Example 2: Dilute Gas of Diatomic Molecules

In the previous example, we showed that the state of a structureless particle is completely defined by the vector ${\bf K}=(Kx, Ky, Kz)$ that specifies the momentum ${\bf K} \hbar$ of the center of mass. Specifying the state of a molecule, however, requires the specification of its internal state besides specifying the translation of the molecule as a whole, since the molecule can vibrate, rotate, or undergo electronic excitations while translating as specified by the vector ${\bf K}$. Contrary to structureless particles, molecules are described by the Hamiltonian

$$\hat{H}=-\frac{\hbar^2}{2m} \nabla_{{\bf R}^2} + \hat{H}_{el}({\bf R},{\bf r}),$$ (170)

where, ${\bf R}$ and ${\bf r}$ are the nuclear and electronic coordinates.

A simple expression for the cononical partition function of an ideal gas of diatomic molecules in the ground electronic state can be obtained by factorizing the total energy into translational, rotational and vibrational modes (i.e., assuming that these modes are uncoupled from each other) and then modeling vibrations according to the harmonic approximation and rotations according to the rigid rotor approximation. The resulting partition function is

$$Z=\frac{(\sum_j e^{-\beta E_j})^N}{N!},$$ (171)

Here, $E_j = E_{rot}(J(j))+E_{transl}(n(i))+E_{vib}(\nu(j))$, where $J(j)$ specifies the rotational quantum number, $\nu(j)$ the vibrational quantum number and $n(j)$ the translational modes of particle $j$. Therefore,

$$Z= \frac{(q_{transl}*q_{int})^N}{N!}, \hspace{.1cm} q_{int} = q_{rot}*q_{vib},$$ (172)

with

$$q_{rot}= \sum_{J=0}^{\infty}(2J+1) e^{-\beta \frac{\hbar^2}{2I_0} J(J+1)},$$ (173)

$$q_{vib} =\sum_{\nu=0}^{\infty} e^{-\beta \hbar \omega_0 (1/2 + \nu) } = \frac{e^{-\beta \hbar \omega_0 /2}}{1-e^{-\beta \hbar \omega_0 /2} },$$ (174)

and according to Eq. (167),

$$q_{transl}= \frac{V}{\hbar^3} \Bigg ( \frac{m}{\beta} \Bigg )^{3/2}.$$ (175)

Note that for simplicity we have ignored the internal structure of nuclei and the degeneracy factor associated with the permutation of indistinguishable nuclei.