The Maximum Hardness Principle

The best values of the C /s define the best wave function, and the best value of p, that can be obtained from the selected basis set of AOs. Any change from the best values will cause the HOMO to rise in energy, or be unchanged, and the LUMO will fall in energy, or be unchanged. Thus the energy gap between them is a maximum for the best values of the coefficients, or the best electron density function. Usually, of course, this will not be the true density function. 

Provided the self-consistent field (SCF) condition is met, calculations at the Hartree-Fock level also obey the mathematics of Equations (4.3) and (4.4). Therefore the HOMO-LUMO gap should also be a maximum in these cases. Because the solutions are normalized and orthogonal, and because the atomic orbitals are conserved, the coefficients for different MOs are not independent. Therefore wrong coefficients in one orbital will usually lead to wrong coefficients in all orbitals. 

Their proof is based on a combination of statistical mechanics and the 

The Hohenberg and Kohn theorem applies to ground states at the absolute zero of temperature. Fortunately there is a finite-temperature version of DFT, first proved by Mermin. The equilibrium properties of a grand canonical ensemble are determined by the grand potential, Q, which is defined as follows  

N and /z have their usual meaning, or to the case where E is the electronic energy, N the number of electrons and p the electronic chemical potential. In either case Mermin showed that the grand potential is a unique functional of the density for a system at finite temperature. Also, the correct density for the system will give a minimum value of Q. Thus we have a DFT for finite temperature by taking a grand canonical ensemble of the system of interest and calculating its properties. 

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As already mentioned, through DFT, it has been possible to explain the electronegativity equalization principle [1,7,10-13] and the hard and soft acids and bases principle [12,15-22] and, additionally, it has also been possible to introduce new ones like the maximum hardness principle [52,53] and the local hard and soft acids and bases principle [20,54—56]. 

These descriptors have been widely used for the past 25 years to study chemical reactivity, i.e., the propensity of atoms, molecules, surfaces to interact with one or more reaction partners with formation or rupture of one or more covalent bonds. Kinetic and/or thermodynamic aspects, depending on the (not always obvious and even not univoque) choice of the descriptors were hereby considered. In these studies, the reactivity descriptors were used as such or within the context of some principles of which Sanderson s electronegativity equalization principle [16], Pearson s hard and soft acids and bases (HSAB) principle [17], and the maximum hardness principle [17,18] are the three best known and popular examples. 

Chattaraj, P. K. 1996. The maximum hardness principle An overview. Proc. Indian Natn. Sci. Acad., Part A 62 513-519. 

The Maximum Hardness principle [41] further extends HSAB principle by stating that molecules try to arrange themselves to be as hard as possible . 

The performance of the method proposed above in the calculation of absolute hardness values of a set of neutral atoms and molecules is investigated. The Fukui indices and the polarization functions for the a-bonds of test molecules are also reported. Finally, the maximum hardness principle was checked by studying the "hardness profile" along the reaction path for the isomerization of HCN and 03H+ systems. 

The reported results show that the inclusion of the gradient corrected nonlocal effects is recommended to obtain data consistent with the maximum hardness principle. In fact, in the case of isomerization of HSiN the calculated hardness value for TS is higher than that of the minimum when local VWN potential is used. The introduction of the nonlocal corrections removes this error. The results for all... 

In this chapter, we review our latest results on the validity of the maximum hardness and minimum polarizability principles in nontotally symmetric vibrations. These nuclear displacements are particularly interesting because they keep the chemical and external potentials approximately constant, thus closely following the two conditions of Parr and Chattaraj required for the strict compliance with the maximum hardness principle. We show that, although these principles are obeyed by most nontotally symmetric vibrations, there are some nontotally symmetric displacements that refuse to comply with them. The underlying physical reasons for the failure of these two principles in these particular nuclear motions are analyzed. Finally, the application of this breakdown to detect the most aromatic center in polycyclic aromatic hydrocarbons is discussed. 

