been discussed in Refs. , the lower limit of M can be determined using the Molien's series of the complete nuclear permutation and inversion (CNPI) group of the system. Practically, the required PIPs under the order of M can be obtained by the software, MAGMA only a reduced permutational symmetry is included, the PIPs contain 8040 terms at a maximum polynomial order of 3, and 95 965 at a maximum polynomial order of 4. One can use only the nonredundant PIPs as the input, as many PIPs are not independent. For the H + H2S system, there are 9 non-redundant PIPs from 23 3 th -order PIPs. 170 For a molecular system with a given symmetry, all invariant polynomials can be derived from a finite generating invariant polynomials, 195 which are called fundamental invariants (FIs) by Prof. Zhang and his group. Correspondingly, FIs can be used as the input of NN and the approach is FI-NN. The size of the FIs is much smaller than that of PIPs, making FI-NN more efficient in evaluation than PIP-NN. It is extremely hard to apply the polynomial-based NN methods, PIP-NN, non-redundant PIP-NN, and FI-NN, to a molecular system with about 10 atoms. This "10 atom limit" is broken by the fragmented PIP method, which relies on the fact that the Morse variables are close to zero at large internuclear distances and thus can be discarded. It's hard to remove those basis polynomials with small Morse variables from the total basis as generating the full fitting basis is prohibitive for large systems. Alternatively, one can directly fragment the large target molecule into a few smaller parts for which the PIP basis can be efficiently generated and then the union of these fragmented bases was used as the entire basis of the system. Details of the fragmented PIP method can be found in the literature. The fragmented PIP method significantly decreases the sizes of PIPs, which can be adopted in the PIP-NN or FI-NN methods for fitting PESs of large complex molecules but awaits further validation in the future work.