A more flexible neural network (NN) method using the fundamental invariants (FIs) as the input vector is proposed in the construction of potential energy surfaces for molecular systems involving identical atoms. Mathematically, FIs finitely generate the permutation invariant polynomial (PIP) ring. In combination with NN, fundamental invariant neural network (FI-NN) can approximate any function to arbitrary accuracy. Because FI-NN minimizes the size of input permutation invariant polynomials, it can efficiently reduce the evaluation time of potential energy, in particular for polyatomic systems. In this work, we provide the FIs for all possible molecular systems up to five atoms. Potential energy surfaces for OH3 and CH4 were constructed with FI-NN, with the accuracy confirmed by full-dimensional quantum dynamic scattering and bound state calculations.

A global potential energy surface for the H2 + OH ↔ H2O + H reaction has been constructed using the neural networks method based on ~17,000 ab initio energies calculated at UCCSD(T)-F12a/AVTZ level of theory. Time-dependent wave packet calculations showed that the new potential energy surface is very well converged with respect to the number of ab initio data points, as well as to the fitting process. Various tests revealed that the new surface is considerably more smooth and accurate than the existing YZCL2 and XXZ surfaces, representing the best available potential energy surface for the benchmark four-atom system. Equally importantly, the number of ab initio energies required to obtain the well converged potential energy surface is rather limited, indicating the neural network fitting is a powerful method to construct accurate potential energy surfaces for polyatomic reactions.