# Why is expected bond energy the *geometric* mean of two pure bonds?

Suppose we are calculating the electronegativity of a bond A—B. In my textbook (Chemical Principles) calculation of electronegativity, they calculate electronegativity in terms of a variable difference in bond energies $$\Delta$$, which they define as $$\ce{\Delta = (A-B)_{actual} - (A-B)_{expected} }$$

and that discrepancy is used to calculate electronegativity. Fair. But I'm confused by their formula for the expected energy of $$\ce{A-B}$$: $$\ce{Expected A-B bond energy = \sqrt{\ce{(A-A bond energy) \times (B-B bond energy)}} }$$

which is the geometric mean of the two. But why the geometric mean, and not, say, an arithmetic or harmonic mean? Why even use a mean at all? What's the intuitive rationale for this?

• Relevant but not really an answer: en.wikipedia.org/wiki/Combining_rules Nov 24, 2023 at 5:45
• Nov 24, 2023 at 9:37
• Yea, like these articles are relevant because of the similar kind of formulae used? As for the answer, I think it is because bond energy is dependent on the orbital overlap. As for the GM, it's used to get a balanced contribution from each atom as the bond strength is proportional to the product of the combining factors. GM is taken when things are multiplied and so yea, we get it Nov 28, 2023 at 1:21
• Am I right everyone? Nov 28, 2023 at 1:23
• If pushed and threatened with violence I would probably wave my hands very vigorously about second order perturbation theory grossly approximated, possibly via the closure approximation. But really I don't think there is a right answer in the sense that the underlying exact theory behind chemistry is very, very difficult and often insoluble, so as such quite often we end up using things which just empirically work - and this is one of them. Nov 28, 2023 at 9:16

For example, let $$X=3$$ and $$Y=7$$, the means are $$HM = \frac{1}{2}(\frac{1}{3} + \frac{1}{7}) = 4.2$$ $$GM = (3\times 7)^{\frac{1}{2}} \approx 4.58$$ $$AM = \frac{3+7}{2} =5$$ There are other types of mean that can also be use (i.e., root mean square).
Now, I believe that the electronegativity that you are referring to are Pauling's electronegativity. I think the main reason that Pauling pick $$GM$$ over the other is due to the fact that energy may come in many order of magnitude. Let's say $$X=1$$ and $$Y=10000$$ $$HM = \frac{1}{2}(\frac{1}{1} + \frac{1}{10000}) = 20000/10001 \approx 1.9998$$ $$GM = (1\times 10000)^{\frac{1}{2}} =100$$ $$AM = \frac{1+10000}{2} =5000.5$$
If you see this, which of the 3 means ($$HM$$, $$GM$$, $$AM$$) is more in the middle. It would be more intuitive to consider $$GM$$ to be more in the middle in this case.