obviously not the total symmetric representation of . Therefore,
Rule 2 | If a function transforms according to an irreducible representation that is not the totally symmetric representation of a group, its integral over all space is necessarily zero |
Next, we look at the quantum-mechanical integral , where and are wavefunctions and is a quantum-mechanical operator, which is also a function.
Question
Why is an operator a function?
Answer
A function is a mapping of each element of the set in one vector space of functions to one element of the set in another vector space of functions , which may be the same space as . Similarly, an operator maps each element (e.g. a wavefunction) of a vector space of functions to another element in another vector space of functions , which may be the same space as .
Due to rule 1, is necessarily zero if and transform according to two different non-equivalent irreducible representations. Let’s consider the integral , where , and transform according to the irreducible representations , and respectively of a group . Since the non-relativistic Hamiltonian is invariant under any symmetry operation, it transforms according to the totally symmetric irreducible representation of . With reference to eq76 and eq77i, consequently transforms according to . Therefore, if .
Question
Do the integrals and vanish in a molecule?
Answer
is totally symmetric, while and transform according to two different non-equivalent irreducible representations and respectively. Therefore, . Using eq76, transforms according to . Hence, is not necessarily zero.