\(\rho_{\text{AB}}^{part-A}\left(\mathbf{r}\right)=\sum_{I\in A}{\rho^{I}(\mathbf{r})}+\sum_{I<J\in A}{{\rho}^{\text{IJ}}\left(\mathbf{r}\right)+\frac{1}{2}}\sum_{I\in A}{\sum_{J\in B}{{\rho}^{\text{IJ}}(\mathbf{r})}},\) (2)
\(\rho_{\text{AB}}^{part-B}\left(\mathbf{r}\right)=\sum_{I\in B}{\rho^{I}(\mathbf{r})}+\sum_{I<J\in B}{{\rho}^{\text{IJ}}\left(\mathbf{r}\right)+\frac{1}{2}}\sum_{I\in A}{\sum_{J\in B}{{\rho}^{\text{IJ}}(\mathbf{r})}}.\) (3)