Knowledge Bank

The molecular energy expression involving cartesian displacement coordinates $si\alpha (i=1,\dots ,n;\alpha =x,y,z$) has two structures, first recognized by Wigner [G$o¯$ttinger Nachrichten (1930), p. 133]. One is a permutation structure: group operators R move equilibrium position $ri0$ to $ri0=Rri0=$, which can be viewed as a permutation of the labels i.j of equivalent atoms. The second is a vector structure: $Rs\alpha =\Sigma R\alpha \beta S;\beta$ where $R\alpha \beta =D(v)(R)\alpha \beta$ is a rotation matrix. The representation of operators R whose basis is the displacement coordinates (suitably ordered) is the direct product $\Delta \times D(v)$ of a permutation matrix $\Delta$ and the vector representation $D(v)$. If the molecule possesses non-equivalent atoms, the permutation matrix is partially reduced. Similarity transformation of $\Delta \times D(v)$ by $u\times v$, where u fully reduces $\Delta$ and v (possibly the identity) fully reduces $D(v)$, produces an at least partially reduced representation, the diagonal blocks of which are direct products of two irreducible representations of the group. Final transformation by a Wigner matrix W completes the reduction of the representation. The normal mode coordinates are given by the composite transformation $(u\times v)W$.