I was probing for first order Jahn-Teller instability in the coordination complex CrCl4 (2 electrons in E set). Taking the direct product yields the states A1 + A2 + E, so I took the symmetric and antisymmetric direct products too to see which states represented a singlet and triplet configurations.

To my surprise, the symmetric direct product yields 2A1 + E while the antisymmetric product yields A2 -A1. I believe that I probably made a mistake somewhere (but can’t find it), so did I make a mistake somewhere in my working, or is there an actual meaning behind a negative irreducible representation?


  • $\begingroup$ have you tried to use a calculator for this? A negative irrep shouldn't be possible. Using a calculator show whether you made a mistake in determining irreps or counting symmetry elements. $\endgroup$ – Tyberius Jul 2 '19 at 9:32
  • $\begingroup$ I’ve tried it for point group Td, and it really does result in -A1. $\endgroup$ – ANZGC FlyingFalcon Jul 3 '19 at 5:17
  • $\begingroup$ than that suggests to me that you have something wrong with how you counted symm elements. As far as I'm aware, for anything in chemistry having negative irreps is unfeasible. $\endgroup$ – Tyberius Jul 3 '19 at 18:47
  • $\begingroup$ Could you try and evaluate the direct product when you have the time? I feel like I probably got too rusty and made a mistake somewhere, but can’t find it. $\endgroup$ – ANZGC FlyingFalcon Jul 5 '19 at 16:19
  • $\begingroup$ Based on this direct product table the antisymmetric part should be A2. $\endgroup$ – Tyberius Jul 5 '19 at 16:27

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