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When we construct symmetry adapted linear combinations of orbitals (aka SOs) to build an MO diagram, we choose the z axis as the axis of highest symmetry and all other axis to make symmetry operations of a group easier to perform.

If we choose z-axis to point in some other direction, we will get a representation that is clearly not irreducible, but yet we would be unable to reduce it with the chosen basis because all orbitals mix (e.g. considering the case of 3d orbitals of a TM octahedral complex and having $d_{z^2}$ at a slant to the z-axis). I wonder if someone could explicitly explain why in this case the approach fails and highlight its limitations in general.

Thank you!

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    $\begingroup$ If you choose another axis, say the 3-fold instead of the z or 4-fold axis in an octahedral complex you should still be able to form a linear combination but of mixtures of $e_g$ symmetry orbitals and mixtures of $t_{2g}$ symmetry orbitals (but not of both as these orbitals are not degenerate) and these mixtures will be different (but equivalent) to those obtained from choosing the 4-fold axis. (The maths may be far harder in one choice than the other.) $\endgroup$ – porphyrin Feb 3 '17 at 10:50
  • $\begingroup$ @porphyrin Ok, thank you! I see: in this case to form an IR in this basis, we need non-integer coefficients, unlike all the cases I've dealt with before. Is this correct? $\endgroup$ – GingerBadger Feb 4 '17 at 14:47
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    $\begingroup$ yes there is nothing special about integer coefficients $\endgroup$ – porphyrin Feb 5 '17 at 14:25
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In general, when dealing with characters tables and with group-operations, it is crucial to choose the right orientation of axes in order to get sensible results. I say in general because it can also occur that, for highly symmetric molecules the "wrong" choice (or better say unconventional) can give the right result. Almost the 90% of characters tables follow the "Mulliken" conventions stated in Mulliken's paper "Report on Notation for the Spectra of Polyatomic Molecules" (The Journal of Chemical Physics 23, 1997 (1955)). Whenever you orient your axes not in agreement with the characters table you are referring for performing operations, there is a high risk to fail and face strange results as you did (all orbitals mixing etc..). I hope my answer is general enough and helpful.

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