In conceptual density functional theory, reactivity indexes as the Fukui function, the global hardness/softness, and hardness/softness kernels are fundamental linear responses extensively studied to predict the nucleophilic and electrophilic propensities of atoms in molecules. We demonstrate that the hardness/softness kernels of an isolated system can be expanded in eigenmodes, solutions of a variational principle. These modes are divided into two groups: the polarization modes and the charging modes. The eigenvectors of the polarization modes are orthogonal to the Fukui function and can be interpreted as densities induced at a constant chemical potential. The charging modes of an isolated system are associated with virtual charge transfers weighted by the Fukui function and obey an exact nontrivial sum rule. The exact relation between these charging eigenmodes and those of the polarizability kernel is established. The physical interpretation of the modes is discussed. Applications of the present findings to the Thomas–Fermi and von Weizacker kinetic energy functionals are presented. For a confined free quantum gas, described by the von Weizacker kinetic energy functional, we succeed to derive an approximate analytical solution for the Fukui function and for hardness/softness and polarizability kernels. Finally, we indicate how numerical calculations of the hardness kernel of a molecule could be performed from the Kohn–Sham orbitals.

中文翻译：

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概念密度泛函理论中反应本征模式的变分原理

在概念密度泛函理论中，作为Fukui函数，整体硬度/柔软度和硬度/柔软度内核的反应性指数是基础线性响应，其被广泛研究以预测分子中原子的亲核和亲电特性。我们证明了一个孤立的系统的硬度/柔软度内核可以以本征模，变分原理的解来扩展。这些模式分为两组：极化模式和充电模式。偏振模的本征矢量与Fukui函数正交，可以解释为在恒定化学势下诱导的密度。隔离系统的计费模式与通过Fukui函数加权的虚拟电荷转移相关联，并遵守确切的非平凡求和规则。建立了这些电荷本征模式与极化率内核的精确关系。讨论了模式的物理解释。介绍了本研究结果在托马斯·费米和冯·韦扎克动能泛函上的应用。对于由冯·韦扎克动能函数描述的受限自由量子气体，我们成功地得出了关于福库函数以及硬度/柔软度和极化率内核的近似解析解。最后，我们指出如何从Kohn-Sham轨道进行分子硬度核的数值计算。介绍了本研究结果在托马斯·费米和冯·韦扎克动能泛函上的应用。对于由冯·韦扎克动能函数描述的受限自由量子气体，我们成功地得出了关于福库函数以及硬度/柔软度和极化率内核的近似解析解。最后，我们指出如何从Kohn-Sham轨道进行分子硬度核的数值计算。介绍了本研究结果在托马斯·费米和冯·韦扎克动能泛函上的应用。对于由冯·韦扎克动能函数描述的受限自由量子气体，我们成功地得出了关于福库函数以及硬度/柔软度和极化率内核的近似解析解。最后，我们指出如何从Kohn-Sham轨道进行分子硬度核的数值计算。