Associated with these properties, important chemical reactivity principles have been rationalized within the framework of conceptual DFT the hard and soft acids and bases principle (F1SAB) [9], the Sanderson electronegativity equalization principle (EEP) [11], the maximum hardness principle (MF1P) [9,12,13], and the minimum polarizability principle (MPP) [14], The aim of this chapter is to revise the validity of the last two principles in nontotally symmetric vibrations. We start with a short section on the fundamental aspects of the MF1P and MPP (section 2). Section 3 focuses on the breakdown of these principles for nontotally symmetric vibrations, while section 4 analyses the relationship between the failure of the MF1P and the pseudo-Jahn-Teller (PJT) effect. A mathematical procedure that helps to determine the nontotally symmetric distortions of a given molecule that produce the maximum failures of the MPP or the... 

The maximum hardness principle also demands that hardness will be minimum at the transition state. This has been found to be true for different processes including inversion of NH3 [147] and PH3 [148], intramolecular proton transfer [147], internal rotations [149], dissociation reactions for diatomics [150,151], and hydrogen-bonded complexes [152]. In all these processes, chemical potential remains either constant or passes through an extremum at the transition state. The maximum hardness principle has also been found to be true (a local maximum in hardness profile) for stable intermediate, which shows a local minimum on the potential energy surface [150]. The energy change in the dissociation reaction of diatomic molecules does not pass through a... 

The study of reaction paths, in DFT, is not a new 1114,115. Thus we have chosen to explore the potential energy surfaces (PES) introducing the possibility to rationalize the results through the computations of the global hardnesses along the whole reaction path, with the aim to verify if, for the studied processes, the maximum hardness principle (MHP) 53 is satisfied. 

In the present article we review the work on the hardness and related concepts done at the University of North Carolina at Chapel Hill. Section 2 introduces the global chemical hardness and softness. Several related local quantities arc described in Sect. 3. The Maximum Hardness Principle and the HSAB Principle are stated and proved in Sect. 4. Various applications of the hardness and related concepts in understanding chemical problems are described in Sect. 5. Finally, Sect. 6 contains a summary and some comments on the future. 

Before turning to specific uses of the various hardness and softness quantities, we state and outline the proofs of two important rules of nature, the Maximum Hardness Principle and the HSAB Principle. 

The maximum hardness principle requires the hardness of the equilibrium state to be maximum. In other words, it amounts to the following inequality ... 

Two formal proofs of the HSAB principle with a restriction of common chemical potential for the partners have been provided very recently [15]. The first proof makes use of the maximum hardness principle. The energy change (to first order) associated with the charge transfer process described above is given by... 

Perhaps little did Ralph Pearson realize when he proposed the hardness concept that it would encompass such a multitude of physico-chemical problems and that it would spawn so many new concepts. The Maximum Hardness Principle and the HSAB Principle, if they prove out, should be cornerstones for understanding molecular structure and molecular reactivity. The complex of ideas related to hardness and softness deserve extensive further application and careful further theoretical study. 

This is not to say that the subject is not without significant unanswered theoretical questions. How does one justify applying statistical mechanics to a system of so few particles as the electrons in a molecule How docs one refine the maximum hardness principle to cover the prediction of equilibrium nuclear configurations in a molecule Isolated atoms have integral numbers of electrons, but in a molecule an atom can bear a noninlegral number of electrons how then does one best describe the process of molecule formation from constituent atoms Where is the simple model of chemical bonding itself, which for example explains why the covalent radius of an atom can be determined from the electron density of the isolated atom in the simple way we have proposed And so on. 

Recently, the maximum hardness principle [5-9] has been added to the list of practical concepts, and it is expected to become a powerful tool in the analysis of chemical behavior. 

Chattaraj, P. K., Liu, G. H., Parr, R. G. (1995). The maximum hardness principle in the Gyftpoulos-Hatsopoulos three-level model for an atomic or molecular species and its positive and negative ions. Chem. Phys. Lett. 237,171-176. 

The hardness and shell stmcture of atoms and molecules must be inter connected. Parr and Zhaou (Parr and Yang 1989) discovered that the absolute hardness is a unifying concept for identifying shells and sub shells in nuclei, atoms. Molecules, and Metallic Clusters. In their consideration, the maxim of the maximum hardness principle is related to the close structure of shells and sub shells. 

