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

$Q_{vib}(T)=\prod _{j}{\sum _{n}{e^{-{\frac {E_{j,n}}{k_{B}T}}}}}$

where $T$ is the absolute temperature of the system, $k_{B}$ is the Boltzmann constant, and $E_{j,n}$ is the energy of j'th 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) equals 3n − 5 or 3n − 6 dependent upon whether the molecule is linear or nonlinear respectively.^[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:

$E_{j,n}=\hbar \omega _{j}(n_{j}+{\frac {1}{2}})$

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

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

where $E_{ZP}={\frac {1}{2}}\sum _{j}\hbar \omega _{j}$ 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^[2] 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

$Q_{vib}(T)=e^{-{\frac {E_{ZP}}{k_{B}T}}}\prod _{j}{\frac {1}{1-e^{-{\frac {hc{\tilde {\nu }}_{j}}{k_{B}T}}}}}$

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)