The reduction formula, the vital, simple, but painfully long-winded calculation that is taught in all physical chemistry (and related areas) group theory courses, and provides insight into IR spectra and much more besides. Here it is shown below:
n(Γ) = 1/h Σ χ(r) ∙ χ(i) ∙ N
where:
n(Γ) - is the number of times a given reducible representation, Γ, occurs in the sum.
h - is the order of the group (i.e. the number of symmetry operations in the group, not just the number of classes)
χ(r) - is the character in the reducible representation
χ(i) - is the character in the irreducible representation
N - is the number of symmetry operations in a given class
If youve been through such a course you will appreciate that getting an answer out of this can take a little more time than youd like, and if you are going through such a course youll often find practice problems without answers you can check against your own.
This app solves these issues. Simply select the point group, type in the reducible representation you have worked out, and the app will reduce it for you.
And all for free, how lovely.