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## Journal of the Chilean Chemical Society

##
*versión On-line* ISSN 0717-9707

### J. Chil. Chem. Soc. v.49 n.2 Concepción jun. 2004

#### http://dx.doi.org/10.4067/S0717-97072004000200001

J. Chil. Chem. Soc., 49, N 2 (2004), pags.:107-111
Dirección para correspondencia
We describe how modern concepts of chemical reactivity and selectivity, defined as response functions in the conceptual density functional theory (DFT), may be implemented within a simple Huckel Molecular Orbital (HMO) formalism. Concepts like electrophilicity may be easily explained using the electronic chemical potential. Intramolecular and intermolecular selectivity may in turn be conveniently described by the Fukui function, which is formulated in terms of single coefficients of the frontier molecular orbitals within the HMO frame. The conceptual value obtained by merging a powerful formalism based on response functions with a simple model of electronic structure is the transparent interpretation of modern concepts of reactivity in terms of classical chemical quantities. The model is illustrated for the reactivity and intermolecular selectivity of Diels-Alder reactions.
Chemical reactivity indexes have been used to rationalize the reactivity and selectivity patterns of molecules from the dawn of theoretical chemistry In density functional theory, the electronic energy E is a unique functional of the real space electron density r(r), which integrates to the total number of electrons in the system: . Minimization of the functional E[r] yields the Euler-Lagrange equation defined for a constant external potential m(r), due to the compensating nuclear charges in the system. F functional The electronic chemical potential was given however a simple working expression by Robert Parr One may immediately realize that m becomes the negative of Mulliken´s electronegativity. This important result led several authors to adopt this useful quantity as the natural descriptor of charge transfer, as it measures the escaping tendency (or fugacity) of electrons from the atomic or molecular system. A more useful expression may be obtained using Koopmans theorem. There results where e The general advantage of reactivity indexes formulated as response functions is that they condense, in a single number, most of the electronic information contained in the density matrix of the system. They may therefore be used to establish a quantitative hierarchy of chemical properties that are to be validated against experimental scales of reactivity. A validated theoretical scale is a useful tool to provide quantitative models of reactivity and selectivity. It may for instance be used to predict the reaction feasibility (whether or not a given reaction will take place)
All the information required to perform the HMO reactivity analysis is obtained by solving the secular equation involving an unspecified Hamiltonian H: where C The non trivial solutions are found by solving the following determinant: In the HMO method, a and b parameters are introduced. The former is the Coulomb integral that corresponds to the Hmm matrix elements and the second is the resonance integral associated with the off diagonal matrix elements for neighbor atoms. It is usual to introduce the transformation ; where X are the roots of the determinant of Eq (5). For the HOMO and LUMO levels we obtain: Using Eq (3), the electronic chemical potential m becomes: A convenient simplification may be obtained by defining a relative electronic chemical potential for a conjugated molecule M. m° is the electronic chemical potential of a reference system which we shall define here as ethylene ( for which m° = a) , within a model similar to that leading to the evaluation of the resonance energy in the HMO theory. In this way, the quantity a in Eq (7) is eliminated and the relative electronic chemical potential scale is expressed in terms of the roots of the determinant of Eq (7) in b units.
where q where are the atomic Fukui function and the bond Fukui function, respectively. The second order terms in (10) are related to polarizability quantities within Coulson´s theory of mobile electrons
If we now use Parr´s interpretation of the Coulomb parameter as an atomic electronegativity c or equivalently, Equation (13b) gives a partition of the electronic chemical potential similar to that proposed by Parr et al The electronic chemical potential gives an approximate criterion to assign the electrophilicity of a molecule. This is simply done by comparing the electronic chemical potential of two molecules, A and B . For instance, if m The main difference with the DFT definition of the electrophilicity index E+ is that the effect of the hardness h, as a resistance to the exchange of electronic charge with the environment, has been neglected (see the paragraph after Eq (10)). Incorporation of a second order quantity like hardness within the HMO theory can only be achieved by modifying the a Let now return to the charge dependent second order terms that render the HMO model a non linear problem. For the a which may be rearranged, using the chain rule, to give: and thereby introducing the atom-atom polarizability and the bond-bond polarizability order terms introduces the inverse of both quantities. Since polarizability is related to softness
The derivative of the electronic chemical potential with respect to the external potential defines the most important local reactivity index, namely, the Fukui function of the system The most currently used expression however, is that given by the second equality. From this expression, it becomes easy to derive working formulae for f(r). Using a finite difference approach, we may evaluate the derivative of the electron density with respect to N in the direction of increasing (
Subtraction of (18a) and (18b) yields the desired result:
Equation (19) shows that the nucleophilic Fukui function becomes exactly equal to the density of the last occupied molecular orbital HOMO, a result deduced by Senet on a more generalized formalism ^{27,28}.
Organic reactions fall into one of three general categories of processes that include polar, radical and pericyclic reactions. In a bimolecular polar reaction, one component called the nucleophile provides a pair of electrons to the other, called the electrophile, to form a new bond. The pericyclic process, on the other hand, shares the feature of having cyclic transition states, with a concerted movement of electrons simultaneously breaking and forming bonds. In a pericyclic reactions neither component may be associated with the supply of electrons to any of the new bonds formed during the concerted process. Within the wide range of pericyclic processes, the cycloaddition reactions are the largest class. Cycloaddition reactions involve the approach of two components presenting a p system to form two new sigma bonds within a cycle structure. The range of reactions, the stereochemistry, and regioselectivity present in these processes are by far the most abundant and useful of all pericyclic reactions
In general, the DA reactions require opposite electronic features in the substituents at the diene and the dienophile to be reasonably fast, changing the reaction mechanism from a synchronous concerted step to a polar stepwise pathway Another relevant aspect of the DA cycloadditions concerns their regioselectivity. Selectivity may be handled in terms of local reactivity parameters
We have shown how reactivity and selectivity concepts may be implemented within simple HMO theory. An electrophilicity scale based on relative electronic chemical potential with reference to ethylene, and framed on an effective electronegativity equalization principle was defined. Second order energy quantities may be identified with non local response functions introducing atom-atom and bond-bond polarizabilities. The empirical electrophilicity scale correctly accounts for the pericyclic nature of the ethylene + butadiene DA reaction. The change from pericyclic-concerted to polar-stepwise mechanisms induced by chemical substitution of electron withdrawing groups at the dienophile and electron releasing groups at the diene is also assessed. Regioselectivity and site activation patterns induced by chemical substitution is correctly described by a simple formulation of the electrophilic and nucleophilic Fukui functions.
Work supported by UTFSM project N 130223, and Fondecyt grant N 1030548.
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Received: july 15, 2003 - Acepted: September 11, 2003 |