Question

1 (b) Consider the character table for the \( C_{2 v} \) point group : \begin{tabular}{llrrrcl} \hline \begin{tabular}{l} \( C_{2 v} \) \\ \( (2 m m) \) \end{tabular} & \( E \) & \( C_{2} \) & \( \sigma_{v}(x z) \) & \( \sigma_{\mathrm{v}}^{\prime}(y z) \) & \begin{tabular}{l} linear, \\ rotations \end{tabular} & quadratic \\ \hline \( \mathrm{A}_{1} \) & 1 & 1 & 1 & 1 & \( z \) & \( x^{2}, y^{2}, z^{2} \) \\ \( \mathrm{~A}_{2} \) & 1 & 1 & -1 & -1 & \( R_{z} \) & \( x y \) \\ \( \mathrm{~B}_{1} \) & 1 & -1 & 1 & -1 & \( x, R_{y} \) & \( x z \) \\ \( \mathrm{~B}_{2} \) & 1 & -1 & -1 & 1 & \( y, R_{x} \) & \( y z \) \\ \hline \end{tabular} (i) Find the representation of the \( C_{2 v} \) point group to which a \( p_{z} \) orbital belongs. Give reasons for your answer. A pz orbital (ii) Find the representation of the \( C_{2 v} \) point group to which a \( 3 d_{x y} \) orbital belongs. Give reasons for your answer. (3)

          1 (b) Consider the character table for the \( C_{2 v} \) point group :
\begin{tabular}{llrrrcl}
\hline \begin{tabular}{l}
\( C_{2 v} \) \\
\( (2 m m) \)
\end{tabular} & \( E \) & \( C_{2} \) & \( \sigma_{v}(x z) \) & \( \sigma_{\mathrm{v}}^{\prime}(y z) \) & \begin{tabular}{l} 
linear, \\
rotations
\end{tabular} & quadratic \\
\hline \( \mathrm{A}_{1} \) & 1 & 1 & 1 & 1 & \( z \) & \( x^{2}, y^{2}, z^{2} \) \\
\( \mathrm{~A}_{2} \) & 1 & 1 & -1 & -1 & \( R_{z} \) & \( x y \) \\
\( \mathrm{~B}_{1} \) & 1 & -1 & 1 & -1 & \( x, R_{y} \) & \( x z \) \\
\( \mathrm{~B}_{2} \) & 1 & -1 & -1 & 1 & \( y, R_{x} \) & \( y z \) \\
\hline
\end{tabular}
(i) Find the representation of the \( C_{2 v} \) point group to which a \( p_{z} \) orbital belongs. Give reasons for your answer.

A pz orbital
(ii) Find the representation of the \( C_{2 v} \) point group to which a \( 3 d_{x y} \) orbital belongs. Give reasons for your answer.
(3)

1 (b) Consider the character table for the C2 v point group :

C2 v

(2 m m)
E C2 σv(x z) σv^'(y z)
linear,

rotations
quadratic

A1 1 1 1 1 z x^2, y^2, z^2

A2 1 1 -1 -1 Rz x y

B1 1 -1 1 -1 x, Ry x z

B2 1 -1 -1 1 y, Rx y z

(i) Find the representation of the C2 v point group to which a pz orbital belongs. Give reasons for your answer.

A pz orbital
(ii) Find the representation of the C2 v point group to which a 3 dx y orbital belongs. Give reasons for your answer.
(3)

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