Vibrational partition function explained

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 byQ_\text(T) = \prod_j where

T

is the absolute temperature of the system,

kB

is the Boltzmann constant, and

Ej,n

is the energy of the 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) 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:E_ = \hbar\omega_j\left(n_j + \frac\right)where j runs over vibrational modes and

nj

is the vibrational quantum number in the j-th mode,

\hbar

is Planck's constant, h, divided by

2\pi

and

\omegaj

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(T) =\prod_j= \prod_j e^ \sum_n \left(e^ \right)^n= \prod_j \frac= e^ \prod_j \frac

where E_\text = \frac \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 and also often misnamed frequency. One can convert to angular frequency by using

\omega=2\pic\tilde{\nu}

where c is the speed of light in vacuum. In terms of the vibrational wavenumbers we can write the partition function asQ_\text(T) = e^ \prod_j \frac

It is convenient to define a characteristic vibrational temperature \Theta_ = \frac 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(T) = \prod_^f \frac

See also

Notes and References

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