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 $$Q_\text{vib}(T) = \prod_j{\sum_n{e^{-\frac{E_{j,n}}{k_\text{B} T}}}} $$ where $$ T $$ is the absolute temperature of the system, $$ k_B $$ is the Boltzmann constant, and $$ E_{j,n} $$ is the energy of the j-th mode when it has vibrational quantum number $$ n = 0, 1, 2, \ldots $$. For an isolated molecule of N atoms, the number of vibrational modes (i.e. values of j) is 3N − 5 for linear molecules and 3N − 6 for non-linear ones. In crystals, the vibrational normal modes are commonly known as phonons.

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: $$E_{j,n} = \hbar\omega_j\left(n_j + \frac{1}{2}\right)$$ where j runs over vibrational modes and $$ n_j $$ is the vibrational quantum number in the j-th mode, $$ \hbar $$ is Planck's constant, h, divided by $$ 2 \pi $$ and $$ \omega_j $$ is the angular frequency of the j'th mode. Using this approximation we can derive a closed form expression for the vibrational partition function.

$$Q_\text{vib}(T) =\prod_j{\sum_n{e^{-\frac{E_{j,n}}{k_\text{B} T}}}} = \prod_j e^{-\frac{\hbar \omega_j}{2 k_\text{B} T}} \sum_n \left( e^{-\frac{\hbar \omega_j}{k_\text{B} T}} \right)^n = \prod_j \frac{e^{-\frac{\hbar \omega_j}{2 k_\text{B} T}}}{ 1 - e^{-\frac{\hbar \omega_j}{k_\text{B} T}} } = e^{- \frac{E_\text{ZP}}{k_\text{B} T}} \prod_j \frac{1}{ 1 - e^{-\frac{\hbar \omega_j}{k_\text{B} T}} }$$

where $ E_\text{ZP} = \frac{1}{2} \sum_j \hbar \omega_j $ is total vibrational zero point energy of the system.

Often the wavenumber, $$\tilde{\nu}$$ 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 $$ \omega = 2 \pi c \tilde{\nu} $$ where c is the speed of light in vacuum. In terms of the vibrational wavenumbers we can write the partition function as $$Q_\text{vib}(T) = e^{- \frac{E_\text{ZP}}{k_\text{B} T}} \prod_j \frac{1}{ 1 - e^{-\frac{ h c \tilde{\nu}_j}{k_\text{B} T}} }$$

It is convenient to define a characteristic vibrational temperature $$ \Theta_{i,\text{vib}} = \frac{h \nu_i}{k_\text{B}} $$ where $$ \nu $$ is experimentally determined for each vibrational mode by taking a spectrum or by calculation. By taking the zero point energy as the reference point to which other energies are measured, the expression for the partition function becomes $$ Q_\text{vib}(T) = \prod_{i=1}^f \frac{1}{1-e^{-\Theta_{\text{vib},i}/T}} $$