A thermal vibrations in a crystalline solid produce atomic displacements, which in turn can be resolved into different states of polarization such that vibrations parallel to the wave vector are longitudinal waves and the two directions at right angles to the wave vector are transverse waves.
As the rules of quantum mechanics apply to all the different atomic vibrations in the crystal, the lattice pulsates as a complete assembly in discrete energy steps of ħω(phonons).
The phonon is related to both the frequency of vibration and the temperature. If the temperature is raised, the amplitude of atomic vibration increases, and in quantum terms, this is considered as an increase in the number of phonons in the system.
The concepts of temperature and thermal equilibrium associated with crystal solids are based on individual atoms in the system possessing vibrational motion.
A molecular vibration occurs when atoms in a molecule are in periodic motion while the molecule as a whole has constant translational and rotational motion.
The frequency of the periodic motion is known as a vibration frequency, and the typical frequencies of molecular vibrations range from less than 1013 to approximately 1014 Hz, corresponding to wavenumbers of approximately 300 to 3000 cm−1.
In general, a non-linear molecule with N atoms has 3N – 6 normal modes of vibration, but a linear molecule has 3N – 5 such modes, because rotation about its molecular axis cannot be observed.
A diatomic molecule has one normal mode of vibration. The normal modes of vibration of polyatomic molecules are independent of each other but each normal mode will involve simultaneous vibrations of different parts of the molecule such as different chemical bonds.
A molecular vibration is excited when the molecule absorbs a quantum of energy, E; E = hν (where h is Planck's constant).
A fundamental vibration is excited when one such quantum of energy is absorbed by the molecule in its ground state. When two quanta are absorbed the first overtone is excited, and so on to higher overtones.
Therefore the thermal vibrations and molecular vibrations are terms representing two different situations and mechanisms, however, the thermal state of the physical system can have exchanges/sharing of thermal energy quanta between the two states.