Vibrational partition function

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 $${\displaystyle Q_{\text{vib}}(T)=\prod _{j}{\sum _{n}{e^{-{\frac {E_{j,n}}{k_{\text{B}}T}}}}}}$$ where ${\displaystyle T}$ is the absolute temperature of the system, ${\displaystyle k_{B}}$ is the Boltzmann constant, and ${\displaystyle E_{j,n}}$ is the energy of the jth mode when it has vibrational quantum number ${\displaystyle n=0,1,2,\ldots }$. 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. ^[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: $${\displaystyle E_{j,n}=\hbar \omega _{j}\left(n_{j}+{\frac {1}{2}}\right)}$$ where j runs over vibrational modes and ${\displaystyle n_{j}}$ is the vibrational quantum number in the jth mode, ${\displaystyle \hbar }$ is the Planck constant, h, divided by ${\displaystyle 2\pi }$ and ${\displaystyle \omega _{j}}$ is the angular frequency of the jth mode. Using this approximation we can derive a closed form expression for the vibrational partition function. $${\displaystyle Q_{\text{vib}}(T)=\prod _{j}{\sum _{n}{e^{-{\frac {E_{j,n}}{k_{\text{B}}T}}}}}=\prod _{j}e^{-{\frac {\hbar \omega _{j}}{2k_{\text{B}}T}}}\sum _{n}\left(e^{-{\frac {\hbar \omega _{j}}{k_{\text{B}}T}}}\right)^{n}=\prod _{j}{\frac {e^{-{\frac {\hbar \omega _{j}}{2k_{\text{B}}T}}}}{1-e^{-{\frac {\hbar \omega _{j}}{k_{\text{B}}T}}}}}=e^{-{\frac {E_{\text{ZP}}}{k_{\text{B}}T}}}\prod _{j}{\frac {1}{1-e^{-{\frac {\hbar \omega _{j}}{k_{\text{B}}T}}}}}}$$ where ${\textstyle E_{\text{ZP}}={\frac {1}{2}}\sum _{j}\hbar \omega _{j}}$ is total vibrational zero point energy of the system.

Often the wavenumber, ${\displaystyle {\tilde {\nu }}}$ 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 ${\displaystyle \omega =2\pi c{\tilde {\nu }}}$ where c is the speed of light in vacuum. In terms of the vibrational wavenumbers we can write the partition function as $${\displaystyle Q_{\text{vib}}(T)=e^{-{\frac {E_{\text{ZP}}}{k_{\text{B}}T}}}\prod _{j}{\frac {1}{1-e^{-{\frac {hc{\tilde {\nu }}_{j}}{k_{\text{B}}T}}}}}}$$

It is convenient to define a characteristic vibrational temperature $${\displaystyle \Theta _{i,{\text{vib}}}={\frac {h\nu _{i}}{k_{\text{B}}}}}$$ where ${\displaystyle \nu }$ 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 $${\displaystyle Q_{\text{vib}}(T)=\prod _{i=1}^{f}{\frac {1}{1-e^{-\Theta _{{\text{vib}},i}/T}}}}$$

References

1. Donald A. McQuarrie, Statistical Mechanics, Harper & Row, 1973
2. G. Herzberg, Infrared and Raman Spectra, Van Nostrand Reinhold, 1945

See also

• Partition function (mathematics)