4.6 Vibrational and rotational energy of a diatomic molecule

So far we have only looked at two-level systems such as the paramagnet. More usually there are many or even infinitely many levels, and hence terms in the partition function. In some special cases the partition function can still be expressed in closed form.

Vibrational energy of a diatomic molecule

The energy levels of a quantum simple harmonic oscillator of frequency $\omega$ are

$$\varepsilon_n=(n+{\textstyle \frac 1 2})\hbar\omega\qquad\hbox{$n=0,1,2\ldots$}$$

so

$$\begin{eqnarray*} Z_1\!\!\!&=&\!\!\!\sum_{n=0}^\infty = e^{-\varepsilon_n\beta}=... ...\Bigl(2 \sinh({\textstyle \frac 1 2}\hbar\omega\beta)\Bigr)^{-1} \end{eqnarray*}$$

where we have used the expression for the sum of a geometric series, $\sum_n x^n=(1-x)^{-1}$, with $x=e^{-\hbar\omega\beta}$.

From this we obtain

$$\left\langle E_1 \right\rangle = -{\partial \ln Z_1\over\par... ...\omega {\rm coth} ({\textstyle \frac 1 2}\hbar\omega\beta).$$

The low temperature limit of this ( $k_{\scriptscriptstyle B}T \ll\hbar\omega$; $\hbar\omega\beta\to \infty$) is ${\textstyle \frac 1 2}\hbar \omega$, which is what we expect if only the ground state is populated. The high temperature limit ( $k_{\scriptscriptstyle B}T \gg \hbar\omega$ ; $\hbar\omega\beta\to 0$) is $k_{\scriptscriptstyle B}T$, which should ring bells! (See here for more on limits.)

Typically the high temperature limit is only reached around 1000 K

Rotational energy of a diatomic molecule

The energy levels of a rigid rotor of moment of inertia $\cal I$ are

$$\varepsilon_l={l(l+1)\hbar^2\over 2\cal I }\qquad\hbox{$l=0,1,2\ldots$}$$

but there is a complication; as well as the quantum number $L$ there is $m_l$, $-l\le m_l\le l$, and the energy doesn't depend on $m_l$. Thus the $l$th energy level occurs $2l+1$ times in the partition function, giving

$$Z_1= \sum_{l=0}^\infty \sum_{m_l=-l}^l e^{-l(l+1)\hbar^2\b... ... = \sum_{l=0}^\infty (2l+1) e^{-l(l+1)\hbar^2\beta/2\cal I}.$$

The term $2l+1$ is called a degeneracy factor since ``degenerate'' levels are levels with the same energy. (I can't explain this bizarre usage, but it is standard.) For general $\beta$ this cannot be further simplified. At low temperatures successive term in $Z_1$ will fall off quickly; only the lowest levels will have any significant occupation probability and the average energy will tend to zero.

At high temperatures, ( $k_{\scriptscriptstyle B}T \gg \hbar^2/2\cal I$) there are many accessible levels and the fact that they are discrete rather than continuous is unimportant; we can replace the sum over $l$ with an integral ${\rm d}l$; changing variables to $x=l(l+1)$ gives

$$\begin{eqnarray*} Z_1\!\!\!&=&\!\!\!{{\cal I} \over \hbar^2\beta}\\ \left\langle E_1 \right\rangle \!\!\!&=&\!\!\!k_{\scriptscriptstyle B}T \end{eqnarray*}$$

Typically $\hbar^2/2\cal I$ is around $10^{-3}$ eV, so the high-temperature limit is reached well below room temperature.

It is not an accident that the high-temperature limit of the energy was $k_{\scriptscriptstyle B}T$ in both cases! These are examples of equipartition which is the subject of a future section.

References

• (Bowley and Sánchez 5.11,5.12)