Theoretical Underpinnings of the Hard-Soft Acid-Base Principle

In 1958, Ahrland introduced the classification of metal cations into Type-A and Type-B based on the stability of the complexes they form for given ligands. Alkali metal cations from group 1 (Li+ to Cs+), alkaline earth metal cations from group 2 (Be2+ to Ba2+), lighter transition metal cations in higher oxidation states (Ti4+, Cr3), and the proton (H+) constitutes Type-A metal cations. The Type-B metal cations include heavier transition metal cations in lower oxidation states (Ni2+, Cu+, Ag+).

Ralph Pearson developed Type-A and Type-B and first proposed the classification of Lewis acids and bases as either hard or soft.

1.The HSAB Principle simply states that,

Hard acids prefer binding to hard bases, soft acids prefer to bind to soft bases.

Let’s revise the characteristics of Lewis hard/soft acid/base.

Use: HSAB concept (also known as the Pearson acid-base concept) is especially used in transitional metal chemistry to determine the relative ordering of the ligands and transition metal ions in terms of their hardness and softness. It is widely used in inorganic chemistry to study the qualitative nature of the predominant factors driving the chemical reactions and properties.

2. Derivation of the HSAB Principle from First-principles

Parr proposed that the chemical hardness can be defined as the second derivative of the energy with respect to the number of electrons.

1a. Definition of Chemical Hardness.

Based on the demand of definition of “strength” by theoretical dealing with the HSAB principle (despite many experimental methods giving relative strength of acids and bases), it is proposed that,

1b. The definition of “intrinsic strength” of Lewis acids and Lewis bases, the electronic chemical potential, based on density functional theory.

The research paper by Parr and et al. shows the concept of electronegativity given by the negative of the chemical potential in the Hohenberg-Kohn density functional theory of the ground state.

1c. The relationship between electronegativity, chemical potential, and energy derivative.

Consider an electron transfer reaction, where electrons are transferring from hard/soft A (Lewis acid) to a hard/soft B (Lewis base).

2. Charge transfer reaction between Lewis acid A and Lewis base B.

Using the quadratic energy model, the energy of each reactant can be written as,

3a. Energy of reactant A after charge transfer takes place.

3b. Energy of reactant B after charge transfer takes place.

According to charge neutrality condition,

4. Charge neutrality condition.

where N⁰ represents the initial electron number of the corresponding reactant (given in subscript) and N(along with subscript) represents electron number after charge transfer. ΔN is the change in electron numbers. The electrons numbers, N_A and N_B, are to be determined so that the chemical potential of A and B are equal in the molecule.

5. Chemical potential of reactants A and B are equal after charge transfer takes place.

The minimization of the energy of acid and base w.r.t. the extent of electron transfer ΔN, we get the transfer of,

6. The obtained equation for a total change in electron numbers.

the number of electrons from the base (B) to the acid (A).

Thus the total change in the energy of the reactants after the charge transfer reactions can be given as,

7. Total change in energy of reactants after the charge transfer reaction.

which leads to an important result showing the relationship between the total change in the energy, chemical potential, hardness, and electronegativity of the species. Substituting eq. 4 & 6 in eq. 7 leads to the following expression which gives the contribution by electron transfer to the reaction energy.

8. The contribution of electron transfer to reaction energy, in terms of intrinsic strength, hardness, and electronegativity.

The study of the fundamental reaction, given by (2), helps to discern the influence of the chemical hardness on the energy of the reaction. The above equation shows: (a) the lowering in energy resulting from the electron transfer. (b) the differences in electronegativity drive the electron transfer (c) the sum of the absolute hardness parameters inhibits electron transfer.

The term, η_A + η_B, will be a small number if both the Lewis acid and Lewis base are soft. Then for a reasonable difference in electronegativities, the reaction energy ΔE will be substantial and stabilizing. This partially explains the HSAB principle; soft prefer soft. But in the other case for a given difference in electronegativities, where both acids and bases are hard, there will be very less energy stabilization from electron transfer. Thus, for convenience, the HSAB principle is usually discussed in relation to the exchange reaction.

Here, the main advantage of using the following exchange reaction above the fundamental acid-base reaction (2), is that many effects not related to the hardness of the reagents tend to “cancel out” between the two sides of the reaction.

9. Exchange reaction between hard/soft acid/base.

As per the HSAB principle, the equilibrium in the above reaction lies to the right. Since the hardness of the reagents is one of the factors that influence the enthalpy of the reactants, we cannot separate the effects of changing the chemical hardness on the reaction profile. We may infer that the above reaction (9) is exothermic (assuming the entropy of reactants and products will have similar entropy). That means,

10. Enthalpy condition inferring that reaction (9) is exothermic.

After mathematical rearrangement by adding and subtracting the same term, the above equation can be written as,

11. Equality in terms of molecular complex formation in the standard fashion.

Pearson emphasized the idea that electronic chemical potential is a first-order quantity whereas chemical hardness is a second-order quantity. Thus, it is important to have negligible effects on the differences in the electron chemical potential. Hence, one can set,

12. Set the chemical potential of hard and soft species equal.

In order to obtain the desirability of softness and hardness matching, it is necessary to have,

13. The condition that soft(hard) acid and base have the same hardness, to characterize the desirability of softness and hardness matching.

In the equality given in eq. (11), the expressed energy of the molecular complex formation can be calculated from eq. (8). Thus using equations (8), (12), & (13) in the expression of equality given in eq. (11) gives,

14. ΔU of the exchange reaction (eq. 9) with assumed conditions on chemical potentials and hardness of reactants.

The above expression of ΔU will always be <0 since η>0 and ξ>1. This proves that the HSAB exchange reaction (9), is always exothermic and so the HSAB principle holds.

3. When HSAB Principle does not hold?

We had considered the exchange reaction due to the advantage that many chemical effects of reagents that are not related to their chemical hardness will tend to cancel each other between two sides of the reaction. Specially, all those “universal” effects of the various properties of the reactants, insensitive to the detailed structure of its reaction partners) will cancel out in reaction (9). Some of these omitted terms in the analysis of the energy of the reaction may play a decisive role. Such as electrostatic effects, polarization, differences in the intrinsic strengths of the chemical species, etc. are not accommodated in the above analysis. These are some of the rare instances where the HSAB principle does not hold.

Also, we presumed that the entropy of the reactants and products will be similar. Here, we are ignoring the significant difference in entropy of solvation. Such effects were studied in further studies.

4. Closure

The theoretical analysis along with mathematical work presented here shows that,

The fundamental driving force behind the HSAB principle is electron transfer.

The very foundation of this conclusion lies in the definition of chemical hardness, it is dependent on the energy w.r.t. the number of electrons. Thus the presented inference here is consistent with the intuitive expectation that the reactivity preferences of hard and soft acids and bases should be dominated by the effects of charge transfer.

References:

1. P. W. Ayers, J. Chem. Phys. https://doi.org/10.1063/1.1897374 122, 141102 (2005).

2. P. W. Ayers, R. G. Parr, and R. G. Pearson, J. Chem. Phys. https://doi.org/10.1063/1.2196882 124, 194107 (2006).

3. R. G. Parr, R. A. Donnelly, M. Levy, and W. E. Palke, J. Chem. Phys. https://doi.org/10.1063/1.436185 68, 3801 (1978).

4. R. G. Parr, and R. G. Pearson J. Am. Chem. Soc. https://doi.org/10.1021/ja00364a005 105, 26, 7512–7516 (1983)