Vibrational partition function

The vibrational partition function traditionally refers to to the partition function corresponding to a canonical ensemble where the energy levels arise from those of the quantum harmonic oscillator. It is a first order approximation to the partition function which allows one to calculate the contribution of the vibrational degree of freedom of molecules towards thermodyanmic variables.

The most common approximation to the vibrational partition function uses a model in which the vibrational eigenmodes or vibrational normal modes of the system are considered to be a set of uncoupled quantum harmonic oscillators, with energy spectrum:

$E_{j,i}=\hbar \omega _{j}(i+{\frac {1}{2}})$

where j is an index representing the vibrational mode, and i is the quantum number for each energy level of the jth vibrational mode. The vibrational partition function is then calculated as:

$Z_{vib}=\prod _{j}{\sum _{i}{e^{-{\frac {E_{j,i}}{kT}}}}}$