Vibrational Partition Function

The vibrational partition functionDonald A. McQuarrie, ''Statistical Mechanics'', Harper & Row, 1973 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
$Q_\text(T) = \prod_j$
where $T$ is the absolute temperature of the system, $k_B$ is the Boltzmann constant, and $E_$ 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'') is 3''n'' − 5 for linear molecules and 3''n'' − 6 for non-linear ones.G. Herzberg, ''Infrared and Raman Spectra'', Van Nostrand Reinhold, 1945 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. 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 $n_j$ 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 ''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, $\tilde$ 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 $\omega = 2 \pi c \tilde$ where ''c'' is the speed of light in vacuum. In terms of the vibrational wavenumbers we can write the partition function as
$Q_\text(T) = e^ \prod_j \frac$

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
$Q_\text(T) = \prod_^f \frac$

References

See also

* Partition function (mathematics)

{{Statistical mechanics topics

Partition functions