# Why CO2 has 4 vibrational degrees of freedom?

If I have a diatomic molecule, so a linear molecule, $$3N-5=6-5=1$$ holds, since this vibrational degree of freedom corresponds to the stretching coordinate beetwen the two nuclei.

If I consider water for example, I have a nonlinear triatomic molecule, so the formula says $$3N-6=9-6=3$$ vibrational degrees of freedom. Also in this case it makes sense since they correspond to the bending angle and to the $$2$$ stretching lenghts.

While, If I consider $$\ce{CO2}$$, in this case I have a triatomic linear molecule, so $$3N-5=9-5=4$$ vibrational degrees of freedom. Of course $$2$$ of these correspond to the two stretching lenghts, but I really don't understand what the other two correspond to. Since the molecule is linear, there shouldn't be angle to consider, otherwise it would not be linear anymore.

• The molecule is linear in the ground state. There are two bending motions in addition to the symmetric stretch and anti-symmetric stretch. – MaxW Dec 17 '18 at 20:53

For a full derivation of the normal modes of CO$$_2$$ there are many sites, but this one looked good: http://www.colby.edu/chemistry/PChem/notes/NormalModesText.pdf