I am attempting to use the reduction formula to find the irreducible representation of $\ce{XeF4}$ to determine it's IR streching vibrations. I know the point group of $\ce{XeF4}$ is $D_{4h}$ and have the character table but I am having a hard time understanding how to get the characters for the reducible representation. I was taught to do it by considering moved/unmoved vectors emanating from the point of the molecule giving values of 1, 0 and -1 for each vector after the symmetry operations have transformed them. Using this method I to find 2$E_u$ irreps, but I found in a paper (http://sces.phys.utk.edu/~moreo/mm08/penchoff.pdf) that $\ce{XeF4}$ also has other irreps including an Au irrep involved in IR activity. How do I find the irreps that I missed? I noticed the paper uses different reducible representations ($\Gamma_{xyz}$, $\Gamma_{unmoved}$ and $\Gamma_{Tot}$) and I don't understand why.

I am attempting to use the reduction formula to find the irreducible representation of $\ce{XeF4}$ to determine its IR stretching vibrations. I know the point group of $\ce{XeF4}$ is $D_{4h}$ and have the character table but I am having a hard time understanding how to get the characters for the reducible representation. I was taught to do it by considering moved/unmoved vectors emanating from the point of the molecule giving values of 1, 0 and -1 for each vector after the symmetry operations have transformed them. Using this method I to find 2$E_u$ irreps, but I found in a paper (http://sces.phys.utk.edu/~moreo/mm08/penchoff.pdf) that $\ce{XeF4}$ also has other irreps including an Au irrep involved in IR activity. How do I find the irreps that I missed? I noticed the paper uses different reducible representations ($\Gamma_{xyz}$, $\Gamma_{unmoved}$ and $\Gamma_{Tot}$) and I don't understand why.

I am attempting to use the reduction formula to find the irreducible representation of XeF4 to determine it's IR streching vibrations. I know the point group of XeF4 is D4h and have the character table but I am having a hard time understanding how to get the characters for the reducible representation. I was taught to do it by considering moved/unmoved vectors emanating from the point of the molecule giving values of 1, 0 and -1 for each vector after the symmetry operations have transformed them. Using this method I to find 2Eu irreps, but I found in a paper (http://sces.phys.utk.edu/~moreo/mm08/penchoff.pdf) that XeF4 also has other irreps including an Au irrep involved in IR activity. How do I find the irreps that I missed? I noticed the paper uses different reducible representations ($\Gamma_{xyz}$, $\Gamma_{unmoved}$ and $\Gamma_{Tot}$) and I don't understand why.

I am attempting to use the reduction formula to find the irreducible representation of $\ce{XeF4}$ to determine it's IR streching vibrations. I know the point group of $\ce{XeF4}$ is $D_{4h}$ and have the character table but I am having a hard time understanding how to get the characters for the reducible representation. I was taught to do it by considering moved/unmoved vectors emanating from the point of the molecule giving values of 1, 0 and -1 for each vector after the symmetry operations have transformed them. Using this method I to find 2$E_u$ irreps, but I found in a paper (http://sces.phys.utk.edu/~moreo/mm08/penchoff.pdf) that $\ce{XeF4}$ also has other irreps including an Au irrep involved in IR activity. How do I find the irreps that I missed? I noticed the paper uses different reducible representations ($\Gamma_{xyz}$, $\Gamma_{unmoved}$ and $\Gamma_{Tot}$) and I don't understand why.