To describe excited states, the electron density alone being insufficient, we use the noninteracting reference density matrix $γ_{s}({\bf x},{\bf x}')$ based on the recently established foundation for the $ΔSCF$ theory, in which ground and excited state energies and densities are obtained from the minimum and stationary solutions of the same functional. We now extend the theory to fractional charges. Based on the exact properties of degeneracy and size consistency, we show that the exact energy functional for fractional charges, expressed as a linear combination of the $γ_{s}$ of an $N-$electron and that of an $\left(N+1\right)-$electron excited state, is a straight line interpolating the energies at integers. We introduce the concepts of excited-state chemical potentials to describe the slopes of these linear lines. Numerical calculations reveal the excited-state delocalization error with common approximate functionals but good performance of corrected functionals on the proven linear conditions.
Excited States from Quasiparticle Hamiltonian Based on Density Functional Theory
Duke University（杜克大学）；Department of Chemistry, Duke University（杜克大学化学系）；Department of Chemistry and Department of Physics, Duke University（杜克大学化学系和物理系）
Excited state orbital optimization via minimizing the square of the gradient: General approach and application to singly and doubly excited states via density functional theory
Fully numerical calculations on atoms with fractional occupations. Range-separated exchange functionals
最近发展的一种全数值原子结构计算的有限元方法[S. Lehtola，Int. J. Quantum Chem. 119，e25945（2019）]扩展到通过分数占据轨道描述具有球对称密度的原子。专门版本的Hartree-Fock以及局域密度和广义梯度近似密度泛函的开发，允许非常快速的计算在基组限制的地面和低洼的激发态，甚至重原子。还描述了基于Yukawa或互补误差函数（erfc）内核的范围分离的实现，允许现代范围分离混合泛函的整数或分数占用数的完整基组基准。最后，在局域密度或广义梯度近似水平上，计算了原子有效势的叠加原子势方法[S。Lehtola，J. Chem. Theory Comput. 15，1593（2019）]，其已被证明是初始化电子结构计算的简单且有效的方式。本数值方法提供超越microhartree精度与少量的数值基函数，并重现文献结果的基态原子和它们的阳离子$1 \leq Z \leq 86 $。我们的研究结果表明，文献值偏离了10 μEh从完整的基组极限。数值方案的erfc内核的工作比较从大高斯基组计算的结果从文献中。自旋限制基态的报告Hartree-Fock和Hartree-Fock-Slater计算与分数占用$1 \leq Z \leq 118$。