Table 13.2 Verification of hard soft acid base rule entailing the maxim of the maximum hardness principle using the sets of hardness data of the present work and those computed through the ansatz and operational and approximate formula of Parr and Pearson...

This identification has paved the way for enhanced understanding of the hard/soft acid/base principle [15-18] and the maximum hardness principle [19-23]. The second term in Eq. (7) is called the linear response function [3, 24, 25]... 

Another important electronic structure principle is the maximum hardness principle " (MHP) which may be stated as, There seems to be a rule of nature that molecules arrange themselves to be as hard as possible . Numerical verification of this principle has been made in several physico-chemical problems such as molecular vibrations , internal rotations , chemical reactions" , isomer stability , pericyclic reactions and Woodward-Hoffmann rules , stability of magic clusters , stability of super atoms ", atomic shell structure" , aromaticity , electronic excitations , chaotic ionization, time-dependent problems like ion-atom collision and atom-field interaction " etc. 

A softer species is more polarizable" " " and more magnetizable" " . The inverse relationship" " between hardness and polarizability/ magnetizability provides two other electronic structure principles to complement the maximum hardness principle. The minimum polarizability principle" " " (MPP) states that, The natural direction of evolution of any system is towards a state of minimum polarizability and the statement of the minimum magnetizability principle" " (MMP) is, A stable config-uration/conformation of a molecule or a favorable chemical process is associated with a minimum value of the magnetizabihty . 

In a chemical reaction, a more stable transition state, measured by the magnitude of the activation energy, implies an easier chemical reaction. Aromatic transition states are also known to facilitate the chemical reaction. Zhou and Parr defined the activation hardness as the hardness difference of the products and the transition state and found, in the case of electrophilic aromatic substitution, that the smaller the activation hardness, the faster the reaction is. For this specific reaction they also found a correlation of the activation hardness and Wheland s cation localization energy, also proposed as an indicator of aromaticity. This finding can indeed be interpreted as a manifestation of the maximum hardness principle. A transition state with a high hardness is more stable than one with a smaller hardness and is therefore easier to reach energetically. The same can be said about two transition states with different aromaticity. Again, hardness and aromaticity parallel each other. The activation hardness has been used in numerous applications for the prediction of site selectivity in chemical reac-... 

Since (I-A) is a measure of hardness according to the maximum hardness principle, the stability of a system or the favorable direction of a physicochemical process is often dictated by this quantity. Because aromatic systems are much less reactive, especially toward addition reactions, I -A may be considered to be a proper diagnostic of aromaticity. Moreover, (/ - A) has been used in different other contexts, such as stability of magic clusters, chemical periodicity, molecular vibrations and internal rotations, chemical reactions, electronic excitations, confinement, solvation, dynamics in the presence of external field, atomic and molecular collisions, toxicity and biological activity, chaotic ionization, and Woodward-Hoffmann rules. The concept of absolute hardness as a unifying concept for identifying shells and subshells in nuclei, atoms, molecules, and metallic clusters has also been discussed by Parr and Zhou. ... 

The relative stabilities of the [MAU]" (M = Li" ", Na" ", K+, Rb" ", Cs" ", Cu" ", Ag+, Au+) isomers have been investigated on the grounds of the minimum polarizability principle [161]. According to the minimum polarizability and maximum hardness principles, the stability of the pyramidal isomer decreases with the augmentation of the atomic number. It is no wonder that for the [AuAU]" the planar isomer is finally more stable than the pyramidal structure. The minimum polarizability and the maximum hardness principles have also been examined to describe the relative stability of various isomers of the [ c-Cu4 Na] , [ c-Cu4 Li] , [ c-Al4 Cu] , [ c-Ag4 Li] , [ c-Au4 Li] , [ c-Ag4 Na] , [ c-Au4 Na] , [ c-Al4 Ag] , and [ c-Al4 Au] using MP2 calculations [162]. The results showed that the pyramidal structures are more stable than the planar ones. 

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