Application of the neural networks for developing new parametrization of the Tersoff potential for carbon

Penta-graphene (PG) is a 2D carbon allotrope composed of a layer of pentagons having sp2- and sp3- bonded carbon atoms. A study carried out in 2018 has shown that the parameterization of the Tersoff potential proposed in 2005 by Ehrhart and Able (T05 potential) performs better than other potentials available for carbon, being able to reproduce structural and mechanical properties of the PG. In this work, we tried to improve the T05 potential by searching for its parameters giving a better reproduction of the structural and mechanical properties of the PG known from the ab initio calculations. We did this using Molecular Statics (MS) simulations and Neural Network (NN). Our test set consisted of the following structural properties: the lattice parameter a; the interlayer spacing h; two lengths of C-C bonds, d1 and d2 respectively; two valence angles, θ1 and θ2, respectively. We also examined the mechanical properties by calculating three elastic constants, C11, C12 and C66, and two elastic moduli, the Young’s modulus E and the Poisson’s ratio v. We used MS technique to compute the structural and mechanical properties of PG at T =0 K. The Neural Network used is composed of 2 hidden layers, with 20 and 10 nodes for the first and second layer, respectively. We used an Adams optimizer for the NN optimization and the Mean Squared Error as the loss function. We obtained inputs (about 80 000 different sets of potential parameters) for the Molecular Statics simulation by using randomly generated numbers. The outputs from these simulations became the inputs to our Neural Network. The Molecular Statics simulations were done with LAMMPS while the Neural Network and other computations were done with Python, Pytorch, Numpy, Pandas, GNUPLOT and Bash scripts. We obtained a parameterization which has a slightly better accuracy (lower relative errors of the calculated structural and mechanical properties) than the original parameterization.