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When we have a molecule say (Co(NH2)6) hypothetically any octahedral molecule it has the point group of Oh and the character table shows the quadratic function such as ($z^2,2z^2-x^2-y^2$) ($xy, xz, xy$). These let you know about the breaking of degeneracy of orbitals. But for some groups like $C_2v$ there are functions like $x^2,y^2$ which don't really correlate to any orbital. What is the significance of such functions?

Moreover to aforementioned $C_2v$ group doesn't have $x^2-y^2$ function so we don't know how the degeneracy breaks. Is there any method to find it using group theory?

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  • $\begingroup$ These are operators that show how the sym species behave under the action of these function. For example x, y, z (linear translations) show how species react to optical (dipole) transitions and squared terms, xy, x^2 etc. for Raman transitions. You should consult a text book to get examples. $\endgroup$
    – porphyrin
    Jul 12, 2023 at 12:44
  • $\begingroup$ @porphyrin I was able to get some info of Raman Transitions and all but the thing that I'm looking for the is the relation of these functions to the corresponding d orbitals, such as I've mentioned $d_{z^2}$ and all. I wanted to know the correlation between d orbitals and the functions of $x^2$ and $y^2$ $\endgroup$ Jul 12, 2023 at 13:29
  • $\begingroup$ Some of these function will correlate to the orbital symmetry, if there is a d orbital in C2v, Td, Oh, etc. it would transform as e.g. z^2, xy, etc., but the functions x^2 etc. do not have to correlate to orbitals but can do to other things that transform as a squared function such as Raman scattering. $\endgroup$
    – porphyrin
    Jul 12, 2023 at 14:17

1 Answer 1

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For the $\mathrm{d}_{x^2 - y^2}$ orbital, note that both $x^2$ and $y^2$ transform as $\mathrm{A_1}$ under $C_\mathrm{2v}$ symmetry: http://symmetry.jacobs-university.de/cgi-bin/group.cgi?group=402&option=4

Any linear combination of $x^2$ and $y^2$, including $x^2 - y^2$, thus also transforms as $\mathrm{A_1}$.

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