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In the book Principles of Structure and Reactivity by Huheey, Chapter 14, the author has taken linear combination of ligand orbitals to form ligand group orbitals and then further combination of LGOs(Ligand Group Orbital) with corresponding orbital of metal which has the same symmetry as that of the LGO.

I do not understand how they calculated the wave function for the Ligand Group Orbitals. An example of wave function for the ligand group orbital with the same symmetry as the a1g irreducible representation

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    $\begingroup$ For this you need Group Theory to understand what exactly is going on. The symmetries of your molecule, e.g. mirror plane or rotational axis etc., give certain restrictions on how the molecular orbitals of the molecule can look, i.e. they will have to reflect the symmetries of the molecule in certain ways (the details of which you can be determined using methods from Group Theory). $\endgroup$ – Philipp Mar 9 at 19:13
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    $\begingroup$ There is something called the projection operator / formula in group theory which helps do this in a rigorous manner. Sometimes it is possible to figure them out "by inspection", but it can be tricky (or even impossible) with highly symmetric groups or degenerate irreps. $\endgroup$ – orthocresol Mar 9 at 19:44
  • $\begingroup$ I have knowledge of group theory and SALCs, but nevertheless I fail to understand how to apply that here. To what function are we applying the projection operator? Like normally, we would apply it to a particular atom, bond, or some unit vector. $\endgroup$ – Raghav Arora Mar 9 at 20:05

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