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

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

where } T is the absolute temperature of the system, \( k_{B} \) is the Boltzmann constant, and \( {\displaystyle E_{j,n}} \) is the energy of j'th 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:

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

where j runs over vibrational modes and \( {\displaystyle n_{j}} \) is the vibrational quantum number in the j 'th mode, \( \hbar \) is Planck's constant, h, divided by \( 2\pi \) 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.

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

where \( {\displaystyle E_{ZP}={\frac {1}{2}}\sum _{j}\hbar \omega _{j}} \) 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 \( {\displaystyle \omega =2\pi c{\tilde {\nu }}} \) where c is the speed of light in vacuum. In terms of the vibrational wavenumbers we can write the partition function as

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

References

Donald A. McQuarrie, Statistical Mechanics, Harper & Row, 1973

G. Herzberg, Infrared and Raman Spectra, Van Nostrand Reinhold, 1945