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This video from MIT OpenCourseWare says that pi bonds and anti-bonds are for 2px and 2py, i.e., $\pi_{2px}$, $\pi_{2py}$ and $\pi_{2px}^*$, $\pi_{2py}^*$ and sigma bonds and anti-bonds are for 2pz i.e., $\sigma_{2pz}$ and $\sigma_{2pz}^*$.

But in my reference books it says pi bonds and anti-bonds are formed for 2py and 2pz i.e., $\pi_{2py}$, $\pi_{2pz}$ and $\pi_{2py}^*$, $\pi_{2pz}^*$ and sigma bonds and anti-bonds are for 2pz i.e., $\sigma_{2px}$ and $\sigma_{2px}^*$.

Which is actually the right one? Please explain.

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    $\begingroup$ It depends on the choice of the bond axis: I guess it was chosen to be $z$-axis in the MIT lecture, but $x$ in your book. Hence the difference. $\endgroup$ – Wildcat Aug 8 '16 at 17:23
  • $\begingroup$ Representing it in either way is correct? If so, can we choose $y$-axis as the bond axis? Add your answer below so that I can mark it as the correct answer. $\endgroup$ – Siddharth Venu Aug 8 '16 at 17:25
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    $\begingroup$ Right. In principle, you're free to choose any direction as the bond axis. But the convention is to choose $z$-axis for this purpose. $\endgroup$ – Wildcat Aug 8 '16 at 17:28
  • $\begingroup$ Add your answer below so that I can mark it as the correct answer. $\endgroup$ – Siddharth Venu Aug 8 '16 at 17:31
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Although we could not see what is written in OP's "reference books", the mentioned difference most likely is the result of different choices for the bond axis.

Conventionally, and as it is (implicitly?) done in the MIT video lecture, it is chosen to be the $z$-axis, in which case $\pi$ bonds are indeed formed from $\mathrm{p}_x$ and $\mathrm{p}_y$ orbitals while $\mathrm{p}_z$ can take part in $\sigma$-bonding. If, however, the bond axis is chosen to be the $x$-axis, then $\pi$ bonds are formed from $\mathrm{p}_y$ and $\mathrm{p}_z$ orbitals while $\mathrm{p}_x$ takes part in $\sigma$-bonding.

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As other people says x y and z directions are arbitrary, However, the tradition that open courseware uses comes from the diatomic molecules. There the atom-atom axis is assumed to be the z direction and then the y and z, being perpendicular, correspond to the direction of pi bonds. A similar use is to design sigma to the atom-atom direction and pi to the perpendicular ones. This tradition of diatomic molecules is then exported to polliatomic examples. See any good book on molecular spectrocopy (Jeanne L McHale, Molecular Spectroscopy) for instance.

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