Local volume effects in the generalized pseudopotential theory

The generalized pseudopotential theory (GPT) is a powerful method for deriving real-space transferable interatomic potentials. Using a coarse-grained electronic structure, one can explicitly calculate the pair ion-ion and multi-ion interactions in simple and transition metals. While successful in determining bulk properties, in central force metals the GPT fails to describe crystal defects for which there is a significant local volume change. A previous paper [J. A. Moriarty and R. Phillips, Phys. Rev. Lett. 66, 3036 (1991)] found that by allowing the GPT total energy to depend upon some spatially averaged local electron density, the energetics of vacancies and surfaces could be calculated within experimental ranges. In this paper, we develop the formalism further by explicitly calculating the forces and stress tensor associated with this total energy. We call this scheme the adaptive GPT (aGPT) and it is capable of both molecular dynamics (MD) and molecular statics. We apply the aGPT to vacancy formation, divacancy binding, and stacking faults in hcp Mg. We also calculate the local electron density corrections to the bulk elastic constants and phonon dispersion for which there is refinement over the baseline GPT treatment. In addition, we demonstrate aGPT-MD simulation through the calculation of thermal expansion in magnesium to 700 K.

Figure 1

GPT cohesion curve $E_{\mathrm{coh}}=E_{\mathrm{tot}}/N$ and volume term $E_{\mathrm{vol}}$ for Eqs. (1) and (2), as calculated from first principles for the simple metal Mg. Here ${\mathrm{\Omega }}_0=156.8\mathrm{a}.\mathrm{u}.$ is the observed equilibrium volume and $n_0=Z/{\mathrm{\Omega }}_0$ is the corresponding average valence electron density for the bulk, with $Z=2$.

Figure 2

The radial valence electron density $u\left(r\right)=4\pi r^{2}n\left(r\right)$ for the $3s$ and $3p$ bands in Mg for the pseudoatom $n=n_{\mathrm{pa}}$ (blue) in the bulk metal at $\mathrm{\Omega }=156.8{\mathrm{a}.\mathrm{u}.}^3$ and also for the corresponding free atom $n=n_{\mathrm{fa}}$ (checked). The pseudoatom valence electron density replicates the inner-core density oscillations of the free atom. At larger distances from the ion, the pseudoatom valence electron density is pushed outward relative to the free-atom, and has the familiar Friedel long-range screening oscillations. Also shown are the real-space screening density $n_{\mathrm{scr}}$ (green) and orthogonalization-hole density (red).

Figure 3

The spatially averaged pseudoatom density ${\overline{n}}_{\mathrm{pa}}$ (red) is calculated for Mg at the experimental room-temperature atomic volume $\mathrm{\Omega }=156.8\mathrm{a}.\mathrm{u}$. The radial derivatives (green and blue) of the spatially averaged pseudoatom density were calculated using Lagrange interpolation polynomials. All of these quantities are calculated at $\alpha =25$ and for two values of the averaging sphere radius $R_a=1.8R_{\mathrm{WS}}$ in (a) and $R_a=3.4R_{\mathrm{WS}}$ in (b).

Figure 4

Phonon dispersion relation for mechanically unstable bcc Mg at the equilibrium volume. The averaging sphere radius was taken to be $R_a=1.8R_{\mathrm{WS}}$. The aGPT phonon band structure (red) is quantitatively similar to the GPT (blue) deviating only in the imaginary sector between high-symmetry points $\mathrm{\Gamma }$ and $N$. The DFT data (black points) is in good agreement with the aGPT/GPT results. The small qualitative difference in the imaginary sector along $\mathrm{\Gamma }-N$ is associated with subtle differences in pressure.

Figure 5

Phonon dispersion relation for hcp Mg at the equilibrium volume and $c/a$ ratio. The averaging sphere radius was taken to be $R_a=1.8R_{\mathrm{WS}}$. The aGPT phonon bandstructure (red) is quantitatively similar to the GPT (blue). Both the aGPT and GPT are in good qualitative agreement with the DFT data (black dots).

Figure 6

The $\gamma$ line calculated for hcp Mg using the GPT (green), aGPT (blue), and compared to the DFT results (red) of Yin et al. [32]. For comparison, the GPT and aGPT was calculated using the DFT zero-temperature lattice parameters. The crystal was tilted along the $\left[1\overline{1}00\right], 1/3\left[\overline{1}2\overline{1}0\right], 1/2\left[\overline{1}102\right]$, and $1/3\left[\overline{2}113\right]$ in the Basal, Prism I, Pyramidal I, and II crystallographic planes, respectively. The aGPT and GPT are in agreement for the stable stacking fault energy. The aGPT increases the value of the unstable stacking fault relative to the GPT.

Figure 7

Calculation of the hcp Mg thermal expansion at fixed axial ratio by molecular dynamics. The volume for hcp Mg was averaged over a canonical isobaric isothermal ($\mathit{NPT}$) ensemble using the GPT and aGPT at three temperatures. The graph shows the change in atomic volume relative to the zero temperature value. The error bars for the 95% confidence interval were found using a bias-free estimator of the variance.