If the value of the integral is greater than a given tolerance T CU T DO , then the PAO µ is included in the domain of LMO i. The PAOs included in the domain of a pair ij are the union of the PAOs included in the domain of LMO i combined with the PAOs in the domain of LMO j. After the PAOs in the pair domain of pair ij are determined, linear dependencies are removed through an algorithm like Gram-Schmidt or Partial Cholesky, and then the resulting space is transformed into a canonical basis (forming a diagonal Fock matrix, and thus orbital energies for these transformed versions of the virtual orbitals for LMO pair ij).