Systematically Improvable Numerical Atomic Orbital Basis Using Contracted Truncated Spherical Waves

Abstract

To solve the Kohn-Sham equation within the framework of density functional theory, we develop a scheme to construct numerical atomic orbital (NAO) basis sets by contracting truncated spherical waves (TSWs). The contraction minimizes the trace of the kinetic operator in the residual space, generalizing the spillage minimizing scheme [M. Chen et al., J. Phys. Condens. Matter 22, 445501 (2010); P. Lin et al., Phys. Rev. B 103, 235131 (2021)]. In addition to the systematic improvability inherited from previous schemes, the use of TSW instead of plane waves as the expansion basis bridges reference states and NAOs more effectively, and eliminates spurious interactions between periodic images, thereby enabling better transferability through the inclusion of extensive reference states. Benchmarks demonstrate that the constructed NAO achieves satisfactory precision for various properties of both molecules and bulk systems, including total energy, bond length, atomization energy, lattice constant, cohesive energy, band gap, and energy-level alignment. By incorporating unoccupied states, the improved transferability in describing the conduction band is demonstrated to be effective and substantial.

Figures (9)

• Figure 1: The workflow of NAOs generation. The total workflow is divided into 4 steps. In the first step, a PW basis set is prepared, and a set of structures is chosen. In the second step, the superparameters $r_\mathrm{c}$ and $l_{\max}$ are chosen so that the energy difference between PW and NSW $\epsilon^\mathrm{PW}_\mathrm{NSW}$ can be smaller than a given threshold. In the third step, series of SCF calculations are performed with converged NSW, yield the overlap $S_{\mu\nu}(\mathbf{k})$, kinetic energy $T_{\mu\nu}(\mathbf{k})$ matrices, and wavefunction $C_{n\mu}(\mathbf{k})$, in which $\mu$ and $\nu$ run over all NSW, $n$ denotes eigenstates, and $\mathbf{k}$ distinguishes different $\mathbf{k}$-points. In the last step, the generalized spillage (eqn. \ref{['eq:generalized-spillage']}) is minimized, during which the contraction of NSW is done.
• Figure 2: Systematically improbability of truncated spherical wave basis. The joint convergence of energy with respect to PW $\epsilon^\mathrm{PW}_\mathrm{NSW}(r_\mathrm{c}, l_{\max})$ on truncation cutoff radius $r_\mathrm{c}$ and maximal included angular momentum $l_{\max}$ of spherical waves tested in (a) Na, (b) S and (c) F diatomic molecule systems. The data points and error bars show the average values and standard deviations of $\epsilon^\mathrm{PW}_\mathrm{NSW}(r_\mathrm{c}, l_{\max})$ among a set of bond lengths. To distinguish between trajectories with different $l_{\max}$, lines are plotted with different markers, colors, and line styles. Red dashed with square: $l_{\max}=0$; Purple dotted with circle: $l_{\max}=1$; Blue solid with circle: $l_{\max}=2$; Orange solid with square: $l_{\max}=3$; Green dashed with circle: $l_{\max}=4$.
• Figure 3: Molecule properties prediction benchmarks against the NSW and PW basis. For all violin plots (a-d), the left half shows the distribution of errors of NAOs with respect to the NSW basis sets, and the right half shows the distribution of errors of NAOs with respect to the PW basis sets. The extrema and the average value are indicated by black whiskers on both sides of the violin. (a) total energy error in eV/atom, which reveals the basis set completeness. (b) bond length error in Å. (c) atomization energy error in eV/atom. (d) $\eta$-metrics in eV. (e) mean-square error with respect to the NSW and PW basis set. For figures, NAOs and NSW basis sets are distinguished by colors: blue: pVDZ; orange: pVTZ$^-$; green: pVTZ; red: pVQZ$^=$; purple: NSW. For (e), NAOs and NSW are additionally distinguished by hatches.
• Figure 4: Bulk property prediction accuracy benchmark against the PW basis. The distribution of NAO error with respect to the PW basis sets is shown in violin plots (a-g), in which the extrema and the mean error are indicated by black whiskers. (a) total energy error in eV/atom, which reveals the basis set completeness. (b) lattice constant error in Å. (c) bulk modulus error in GPa. (d) $\Delta$-metrics in meV/atom. (e) cohesive energy error in eV/atom. (f) bandgaps error in eV. (g) $\eta$-metrics in eV. (h) mean relative error statistics for properties including lattice constant, bulk moduli, cohesive energy, and bandgap. For figures (a-g), NAOs are distinguished by positions of violins and colors; for Figure (h), NAOs are distinguished by hatches and colors. Colors: blue: pVDZ; orange: pVTZ$^-$; green: pVTZ; red: pVQZ$^=$.
• Figure 5: Distribution of $\eta_{10}$ metrics with respect to PW (denoted as $\eta^\mathrm{PW}_{10}$) of pVDZ (blue), pVTZ$^-$ (orange), pVTZ (green), pVTZ-2V (purple), pVTZ-3V (brown) and pVQZ$^=$ (red) basis sets, from the bottom to the top, respectively. The extrema and average are indicated with black whiskers. Detailed data can be found in Table \ref{['tab:crystal-eta10']}.