The square planar ion has its four chlorines at , , , (distances in units of the…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Linear algebra and vector spaces · Group theory and symmetry

Invariance, Ch Groups: the algebra of symmetry operations

Problem

The square planar ion

\mathrm{PtCl_4^{2-}}

has its four chlorines at

1=(1,0,0)

2=(0,1,0)

3=(-1,0,0)

4=(0,-1,0)

(distances in units of the Pt-Cl bond length). The

C_4

rotation of the ion about the

z

axis is represented, by the standard rotation matrix of Chapter 1, as

\mathsf{R}(90^\circ,z) = \begin{pmatrix}0&-1&0\\1&0&0\\0&0&1\end{pmatrix}.

((a)) State

\det\mathsf{R}

(Chapter 1) and say what this tells you about the operation. ((b)) Apply

\mathsf{R}

to the four ligand positions and write the induced permutation of

\{1,2,3,4\}

in cycle notation. ((c)) Give the parity of this permutation and compare it with

\det\mathsf{R}