The Perturbation Theory of Reactivity

The formation of the bonding orbital is, as usual, exothermic ( )), but the formation of the antibonding orbital is endothermic (E2), because there are two electrons which must go into it. The energy put into the antibonding combination is greater than that released from the bonding combination, and similarly for all combinations of fully occupied orbitals, the sum of which provide the repulsion experienced by one molecule when it is brought close to another. Combinations of 

The interaction of the HOMO of one molecule with the HOMO of another 

The interactions which do have an important energy-lowering effect are the combinations of filled orbitals with unfilled ones. Thus, in Fig. 2-18 and Fig. 2-19, we have such combinations, and in each case we see that the energylowering in the bonding combination is the usual one, and that the rise in energy of the antibonding combination is without effect on the actual energy of the system, because there are no electrons to go into that orbital. 

We can also see on Fig. 2-18 that it is the interaction of the HOMO of the left hand molecule and the LUMO of the right hand molecule that leads to the largest drop in energy (2EA 2EB). The interaction of other occupied orbitals with other unoccupied orbitals—as in Fig. 2-19—is less effective, because the closer the interacting orbitals are in energy, the greater is the splitting of the levels (see p. 15). Now we can see why it is the HOMO/LUMO interaction 

---

Fig. 2-16 The energy along two possible reaction coordinates 2.2.2 The Perturbation Theory of Reactivity...

An attempt has been made to predict the sites of nucleophilic attack on [M(CO)3(fl--hydrocarbon)] complexes using the perturbation theory of reactivity. For the model allyl substrate [Co(CO)3( j -C3H5)] the site preference CO > M > C3H5 was predicted for reaction with hard nucleophiles in polar solvents. On the other hand, with soft nucleophiles initial attack at the ir-allyl ligand was favored. Mechanistic studies have suggested only a small energy difference between attack by alkoxide ions on the allyl ligand and the metal in related ( Tr-allyl) palladium(II) complexes. ... 

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. ... 

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... 

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. 

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. 

These points need emphasizing, for the use of free valence as a measure of reactivity was first introduced on intuitive grounds, based on the Thiele theory of partial valence. On this basis there seems no obvious reason why free valence should serve as a measure of reactivity only for compounds of one type. The perturbational treatment given above shows that the validity of the correlation is due merely to a fortuitous coincidence which holds only for alternant hydrocarbons. 

The reactivity in zeolites can be explained on the bases of the perturbation theory, in where the change of energy during the reaction is given by ... 

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. 

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. 

Comparing calculated and experimentally determined reactivity worth enables verification of the accuracy qf nuclear data and the adequacy of computational methods used. For a meaningful comparison of theory and experiment, it is essential that the perturbation theory expressions used for the calculations apply to exactly the same parameter as that deduced fi-om the experiments, and that these expressions are evaluated accurately. This paper reviews three aspects of accurate determination of reactivity (1) the definition of reactivity, (2) high-order perturbation theory expressions for reactivity, and (3) the accuracy of computational techniques based on the multigroup approximation. 

This review is essentially organized into four parts (1) the definition of reactivity (Section II), (2) perturbation theory formulations for reactivity as well as for other integral and differential parameters (Sections III-V), (3) applications for some of the new formulations (Section VI and Section V, E), and (4) problems encountered in practical implementation of some of the perturbation theory formulations (Sections VII and VIII). 

The perturbation theory we consider is for the static reactivity pertaining to a perturbation of the reactor from a reference critical state. We use the general form of the transport equations with spelled-out notations and a continuous representation of the phase-space variables (r, E, 1). The subscript 0 is omitted from the parameters corresponding to the critical reactor. Instead, a bar denotes perturbed parameters. We take 7=1. 

The first term in the decomposition of the supermolecular interaction energy E s has the same formal expression as the corresponding term in the perturbation theory (Eq. 5). The basic expression for the use of the electrostatic potential as an index of chemical reactivity does not depend upon the theory adopted in describing molecular interactions. 

The main goal of the reewtivity theory is to predict the outcome of chemical reactions fi om properties of reactants, including flie effects of their interactions in a given reactive system. This calls for the quantum theory of subsystems, e.g., within tile Hartree-Fock or Kohn-Sham approaches. Sudi a "subsystem theory" is vital for generating the truly two-reactant reactivity criteria of the charge sensitivity analysis, which reflect the subsystem responses to perturbations due to the presence of the other, complementary subsystem. 

CNIX)/2 Wavefunctions of thiophen have been used as a basis in a consideration of the concerted Diels-Alder addition to thiophen, thiophen dioxide, and furan Lert and Trindle were able to rationalize the observed order of reactivity with the aid of qualitative ideas from 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... 

---