Molecular Symmetry and Group Theory |

Find the number and symmetries of the Raman and IR active vibrations of the fumarate ion (shown). |

- Determine the point group (The fumarate ion is C
_{2h})and using the basis of three arrows (along*x*,*y*and*z*directions) on each atom, obtain a representation. - Reduce the representation
- Remove the irreducible representations that correspond to translation of the molecule (labelled
*x*,*y*and*z*in the character table) and rotation of the molecule (*R*,_{x}*R*and_{y}*R*in the character table) the remaining irreducible representations correspond to vibrational modes._{z}C _{2h}E C _{2}*i*σ _{h}A _{g}1 1 1 1 *R*_{z}*x*^{2}+*y*^{2},*z*^{2}B _{g}1 -1 1 -1 *R*,_{x}*R*_{y}*xz*,*yz*A _{u}1 1 -1 -1 *z*B _{u}1 -1 -1 1 *x*,*y*

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