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While studying about bonding there was one statement that "pi bonds can only be formed only with sigma bonds" as we know that in double bond there is 1 sigma bond and 1 pi bond but then one question arises: Why is a pi bond formed only when a sigma bond is formed? Is it possible to form a pi bond without any formation of a sigma bond?

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A $\pi$ bond has a plane of symmetry along the bond axis. It cannot be formed by s-orbitals; it needs at least p-orbitals to be created. $90\,\%$ of all bonds described some time or another are somehow involving carbon, nitrogen or oxygen. (In fact, I probably underestimated). But these elements can only use p-orbitals to create $\pi$ bonds. To do that, one needs a p-orbital that is ortohogonal to the bond axis. So you run into the problem that you have an orbital pointing in one direction, but want to bond into another direction — hardly optimal, especially since there likely is already another orbital pointing in the direction you need to give a $\sigma$ bond.

Transition metals can use d-orbitals for $\pi$ bonding. They can actually point towards the atom they want to bond with so there is a greater chance of using them due to higher overlap. However, there will usually also be a different orbital pointing directly in the bonding direction which again will bond earlier and would give a $\sigma$ bond.

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$\sigma$-bonding MOs tend to have lower energy than $\pi$-bonding MOs so they will be formed first. One explanation is that the $\sigma$-bonding MOs have a lot of $s$-AO character and $s$-AOs have lower energy than $p$-AOs.

I don't know of any double bond that is purely $\pi$-bonding. Basic MO theory suggest that $\ce{C2}$ should have a double bond made of two $\pi$-bonding MOs because the $\sigma$-anti-bonding MO is lower in the energy than the $\pi$-bonding MOs. However, experiment tells us that $\ce{C2}$ has two unpaired electrons so this simple picture cannot be right. This is explained in greater detail here and here.

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    $\begingroup$ In transition metal complexes with 2 metal centers that are bound to each other you can have the situation that the $\pi$-MOs (and sometimes also the $\delta$-MOs) lie energetically below the bonding $\sigma$-MO (all formed from d-orbitals). $\endgroup$ – Philipp Jun 7 '15 at 11:03
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    $\begingroup$ I have a problem with your statement "One explanation is that the $\sigma$-bonding MOs have a lot of $s$-AO character and $s$-AOs have lower energy than $p$-AOs." While this is true for some cases it is not the main reason for $\sigma$ orbitals being lower in energy than their $\pi$ counterparts (sometimes, like in $\ce{C2}$, it is even the other way round: sp-mixing raises the energy of the higher lying $\sigma$ orbitals so that they lie above the $\pi$ orbitals). The more important reason is the better overlap in $\sigma$-type interactions compared to $\pi$-type interactions. $\endgroup$ – Philipp Jun 7 '15 at 11:14
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