+ Write down the potential energy for the instantaneous positions
+ ${\bf r}({\bf R})$, with ${\bf u}({\bf R})={\bf r}({\bf R})-{\bf R}$.
+ Apply Taylor approximation to $\Phi({\bf r}+{\bf a})$ with
+ ${\bf r}={\bf R}-{\bf R'}$ and
+ ${\bf a}={\bf u}({\bf R})-{\bf u}({\bf R'})$
+ and only retain terms quadratic in $u$.
+ \item Use the evaluated potential to calculate the energy density
+ (do not forget the kinetic energy contribution) and
+ the specific heat $c_{\text{V}}$.
+ {\bf Hint:}
+ Use the following change of variables
+ \[
+ {\bf u}({\bf R})=\beta^{-1/2}\bar{{\bf u}}({\bf R}), \qquad
+ {\bf P}({\bf R})=\beta^{-1/2}\bar{{\bf P}}({\bf R})
+ \]
+ to extract the temperature dependence of the integral.
+ Does this also work for anharmonic terms?
+ Which parts of the integral do not contribute to $w$ and why?