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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 ($\mathrm{EBO}$). $$\mathrm{EBO} = \frac{1}{2}\left(\sum_{i=1}^N \eta_{b,i}-\eta_{a,i}\right)$$ ... 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 Ruipérez, 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 ($\mathrm{EBO}$) of the $\ce{MoU}$ dimer (5.5) is higher than that for the tungsten dimer (5.2), known to date as the molecule with the highest $\mathrm{EBO}$.

So somewhat arbitrarily, $\ce{MoU}$ seems to be one of the best candidates for the highest bond order.


References:

$^{\ast1}$. Björn O. Roos, Antonio C. Borin, Laura Gagliardi, "Reaching the Maximum Multiplicity of the Covalent Chemical Bond," Angew. Chem. Int. Ed. 2007, 46(9), 1469–1472 (https://doi.org/10.1002/anie.200603600).

$^{\ast2}$. Fernando Ruipérez, Gabriel Merino, Jesus M. Ugalde, Ivan Infante, "Molecules with High Bond Orders and Ultrashort Bond Lengths: $\ce{CrU, MoU,}$ and $\ce{WU}$," Inorg. Chem. 2013, 52(6), 2838–2843 (https://doi.org/10.1021/ic301657c).

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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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Bond order is the number of electrons shared between two atoms divided by two. There are a few things that limit how high a bond order can possibly go, however.

First, atoms can usually only form bonds until their valence electron shells are filled (any more electron would be unstable). The heaviest elements, those of periods 6 and 7, have at most 32 valence electrons (2 s, 6 p, 10 d, and 14 f), so one could imagine 16 as the highest bond order possible with the elements we've discovered so far.

However, atoms do not evenly split their electrons like this. An atom's electrons are in certain, limited allowed energy states called orbitals, and for a bond to form, (at least) two atomic orbitals from the two atoms have to mix with each other to form two molecular orbitals. Some of these atomic orbitals are already full and don't mix. For example, carbon atoms have 4 valence electrons and almost always form 4 bonds to gain a full octet, but there is no such thing as a carbon-carbon quadruple bond. The reason is because one of the four valence orbitals for a carbon atom, the 2s orbital, is already full, so the 2s orbitals of the two carbon atoms cannot mix with each other to form a bond (or, more accurately, they mix to form bonding and antibonding orbitals which are both filled, contributing a net 0 to the bond order). Only the three half-filled or empty p orbitals can mix, so the highest bond order achieved by carbon is 3. Based on this reasoning, we can imagine the element Gd, which has 8 half-filled orbitals (a 5d orbital and all seven 4f orbitals), would be the absolute limit with a bond order of 8.

However, there are yet more limits on how high bond order can go. First of all, in order for atomic orbitals to mix, they have to overlap in space, but electron density decreases with increasing distance from the center of the atom. So the atoms that have the largest valence shells also tend to be too far away from each other for their orbitals to overlap and actually form bonds, so their multiple bonds tend to be weaker. Also, as elements get heavier, their nuclear charge increases, and as a result some of their electrons are moving at speeds approaching the speed of light. This leads to relativistic effects that change the behavior of the valence electrons from what we would expect; for instance the 7 valence f electrons of Gd actually behave more like core electrons.

On a final note, electrons repel each other, so it is actually destabilizing to concentrate huge numbers of them between two atoms. This can be seen with carbon, for example - a carbon carbon triple bond is stronger than a double bond, and a double bond is stronger than a single bond, but a triple bond is less stable than three single bonds or a double bond and a single bond. This is another reason why multiple bonding is limited, and why the highest bond orders (such as 6) are between atoms with very large nuclei (and thus high nuclear charges), like U and Mo.

As you go down the periodic table and get to heavier and heavier elements, relativistic effects become more important, so we are still not certain how the electrons of the heaviest elements arrange themselves and what their maximum bond orders could be.

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