Basics

Warning

Unless stated otherwise, \(\mathcal{G}\) denotes a unitary-represented symmetry group, which can be a unitary symmetry group or a magnetic symmetry group in which antiunitary operators are represented unitarily. Projection operators for magnetic-represented groups are not yet supported.

Let \(V\) be a linear space and \(\mathbfit{w} \in V\) be an element to be projected by QSym² onto a particular irreducible representation \(\Gamma\) of a unitary group \(\mathcal{G}\). This amounts to constructing the orbit

\[ \mathcal{G} \cdot \mathbfit{w} = \{ \hat{g}_i \mathbfit{w} \ :\ g_i \in \mathcal{G} \} \]

and then computing the sum

\[ \hat{\mathscr{P}}^{(\Gamma)} \mathbfit{w} = \frac{d_{\Gamma}}{|\mathcal{G}|} \sum_{i = 1}^{|\mathcal{G}|} \chi^{(\Gamma)}(g_i)^* (\hat{g}_i \mathbfit{w}), \]

where \(d_{\Gamma}\) is the dimension of the irreducible representation \(\Gamma\) and \(\chi^{(\Gamma)}\) its character function.

To perform symmetry projection in QSym², the atomic-orbital basis angular order information for the underlying electronic-structure calculation is required.