# Radical addition to the vinyl C=C bond: Quantum chemistry model of the reaction

## Introduction and Retrospective

Radical addition to the unsaturated C=C bond is an important part of radical chemistry. Only the experimental data of radical polymerization and oxidation of unsaturated compounds are an extremely large block of information [1]. So, the question of reactivity is not discussed once. There were proposed both descriptive empirical schemes of relative reactivity and theoretical approaches. The main characteristic of this class of reactions is that functional groups at the reaction center cannot be considered just substituents. So, the usual linear correlations of Hammett type [2] had no success in the general case.

The first successful scheme of that kind was the *Q–e* scheme of Alfrey-Price for bulk copolymerization. Its equation for rate constant *k _{RM}* of addition of polymer radical

*R*to monomer

*M*looks like this:

*ln k*[3]. Although parameters

_{RM}= ln P_{R}+ ln Q_{M}− e_{R}e_{RM}*P/Q*were assigned the meaning of energies of stabilization, and

*e*the measure of polar factor, this equation could be considered a 2-D variant of the usual Hammett equation: Fix parameter

*e*of the radical and get a linear equation for monomer reactivity; and the same but with radicals.

Hoyland proposed another scheme for this class of reactions, known as the model of electronegativities [4]. He used another mathematical representation of polar factor to improve agreement with experimental data:

*ln k _{RM} = ln L_{R} + ln L_{RM} + |χ_{R} + χ_{RM}|*

Another general descriptive scheme of radical reactions was proposed by Denisov [5], who made (and still does) significant work of classifying kinetic data of radical substitution and addition. His model, known as parabolic, was intended for calculating rate constants and activation energies of radical H-abstraction; it later was transferred on radical addition reactions. The author of this article would rather consider it another empirical scheme^{*}.

Since the pioneer works of Fukui [6], the main approach to discussion of radical reactivity became the method of boundary molecular orbitals (MOs), and its extreme variant of using ionization potential (or electron affinity, depending on the relation of donor–acceptor properties of reagents) as an index of reactivity of monomers/radicals. For example, this approach was good enough for studies of reactivity of alkyl radicals toward vinyl monomers [7].

Now, this study is another ambitious attempt to build some universal model of reactivity of vinyl monomers (`CH`

) in radical addition reactions, using for the basis a theoretical investigation of reactivity of the _{2}=CXY`C=C`

bond. **For the basic concept of this approach we assumed that criterion of reactivity of reagents should be not experimental data but directly calculated activation barriers** (experimental activation energies are not numerous and reliable enough). This enabled us to reveal true electronic factors of reactivity and avoid effects caused by differences of experimental conditions, for example, influence of solvents. For this purpose there were performed a series of calculations of reactions of C- and O-centered radicals `CH`

, _{3}^{•}`CH`

, _{3}O^{•}`CCl`

, _{3}^{•}`CF`

, and _{3}^{•}`HOO`

and vinyl monomers ^{•}`CH`

, both ordinary ones as vinyl acrylate and with conjugated double bonds (butadiene, styrene, etc.) [8]._{2}CHX

A special part of this article is the new model of electronegativities that was derived from the perturbated MO (PMO) theory equation as its alternative. It has general character and can be applied in any studies of reactivity where the PMO approach could be used.

Calculations of all electronic structures in this study were performed by means of the MNDO method implemented in the AMPAC package, UHF and gas-phase approximations being assumed. Geometry of all transition states (TSs) were found by means of standard routine and checked for a presence of one negative force constant.

^{*} That model has a considerable number of parameters and its equation is based on the idea that the reaction pathway and so the value of the activation barrier of reaction can be obtained by simple superposition of two parabolic potentials of harmonic vibrations of two bonds: the one being formed and the one being broken. And, the picture of crossing of two potential surfaces is used for this different case. It seems that in a model of H-atom movement between two attractors its total energy must be gained by ordinary summation of potential energies. But, the sum of two parabolic functions is another parabola with a minimum somewhere around the place we want to find a barrier.