The Principle of Maximum Hardness

From what has been said already, it is obvious that the hardness of a chemical system plays a major role in determining its stability or reactivity. Alternatively, we can use the HOMO-LUMO energy gap as a criterion. If stability is desired, then it is advantageous to have a large energy gap, or a high value of the hardness. If reactivity is desired, then a small gap or hardness is desirable. 

Copyright 1997 WILEY-VCH Verlag GmbH, Weinheim ISBN 3-527-29482-1 

This is the result for orbital interactions, or covalent bonding. What about ionic bonding Consider an anion and a cation approaching each other, with a 

Borane anion Idealized geometry AEgap [eV] Borane anion Idealized geometry AEgap [eV] 

There is another test that can be applied. Since experimental values of 77 are available for many systems, we can check to see if 7/ does increase on going from atoms or radicals to stable molecules. Table 4.2 shows some sample results. As expected, the hardness increases on forming the stable product from the unstable reactants. These are not isolated cases an examination of a large number of reactions shows similar behavior.  

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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. 

Pearson35,36 and Parr and co-workers366 c 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 rj and its reciprocal, softness a ... 

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. 

The present review chiefly deals with one such chemical concept, hardness [3-7], which is a helpful concept for describing a variety of acid-base reactions. From the time hardness was first defined within DFT [6], various related concepts like softness [8], local hardness [9,10] and local softness [8], hardness and softness kernels [11], relative hardness [12], etc. have emerged. These new ideas contain valuable information about other hitherto unknown or not-clearly-known concepts in chemistry. They also provide insights into various phenomena occurring in fields other than chemistry. Two major principles associated with hardness, viz., the Principle of Maximum Hardness [13,14] and the Hard-Soft-Acid-Basc (HSAB) Principle [4, 5, 15] are important for understanding of molecular electronic structure and generalized acid-base reactions. 

The principle of HSAB, and the principle of maximum hardness need more elaboration in order to be theoretically justified. The latter was originally stated by Pearson, who concluded that [5] there seems to be a rule of nature that molelculcs arrange themselves so as to be as hard as possible . Later on, Zhou and Parr [6-8] found some evidence about the validity of this principle under conditions of constant temperature and chemical potential, and recently Parr and ChattaraJ [9] have provided a formal proof using statistical mechanics. Here this principle will be analyzed from the point of view of the changes that take place in the electronic structure of two systems that come into interaction. 

This relation leads to the principle of maximum hardness, because if one assumes that the interaction between A and B leads to an equilibrium state of lower energy than the total energy of A and B when they are far apart from each other, then Bab > Bab, AE, < 0. The state of maximum hardness can be associated to the state of minimum energy. On the other hand, if the equilibrium state of AB occurs when A and B are very far apart, then it means that Bab > Bab which also implies a state of maximum hardness. All together, Eq. (SI) establishes that under conditions of constant chemical potential, the system AB evolves towards a state of maximum hardness. 

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