I came across a question which involved non linear tetra atomic gasses. In it, the value of specific heat capacity at constant volume was given as $9R,$ added up as
$$\frac 3 2 R + \frac 3 2 R + 6R = 9R$$
Now I know $6R$ is $C_V$ for triatomic gasses, but why do we add the two $\frac 3 2 R$ terms?
Also, how does being linear or non-linear affect the specific heat capacity of a gas? Explanations with diagrams showing it in 3-D coordinate system would be helpful.