High accuracy and transferability of a neural network potential through charge equilibration for calcium fluoride

We investigate the accuracy and transferability of a recently developed high-dimensional neural network (NN) method for calcium fluoride, fitted to a database of ab initio density functional theory (DFT) calculations based on the Perdew-Burke-Ernzerhof (PBE) exchange correlation functional. We call the method charge equilibration via neural network technique (CENT). Although the fitting database contains only clusters (i.e., nonperiodic structures), the NN scheme accurately describes a variety of bulk properties. In contrast to other available empirical methods the CENT potential has a much simpler functional form, nevertheless it correctly reproduces the PBE energetics of various crystalline phases both at ambient and high pressure. Surface energies and structures as well as dynamical properties derived from phonon calculations are also in good agreement with PBE results. Overall, the difference between the values obtained by the CENT potential and the PBE reference values is less than or equal to the difference between the values of local density approximation (LDA) and Born-Mayer-Huggins (BMH) with those calculated by the PBE exchange correlation functional.

Figure 1

Distribution of the errors in the total energies per atom with respect to reference DFT calculations for all data points in the training and validation sets. The number of data points are normalized such that the total number of data points gives 100%. In total, the training and validation sets contain 2379 and 419 structures, respectively.

Figure 2

Unit cell of the four different ${\mathrm{CaF}}_2$ polymorphs: (a) cubic fluorite $\left(Fm\overline{3}m\right)$, (b) $P4/mmm$, (c) $Pmc{2}_1$, and (d) $Pnma$. The blue (large) and purple (small) spheres denote Ca and F atoms, respectively.

Figure 3

Enthalpy differences per atom with respect to the $Fm\overline{3}m$ phase as a function of pressure for three different polymorphs, using CENT (a), PBE (b), LDA (c), and BMH (d).

Figure 4

Three different low-index surfaces of ${\mathrm{CaF}}_2$ shown from the side along the $a$ axis (denoted by a) and the top along the $c$ axis (denoted by b) after a local relaxation. The blue (large) and purple (small) spheres denote Ca and F atoms, respectively.

Figure 5

Heat capacity of ${\mathrm{CaF}}_2$ in four different phases: (a) cubic fluorite $\left(Fm\overline{3}m\right)$, (b) $\left(Pnma\right)$, (c) $Pmc{2}_1$, and (d) $P4/mmm$.

Figure 6

Free energy of ${\mathrm{CaF}}_2$ in four different phases: (a) cubic fluorite $\left(Fm\overline{3}m\right)$, (b) $\left(Pnma\right)$, (c) $Pmc{2}_1$, and (d) $P4/mmm$.

Figure 7

Phonon dispersion of ${\mathrm{CaF}}_2$ in the cubic fluorite $\left(Fm\overline{3}m\right)$ structure. The results from the CENT, LDA, and BMH are shown in the panels (a), (b), and (c), respectively. The red (dashed) lines correspond to the reference data obtained with PBE.