Vibrational partition function
title: "Vibrational partition function" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["partition-functions"] topic_path: "general/partition-functions" source: "https://en.wikipedia.org/wiki/Vibrational_partition_function" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0
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 jth 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 jth mode, \hbar is the Planck constant, h, divided by 2 \pi and \omega_j is the angular frequency of the jth 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}}
References
References
- Donald A. McQuarrie, ''Statistical Mechanics'', Harper & Row, 1973
- G. Herzberg, ''Infrared and Raman Spectra'', Van Nostrand Reinhold, 1945
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