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The bond order of H2O is 2, and that of NH3 is 3, which makes sense, when considering the number of bonds they have.

However, if using basic group theory, you get the MO diagrams as presented in this paper:

http://www.scielo.br/pdf/qn/v35n7/v35n7a32.pdf

Upon examination, we see that in the case of H2O, there are two orbitals that are strongly bonding, one weakly bonding orbital, and one non-bonding orbital. If we then tried to apply the bonding order equation to this MO diagram, would we not end up with a bond order of 3 - despite the a1 orbital being only weakly bonding, its occupation still stabilizes the molecule relative to its constituent atoms. Is it the case that the concept of bonding order breaks down using these kinds of MO diagrams?

A further example is given using the NH3 MO diagram. Here, they calculate the bond order as 3, ignoring the fact that the NH3 a1 orbital is weakly bonding. If we were to in theory calculate the MO diagram for NH3 in a trigonal planar geometry (same as for BH3), we would also get the bond order as 3. Of course, the occupied a1 orbital is weakly bonding in the trigonal pyrimidal NH3 geometry, leading to a preference over the trigonal planar geometry, but why is this not reflected in the bond order?

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It depends on your definition of bond order. If you are using the simple formula

$$\text{Bond order} = \frac{\text{Number of bonding electrons} - \text{Number of antibonding electrons}}{2}$$

then, yes, it won't be able to capture the subtleties that you describe. After all, when you use that, you're categorising electrons into bonding vs antibonding in a black and white manner. So it's really no surprise that it doesn't reflect "weakly bonding" MOs.

You gave the example of water. Well, for that weakly bonding MO, you need to make a choice: is that considered a bonding MO, leading to a bond order of 1.5 per O–H bond in water (doesn't make much sense)? Or is it considered a nonbonding MO, leading to a bond order of 1 (makes more sense but then you have this problem you brought up)? Or is it somewhere in between, leading to a bond order between 1 and 1.5 – and crucially, if this is the route you want to go down, how do you quantify the bonding character of the MO?

I don't have any experience using different definitions of bond order, but if you used one of these, I would not be surprised to see an O–H bond order that deviates ever so slightly from 1.

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