LG(OI) ω . However, the question remains whether there are alternative choices of origin and molecular orientation where β αα = βαα for all diagonal elements. This would require that the second terms in each equation in Eq. 10 are simultaneously zero, which is a problem with six unknowns and three equations. To seek out these alternative spatial orientations of the molecules, we wrote a Python script that attempts the search numerically. The script uses a Broyden-Fletcher-Goldfarb-Shannon (BFGS) optimization module in SciPy 74 to determine the necessary shift and rotation. This algorithm seeks to minimize the root mean squared difference (RMSD) between the diagonal elements and the trace of the LG and LG(OI) β tensors by performing consecutive rotations and shifts of the coordinate system. Gradients of the RMSD are calculated with numerical differentiation of the four target variables (i.e., the tensor diagonal elements and trace) with respect to rigid translations and rotations. After some initial testing, the displacement for the numerical derivative was set to 10 -4 Å for translations and 10 -4 rad for rotations. The trust radius for the displacements during the optimization was set to 10 -1 Å and 10 -1 rad at the beginning of the procedure, and incrementally decreased close to convergence. Convergence was reached when the RMSD for each variable was below 10 -8 . We performed two sets of orientation optimization, one starting from the origin at the center of mass (COM) and orientation equivalent to that at O An , and a second one starting from the same orientation but arbitrarily translating the COM by 10 Å in each Cartesian direction. In the following, we call the origin found by the numerical procedure with the two starting points O Num and O Far , respectively. OR values are computed in the LG and LG(OI) for a set of 15 rigid, organic molecules (see Figure ) using CAM-B3LYP and the aug-cc-pVDZ basis set. The initial geometries were obtained from Refs. 55,56,58. All calculations are performed in the gas-phase. The OR is calculated with different choices of origin: O An , O Num , O Far , O COM , and O COC . The first three are described in Section II. O COM indicates that the COM is placed at the origin and the molecular orientation is obtained from the analytical procedure; O COC indicates that the center of nuclear charge (which is obtained as the COM but using the nuclear charges rather than masses) is placed at the origin and the orientation is the same as for O COM .