基于埃尔米特插值多项式的原子电子结构计算
Atomic electronic structure calculations with Hermite interpolating polynomials
AI总结：
本文研究原子电子结构计算中形函数的选择，提出一阶及高阶埃尔米特插值多项式（HIPs）的稳定实现，证明其优于拉格朗日插值多项式且能避免局域动能的不连续性，并探讨了其对meta-GGA泛函计算和数值原子轨道基组构造的影响。
摘要：
我们最近描述了在有限元方法中实现原子电子结构计算的工作，采用形式为 $χ_μ(r)=r^{-1}B_μ(r)$ 的数值径向基函数，其中高阶拉格朗日插值多项式（LIPs）被用作形函数 $B_μ(r)$ [S. Lehtola, Int. J. Quantum Chem. 119, e25945 (2019)]。在本工作中，我们讨论了如何在较小的 $r$ 处以稳定的方式计算 $χ_μ(r)$，并重新审视了形函数 $B_μ(r)$ 的选择。我们考虑了三种形函数：除了 $\mathcal{C}^{0}$ 连续的 LIPs 之外，我们还考虑了一阶埃尔米特插值多项式（HIPs）的解析实现，这些 HIPs 是 $\mathcal{C}^{1}$ 连续的，以及以底层高阶 LIP 基表示的 $n$ 阶（$\mathcal{C}^{n}$ 连续）HIPs 的数值实现。借助新的实现，我们证明了一阶 HIPs 即使在节点数量很大的情况下也是可靠的，并且它们也适用于非均匀单元网格，在原子电子结构计算中比具有相同基函数总数的 LIPs 提供了更好的结果。我们证明了在较小的 LIP 基组中，自旋-$σ$ 局域动能 $τ_σ$ 中可以观察到不连续性，而 HIP 基组则不存在此类问题；然而，使用任一基组都可以达到具有平滑 $τ_σ$ 的完备基组极限。此外，我们讨论了 HIPs 对使用多种近期 meta-GGA 泛函进行 meta-GGA 泛函计算的影响，并发现大多数 Minnesota 泛函表现不佳。我们还研究了 HIPs 中对导数的显式控制在构造数值原子轨道基组方面的潜在用途，但发现约束势可能仍然是更好的选择。
英文摘要：
We have recently described the implementation of atomic electronic structure calculations within the finite element method with numerical radial basis functions of the form $χ_μ(r)=r^{-1}B_μ(r)$, where high-order Lagrange interpolating polynomials (LIPs) were used as the shape functions $B_μ(r)$ [S. Lehtola, Int. J. Quantum Chem. 119, e25945 (2019)]. In this work, we discuss how $χ_μ(r)$ can be evaluated in a stable manner at small $r$ and also revisit the choice of the shape functions $B_μ(r)$. Three kinds of shape functions are considered: in addition to the $\mathcal{C}^{0}$ continuous LIPs, we consider the analytical implementation of first-order Hermite interpolating polynomials (HIPs) that are $\mathcal{C}^{1}$ continuous, as well as numerical implementations of $n$-th order ($\mathcal{C}^{n}$ continuous) HIPs that are expressed in terms of an underlying high-order LIP basis. Furnished with the new implementation, we demonstrate that the first-order HIPs are reliable even with large numbers of nodes and that they also work with non-uniform element grids, affording even better results in atomic electronic structure calculations than LIPs with the same total number of basis functions. We demonstrate that discontinuities can be observed in the spin-$σ$ local kinetic energy $τ_σ$ in small LIP basis sets, while HIP basis sets do not suffer from such issues; however, either set can be used to reach the complete basis set limit with smooth $τ_σ$. Moreover, we discuss the implications of HIPs on calculations with meta-GGA functionals with a number of recent meta-GGA functionals, and find most Minnesota functionals to be ill-behaved. We also examine the potential usefulness of the explicit control over the derivative in HIPs for forming numerical atomic orbital basis sets, but find that confining potentials are still likely a better option.
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