Past exam of the chemistry A course of the University of Cambridge / 2024 / Paper 1 / A4 / b

A symmetry-closed orbital basis, often called a complete basis in this context, is a linearly independent set whose span is carried into itself by every group operation. No transformed basis function requires a function outside that span. This is completeness for the chosen group representation, not a claim that a small atomic-orbital set spans the full electronic Hilbert space.

A representation matrix $D(g)$ specifies the linear transformation produced by a molecular symmetry operation $g$. With the column convention,Its character of a representation is the matrix trace,The character is unchanged by a simultaneous change of basis and is constant on a conjugacy class, which is why character tables have one column per class.

Use the ordered basis $(p_x,p_y,p_z)$ at the origin. Choose $z$ as the principal improper-rotation axis, $x,y$ along the two $C_2^{\prime }$ axes, and the diagonal mirror planes as $x=\pm y$. Use active operations with $\hat{U}(g)f(\mathbf{r})=f(g^{-1}\mathbf{r})$, so the matrix columns give the transformed orbitals. For these real p orbitals the matrices are the ordinary vector transformation matrices.

One representative from each D2d molecular point group class isFor example $S_4^{+}$ sends $p_x\mapsto p_y$, $p_y\mapsto -p_x$ and $p_z\mapsto -p_z$. Choosing the opposite sense of $S_4$ transposes its xy block but leaves every character and the final decomposition unchanged.

Taking the matrix trace of those matrices givesThe one-dimensional $p_z$ subspace has characters $(1,-1,1,-1,1)$, identifying $B_2$. The pair $(p_x,p_y)$ has characters $(2,0,-2,0,0)$, identifying $E$. ThereforeThe dimensions add to three, as required. The $p_x,p_y$ pair must be retained together: a ninety-degree improper rotation mixes them, so either one alone is not a complete symmetry basis.^[ref]