Department of Chemistry and Chemical Biology, McMaster University, Hamilton, Ontario, Canada L8S 4M1.
Phys Chem Chem Phys. 2011 Nov 21;13(43):19594-600. doi: 10.1039/c1cp21646e. Epub 2011 Oct 10.
Developing a mathematical approach to the local hard/soft acid/base principle requires an unambiguous definition for the local hardness. One such quantity, which has aroused significant interest in recent years, is the unconstrained local hardness. Key identities are derived for the unconstrained local hardness, δμ/δρ(r). Several identities are presented which allow one to determine the unconstrained local hardness either explicitly using the hardness kernel and the inverse-linear response function, or implicitly by solving a system of linear equations. One result of this analysis is that the problem of determining the unconstrained local hardness is infinitely ill-conditioned because arbitrarily small changes in electron density can cause enormous changes in the chemical potential. This is manifest in the exponential divergence of the unconstrained local hardness as one moves away from the system. This suggests that one should be very careful when using the unconstrained local hardness for chemical interpretation.
发展局部硬/软酸碱原理的数学方法需要对局部硬度进行明确的定义。近年来，一种引起广泛关注的量是无约束局部硬度。本文推导出了无约束局部硬度的关键恒等式 δμ/δρ(r)。本文提出了几个恒等式，允许人们通过使用硬度核和逆线性响应函数来显式地确定无约束局部硬度，或者通过求解线性方程组来隐式地确定无约束局部硬度。这种分析的一个结果是，确定无约束局部硬度的问题是无穷病态的，因为电子密度的任意小变化都会导致化学势的巨大变化。这在无约束局部硬度随着远离系统而呈指数发散时表现出来。这表明在进行化学解释时，应该非常小心地使用无约束局部硬度。
