Vibrational partition function
# 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 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.

## 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:

$E_{j,n}=\hbar \omega _{j}\left(n_{j}+{\frac {1}{2}}\right)$

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.

$Q_{\text{vib}}(T)=\prod _{j}{\sum _{n}{e^{-{\frac {E_{j,n}}{k_{\text{B}}T}}}}}=\prod _{j}e^{-{\frac {\hbar \omega _{j}}{2k_{\text{B}}T}}}\sum _{n}\left(e^{-{\frac {\hbar \omega _{j}}{k_{\text{B}}T}}}\right)^{n}=\prod _{j}{\frac {e^{-{\frac {\hbar \omega _{j}}{2k_{\text{B}}T}}}}{1-e^{-{\frac {\hbar \omega _{j}}{k_{\text{B}}T}}}}}=e^{-{\frac {E_{\text{ZP}}}{k_{\text{B}}T}}}\prod _{j}{\frac {1}{1-e^{-{\frac {\hbar \omega _{j}}{k_{\text{B}}T}}}}}$

where $E_{ZP}=1/2$ $∑_jℏω_j{\textstyle$ 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

$Q_{\text{vib}}(T)=e^{-{\frac {E_{\text{ZP}}}{k_{\text{B}}T}}}\prod _{j}{\frac {1}{1-e^{-{\frac {hc{\tilde {\nu }}_{j}}{k_{\text{B}}T}}}}}$

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

$Q_{\text{vib}}(T)=\prod _{i=1}^{f}{\frac {1}{1-e^{-\Theta _{{\text{vib}},i}/T}}}$

