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I'm currently trying to do a problem where it asks me to "show that the irreducible representations spanned by the atomic x-, y-, and z- coordinates is independent of the choice of local coordinate system. Use the diagrams below."

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I know the general process is to determine the irreps for the separate coordinates. So, when I attempted to do the first one, by doing the symmetry operations to the x vectors from the D3h character table. However, it did not work as well. So now I don't know where to go with it. Please help and thank you.

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Have a look at the method given in this answer. In your example subject each vector on each atom to the operation in the point group, e.g rotation, reflection etc. Count 0 if the vector is moved else count 1 if it remains unchanged and -1 if it is unchanged but just points in the other direction. Now you have the reducible representation. Use the method suggested above to get the irreps.

If the angles are not 180 then it is necessary to use a rotation matrix and determine the trace (sum of diagonal elements). (This is explained in a good symmetry text book such as R. Carter, 'Molecular Symmetry and Group Theory'. Of course this has all been worked out and so it is necessary only to use a formula for $C_n$ and $S_n$ operations.

These are $\displaystyle \chi(C_n)=1+2\cos\left(\frac{2\pi}{n}\right) $ for $n$ fold rotations and $\displaystyle \chi(S_n)=-1+2\cos\left(\frac{2\pi}{n}\right) $ for rotation reflections, e.g. $\chi(S_3)=-2\;; \chi(C_5)=(1+\sqrt{5})/2$,

Excepting rotation, for each atom each operation contributes an amount as shown below.

$$\begin{array}{ccc} \text{operation} & E & \sigma & i \\ \chi(n) & 3 & 1 & -3 \end{array}$$

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