Perturbation theory, chemical reactivity

These concepts play an important role in the Hard and Soft Acid and Base (HSAB) principle, which states that hard acids prefer to react with hard bases, and vice versa. By means of Koopmann s theorem (Section 3.4) the hardness is related to the HOMO-LUMO energy difference, i.e. a small gap indicates a soft molecule. From second-order perturbation theory it also follows that a small gap between occupied and unoccupied orbitals will give a large contribution to the polarizability (Section 10.6), i.e. softness is a measure of how easily the electron density can be distorted by external fields, for example those generated by another molecule. In terms of the perturbation equation (15.1), a hard-hard interaction is primarily charge controlled, while a soft-soft interaction is orbital controlled. Both FMO and HSAB theories may be considered as being limiting cases of chemical reactivity described by the Fukui ftinction. 

Klopman, G. "The Generalized Perturbation Theory of Chemical Reactivity and Its Applications , in "Chemical Reactivity and Reaction Paths", G. Klopman, Ed. Wiley-Interscience New York, 1974, pp. 55-166... 

This chapter is intended to provide basic understanding and application of the effect of electric field on the reactivity descriptors. Section 25.2 will focus on the definitions of reactivity descriptors used to understand the chemical reactivity, along with the local hard-soft acid-base (HSAB) semiquantitative model for calculating interaction energy. In Section 25.3, we will discuss specifically the theory behind the effects of external electric field on reactivity descriptors. Some numerical results will be presented in Section 25.4. Along with that in Section 25.5, we would like to discuss the work describing the effect of other perturbation parameters. In Section 25.6, we would present our conclusions and prospects. 

Klopman, G. 1974. The generalized perturbational theory of chemical reactivity and its applications. In Chemical Reactivity and Reaction Paths, (Ed.) G. Klopman, pp. 55-165. New York Wiley-Interscience. 

One of the primary aims of the research program described in this review has been to formulate adequate two-reactant reactivity concepts, and the underlying coordinate systems, which can be used to diagnose reactivity and selectivity trends in systems of very large donor/acceptor reactants, e.g., chemisorption systems. The CSA approach [52], which provides the basis for the present work, is both relevant and attractive from the chemist s point of view, since many branches of chemistry—the theory of chemical reactivity in particular—consider responses of chemical species to perturbations of the external potential and... 

We have now seen that the effort of Parr and collaborators [8-12] to put Fukui s frontier-orbital concept of chemical reactivity on sound footing in density-functional theory through the definition of the Fukui function and the local and global softness works only for extended systems. This restriction to extended systems raises a sixth issue. In both the local softness and the Fukui function, Eqs. (54) and (53a), the orbitals at the chemical potential represent both the LUMO and the HOMO in the Fukui sense. However, there is a continuum of unoccupied KS states above the chemical potential accessible even to weak chemical perturbations any linear combination of which could in principle be selected as the LUMO, and similarly for states below fi and the HOMO. This ambiguity in the frontier-orbital concept obviously applies as well to localized systems when there is more than one KS state significantly affected by a chemical perturbation. 

The softness kernels are relevant to the remaining cases of two or more interacting systems. However, they do not by themselves provide sufficient information to constitute a basis for a theory of chemical reactivity. Clearly, the chemical stimulus to one molecule in a bimolecular reaction is provided by the other. That being the case, an eighth issue arises. Both the perturbing system and the responding system have internal dynamics, yet the softness kernel is a static response function. Dynamic reactivities need to be defined. 

It is clear that there are quite a few possible theoretical approaches to the formulation of a comprehensive model for the adsorption processes discussed above. The most fundamental one would be based on the perturbation molecular orbital (PMO) theory of chemical reactivity [730,731] in which the wave functions of the products are approximated using the wave functions of the reactants. A key issue in the use of Klopman s PMO theory is the relative importance of the two terms in the expression for the total energy change of the system, Afpen. which is taken to be a good index of reactivity, [732,733] ... 

Index of chemical reactivity calculated from the perturbation theory as ... 

More recently, molecular orbital theory has provided a basis for explaining many other aspects of chemical reactivity besides the allowedness or otherwise of pericyclic reactions. The new work is based on the perturbation treatment of molecular orbital theory, introduced by Coulson and Longuet-Higgins,2 and is most familiar to organic chemists as the frontier orbital theory of Fukui.3 Earlier molecular orbital theories of reactivity concentrated on the product-like character of transition states the concept of localization energy in aromatic substitution is a well-known example. The perturbation theory concentrates instead on the other side of the reaction coordinate. It looks at how the interaction of the molecular orbitals of the starting materials influences the transition state. Both influences on the transition state are obviously important, and it is therefore important to know about both of them, not just the one, if we want a better understanding of transition states, and hence of chemical reactivity. 

The paucity of information on the mechanism of reactions, and on the structure of the transition state, and the role of the anomeric effect in its stabilization, constitutes the main reason why qualitative interpretation of reactivity as shown in the aforementioned examples is still very rare. An alternative, more-popular estimation of the relative reaction-rates of con-formers is based on the lone-pair orbital interactions, and their symmetry and energy in the ground state, and could be loosely associated with the perturbation theory of chemical reactivity. ... 

Up to a few years ago chemical reactivity was discussed in term of reactivity indexes. These approaches, although valuable, will not be discussed here, since they have been frequently reviewed in the past40-44). Nor will we discuss the perturbation molecular orbital theory for reactants, which has been the subject of extensive reviews 45—47) Extensions of this method can be found in papers by Klopman 48 5°) and Dougherty 51). I shall now mention some methods which have not yet found wide popularity but seem very promising. I mean the criterion of maxi-... 

This perturbation theory of chemical reactivity is based upon an early stage of the reactant mutual approach, when the molecules are still distinct though close enough for the MO description of the combined reactive system to be valid, say separated by a distance of the order of 5-10 a.u. The implicit assumption is that the reaction profiles for the compared reaction paths are of similar shape, so that the trend of the predicted energy differences at an early point on the reaction coordinate is expected to reflect the difference in the activation energy. 

The self-atom and atom-atom polarizabilities (nAA,nAB) defined using perturbation theory have been also employed to describe chemical reactivity [44], These quantities represent the effect of an electric field perturbation at one atom on the electronic charge at the same (nAA) or another atom (nAB), respectively. 

In QSAR of enzyme inhibition reactions, quantum-chemically calculated electrostatic or MO-related descriptors have been widely used. The former are expected to describe the complex formation between enzyme and the substrate, whereas the latter reflect the chemical reactivity of the substrate at the site. Already in 1967, Klopman and Hudson [83] developed a polyelectronic perturbation theory, according to which the drug-receptor interactions can be under either charge or orbital control. Thus the net atomic... 

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