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 j-th mode, ${\displaystyle \hbar }$ is Planck's constant, h, divided by ${\displaystyle 2\pi }$ and ${\displaystyle \omega _{j}}$ is the angular frequency of the j'th mode. Using this approximation we can derive a closed form expression for the vibrational partition function.

$${\displaystyle Q_{\text{vib}}(T)=\prod _{j}{\sum _{n}{e^{-{\frac {E_{j,n}}{k_{\text{B}}T}}}}}=\prod _{j}e^{-{\frac {\hbar \omega _{j}}{2k_{\text{B}}T}}}\sum _{n}\left(e^{-{\frac {\hbar \omega _{j}}{k_{\text{B}}T}}}\right)^{n}=\prod _{j}{\frac {e^{-{\frac {\hbar \omega _{j}}{2k_{\text{B}}T}}}}{1-e^{-{\frac {\hbar \omega _{j}}{k_{\text{B}}T}}}}}=e^{-{\frac {E_{\text{ZP}}}{k_{\text{B}}T}}}\prod _{j}{\frac {1}{1-e^{-{\frac {\hbar \omega _{j}}{k_{\text{B}}T}}}}}}$$

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

$${\displaystyle Q_{\text{vib}}(T)=\prod _{i=1}^{f}{\frac {1}{1-e^{-\Theta _{{\text{vib}},i}/T}}}}$$