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.
In the present work, we develop a highly accurate, fifteen-dimensional potential energy surface (PES) of CH4 interacting on a rigid flat Ni(111) surface with the methodology of neural network (NN) fit to a database consisted of about 194 208 ab initio density functional theory (DFT) energy points. Some careful tests of the accuracy of the fitting PES are given through the descriptions of the fitting quality, vibrational spectrum of CH4 in vacuum, transition state (TS) geometries as well as the activation barriers. Using a 25-60-60-1 NN structure, we obtain one of the best PESs with the least root mean square errors: 10.11 meV for the entrance region and 17.00 meV for the interaction and product regions. Our PES can reproduce the DFT results very well in particular for the important TS structures. Furthermore, we present the sticking probability S0 of ground state CH4 at the experimental surface temperature using some sudden approximations by Jackson's group. An in-depth explanation is given for the underestimated sticking probability.
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