If we then introduce the harmonic oscillator one-body Hamiltonian $$ H_0= \sum_{i=1}^A\left(\frac{\boldsymbol{p}_i^2}{2m}+ \frac{1}{2}m\omega^2\boldsymbol{r}_i^2\right), $$ with \( \omega \) the oscillator frequency, we can rewrite the latter as $$ H_{\mathrm{HO}}= \frac{\boldsymbol{P}^2}{2mA}+\frac{mA\omega^2\boldsymbol{R}^2}{2} +\frac{1}{2mA}\sum_{i < j}(\boldsymbol{p}_i-\boldsymbol{p}_j)^2 +\frac{m\omega^2}{2A}\sum_{i < j}(\boldsymbol{r}_i-\boldsymbol{r}_j)^2. \tag{4} $$