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

$Q_{vib}(T) =\prod_j{\sum_n{e^{-\frac{E_{j,n}}{k_B T}}}}$

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) equals 3n − 5 or 3n − 6 dependent upon whether the molecule is linear or nonlinear respectively. 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:

$E_{j,n}=\hbar\omega_j(n_j +\frac{1}{2})$

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.

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

where $E_{ZP} = \frac{1}{2} \sum_j \hbar \omega_j$ 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

$Q_{vib}(T) = e^{- \frac{E_{ZP}}{k_B T}} \prod_j \frac{1}{ 1 - e^{-\frac{ h c \tilde{\nu}_j}{k_B T}} }$

See also

• Partition function (mathematics)