In our working equations for DLPNO-CCSD, the terms have a slightly different structure than the original set of working equations from canonical T1-transformed DF-CCSD. One notable modification is the expansion of the T1-dressed integrals and the removal of the leading two-virtual integrals in equations 83 and 84 from their canonical counterparts and expressing their contributions explicitly in equations 78 and 79. This is done since PNO projection errors are greatest in terms containing linear doubles amplitude contributions to the doubles residual. In addition, even though the R ab ij and R a i residuals are only updated over strong pairs, weak MP2 pairs also contribute to the residual of strong pairs. Because of this, the DF integrals from equations 91-93 are only constructed over strong pairs to save memory costs. Therefore, the T1-dressed integrals are expanded explicitly in other parts of the working equations as well, such as in equations 88 and 89. Certain Fock matrix contributions also have a unique form. Equation 95 is not constructed from equation 99, since the former is looped over strong pairs and the latter is looped over both strong pairs and weak LMP2 pairs. Finally, equation 101 is built with the explicit DF integrals, and not through projecting integrals of type (kc kk |a kk b kk ) (errors too large) or building integrals of type (kc kk |a ij b ij ) (too expensive to store) as a trade off between speed and accuracy. For the sake of absolute clarity, and reproducibility, the energy expression we used was