Neural networks offer an unbiased and numerically very accurate approach to represent high-dimensional ab initio potential-energy surfaces. Once constructed, neural network potentials can provide the energies and forces many orders of magnitude faster than electronic structure calculations, and thus enable molecular dynamics simulations of large systems. However, Cartesian coordinates are not a good choice to represent the atomic positions, and a transformation to symmetry functions is required. Using simple benchmark systems, the properties of several types of symmetry functions suitable for the construction of high-dimensional neural network potential-energy surfaces are discussed in detail. The symmetry functions are general and can be applied to all types of systems such as molecules, crystalline and amorphous solids, and liquids.

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\begin{equation*}f_{\mathrm{c}}(R_{ij})=\Bigg \lbrace \begin{array}{lc@{\hskip-6pt}c}\tanh ^3\left[1-\frac{R_{ij}}{R_{\mathrm{c}}}\right] &\quad \mbox{for} &\quad R_{ij}\le R_{\mathrm{c}}\\[6pt]0 &\quad \mbox{for} &\quad R_{ij} > R_{\mathrm{c}}. \end{array}\end{equation*}
fc(Rij)={tanh31RijRcforRijRc0forRij>Rc.
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