For O3, since we want many-body part of the DPEM elements to decay to zero, 𝑟 D.!!,% should be defined such that when the system becomes O2 + O, in which one of the atoms is far from the other two, 𝑓 & 9𝑟 D.!!,% ;𝑓 ' 9𝑟 D.!!,% ; approaches zero. We previously 4,28,39 defined 𝑟 D.!!,% as the second largest of the three diatomic distances. This works because when O is dissociated from O2, two of the three O-O distances should be big. Hence, using the second largest distance is a natural choice to indicate if the system has entered the asymptotic region. However, such a choice makes 𝑟 D.!!,% a discontinuous function. Therefore, in the present work we have re-defined