The vibrational partition function[1] 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
![{\displaystyle Q_{\text{vib))(T)=\prod _{j}{\sum _{n}{e^{-{\frac {E_{j,n)){k_{\text{B))T))))))](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a4b2d56eac08238c9c92876a17143dc0144a684)
where
is the absolute temperature of the system,
is the Boltzmann constant, and
is the energy of the j-th mode when it has vibrational quantum number
. 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.[2] 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.[1] A quantum harmonic oscillator has an energy spectrum characterized by:
![{\displaystyle E_{j,n}=\hbar \omega _{j}\left(n_{j}+{\frac {1}{2))\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/93a67cfa3eb5770022ffbd85bd534c3524fc728d)
where j runs over vibrational modes and
is the vibrational quantum number in the j-th mode,
is Planck's constant, h, divided by
and
is the angular frequency of the j'th mode. Using this approximation we can derive a closed form expression for the vibrational partition function.
![{\displaystyle 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))))))](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a119e5512ea7679d8ac9a0ace76b4f12a0f7aad)
where
is total vibrational zero point energy of the system.
Often the wavenumber,
with units of cm−1 is given instead of the angular frequency of a vibrational mode[2] and also often misnamed frequency. One can convert to angular frequency by using
where c is the speed of light in vacuum. In terms of the vibrational wavenumbers we can write the partition function as
![{\displaystyle 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))))))](https://wikimedia.org/api/rest_v1/media/math/render/svg/e41425cd1b7f660b9ee138378f75dd002cf7d41a)
It is convenient to define a characteristic vibrational temperature
![{\displaystyle \Theta _{i,{\text{vib))}={\frac {h\nu _{i)){k_{\text{B))))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4990f56f2da281e64c5271ee0b878f92880866d8)
where
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
![{\displaystyle Q_{\text{vib))(T)=\prod _{i=1}^{f}{\frac {1}{1-e^{-\Theta _((\text{vib)),i}/T))))](https://wikimedia.org/api/rest_v1/media/math/render/svg/dc8921c5870dd4dc186a917c29c5472ee018b29a)