The simplest theoretical model of molecular vibrations involves a linear restraining force, Eqn. \eqref{eq:force}, where k is the proportionality constant and r-r0 the displacement from an equilibrium position, r0.
\begin{equation}
F = k\left( {r - {r_0}} \right)
\label{eq:force}
\end{equation}
This results in a Hooks Law harmonic potential, Eqn. \eqref{eq:harmonic} where k is called the vibrational force constant.
\begin{equation}
V = {\textstyle{1 \over 2}}k{\left( {r - {r_0}} \right)^2}
\label{eq:harmonic}
\end{equation}
Solving the quantum mechanical problem results in a set of evenly spaced energy levels or frequencies ν, Eqn. \eqref{eq:energies}.
\begin{equation}
E = \nu \left( {n + {\textstyle{1 \over 2}}} \right){\rm{ where }} \nu = \frac{h}{{2\pi }}\sqrt {\frac{k}{\mu }}
\label{eq:energies}
\end{equation}
At finite temperatures these energy levels are populated according to a Boltzman population distribution, Eqn. \eqref{eq:pops}
\begin{equation}
{P_i} = \frac{{{e^{-iE/{k_B}T}}}}{Q}
\label{eq:pops}
\end{equation}
where Q is the partition function, Eqn. \eqref{eq:part_func}.
\begin{equation}
Q = \sum\limits_{j = 1}^n {{e^{ - jE/{k_B}T }}}
\label{eq:part_func}
\end{equation}
The Boltzman populations can be combined with the dependences of thermodyanmic quantities on Q to develop the temperature dependence of the average energy, Eqn. \eqref{eq:ave_e},
\begin{equation}
\left\langle E \right\rangle = \sum\limits_s {{E_s}{P_s}} = \frac{1}{Z}\sum\limits_s {{E_s}{e^{ - {E_s}/{k_B}T}}}
\label{eq:ave_e}
\end{equation}
the enthalpy Eqn. \eqref{eq:enthalpy},
\begin{equation}
\left\langle H \right\rangle = \left\langle E \right\rangle + nRT
\label{eq:enthalpy}
\end{equation}
the entropy, Eqn. \eqref{eq:entropy}
\begin{equation}
S = - {k_B}\sum\limits_s {{P_s}\ln {P_s}}
\label{eq:entropy}
\end{equation}
the free energy, Eqn. \eqref{eq:free}
\begin{equation}
\Delta G = \Delta H - T\Delta S
\label{eq:free}
\end{equation}
and the heat capacity, Eqn. \eqref{eq:cp}.
\begin{equation}
{C_V} = \frac{{\partial \left\langle E \right\rangle }}{{\partial T}} = \frac{1}{{{k_B}T}}\left\langle {{{\left( {\Delta E} \right)}^2}} \right\rangle ;{\rm{ }}{C_p} = {C_V} + nR
\label{eq:cp}
\end{equation}
Figure 1. Harmonic Potential Energy curve with vibrational energy levels and thermal populations superimpsoed.
