A critical comparison of neural network potentials for molecular reaction dynamics with exact permutation symmetry

The availability of accurate full-dimensional potential energy surfaces (PESs) is a mandatory condition for efficient computer simulations of molecular systems. Much effort has been devoted to developing reliable PESs with physically sound properties, such as the invariance of the energy with respect to the permutation of chemically identical atoms. In this work, we compare the performance of four neural network (NN)-based approaches with a rigorous permutation symmetry for fitting five typical reaction systems: OH + CO, H + H2S, H + NH3, H + CH4 and OH + CH4. The methods can be grouped into two categories, invariant polynomial based NNs and high-dimensional NN potentials (HD-NNPs). For the invariant polynomial based NNs, three types of polynomials, permutation invariant polynomials (PIPs), non-redundant PIPs (NRPIPs) and fundamental invariants (FIs), are used in the input layer of the NN. In HD-NNPs, the total energy is the sum of atomic contributions, each of which is given by an individual atomic NN with input vectors consisting of sets of atom-centered symmetry functions. Our results show that all methods exhibit a similar level of accuracy for the energies with respect to ab initio data obtained at the explicitly correlated coupled cluster level of theory (CCSD(T)-F12a). The HD-NNP method allows study of systems with larger numbers of atoms, making it more generally applicable than invariant polynomial based approaches, which in turn are computationally more efficient for smaller systems. To illustrate the applicability of the obtained potentials, quasi-classical trajectory calculations have been performed for the OH + CH4 → H2O + CH3 reaction to reveal its complicated mode specificity.