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I am looking at the normal modes for methane and have attached a figure of them below:

enter image description here

The only IR active modes are $\nu_3$ and $\nu_4$ which I understand because the other two modes do not change the dipole moment of the molecule and hence do not interact with IR.

Now I have also found in my research that there exists a combination band $\nu_1 + \nu_4$. I understand that combination bands are created when two different normal modes are excited by a given photon.

However, the $\nu_1$ mode of methane is IR inactive so how can a photon excite it to give the aforementioned combination band?

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In a combination band you need to make the direct product and reduce it to its irreducible representation (irreps) and see if any symmetry species are ir active by looking for x, y, z in the point group table. If you make the direct product in $T_d$ of $A_1 $ (totally sym. so changes nothing) and $T_3$ you get $T_3$ which is ir active.

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