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We all learned about single vs double vs triple bonds in high school chemistry and biology. We learned that unsaturated fats have double bonds whereas saturated fats have only single bonds, which makes unsaturated fats liquid (vs. solid) and easier to break/process by the body.

I was impressed when this Stack Exchange first went live though, to learn about the existence of about a dozen different definitions of bond order.

I wonder if/how these have led to useful insights such as the textbook example I gave in the first paragraph of this question?

I can imagine studying some large un-studied molecule in which the bond orders are not known. Once I have an approximation of the wavefunction, I can calculate one of the many types of bond orders mentioned in the aforementioned question (such as the Wiberg bond index explained here, which only needs the orbital coefficients from the LCAO model of a wavefunction); or if I have calculated the electron density, I can use one of the many techniques of "conceptual DFT" from the QTAIM framework or extensions, such as in this answer. But the existence of so many of these bond-order definitions, suggests that the very simple-to-calculate Wiberg bond index is not accurate enough for some people's liking (hence the need to formulate more complicated bond-order definitions).

  • If we wish to know the length of a bond and we are using some advanced wavefunction-based or DFT software, I suppose that we would just do geometry optimization in our quantum chemistry software anyway, which would give us an even better description of the bond lengths than some "heuristic" method such as assigning a bond-order.
  • If we wish to know the strength of a bond (remembering Tyberius's famous question from 3 years ago) we could just calculate the force constant for a particular bond, as MSwart said in this answer. This may add a bit to the cost of the quantum chemistry calculation, compared to just using a simple formula like the Wiberg bond-order formula, but probably not more than all of the other painful effort that goes into doing a quantum chemistry calculation (e.g. preparing a ZMAT/XYZ file, doing geometry optimization, converging an SCF, converging with respect to basis set and correlation-treatment, accounting for relativity and Born-Oppenheimer breakdown, etc.), and it seems it would be much more accurate (clearly the Wiberg formula is too simple because so many more sophisticated bond-order methods have been devised afterwards).
  • I appreciate that we might want to draw the chemical structure with all its single and double bonds, but the desire to do this hinges on the bond-order actually being truly useful, which wouldn't seem to always be the case with, for example, the Wiberg bond-order which seems to have needed added levels of seemingly un-ending further sophistication in dozens of subsequent papers.

MSwarts's answer to the previous question ends with:

"Of course you could also compute the intrinsic force constant for that particular bond, with all kind of ifs and buts. But that would be opening up a whole new discussion."

Perhaps we can now have that discussion.

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Why Bond Order?

Bond order isn't terribly useful to a computationalist directly; however, it can be invaluable for translating Quantum Mechanical results into a framework thats readily understood by experimentalists.

Experimental chemists tend to make predictions about material properties, reaction feasibility, molecular structure, etc. based on a combination of empirical heuristics (e.g. this synthesis route worked for a similar looking molecule, this type of reaction works better with less bulky ligands involved) and simplified, qualitative MO theory (e.g. Ligand Field theory). These simplified MO models give insight into the strength of intramolecular interactions and how these should change when different substituents are involved. They also allow for discussion of the symmetry of a given interaction and whether it favors/disfavors some desired outcome like ligand dissociation. These insights have led to general principles (Pi backbonding,Charge transfer complexes, Green-Davies-Mingos, 18-electron rule) that help to rationalize chemical behavior and predict new types of molecules and reactions.

These simplified MO models basically correspond to a qualitatively produced minimal basis SCF calculation, so while they can be useful in broad strokes, there are bound to be some inaccuracies when they are applied in specific cases. Naturally, we can correct some of these misconceptions with higher level quantum calculations, but this isn't useful to the experimentalist unless we can also translate the results into the framework/heuristic that they are trying to use. These various bond order methods are all attempts to translate to QM results to this more understandable form. By being able to tell the chemist that say, Pi backdonation isn't occurring in some compound, or that using a more donating ligand increases the bond order, they can adjust their heuristics with that new information and come up with a different synthetic pathway or molecular structure to meet their goal.

Further reading:

  1. Adam J. Bridgeman, Germán Cavigliasso, Luke R. Ireland, and Joanne Rothery J. Chem. Soc., Dalton Trans., 2001, 2095–2108

Why Not Force Constants?

There is nothing strictly wrong with using force constants for this and there are some methods that use force constants to get a bond order. I think these methods have there place, but I'll just briefly enumerate some of the negatives of this approach.

  • Cost: For larger molecules/higher levels of theory, the cost of an optimization/frequency can greatly exceed that of the SCF. If you aren't interested in the frequencies for any other reason, it probably isn't worth incurring this extra cost.

  • Detail: The vibrations don't give as clear a picture of what is occurring electronically (e.g. symmetry, covalent v.s. ionic interaction), so they don't fit as well into the way that chemists tend to think about bonding (there are of course cases, like infrared/Raman spectroscopy, where these vibrational details are more important).

  • Ambiguity: One of the difficult things with vibrational modes is they typically aren't localized to a particular bond, but are instead spread throughout the molecule. This can make it difficult to say which force constant, or combination of forces constants, describe the bond you are interested in. There are procedures to localize vibrational modes, but this just adds another layer of complexity, since these localized modes are coupled.

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