Principle of maximum hardness

Pearson and Parr and co-workers developed the principle of maximum hardness, which states that reacting molecules will arrange their electrons so as to be as hard as possible. Chemical equilibrium, then, is the state of maximum hardness. Soft donors prefer soft acceptors because both partners can increase their hardness by reacting with one another—the shared electrons flow to become less polarizable. To implement this theory quantitatively, Pearson et al. introduced scales of absolute hardness rf and its reciprocal, softness cr  

In these relationships, IP and EA refer to free atoms or ions, whole molecules or radicals, rather than atoms within molecules, and so —p bears little relationship to Pauling s x- In terms of molecular orbital (MO) theory, IP is the negative of the energy of the highest occupied MO (-eHOMo)i and EA is the energy of the lowest unoccupied MO (clumo)- Thus, hard molecules 

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Another useful generalization is the principle of maximum hardness. This states that molecular arrangements that maximize hardness are preferred. Electronegativity and hardness detennine the extent of electron transfer between two molecular fragments in a reaction. This can be approximated numerically by the expression... 

The hardness measures the stability of the system. A hard molecule resists changes within itself, or in reaction with others. As a result, a molecule will arrange itself to be as hard as possible, the principle of maximum hardness. This usually is interpreted as the placing of the nuclei. 

The conclusion that the local hardness is given entirely by the variable parts of the kinetic energy is very logical. It is the kinetic energy increase which limits the distribution of electron density in all systems with fixed nuclei. Since the equilibrium state of atoms and molecules is characterized by minimum energy, they will also be marked by maximum kinetic energy because of the virial theorem. This will put them in agreement with the principles of maximum hardness, for which much evidence exists. 

The global parameters help understanding the behavior of a system and lead to applicable and useful principles such as the principle of maximum hardness (MHP) [1], In this chapter, however, our main focus is to introduce the working formula of local reactivity parameters, their actual computations, and practical ways of application to different types of organic reactions. In this process, we mention briefly some of the relevant global reactivity parameters and their calculations as well just to have continuity in the subject matter. 

Theoretical ionization energies are in good agreement with the experimental values. For all the molecules, the HOMO-LUMO gap is larger for the most stable isomers. This confirms previous results that claim that the stability of aromatic hydrocarbons depends on the HOMO-LUMO gap. The principle of maximum hardness establishes that the system would be more stable if the global hardness, related to the HOMO-LUMO gap, is a maximum. As shown in Table 61, the HOMO-LUMO gap correlates well with the expected stability of these molecules and the energy difference between the HOMO and HOMO-1 for benzo[3]thiophene is smaller than for benzo[c]thiophene (Figure 27). Therefore, it is possible to use hardness as a criterion of stability. 

Another important postulate, put forward by Pearson [13], is the principle of maximum hardness, according to which a system tends to attain the maximum rigidity. This principle was based also on experimental observations. According to (10), t] increases with increasing of ionization potential and with decreasing of electron affinity. Thus, the system tends neither to render its own electrons, no to get foreign ones, i.e. to remain stable. 

An extension of the Hohenberg-Kohn theorems to an arbitrary excited electronic state has not been possible till date. It has been possible only for the lowest state of a given symmetry [45] and for the ensemble of states [46], It may be anticipated from the principles of maximum hardness and minimum polarizability that a system would become softer and more polarizable on electronic excitation since it is generally more reactive in its excited state than in the ground state. Global softness, polarizability, and several local reactivity parameters p(r, t), Vp, —V2p,/(r), electrostatic potential, and quantum potential have been calculated [25] for different atoms, ions, and molecules for the lowest energy state of a particular symmetry and various complexions of a two-state ensemble. It has been observed that a system is harder and less polarizable in its ground state than in its excited states, and an increase in the excited state contribution in a two-state ensemble makes the system softer and more polarizable. Surface plots of different local quantities reveal an increase in reactivity with electronic excitation. 

An interesting feature of the Parr-Chattaraj proof of the Principle of Maximum Hardness, is that the specific example of chemical softness is not introduced until the last step. The proof should then be valid for many other observables, provided that certain restrictions are met. One requirement is that the observable always has a positive value (or in some cases always a negative one). 

There seems to be a law of nature that, in an equilibrium system, the chemical hardness and the physical hardness have maximum values, compared with nearby non-equilibrium states. However, it must not be inferred that these maximum principles are being proposed to take the place of estabished criteria for equilibrium. Instead, they are necessary consequences of these fundamental laws. It is very clear that the Principle of Maximum Hardness for electrons is a result of the quantum mechanical criterion of minimum energy. Similarly, Sanchez has recently derived the relationship (dB/dP) = 5 by a straightforward manipulation of the thermodynamic equation of state.The PMPH is a result of the laws of thermodynamics. 

There is, of course, much space allotted to certain Hardness Principles, such as the Principle of Maximum Hardness, or the Principle of Hard and Soft Acids and Bases. An attempt is made to show their wide range of useful application, as well as their limitations. 

It is not yet a well established concept in organic chemistry, but it appears that there is a principle of maximum hardness,235 which says that reactions take place in the direction that increases hardness. We can use the two reactions in Fig. 3.1 to see how this works. The hardness of a pair of starting materials is measured by taking the smaller value of I and the more positive value of A, and using Equation 3.3. The combination of a methyl radical and a fluorine atom has a change from rj = 9.82 — 3.40 = 6.42 to rj = 18.7, whereas the combination of a methyl radical with an iodine atom has a change from rj = 9.82 — 3.06 = 6.76 to rj = 9.30. Thus the former is the reaction with the greater increase in hardness, with methyl fluoride a very hard molecule, and is the more exothermic reaction. 

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