O An O Num O COM O COC O Far xx 0.00 0.00 0.68 0.52 0.02 yy 0.01 0.01 1.94 1.84 0.01 zz 0.00 0.00 1.47 1.09 0.05 The matching of the LG and LG(OI) specific rotation with O An and O Num is not due to cancellation of error, but to the matching of the individual diagonal elements of the β tensor, as shown by the % MURE results in Table II. The errors for the individual molecules are shown in Figure of the SI, and the raw data are collected in Tables S1-S4 of the SI. Furthermore, the analytical and numerical procedures lead to the same position and orientation of the molecules: this is shown in Figure , which reports the distance of O An and O Num from the COM, and in Figure in the SI, which reports the overlaid geometry of each molecule obtained with the two procedures. Even for molecule 15, Figure in the SI shows that the two structures are essentially overlapped. The difference is likely due to a small origin dependence of the OR along the x axis direction, which the numerical procedure is unable to fully capture. In principle, an exact matching could be obtained by improving the numerical algorithm by implementing analytical gradients and further tightening the thresholds for convergence, but this is beyond the scope of this work. In general, these results show that the convergence of the β tensor towards the LG(OI) values is faster than the convergence of the position and orientation of the molecule in space. Note that O An is rather close to the COM (within 1 Å for each molecule), but the latter is not necessarily a good choice of origin for every molecule, as shown in Table and with those in Table ). This is because the elements of the β tensor are signed quantities and a small error in the diagonal elements may lead to a larger error in the trace. Although the numerical procedure starting from the COM consistently leads to an orientation of the molecules that matches that obtained with the analytical procedure, other choices of origin and orientation can also lead to perfect matching of the diagonal elements of the LG β tensor with the corresponding LG(OI) values. These geometries were found by starting the numerical procedure with each molecule far away from the origin of the coordinate system (as explained in Section II, the COM was arbitrarily shifted by 10 Å in every Cartesian direction). The numerical procedure found a new origin and orientation for all 15 molecules with negligible errors in the diagonal β elements, as shown in Table of theory. There are few cases where all choices of origin have small errors: Molecules 3, 4, 10, and 14. However, in other cases the difference is stark. For instance, for B3LYP the relative error for Molecule 13 falls from 25% to 1% passing from O COM /O COC to O An CAM . For CCSD, the change is even more dramatic for molecules 9 (the errors goes from 72% to 1%) and 13 (from 50% with O COM to 47% with O COC to 1% with O An CAM ). Therefore, the choice of position and orientation obtained with the analytical procedure at one level of theory seems transferable to other levels of theory.