What is the total number of degrees of freedom for a molecule containing N atoms?

For a molecule containing N atoms, the concept of degrees of freedom is based on the number of ways in which the atoms can move in space. In three-dimensional space, each atom contributes three translational degrees of freedom, corresponding to movement along the x, y, and z axes. Therefore, if we simply consider the total translational movements of the N atoms, we would initially think it could be 3N.

However, in practice, for molecules that are not free, certain constraints come into play. If the molecule is rigid, it may have additional degrees of freedom due to its overall orientation and internal movements. Specifically, when considering a rigid molecule, we account for the fact that the entire molecule can only rotate about its center of mass.

For a three-dimensional molecule, in addition to the translational degrees of freedom (3N), we must take into account the rotational degrees of freedom. A non-linear molecule typically has 3 rotational degrees of freedom because it can rotate around three axes. However, there are constraints that reduce the total degrees of freedom for the vibrations and the internal motions of the molecule.

When we deduce the total number of degrees of freedom for a molecule that has fixed bond lengths and angles, we arrive at the