Due to the sparsity, as well as the robustness of the PAO space in representing the virtual space, earlier local correlation methods, PAOs are assigned to LMOs based on their overlap. The local nature of PAOs effectively give each LMO a constant number of PAOs with increasing system size, so these methods form the first generation of asymptotically linear-scaling local correlation methods. Though PAO-based local correlation methods have fallen out of favor due to their high crossover points with canonical correlated methods, and have been replaced by PNO-based methods, the formation of PAOs remains a key step in most PNO-based local correlation methods, including DLPNO. In these methods, PAOs are used in a "cascaded" approach where PAOs are assigned to LMOs to form an initial virtual space before PNOs are formed from the union of the PAOs in each pair. This allows for linear-scaling generation of necessary integrals and intermediates, and eliminates the need for using the non-local canonical virtual space altogether. Following the work of Pinski, Riplinger, Valeev, and Neese, PAOs are assigned to each LMO, through the "differential overlap integral" (DOI), a metric for spatial overlap inspired by the Cauchy-Schwarz inequality