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I have some major confusion on irreducible representations. An example of a problem I have is

Based on the character table for the point group of each of the follwing molecules, which irreducible representation would be used to classify the Cu $\mathrm{d}_{xz}$ orbital in $\ce{[CuCl5]^3-}$?

Since this complex probably has a trigonal bipyramidal geometry, I'm starting with the $D_\mathrm{3h}$ character table:

$$\begin{array}{c|cccccc|cc} \hline D_\mathrm{3h} & E & 2C_3 & 3C_2 & \sigma_\mathrm{h} & 2S_3 & 3\sigma_\mathrm{v} & & \\ \hline \mathrm{A_1'} & 1 & 1 & 1 & 1 & 1 & 1 & & x^2+y^2,z^2 \\ \mathrm{A_2'} & 1 & 1 & -1 & 1 & 1 & -1 & R_z & \\ \mathrm{E'} & 2 & -1 & 0 & 2 & -1 & 0 & (x,y) & (x^2-y^2,xy) \\ \mathrm{A_1''} & 1 & 1 & 1 & -1 & -1 & -1 & & \\ \mathrm{A_2''} & 1 & 1 & -1 & -1 & -1 & 1 & z & \\ \mathrm{E''} & 2 & -1 & 0 & -2 & 1 & 0 & (R_x,R_y) & (xz,yz) \\ \hline \end{array}$$

But i don't know what exactly I need to do.

Is the irreducible representation just the row label, i.e. the Mulliken term? Or is it one of the operations?

Looking at the column with quadratic functions, $xz$ is paired with $\mathrm{E''}$, so that's my current best guess.

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marked as duplicate by Philipp, Abel Friedman, LDC3, Martin - マーチン, Freddy Oct 17 '14 at 4:56

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