\begin{equation}
    {\mathbf{p}}_t(\hat{X}_{{\rm pair}}) = {\mathbf{p}}_t(\hat{X}_{{\rm atom}}) \cdot {\mathbf{p}}_t(\hat{X}_{{\rm atom}}) \in \mathbb{R}^{N \times N \times nf \times nf}.
\end{equation}
