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I am studying molecular bond theory, and I see that sometimes this happens. Does this have some significance to the molecule?

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It is one of the man-made tools to conceptualize the nature of a bond between two atoms, to know if there is a single / double / triple / quadruple *) bond between them, and to account for both electrons involved in bonding / antibonding molecular orbitals. Because the nature of electrons is neither "true and only" particle, nor wave alone, hence sometimes escaping our day-to-day experience with macroscopic objects.

In addition, rarely you look at the bond order of two atoms in a molecule without keeping in mind the situation with other atoms present in the same molecule, too.

*) Sorry, I forgot at least the three-centre-two electron bonds in diborane...

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Bond order is non-integral when the difference in the number of electrons in molecular orbital and the number of electrons in anti-molecular orbital is odd.

I don't see anything "significant" in the difference being odd.

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Bonding is more complex than a simple 2-electron picture suggests

In the teaching of chemistry and in the early development of the theory of chemical bonding, bonds are (or were) often thought of as consisting of two shared electrons. This is where the idea of bond order comes from: a bond consisting of two electrons is a single bond (as in a hydrogen molecule). Some elements have enough electrons arranged in the right way to give higher bond orders because more electrons are shared (in nitrogen molecules six electrons are shared between the two nitrogens giving a bond order of 3).

This picture, given how simple it is, does a surprisingly good job of explaining the structure of many chemicals and a great deal of organic chemistry. But it is an overly simplistic picture. Sure, ethene is easy to explain: there is bond between the carbons that has a bond order of two. That bond is shorter than the C-C single bond in ethane but longer than the C-C triple bond in ethyne (acetylene for traditionalists). But this simple picture falls apart for other molecules. In benzene, the archetypal example, we could write the structure as cyclohexatriene with alternating double and single bonds around the six membered ring. That certainly uses up all the electron pairs. But this neither explains the reactivity nor the measured structure where all the bonds are the same.

In reality six electrons in benzene are shared equally among all the bonds ("delocalised" is one way to describe this). Each bond is shorter than a single bond but not as short as a double bond. There is no point in trying to explain this using the simple picture where all bonds share two electrons and have an integral bond order. So we say those bonds have a bond order of 1.5 which is compatible with the actual physical structure.

A full explanation of how those bonds works requires some molecular orbital theory and more complex structures where bond orders can be other fractions are not uncommon. But without getting in the details of more sophisticated theories of bonding, the basic explanation of non-integer bond orders is that electrons are sometimes not very monogamous and share themselves between several atoms in which case we get bond orders that are not integers depending on the number shared and the numbers of atoms involved in the sharing.

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Bond order not being a whole number usually happens when the molecule has different resonant structures....basically what one should understand first and foremost is that a chemical bond is not the stick like thingy we draw...the stick/line is just a representation of the bond...hence a non whole number bond order means a non whole number bond order...which you obtain from 1/2(Nb-Na)..where Nb and Na mean the no of electrons in the bonding and antibonding orbitals respectively Im not really too sure it means anything more than that

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