AI 中文总结
研究在无轨道密度泛函理论中构建依赖密度的核KEDF，通过从KS参考数据学习修正，利用能量匹配等式和惩罚处理相关问题，将径向EL方程表述为单轨道本征问题，在基准测试和特定系统中取得良好结果，能转移壳层模式和半径。
AI 中文摘要
非局部动能密度泛函（KEDFs）可在无轨道密度泛函理论（OFDFT）中编码核壳结构，但自洽性需要精确的泛函导数和欧拉方程的稳定解。我们构建了一个依赖密度的核KEDF，其修正从Kohn-Sham（KS）参考数据中学习。核形状的线性产生解析的欧拉-拉格朗日（EL）响应。通过活动集重复惩罚最小二乘迭代评估拟合，使用精确的能量匹配等式和软二次不等式惩罚来处理违反规定的尾部响应和选定路径能量上升余量的情况。我们将径向EL方程表述为重新排列的单轨道本征问题。在恒定$k_F$的$^{16}$O基准测试中，它无需密度混合即可收敛，并且与虚时演化（ITE）在亚keV能量上一致。自适应步长ITE在报告的计算中给出的最终物种EL残差最多为0.07 MeV，重新排列的对角化达到相同的稳定密度。对于没有自旋-轨道或库仑项的球形$N = Z$系统，对$A = 16$到$140$的特定核拟合再现了壳层模式和半径。在三个核上训练的修正将壳层模式和半径，但不是绝对能量，转移到一个先前未见过的$A = 140$系统。
英文摘要
Nonlocal kinetic-energy density functionals (KEDFs) can encode nuclear shell structure in orbital-free density functional theory (OFDFT), but self-consistency requires accurate functional derivatives and a stable solution of the Euler equation. We construct a density-dependent-kernel KEDF whose correction is learned from Kohn-Sham (KS) reference data. Linearity in the kernel shape yields analytic Euler--Lagrange (EL) responses. The fit uses exact energy-matching equalities and soft quadratic inequality penalties for violations of prescribed tail-response and selected-path energy-rise margins, evaluated by an active-set repeated penalized least-squares iteration. We formulate the radial EL equation as the rearranged one-orbital eigenproblem. In a constant-$k_F$ $^{16}$O benchmark it converges without density mixing and agrees with imaginary-time evolution (ITE) to sub-keV energy. Adaptive-step ITE gives final species EL residuals of at most 0.07 MeV in the reported calculations, and rearranged diagonalization reaches the same stationary densities. For spherical $N=Z$ systems without spin--orbit or Coulomb terms, nucleus-specific fits for $A=16$ to $140$ reproduce shell patterns and radii. A correction trained on three nuclei transfers the shell pattern and radius, but not the absolute energy, to a previously unseen $A=140$ system.
