Reactivity perturbation theory

Many experimental techniques now provide details of dynamical events on short timescales. Time-dependent theory, such as END, offer the capabilities to obtain information about the details of the transition from initial-to-final states in reactive processes. The assumptions of time-dependent perturbation theory coupled with Fermi s Golden Rule, namely, that there are well-defined (unperturbed) initial and final states and that these are occupied for times, which are long compared to the transition time, no longer necessarily apply. Therefore, truly dynamical methods become very appealing and the results from such theoretical methods can be shown as movies or time lapse photography. 

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. 

The reactivity of the carbo-Diels-Alder reaction, as well as the other reactions considered in this chapter, can be accounted for by a simple FMO line of reasoning, i.e., the energy term from second-order perturbation theory... 

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... 

The theoretical interpretation of the results was made (334) in terms of the molecular orbital perturbation theory, in particular, of the FMO theory (CNDO-2 method), using the model of the concerted formation of both new bonds through the cyclic transition state. In this study, the authors provided an explanation for the regioselectivity of the process and obtained a series of comparative reactivities of dipolarophiles (methyl acrylate > styrene), which is in agreement with the experimental data. However, in spite of similar tendencies, the experimental series of comparative reactivities of nitronates (249) toward methyl acrylate (250a) and styrene (250b) are not consistent with the calculated series (see Chart 3.17). This is attributed to the fact that calculation methods are insufficiently correct and the... 

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. 

While there is no complete theory of surface reactivity, an understanding of how reactant, intermediate, and product adsorbates interact with a surface often gives insight into the catalytic properties of a metal. Quantum mechanical theories show that as long as the perturbation due to the interacting systems is small, the interaction of two isolated systems can be estimated using second order perturbation theory ... 

The MO analysis of reactivity in processes A, B and C by the explicit computation of energy profiles for the reaction path as described above for the parent system could be undoubtedly carried out also for the other members of this series. However, considering the large size of these systems and the relative lack of perfection of the computations still practical in such cases there are important advantages in resorting to approximate reactivity analyses which depend on the application of Perturbation Theory. 

Various theoretical methods and approaches have been used to model properties and reactivities of metalloporphyrins. They range from the early use of qualitative molecular orbital diagrams (24,25), linear combination of atomic orbitals to yield molecular orbitals (LCAO-MO) calculations (26-30), molecular mechanics (31,32) and semi-empirical methods (33-35), and self-consistent field method (SCF) calculations (36-43) to the methods commonly used nowadays (molecular dynamic simulations (31,44,45), density functional theory (DFT) (35,46-49), Moller-Plesset perturbation theory ( ) (50-53), configuration interaction (Cl) (35,42,54-56), coupled cluster (CC) (57,58), and CASSCF/CASPT2 (59-63)). 

Various reactivity indices have been derived for benzenoid hydrocarbons from the following purely topological approaches the Huckel model (HMO), first-order perturbation theory (PMO), the free electron MO model (FEMO), and valence-bond structure resonance theory (VBSRT). Since many of the indices that have been known for a long time (index of free valence Fr, self-atom polarizability ir , superdelocalizability Sr, Brown s index Z, cation localization energy Lr+, Dewar reactivity number Nt, Brown s para-localization energy Lp) have been described in detail by Streitwieser in his well-known volume [23] we will refer here only to some more recent developments. 

The similarity between Eqs. (39) and (64) implies that there must be a relationship between free valence number and reactivity number. This is indeed the case, for it can be shown14 at once by first order perturbation theory that... 

The philosophy of perturbation theory is to consider the whole group of reactants undergoing a given reaction as variations on a central theme. We have seen how this principle is applied to heteroatomic systems by regarding them as perturbed forms of the isoconjugate hydrocarbons likewise it is convenient to treat as many such molecules as possible in terms of a fixed kernel with varying substituents attached to it. This is indeed the procedure commonly followed in chemistry, confirming the view that chemistry is in effect an exercise in perturbation theory. Our next problem then is to consider how substituents may influence reactivity. 

The previous section was designed merely to provide some indication of the accuracy and limitations of perturbation methods. However, there is little point in doing approximate calculations by hand when the cheapest computers can give the correct results in seconds. The real utility of perturbation theory lies in qualitative applications. We will begin to explore its potential by asking, for example (1) will an end be more reactive exercise a given electrophile than ethylene and (2) will an electrophile attack the end at C, or at Os ... 

Use perturbation theory to show that 2-azabutadiene is more reactive than 1-azabutadiene. 

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