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In science we learn about single, double, and sometimes triple bonds. From a quick search I have found up to sextuple bonds.

Is there a maximum bond order? If yes/no, what causes this?

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In the paper that you've mentioned$^{\ast1}$, there is something described, which is called effective bond order (EBO). $$\mathrm{EBO} = \frac{1}{2}\left(\sum_{i=1}^N \eta_{b,i}-\eta_{a,i}\right)$$ ... with $\eta_b = 2 − x$ as the occupation number of an occupied orbital and $\eta_a = x$ as the occupation number of the corresponding antibonding orbital. (In multireference quantum chemistry methods, the resulting averaged orbitals can have occupation values of $0 \leq x \leq 2$)

In 2013 Ruiperez et al. calculated the effective bond order for several homo- and heteroatomic dimers.$^{\ast2}$ Based on their very high quality calculations, the

[...] results show that the effective bond order (EBO) of the MoU dimer (5.5) is higher than that for the tungsten dimer (5.2), known to date as the molecule with the highest EBO.

So somewhat arbitrarily, MoU seems to be one of the best candidates for the highest bond order.


$^{\ast1}$ B. O. Roos, A. C. Borin, L. Gagliardi, Angew. Chem. Int. Ed. 2007, 46, 1469–1472
$^{\ast2}$ F. Ruiperez, G. Merino, J. M. Ugalde, I. Infante, Inorg. Chem. 2013, 52, 2838–2843

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It seems that I have answered my own question:

The maximum bond order achieved between two atoms in the periodic table is thus six [sextuple] and is represented by the Mo and W diatoms.

From Reaching the Maximum Multiplicity of the Covalent Chemical Bond, by Bjrn O. Roos, Antonio C. Borin, and Laura Gagliardi

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