\LARGE(\int\prod_{{\bf R}}d\bar{{\bf u}}({\bf R})d\bar{{\bf P}}({\bf R})
\nonumber\\
&&\times \exp\left[
- -\sum\frac{1}{2M}{\bf P}({\bf R})^2
+ -\sum\frac{1}{2M}\bar{{\bf P}}({\bf R})^2
-\frac{1}{4}\sum
[\bar{u}_{\mu}({\bf R})-\bar{u}_{\mu}({\bf R'})]
\Phi_{\mu v}({\bf R}-{\bf R'})
e^{-\beta\hbar\omega_s({\bf k})}(-\hbar\omega_s({\bf k}))}
{(1-e^{-\beta\hbar\omega_s({\bf k})})^2}\nonumber\\
&=&-\frac{1}{V}\sum_{{\bf k}s}\hbar\omega_s({\bf k})
- \frac{e^{-\beta\hbar\omega_s({\bf k})}-
+ \frac{{\color{red}-}e^{-\beta\hbar\omega_s({\bf k})}-
\frac{1}{2}(1-e^{-\beta\hbar\omega_s({\bf k})})}
{1-e^{-\beta\hbar\omega_s({\bf k})}}\nonumber\\
&=&-\frac{1}{V}\sum_{{\bf k}s}\hbar\omega_s({\bf k})
- \frac{\frac{1}{2}e^{-\beta\hbar\omega_s({\bf k})}-\frac{1}{2}}
+ \frac{{\color{red}-}\frac{1}{2}e^{-\beta\hbar\omega_s({\bf k})}-\frac{1}{2}}
{1-e^{-\beta\hbar\omega_s({\bf k})}}
=\frac{1}{V}\sum_{{\bf k}s}\hbar\omega_s({\bf k})\frac{1}{2}
- \frac{1+e^{\beta\hbar\omega_s({\bf k})}}
- {e^{\beta\hbar\omega_s({\bf k})}-1}\nonumber\\
+ \frac{e^{-\beta\hbar\omega_s({\bf k})}+1}
+ {1-e^{-\beta\hbar\omega_s({\bf k})}}\cdot
+ \frac{e^{\beta\hbar\omega_s({\bf k})}}{e^{\beta\hbar\omega_s({\bf k})}}
+ \nonumber\\
&=&\frac{1}{V}\sum_{{\bf k}s}\hbar\omega_s({\bf k})\frac{1}{2}
\frac{1+e^{\beta\hbar\omega_s({\bf k})}}
{e^{\beta\hbar\omega_s({\bf k})}-1}