HSAB theory

Chemical hardness

In 1983 Pearson together with Robert Parr extended the qualitative HSAB theory with a quantitative definition of the chemical hardness (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:

${\displaystyle \eta =\frac{1}{2}{\left(\frac{{\mathrm{\partial }}^{2}E}{\mathrm{\partial }{N}^2}\right)}_Z}$

The factor of one-half is arbitrary and often dropped as Pearson has noted.

An operational definition for the chemical hardness is obtained by applying a three-point finite difference approximation to the second derivative:

${\displaystyle \begin{array}{cc}\eta & \approx \frac{E(N+1)-2E(N)+E(N-1)}{2}\\ & =\frac{(E(N-1)-E(N))-(E(N)-E(N+1))}{2}\\ & =\frac{1}{2}(I-A)\end{array}}$

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,

${\displaystyle \mu ={\left(\frac{\mathrm{\partial }E}{\mathrm{\partial }N}\right)}_Z}$,

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

${\displaystyle \begin{array}{cc}\mu & \approx \frac{E(N+1)-E(N-1)}{2}\\ & =\frac{-(E(N-1)-E(N))+(E(N)-E(N+1))}{2}\\ & =-\frac{1}{2}(I+A)\end{array}}$

which is equal to the negative of the electronegativity (χ) definition on the Mulliken scale: μ = −χ.

The hardness and Mulliken electronegativity are related as

${\displaystyle 2\eta ={\left(\frac{\mathrm{\partial }\mu }{\mathrm{\partial }N}\right)}_Z\approx -{\left(\frac{\mathrm{\partial }\chi }{\mathrm{\partial }N}\right)}_Z}$,

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 Tl³⁺ compared to Tl⁺.

Modifications

If the interaction between acid and base in solution results in an equilibrium mixture the strength of the interaction can be quantified in terms of an equilibrium constant. An alternative quantitative measure is the heat (enthalpy) of formation of the Lewis acid-base adduct in a non-coordinating solvent. The ECW model is quantitative model that describes and predicts the strength of Lewis acid base interactions, -ΔH . The model assigned E and C parameters to many Lewis acids and bases. Each acid is characterized by an E_A and a C_A. Each base is likewise characterized by its own E_B and C_B. The E and C parameters refer, respectively, to the electrostatic and covalent contributions to the strength of the bonds that the acid and base will form. The equation is

-ΔH = E_AE_B + C_AC_B + W

The W term represents a constant energy contribution for acid–base reaction such as the cleavage of a dimeric acid or base. The equation predicts reversal of acids and base strengths. The graphical presentations of the equation show that there is no single order of Lewis base strengths or Lewis acid strengths. The ECW model accommodates the failure of single parameter descriptions of acid-base interactions.

A related method adopting the E and C formalism of Drago and co-workers quantitatively predicts the formation constants for complexes of many metal ions plus the proton with a wide range of unidentate Lewis acids in aqueous solution, and also offered insights into factors governing HSAB behavior in solution.

Another quantitative system has been proposed, in which Lewis acid strength toward Lewis base fluoride is based on gas-phase affinity for fluoride. Additional one-parameter base strength scales have been presented. However, it has been shown that to define the order of Lewis base strength (or Lewis acid strength) at least two properties must be considered. For Pearson's qualitative HSAB theory the two properties are hardness and strength while for Drago's quantitative ECW model the two properties are electrostatic and covalent .

Kornblum's rule

HSAB theory is commonly, but misleadingly, applied to predict the reactions of ambident nucleophiles (nucleophiles that can attack from two or more places). In 1954, Nathan Kornblum et al proposed that the more electronegative atom reacts when the reaction mechanism is S_N1 and the less electronegative one in a S_N2 reaction. Kornblum's rule was later rationalized through HSAB theory, as follows: in a S_N1 reaction the carbocation (a hard acid) reacts with a hard base (high electronegativity) and in a S_N2 reaction tetravalent carbon (a soft acid) reacts with soft bases.

However, Kornblum's theory predicts the actual behavior of ambident nucleophiles quite poorly. Violations occur with cyanide, cyanate, thiocyanate, nitrite, nitronates, amide enaminols, and phenylsulfinate. Instead, the determining factor is whether the reaction exhibits a kinetic barrier. Barrier-free reactions are (initially) unselective or (subsequently) determined by equilibrium thermodynamics. Reactions with a barrier tend to involve attack on atoms from later groups and in accordance with the principle of least motion.