The Great Orthogonality Theorem is one of the core tools of group theory. It provides the mathematical foundation for character tables, irreducible representations, and the procedures used to analyze molecular vibrations, orbitals, and spectroscopy.
At its core, the theorem states that the matrices belonging to different irreducible representations are orthogonal to one another in a precise mathematical sense.
Statement of the Great Orthogonality Theorem
For a finite group with order \(h\), the matrix elements of irreducible representations satisfy
\[ \sum_{R} D^{(\alpha)}_{ij}(R)\, D^{(\beta)}_{kl}(R) = \frac{h}{l_\alpha}\, \delta_{\alpha\beta}\, \delta_{ik}\, \delta_{jl}, \]
where:
- \(R\) runs over all symmetry operations in the group,
- \(D^{(\alpha)}(R)\) is the matrix for operation \(R\) in irreducible representation \( \alpha \),
- \(l_\alpha\) is the dimension of the irreducible representation,
- and the \(\delta\) symbols enforce orthogonality.
While this expression may look intimidating, its physical meaning is simple: different irreducible representations are mutually independent.
Consequence 1: Orthogonality of characters
A particularly useful consequence of the Great Orthogonality Theorem is that the characters of irreducible representations are orthogonal.
For two irreducible representations \( \alpha \) and \( \beta \),
\[ \sum_{\text{classes}} n_R\,\chi^{(\alpha)}(R)\,\chi^{(\beta)}(R) = h\,\delta_{\alpha\beta}, \]
where \(n_R\) is the number of operations in each class. This orthogonality is what allows character tables to be constructed and verified.
Example: \(C_{2v}\)
The point group \(C_{2v}\) has four operations: \(E, C_2, \sigma_v, \sigma_v'\). All irreducible representations are one-dimensional.
Consider two different irreducible representations, for example \(A_1\) and \(B_1\). Their characters are:
\[ \chi^{(A_1)} = (1,\;1,\;1,\;1), \qquad \chi^{(B_1)} = (1,\;-1,\;1,\;-1). \]
Applying the orthogonality condition,
\[ 1(1)(1) + 1(1)(-1) + 1(1)(1) + 1(1)(-1) = 0, \]
confirming that the two irreducible representations are orthogonal.
In contrast, taking the dot product of \(A_1\) with itself gives
\[ 1^2 + 1^2 + 1^2 + 1^2 = 4 = h, \]
which matches the prediction of the theorem.
Example: \(C_{3v}\)
The group \(C_{3v}\) has order \(h = 6\) and three irreducible representations: two one-dimensional representations and one two-dimensional representation \(E\).
Using the characters of the two-dimensional irreducible representation \(E\),
\[ \chi^{(E)} = (2,\;-1,\;0) \quad \text{for classes } (E,\;2C_3,\;3\sigma_v), \]
the orthogonality condition with itself gives
\[ 1(2)^2 + 2(-1)^2 + 3(0)^2 = 4 + 2 + 0 = 6 = h. \]
Orthogonality between \(E\) and a one-dimensional representation (such as \(A_1\)) similarly yields zero.
Why the theorem matters
The Great Orthogonality Theorem guarantees that irreducible representations form a complete, non-overlapping set. This is why any reducible representation can be decomposed uniquely into a sum of irreducible representations.
Big idea: the Great Orthogonality Theorem is the mathematical reason that symmetry analysis works. It ensures that irreducible representations are independent, complete, and uniquely identifiable.