Title: Fully numerical calculations on atoms with fractional occupations. Range-separated exchange functionals Show Chinese title

Title: 原子上具有分数占据的完全数值计算 范围分离的交换泛函

Abstract: A recently developed finite element approach for fully numerical atomic structure calculations [S. Lehtola, Int. J. Quantum Chem. 119, e25945 (2019)] is extended to the description of atoms with spherically symmetric densities via fractionally occupied orbitals. Specialized versions of Hartree-Fock as well as local density and generalized gradient approximation density functionals are developed, allowing extremely rapid calculations at the basis set limit on the ground and low-lying excited states even for heavy atoms. The implementation of range-separation based on the Yukawa or complementary error function (erfc) kernels is also described, allowing complete basis set benchmarks of modern range-separated hybrid functionals with either integer or fractional occupation numbers. Finally, computation of atomic effective potentials at the local density or generalized gradient approximation levels for the superposition of atomic potentials (SAP) approach [S. Lehtola, J. Chem. Theory Comput. 15, 1593 (2019)] that has been shown to be a simple and efficient way to initialize electronic structure calculations is described. The present numerical approach is shown to afford beyond microhartree accuracy with a small number of numerical basis functions, and to reproduce literature results for the ground states of atoms and their cations for $1 \leq Z \leq 86 $. Our results indicate that the literature values deviate by up to 10 {\mu}Eh from the complete basis set limit. The numerical scheme for the erfc kernel is shown to work by comparison to results from large Gaussian basis set calculations from the literature. Spin-restricted ground states are reported for Hartree-Fock and Hartree-Fock-Slater calculations with fractional occupations for $1 \leq Z \leq 118$. Show Chinese abstract

Abstract: 一种最近开发的用于完全数值原子结构计算的有限元方法 [S. Lehtola, Int. J. Quantum Chem. 119, e25945 (2019)] 被扩展以通过分数占据轨道描述具有球对称密度的原子。 专门化的哈特里-福克以及局部密度和广义梯度近似密度泛函被开发出来，使得即使对于重原子，也能在基组极限下对基态和低激发态进行极其快速的计算。 还描述了基于尤卡夫或互补误差函数（erfc）核的范围分离实现，允许使用整数或分数占据数对现代范围分离杂化泛函进行完全基组基准测试。 最后，描述了用于原子势叠加（SAP）方法 [S. Lehtola, J. Chem. Theory Comput. 15, 1593 (2019)] 的原子有效势计算，该方法已被证明是初始化电子结构计算的简单而有效的方法。 目前的数值方法被证明可以在使用少量数值基函数的情况下达到微哈特里精度，并能再现文献中关于原子及其阳离子基态的结果，对于$1 \leq Z \leq 86 $。 我们的结果表明，文献值与完全基组极限之间的偏差可达 10 {\mu }Eh。 通过与文献中大型高斯基组计算结果的比较，展示了 erfc 核的数值方案是有效的。 报告了哈特里-福克和哈特里-福克-斯莱特计算中分数占据情况下的自旋限制基态，针对$1 \leq Z \leq 118$。