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I've heard of single bonds, double bonds, triple bonds, and quadruple bonds between two atoms, but in each of those bonds the two atoms contribute the same number of electrons from each of their valence shells.

My question is, why can't the atoms contribute an uneven number of electrons? For example, why can't there be a situation where a carbon atom contributes two valence electrons while a nitrogen atom only contributes one?

Surely the sigma bonds arising from the overlap of two valence orbitals allow one valence orbital to have two valence electrons while the other only has one. (I don't know about pi-bonds, and I don't know if it is possible for a carbon atom and a nitrogen atom to give two and one valence electrons, respectively, but maybe there is another pair of atoms that would allow this).

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    $\begingroup$ Fractional bond order isn't sth unusual, also there are 2-center-3-electron bonds and other types $\endgroup$ – Mithoron Aug 25 '15 at 11:45
  • $\begingroup$ You can make a "bond" with one electron. It probably is somewhat more stable than the two single entities. Trouble is, what you then have is still a radical (as one of the two partners was before). $\endgroup$ – Karl Aug 25 '15 at 12:14
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One example to consider is benzene, where each carbon contributes 1 electron to a pi-bonding system of 6 electrons in 3 orbitals. The C-C bond lengths in benzene are intermediate between standard single and double bond lengths. Also take a look at the various boranes such as diborane (B2H6) which contain three-centered (or higher?) bonds involving boron and hydrogen.

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the two atoms contribute the same number of electrons from each of their valence shells

While this may not be exactly what you are looking for, bonds in metal complexes such as $\ce{[FeF6]^{3-}}$ are real, although only one of the atoms (an anion, actually) contributed electrons. For other complexes, such as ferrocene or those involving allyl ligands, the accounting can become complicated bordering on hair-splitting.

Other than that, boranes (@iad22agp already mentioned those) and compounds that are often called hypervalent such as $\ce{SF6}$, $\ce{PCl5}$ are examples of three-center bonding.

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