Ab initio quality neural-network potential for sodium

An interatomic potential for high-pressure high-temperature (HPHT) crystalline and liquid phases of sodium is created using a neural-network (NN) representation of the ab initio potential-energy surface. It is demonstrated that the NN potential provides an ab initio quality description of multiple properties of liquid sodium and bcc, fcc, and cI16 crystal phases in the $P\text{−}T$ region up to 120 GPa and 1200 K. The unique combination of computational efficiency of the NN potential and its ability to reproduce quantitatively experimental properties of sodium in the wide $P\text{−}T$ range enables molecular-dynamics simulations of physicochemical processes in HPHT sodium of unprecedented quality.

Figure 1

Experimental phase diagram of sodium (Ref. 1).

Figure 2

Normalized cumulative histogram of the absolute NN errors in the training and test sets. It demonstrates that the fitting error is less than 1 meV/atom for 83% of structures and less than 2 meV/atom for 95% of structures in the test set. The inset shows the ordinary histogram for the same data. Only a few structures in the test and train sets have errors larger than 3 meV/atom.

Figure 3

Diffusion coefficient at $T=490\text{K}$ as a function of the inverse box length. Circles show apparent diffusion coefficients with the linear-fit represented by the solid line. Squares show the diffusion coefficients $D\left(\infty \right)$ corrected for finite-size effects with Eq. (1). The constant dashed line marks the value of $D\left(\infty \right)$.

Figure 4

Pressure dependence of the elastic coefficients. Points (lines) represent NN (DFT) data.

Figure 5

(Color) Pressure dependence of the phonon-dispersion curves. Red (black) lines correspond to NN (DFT) frequencies, triangles represent experimental data (Ref. 47).

Figure 6

(Color) Stability of crystal phases of sodium relative to the fcc phase. Solid (dashed) lines represent NN (DFT) results.

Figure 7

Distortion $x$ for cI16 as a function of compression $U$.

Figure 8

(Color online) Thermal expansion of sodium liquid at zero pressure. The EAM curve is from Ref. 17, the experimental curve is from Ref. 48.

Figure 9

Pressure dependence of the volumetric thermal-expansion coefficient of sodium liquid. The line is calculated using Eq. (2).

Figure 10

Isothermal compressibility of liquid sodium at zero pressure. The experimental curve is from Ref. 50, the dashed lines show the uncertainty in the experimental data.

Figure 11

(Color online) Radial-distribution functions of highly compressed liquid sodium. The lower curve is calculated at $\rho =2.8\text{g}/{\text{cm}}^3$ and $T=1130\text{K}$, the upper curve is obtained at $\rho =3.7\text{g}/{\text{cm}}^3$ and $T=930\text{K}$ and is shifted up by 1. The ab initio results are from Ref. 51.