# Significance of Character tables in d orbital splitting

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?

• 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. Commented Jul 12, 2023 at 12:44
• @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 $y^2$ Commented Jul 12, 2023 at 13:29
• 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. Commented Jul 12, 2023 at 14:17

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 $$\mathrm{A_1}$$.