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 jth mode when it has vibrational quantum number $n=0,1,2,\dots$. For an isolated molecule of N atoms, the number of vibrational modes (i.e. values of j) is Template:Nowrap for linear molecules and Template:Nowrap 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: $$E_{j,n}=\hslash {\omega }_j(n_j+\frac{1}{2})$$ where j runs over vibrational modes and $n_j$ is the vibrational quantum number in the jth mode, $\hslash$ 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. $$Q_{\text{vib}}(T)=\prod _j\sum _n{e}^{-\frac{E_{j,n}}{k_{\text{B}}T}}=\prod _j{e}^{-\frac{\hslash {\omega }_j}{2k_{\text{B}}T}}\sum _n{\left(e^{-\frac{\hslash {\omega }_j}{k_{\text{B}}T}}\right)}^n=\prod _j\frac{e^{-\frac{\hslash {\omega }_j}{2k_{\text{B}}T}}}{1-e^{-\frac{\hslash {\omega }_j}{k_{\text{B}}T}}}=e^{-\frac{E_{\text{ZP}}}{k_{\text{B}}T}}\prod _j\frac{1}{1-e^{-\frac{\hslash {\omega }_j}{k_{\text{B}}T}}}$$ where $E_{\text{ZP}}=\frac{1}{2}{\sum }_j\hslash {\omega }_j$ 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 $${\mathrm{\Theta }}_{i,\text{vib}}=\frac{h{\nu }_i}{k_{\text{B}}}$$ 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 $$Q_{\text{vib}}(T)=\prod _{i=1}^f\frac{1}{1-e^{-{\mathrm{\Theta }}_{\text{vib},i}/T}}$$

See also

• Partition function (mathematics)