In a formal criterion, all the nuclear motions other than the external ones (translation and external rotation) are vibration. Thus, in quantum chemical calculation packages, like Gaussian, the thermodynamic functions are calculated after the frequency analysis with harmonic oscillator (HO) approximation. However, the HO approximation may not be valid espacially for the vibrational modes with small frequencies, which is also warned by the programs such as Gaussian. The vibrational modes with very small frequencies, in many cases, are internal rotations.

In this section, starting from the harmonic oscillator, statistical treatments for hindered, or may be effectively free, rotors will be introduced.

## Keywords

Harmonic oscillator, Vibrational density of states,
Free 1D rotation, Reduced moment of inertia, Hard sphere hindrance,
Sinusoidally hindered rotor

## Lecture note

- Section-4 blackboard note
- Exercise-4

(see lecture note for answers)