Rotational partition function

The rotational partition function relates the rotational degrees of freedom to the rotational part of the energy.

Definition

The total canonical partition function $Z$ of a system of$N$ identical, noninteracting atoms or molecules can be divided into the atomic or molecular partition functions $\zeta$.

$Z={\frac {\zeta ^{N}}{N!}}$

with

$\zeta =\sum _{j}g_{j}e^{E_{j}/k_{B}T}$

Where $g_{j}$ is the degeneracy of the j'th quantum level of an individual particle, $k_{B}$ is the Boltzmann constant, and $T$ is the absolute temperature of system. for molecules, under the assumption that that total energy levels $E_{j}$ can be partitioned into its contributions from different degrees of freedom (weakly coupled degrees of freedom)

$E_{j}=\sum _{i}E_{j}^{i}=E_{j}^{trans}+E_{j}^{ns}+E_{j}^{rot}+E_{j}^{vib}+E_{j}^{e}$

and the number of degenerate states are given as products of the single contributions

$g_{j}=\prod _{i}g_{j}^{i}=g_{j}^{trans}g_{J}^{ns}g_{j}^{rot}g_{j}^{vib}g_{j}^{e},$

where "trans", "ns", "rot", "vib" and "e" denotes translational, nuclear spin, rotational and vibrational contributions as well as electron excitation, the molecular partition functions

$\zeta =\sum _{j}g_{j}e^{-E_{j}/k_{B}T}$

can be written as a product itself

$\zeta =\prod _{i}\zeta ^{i}=\zeta ^{trans}\zeta ^{rot}\zeta ^{vib}\zeta ^{e}.$

Rotational energies are quantized. For a diatomic molecule like CO or HCl or a linear polyatomic molecule like OCS in its ground vibrational state, the allowed rotational energies in the Rigid rotor approximation are

$E_{J}^{rot}={\frac {{\mathbf {J} }^{2}}{2I}}={\frac {J(J+1)\hbar ^{2}}{2I}}=J(J+1)B.$

J is the quantum number for total rotational angular momentum and takes all integer values starting at zero,i.e. $J=0,1,2,\ldots \,\,B={\frac {\hbar ^{2}}{2I}}$ is the rotational constant, and $I$ is the Moment of inertia. Here we are using B in energy units. If it is expressed in frequency units, replace B by hB in all the expression that follow, where h is Planck's Constant. If B is given in units of $cm^{-1}$, then replace B by hcB where c is the speed of light in vacuum.

For each value of J, we have rotational degeneracy, $g_{j}$ = (2J+1), so the rotational partition function is therefore

$\zeta ^{rot}=\sum _{J=0}^{\infty }g_{j}e^{-E_{J}/k_{B}T}=\sum _{J=0}^{\infty }(2J+1)e^{-J(J+1)B/kT}.$

For all but the lightest molecules or the very lowest temperatures, $B<<k_{B}T$. This suggests we can approximate the sum by replacing the sum over J by an integral of J treated as a continuous variable.

$\zeta ^{rot}\approx \int _{0}^{\infty }(2J+1)e^{-J(J+1)B/k_{B}T}dJ={\frac {k_{B}T}{B}}.$

This approximation is known as the high temperature limit. It is also called the classical approximation as this is the result for the canonical partition function for a classical rigid rod.

Using the Euler-Maclaurin formula an improved estimate can be found

$\zeta ^{rot}={\frac {k_{B}T}{B}}+{\frac {1}{3}}+{\frac {1}{15}}\left({\frac {B}{k_{B}T}}\right)+{\frac {4}{3125}}\left({\frac {B}{k_{B}T}}\right)^{2}+{\frac {1}{315}}\left({\frac {B}{k_{B}T}}\right)^{3}\ldots$.

For the CO molecule at $T=300K$, the (unit less) contribution $\zeta ^{rot}$ to $\zeta$ turns out to be in the range of $10^{2}$.

The mean thermal rotational energy per molecule can now be computed by taking the derivative of $\zeta ^{rot}$ with respect to temperature $T$. In the high temperature limit approximation, the mean thermal rotational energy of a linear rigid rotor is $k_{B}T$.

A rigid, nonlinear molecule has rotational energy levels determined by three rotational constants, conventionally written $A,B,$ and $C$, which can often be determined by Rotational spectroscopy. In terms of these constants, the rotational partition function can be written in the high temperature limit as

$\zeta ^{rot}\approx {\frac {\sqrt {\pi }}{\sigma }}{\sqrt {\frac {(k_{b}T)^{3}}{ABC}}}$

with $\sigma$ the rotational symmetry number which equals the number ways a molecule can be rotated to overlap itself in an indistinguishable way, i.e. that at most interchanges identical atoms. The expression works for asymmetric, symmetric and spherical top rotors.

See also

• Translational partition function
• Vibrational partition function
• Partition function (mathematics)