Various post-processing modifications of this PIPs basis are possible. One is what we term "purifying". This is useful for PESs describing fragmentation of a molecule. For such PESs there should be no mathematical coupling of the fragments in the limit of large separation between them. In this limit all intermolecular Morse variables go to zero and so correctly uncouple polynomials in terms of them. For each fragment there are polynomials that contain only the corresponding intramolecular Morse variables; however, there are also some polynomials that contain intramolecular Morse variables for more than one fragment.