Polanco-Ramírez Carlos A, Franco-Pérez Marco, Carmona-Espíndola Javier, Gázquez José L, Ayers Paul W
Departamento de Química, Universidad Autónoma Metropolitana-Iztapalapa, Av. San Rafael Atlixco 186, México, D. F. 09340, Mexico.
Phys Chem Chem Phys. 2017 May 17;19(19):12355-12364. doi: 10.1039/c7cp00691h.
An analysis of the hardness kernel and local hardness is performed to propose new definitions for these quantities that follow a similar pattern to the one that characterizes the quantities associated with softness, that is, we have derived new definitions for which the integral of the hardness kernel over the whole space of one of the variables leads to local hardness, and the integral of local hardness over the whole space leads to global hardness. A basic aspect of the present approach is that global hardness keeps its identity as the second derivative of energy with respect to the number of electrons. Local hardness thus obtained depends on the first and second derivatives of energy and electron density with respect to the number of electrons. When these derivatives are approximated by a smooth quadratic interpolation of energy, the expression for local hardness reduces to the one intuitively proposed by Meneses, Tiznado, Contreras and Fuentealba. However, when one combines the first directional derivatives with smooth second derivatives one finds additional terms that allow one to differentiate local hardness for electrophilic attack from the one for nucleophilic attack. Numerical results related to electrophilic attacks on substituted pyridines, substituted benzenes and substituted ethenes are presented to show the overall performance of the new definition.
对硬度核和局部硬度进行了分析，以提出这些量的新定义，这些定义遵循与表征与软度相关量的模式类似的模式，也就是说，我们已经推导出了新定义，对于这些定义，硬度核在其中一个变量的整个空间上的积分导致局部硬度，而局部硬度在整个空间上的积分导致全局硬度。本方法的一个基本方面是全局硬度保持其作为能量相对于电子数的二阶导数的特性。由此获得的局部硬度取决于能量和电子密度相对于电子数的一阶和二阶导数。当这些导数通过能量的平滑二次插值近似时，局部硬度的表达式简化为梅内塞斯、蒂兹纳多、孔特雷拉斯和富恩特阿尔巴直观提出的表达式。然而，当将一阶方向导数与平滑二阶导数结合时，会发现额外的项，这些项允许区分亲电攻击和亲核攻击的局部硬度。给出了与对取代吡啶、取代苯和取代乙烯的亲电攻击相关的数值结果，以展示新定义的整体性能。
