Inorganic Chemistry: the order of MO diagram

Question

I understand that to create a molecular orbital diagram, the following rules are supposed to be met:

1. number of A.Os = number of M.Os
2. Symmetry of the binding orbitals
3. Similarity in energy

They will eventually create bonding, anti-bonding, and maybe non-bonding orbitals. However, I am having trouble stacking them up in a correct order. For instance, if you refer to the attached diagram below, I don't know how it first starts with 4a1' and goes up to 5a1'. The only clear ones for me are the non-bonding orbitals, which have the same energy as the original group orbitals.

Could you please help me understand this? The coefficients that come in front of letters are also confusing me as well...

Thank you so much!

Answer 1

The numbers that come in front of each molecular orbital irrep are simply referring to the energy order in which the MO's are arranged, with the maximum number (for example, $5a_1'$ for $BF_3$) corresponding to the highest energy MO that has an $a_1'$ symmetry.

For understanding the molecular orbital energy order, you need to explicitly consider the combination of the fluorine ligand group orbitals and the boron AO of the same symmetry. If you do that you'll find $4a_1'$ has perfect bonding interactions between all the ligand p-orbitals involved in $\sigma$-bonding and the boron s orbital (both transform as $A_1'$ in $D_{3h}$). The identity of the p-orbitals would depend on the co-ordinate axis you choose to define as the molecular axis. $3e'$ would arise because of $\sigma$ interactions between the same ligand p-orbitals, this time slightly differently oriented and the boron p-orbitals. The $a_2''$ MO is pi-bonding in nature, so on and so forth.

But without further ado, I'll point you to this excerpt from Purcell and Kotz's Inorganic Chemistry, which is self contained and provides a detailed pictorial description of the various ligand group orbitals very nicely.

Best of luck!