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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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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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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 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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