Editing HSAB theory (section)

== Chemical hardness ==
{|align="right"  class="wikitable" style="margin-left: 1.5em"
|+ align="center"|Chemical hardness in [[electron volt]]<ref name="abshardess" />
|-
!colspan=3 align="center"|Acids||colspan=3 align="center"|Bases
|-
||[[Hydrogen]]|| H<sup>+</sup>||[[Infinity|∞]]||[[Fluoride]]|| F<sup>−</sup>||7
|-
||[[Aluminium]]|| Al<sup>3+</sup>||45.8||[[Ammonia]]|| NH<sub>3</sub>||6.8
|-
||[[Lithium]]|| Li<sup>+</sup>||35.1||[[hydride]]|| H<sup>−</sup>||6.8
|-
||[[Scandium]]|| Sc<sup>3+</sup>||24.6||[[carbon monoxide]]|| CO ||6.0
|-
||[[Sodium]]|| Na<sup>+</sup>||21.1||[[hydroxyl]]|| OH<sup>−</sup>||5.6
|-
||[[Lanthanum]]|| La<sup>3+</sup>||15.4||[[cyanide]]|| CN<sup>−</sup>||5.3
|-
||[[Zinc]]|| Zn<sup>2+</sup>||10.8||[[phosphine]]|| PH<sub>3</sub>||5.0
|-
||[[Carbon dioxide]]|| CO<sub>2</sub>||10.8||[[nitrite]]|| NO<sub>2</sub><sup>−</sup>||4.5
|-
||[[Sulfur dioxide]]|| SO<sub>2</sub>||5.6||[[Hydrosulfide]]|| SH<sup>−</sup>||4.1
|-
||[[Iodine]]|| I<sub>2</sub>||3.4||[[Methane]]|| CH<sub>3</sub><sup>−</sup>||4.0
|}

In 1983 Pearson together with [[Robert Parr]] extended the qualitative HSAB theory with a quantitative definition of the '''chemical hardness''' ([[Eta (letter)|η]]) as being proportional to the second derivative of the total energy of a chemical system with respect to changes in the number of electrons at a fixed nuclear environment:<ref name="abshardess">{{cite journal |author1=Robert G. Parr  |author2=Ralph G. Pearson  |name-list-style=amp | title = Absolute hardness: companion parameter to absolute electronegativity | journal = [[J. Am. Chem. Soc.]] | year = 1983 | volume = 105 |issue = 26 | pages = 7512–7516 | doi = 10.1021/ja00364a005}}</ref>

:<math>\eta = \frac{1}{2}\left(\frac{\partial^2 E}{\partial N^2}\right)_Z</math>

The factor of one-half is arbitrary and often dropped as Pearson has noted.<ref>{{cite journal|author=Ralph G. Pearson|title=Chemical hardness and density functional theory|journal=J. Chem. Sci.|volume=117|issue=5|year=2005|pages=369–377|url=http://www.ias.ac.in/chemsci/Pdf-sep2005/369.pdf|doi=10.1007/BF02708340|citeseerx=10.1.1.693.7436|s2cid=96042488}}</ref>

An operational definition for the chemical hardness is obtained by applying a three-point [[finite difference]] approximation to the second derivative:<ref>{{cite book|last=Delchev|first=Ya. I.|author2=A. I. Kuleff |author3=J. Maruani |author4=Tz. Mineva |author5=F. Zahariev |title=Strutinsky's shell-correction method in the extended Kohn-Sham scheme: application to the ionization potential, electron affinity, electronegativity and chemical hardness of atoms in Recent Advances in the Theory of Chemical and Physical Systems|editor=Jean-Pierre Julien |editor2=Jean Maruani |editor3=Didier Mayou|publisher=Springer-Verlag|location=New York|year=2006|pages=159–177|isbn=978-1-4020-4527-1|url=https://books.google.com/books?id=MxZhcgIg9x0C}}</ref>

:<math>
\begin{align}
\eta &\approx \frac{E(N+1)-2E(N)+E(N-1)}{2}\\
     &=\frac{(E(N-1)-E(N)) - (E(N)-E(N+1))}{2}\\
     &=\frac{1}{2}(I-A)
\end{align}
</math>

where ''I'' is the [[ionization potential]] and ''A'' the [[electron affinity]]. This expression implies that the chemical hardness is proportional to the [[band gap]] of a chemical system, when a gap exists.

The first derivative of the energy with respect to the number of electrons is equal to the [[chemical potential]], ''μ'', of the system,

:<math>\mu= \left(\frac{\partial E}{\partial N}\right)_Z</math>,

from which an operational definition for the chemical potential is obtained from a finite difference approximation to the first order derivative as

:<math>
\begin{align}
\mu &\approx \frac{E(N+1)-E(N-1)}{2}\\
     &=\frac{-(E(N-1)-E(N))-(E(N)-E(N+1))}{2}\\
     &=-\frac{1}{2}(I+A)
\end{align}
</math>

which is equal to the negative of the [[electronegativity]] ([[Chi (letter)|''χ'']]) definition on the [[Mulliken scale#Mulliken electronegativity|Mulliken scale]]: ''μ'' = −''χ''.

The hardness and Mulliken electronegativity are related as

:<math>2\eta = \left(\frac{\partial \mu}{\partial N}\right)_Z \approx -\left(\frac{\partial \chi}{\partial N}\right)_Z</math>,

and in this sense hardness is a measure for resistance to deformation or change. Likewise a value of zero denotes maximum '''softness''', where softness is defined as the reciprocal of hardness.

In a compilation of hardness values only that of the [[hydride]] anion deviates. Another discrepancy noted in the original 1983 article are the apparent higher hardness of [[Thallium|Tl<sup>3+</sup>]] compared to Tl<sup>+</sup>.