Extending Nonlocal Kinetic Energy Density Functionals to Isolated Systems via a Density-Functional-Dependent Kernel

Abstract

The Wang-Teter-like nonlocal kinetic energy density functional (KEDF) in the framework of orbital-free density functional theory, while successful in some bulk systems, exhibits a critical Blanc-Cances instability [J. Chem. Phys. 122, 214106 (2005)] when applied to isolated systems, where the total energy becomes unbounded from below. We trace this instability to the use of an ill-defined average charge density, which causes the functional to simultaneously violate the scaling law and the positivity of the Pauli energy. By rigorously constructing a density-functional-dependent kernel, we resolve these pathologies while preserving the formal exactness of the original framework. By systematically benchmarking single-atom systems of 56 elements, we find the resulting KEDF retains computational efficiency while achieving an order-of-magnitude accuracy enhancement over the WT KEDF. In addition, the new KEDF preserves WT's superior accuracy in bulk metals, outperforming the semilocal functionals in both regimes.

Figures (6)

• Figure 1: Pauli energy $T_{\theta}$ and non-interacting kinetic energy $T_s$ of Gaussian densities with varying electron number $N$ and scaling parameter $\sigma$. (a) $T_\theta / T_{\mathrm{TF}}$ and (b) $T_s / T_{\mathrm{TF}}$ computed via the WT KEDF, where $T_{\mathrm{TF}}$ is the Thomas-Fermi kinetic energy. The estimated boundaries $\rho_{\mathrm{avg}}=\rho_{\mathrm{c}}$ (for $T_\theta$) and $\rho_{\mathrm{avg}}=\rho_{\mathrm{c}}^{\prime}$ (for $T_s$), as determined by characteristic density $\rho_{\mathrm{c}}$ and $\rho_{\mathrm{c}}^{\prime}$, closely align with the transition between positive and negative energies. The critical particle number $N_{\alpha, \beta}$ predicted by Blanc and Cancès demonstrates quantitative agreement with numerical results. (c, d) Corresponding $T_\theta / T_{\mathrm{TF}}$ ($T_s / T_{\mathrm{TF}}$) for the ext-WT KEDF. The consistent satisfaction of $\zeta[\rho] > \rho_{\mathrm{c}}$ ($\zeta[\rho] > \rho_{\mathrm{c}}^{\prime}$) across all $N$ and $\sigma$ ensures non-negative energies. The $\sigma$-invariance of these ratios confirms strict adherence to the scaling law.
• Figure 2: Charge density profiles for selected atoms. (a, d) H and He densities obtained with bare Coulomb potentials, with insets highlighting nuclear cusp behavior. (b, e) Al and Si using BLPS. (c, f) Cu and Zn employing HQLPS. Insets in (b, c, e, f) depict the spatial product $V_{\theta}(\mathbf{r})\rho(\mathbf{r})$.
• Figure 3: The Kato’s nuclear cusp condition of (a) H and (b) He, as obtained by KSDFT, and GE2, LKT, WT, and ext-WT KEDFs. The charge densities obtained by ext-WT KEDF satisfy the condition $\lim_{\mathbf{r}\to \mathbf{R}_i}{\frac{|\nabla\rho(\mathbf{r})|}{2Z_i\rho(\mathbf{r})}} \sim 1$, where $\mathbf{R}_i$ and $Z_i$ denote the nuclear coordinate and atomic numbers, respectively.
• Figure 4: Comparison of the density functional $\zeta[\rho]$, characteristic density $\rho_{\mathrm{c}}$, and average charge density $\rho_{\mathrm{avg}}$ across single-atom systems employing (a) bare Coulomb potentials, (b) BLPS, and (c) HQLPS. Background color maps represent $T_\theta/T_{\mathrm{TF}}$ ratios from WT KEDF calculations across $\rho_0$ values, the dark-blue dashes specify $\rho_{T_\theta=0}$, which render $T_\theta=0$. All data were derive from the charge densities obtained by ext-WT KEDF.
• Figure 5: Dependence of the total energy on (a) the simulation cell size $L$ and (b) the parameter $\kappa$ for an Al atom. In (a), the relative energy change $\Delta E(L) = E(L) - E(L=30)$ is shown for KSDFT, GE2, LKT, WT, and ext-WT KEDFs at $L=$ 30, 32, 34, and 36 a.u. In (b), the total energy of the ext-WT KEDF is plotted across the range $\kappa=0.5$ to 1.5, with the black and yellow lines representing the corresponding results from KSDFT and the WT KEDF, respectively. The red star marks the adopted value $\kappa\approx0.832$.