# 18.7: The Vibrational Partition Function

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There is a great deal of utility for thermodynamic functions calculated from the vibrational normal modes of a molecule. The vibrational energy and entropy depend on the shape a multidimensional potential energy surface. If one performs a conformational search of macromolecule it is one obtains energies and structures but little direct information concerning the shape of the potential energy surface for each conformation. The vibrational entropy gives a means determining whether there are significant entropic differences in the structures and therefore whether certain conformations will be favored based on the entropy.

However, it is possible to take appropriate linear combinations of the coordinates so that the cross terms are eliminated and the classical Hamiltonian as well as the operator corresponding to it contains no cross terms and in terms of the new coordinates, the Hamiltonian can be written as,

$H = \sum_{i=1}^{f} \dfrac{h^2}{2 \mu_i} \dfrac{\partial}{\partial q_i^2}+ \sum_{i=1}^{f} \dfrac{k_i}{2} q_i^2 \label{3.81}$

Here, the degrees of freedom $$f$$ is $$3N - 5$$ for a linear molecule and $$3N - 6$$ for a nonlinear molecule. Here, $$k_i$$ is the force constant and $$μ_i$$ is the reduced mass for that particular vibrational mode which is referred to as a normal mode.

The Equation $$\ref{3.81}$$ represents $$f$$ linearly independent harmonic oscillators and the total energy for such a system is

$\epsilon_{vib} = \sum_{i=1}^{f} \left( v_i + \dfrac{1}{2} \right) h \nu_i \nonumber$

The vibrational frequencies are given by

$\nu_i = \dfrac{1}{2 \pi} \sqrt{\dfrac{k_i}{\mu_i}} \nonumber$

The vibrational partition function is given by the product of $$f$$ vibrational functions for each frequency.

$q_{vib} = \prod_{i=1}^f \dfrac{ e^{-\Theta_{vib,i}/2T} }{1- e^{-\Theta_{vib,i}/T}} \label{Qvib1}$

with

$\Theta_{vib,i} = \dfrac{h\nu_i}{k_B} \nonumber$

As with the previous discussion regarding simple diatomics, $$\Theta_{vib,i}$$ is called the characteristic vibrational temperature. The molar energies and the heat capacities are given by

$\langle E_{vib} \rangle = Nk \sum_{i=1}^f \left[ \dfrac{ \Theta_{vib,i} }{2} + \dfrac{ \Theta_{vib,i} e^{-\Theta_{vib,i}/T} }{1- e^{-\Theta_{vib,i}/T}} \right] \nonumber$

and

$\bar{C}_V = Nk_B \sum_{i=1}^f \left( \dfrac{ \Theta_{vib,i} }{T} \right)^2 \dfrac{ e^{-\Theta_{vib,i}/T} }{\left(1- e^{-\Theta_{vib,i}/T}\right)^2} \nonumber$

##### Example $$NO_2$$

The three characteristic vibrational temperatures for $$\ce{NO2}$$ are 1900 K, 1980 K and 2330 K. Calculate the vibrational partition function at 300 K.

###### Solution

The vibrational partition is (Equation $$\ref{Qvib1}$$)

$q_{vib} = \prod_{i=1}^f \dfrac{ e^{-\Theta_{vib,i}/2T} }{1- e^{-\Theta_{vib,i}/T}} \nonumber$

If we calculate $$q_{vib}$$ by taking the zero point energies as the reference points with respect to which the other energies are measured

\begin{align*} q_{vib} &= \prod_{i=1}^f \dfrac{ 1 }{1- e^{-\Theta_{vib,i}/T}} = \left( \dfrac{ 1 }{1- e^{-1900/300}} \right) \left( \dfrac{ 1 }{1- e^{-1980/300}} \right) \left( \dfrac{ 1 }{1- e^{-2330/300}} \right) \\[4pt] & =(1.0018) ( 1.0014)(1.0004) = 1.0035 \end{align*} \nonumber

The implication is that very few vibrational states of $$\ce{NO2}$$ (other than the ground vibrational state) are accessible at 300 K. This is standard of the vibrations of most molecules.