Many-Body Permutationally Invariant Polynomial Neural Network Potential Energy Surface for N₄

A potential energy surface (PES) for high-energy collisions between nitrogen molecules is useful for modeling chemical dynamics in shock waves and plasmas. In the present work, we fit the many-body (MB) component of the ground-state PES of N4 to an analytic function using neural networks (NNs) with permutationally invariant polynomials (PIPs). The fitting dataset of the N4 system is an extension of one used previously, extended with 4859 new CASPT2 points and 13 new CCSD(T) points to reach a total of 21 406 points. The MB-PIP-NN fit required a very complete coverage of the geometry domain in order to get a physical fit, and we devised several tactical steps to achieve this, including trajectory calculations, the comparison of the NN fit with PIP fits with mixed-exponential Gaussian bond order variables, and searching geometry regions with sparse data coverage. With these efforts, the final dataset is more suitable for a NN fit. The energy range of the dataset is much wider than those used in other systems previously fitted by NNs, and there are more rugged surface regions than usual due to locally avoided crossings. The performance of the new MB-PIP-NN fit is compared to that of another new fit to the same data by least-squares methods not employing NNs, and the mean unsigned deviation from the electronic structure calculations is reduced by a factor of 3, although the computing time to calculate a force for a trajectory calculation is larger by an order of magnitude. The accuracy achievable with the NN fit for this difficult system is very impressive, and we anticipate that the NN method can give useful fits to difficult cases that cannot be achieved by more conventional methods.