This is not always a good approximation, as it assumes that the span of the PNOs of pairs ij is close enough to the span of the PNOs of pairs kl. The error resulting from building integrals from the projection approximation is defined as the "projection error." The projection error decreases as the PNO cutoff is tightened. Interestingly enough, for most terms, using the projection approximation, sometimes even repeatedly, does not induce large errors. However, for contributions to R ab ij that are linear in t ab ij , as well as certain terms involved in dressing the Fock matrix, the projection approximation cannot be applied without bringing large errors. After extensive experimentation, we have determined a set of working equations that best balance speed and accuracy, derived from the original set of equations presented in this work, transformed to the local basis. For terms that explicitly show four-center integrals, the integrals are first computed from the sparse three-center integrals through local density-fitting and LMO/PAO sparsity and stored explicitly in sparse-format as four-index quantities, while for terms that involve three-center integrals, the four-index quantities are never explicitly formed. Select terms in some equations are bolded to ease reader comprehension and highlight design choices that balance accuracy and efficiency.