Vibrational partition function

The vibrational partition function traditionally refers to the component of the canonical partition function resulting from the vibrational degrees of freedom of a system. The vibrational partition function is only well-defined in model systems where the vibrational motion is relatively uncoupled with the system's other degrees of freedom.
Definition.
For a system (such as a molecule or solid) with uncoupled vibrational modes the vibrational partition function is defined by
formula_1
where formula_2 is the absolute temperature of the system, formula_3 is the Boltzmann constant, and formula_4 is the energy of j'th mode when it has vibrational quantum number formula_5. For an isolated molecule of "n" atoms, the number of vibrational modes (i.e. values of j) is 3"n" − 5 for linear molecules and 3"n" − 6 for non-linear ones. In crystals, the vibrational normal modes are commonly known as phonons.
Approximations.
Quantum harmonic oscillator.
The most common approximation to the vibrational partition function uses a model in which the vibrational eigenmodes or normal modes of the system are considered to be a set of uncoupled quantum harmonic oscillators. It is a first order approximation to the partition function which allows one to calculate the contribution of the vibrational degrees of freedom of molecules towards its thermodynamic variables. A quantum harmonic oscillator has an energy spectrum characterized by:
formula_6
where "j" runs over vibrational modes and formula_7 is the vibrational quantum number in the "j" 'th mode, formula_8 is Planck's constant, "h", divided by formula_9 and formula_10
is the angular frequency of the "j"'th mode. Using this approximation we can derive a closed form expression for the vibrational partition function.
formula_11
where
formula_12 is total vibrational zero point energy of the system.
Often the wavenumber, formula_13 with units of cm−1 is given instead of the angular frequency of a vibrational mode and also often misnamed frequency. One can convert to angular frequency by using formula_14 where "c" is the speed of light in vacuum. In terms of the vibrational wavenumbers we can write the partition function as
formula_15