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I was looking at this tutorial about reductions of representations. At first, the basis consists of four orbitals $(s_N, s_1, s_2, s_3)$ and the representations look like this

enter image description here

and we want to reduce the 3D $\Gamma^{(3)}(g)$ representation. Because its matrices aren't block-diagonal, we need to perform some similarity transform to get them into the convenient form. For that, we should choose a new appropriate basis, i.e. the linear combination of our orbitals.

Now the tutorial says that the appropriate new basis looks like this:

enter image description here

And yes, the basis is correct and the new block-diagonal matrices can be further derived using it.

But I have no idea, how did the author come with this specific basis.

So, how can I find the appropriated basis to reduce a reducible representation?

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    $\begingroup$ I suspect from a purely mathematical point of view it’s highly non-trivial (mathoverflow.net/questions/199911). However, I suppose you can find the irreducible representations (using the reduction formula) then use the projection formula to get the form of the MOs. See e.g. slide 13 of goicoechea.chem.ox.ac.uk/resources/s2_presentation.pdf onwards. $\endgroup$ – orthocresol Oct 11 '18 at 23:04
  • $\begingroup$ @orthocresol Thank you! Please, try to write down a more detailed answer, so I could accept it. $\endgroup$ – Eenoku Oct 11 '18 at 23:29

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