If the computed value of the DOI is greater than a given tolerance, parametrized as T CUT DO (default 0.01), then the PAO µ is included in the domain of LMO i. For the pair domain of ij (denoted µ ij ), the PAOs of domain i ( µ i ) are merged with the PAOs of domain j ( µ j ) to form the larger pair domain. Analagously, for triplet ijk, the PAOs of LMOs i, j, and k are merged to form the triples domain. Unfortunately, for PAO-based local correlation methods, there is a high crossover with these methods and their canonical counterparts, as the PAOs do not form the most sparse representation of the virtual space.