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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

BACK TO BASICS: MODERN REACTIVITY CONCEPTS WITHIN THE HMO THEORY FRAMEWORK è..

 

Arie Aizman1* and Renato Contreras2

1Departamento de Química, Universidad Técnica Federico Santa María, Valparaíso
and 2Departamento de Química, Facultad de Ciencias, Universidad de Chile, Santiago, Chile.

Dirección para correspondencia


Abstract

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.


1. Introduction

Chemical reactivity indexes have been used to rationalize the reactivity and selectivity patterns of molecules from the dawn of theoretical chemistry 1-3. Most of them were defined within semiempirical models of the electronic structure of molecules. Atomic charges, bond orders, free valence index, atomic and bond polarizability indexes are just few examples that illustrate well the role that the electronic descriptors of reactivity have played in theoretical chemistry in the past 1-3 . Recently, and after the formulation of the Hohenberg-Kohn theorems 4, chemistry has benefited a lot from the introduction of new concepts defined in the context of the DFT school, mainly developed by Robert Parr 5, who formulated global and local reactivity indexes as response functions of the system. In Chile, this new formalism was introduced by Fernando Zuloaga, who irradiated in a very pedagogical and stimulating way the use of DFT quantities for the scrutiny of reactivity in organic and inorganic chemistry 6-8. The present work is dedicated to him in memoriam, with the intention of acknowledging his important pedagogical contribution to the formation of new generations of Chilean chemists, at both undergraduate and graduate levels.

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 5.

thereby introducing the electronic chemical potential

defined for a constant external potential m(r), due to the compensating nuclear charges in the system. FHK[r] is the universal Hohenberg-Kohn

functional 4.

The electronic chemical potential was given however a simple working expression by Robert Parr 5 in terms of the vertical ionization potential I and electron affinity A as:

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 5:

where eH and eL are the one electron energy levels of the frontier molecular orbitals HOMO and LUMO, respectively. Expression (3) will be used in this work.

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) 9, the nature of the possible reaction mechanisms (concerted vs stepwise 10-12 ), and the inter- and intramolecular selectivity expected for a given reaction 9. All this concepts have been normally introduced in organic chemistry in the form of empirical qualitative reactivity rules. Some of them are the electronegativity equalization principle (EEP) 13, the maximum hardness principle (MHP) 14, the hard and soft acids and bases principle (HSAB) 15, and other empirical rules concerning regioselectivity, like the Markovnikov rule and others 16. All these empirical principles may now be discussed on a more quantitative basis. Another relevant advantage of the reactivity model based on response functions is that the analysis is performed within a single reactant model; while within the frontier molecular orbital (FMO) analysis, the properties of the two interacting system must be taken into account 17. In the following, we will implement these ideas using the simple HMO theory emphasizing the meaning of the DFT quantities within this framework. The model is illustrated for the Diels-Alder reaction, for which experimental data about reactivity and (regio) selectivity is available. We stress however that the present approach is uniquely focused on conceptual rather than methodological aspects, yet the HMO formalism has been already used by Parr et al to predict orientation in electrophilic substitution reactions using the concept of activation hardness 18.

2. Model equations.

All the information required to perform the HMO reactivity analysis is obtained by solving the secular equation involving an unspecified Hamiltonian H:

where Cim are the coefficients of the m carbon atom in the ith MO that results after using a linear combination of the 2pZ atomic orbitals (LCAO approximation). The Ei are the MO energy levels.

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 Xi

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.

3. The meaning of DFT quantities within the HMO theory .

3.1 The electronic chemical potential. Beyond the information we can quickly get from the HMO energy levels, there are fundamental ideas that may be discussed around the concept of electronic chemical potential. Consider for instance the HMO energy expression:

where qr and prs are the charge and bond order respectively We can first obtain an expression for m by directly taking the derivative of E in Eq (9) with respect to the total number of (p) electrons N. There results

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 2,3 and we will return to these contributions later on. Let for the moment neglect the second order terms. The truncated expression of m becomes:

If we now use Parr´s interpretation of the Coulomb parameter as an atomic electronegativity cr 19, namely, ar = cr , and recalling that the electronegativity in DFT is the negative of the electronic chemical potential and the b parameter can be identified as a bond chemical potential, we obtain the following relationships:

or equivalently,

Equation (13b) gives a partition of the electronic chemical potential similar to that proposed by Parr et al 19.. It introduces a monocentric contribution (first term of Eq (13b) and a two center contribution which may be assimilated to a bond contribution to the electronic chemical potential.

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 mA > mB, then the electronic flux will take place from A to B, until a unique equilibrium mAB is reached. In this case the species B will behave as an electrophile and the A species as the nucleophile. This means that up to first order, the electrophilicity scale introduced in terms of the relative electronic chemical potential in Eq (8) becomes an electronegativity scale, framed on an effective electronegativity equalization principle.

The main difference with the DFT definition of the electrophilicity index E+ 20, namely

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 ar parameter to take into account its charge dependence as an electronegativity quantity (electronegativity does depends on the charge 21). Such a correction may be introduced within the so called w-technique proposed by Streitwieser 22. Within this approach, the chemical hardness becomes b times the w parameter of the w- technique 19. This approximation is equivalent of adding higher order derivatives of E in Eq (10), thereby introducing chemical hardness, as we will show below. Therefore, in simple HMO theory the electrophilicity scale contains only information about the electronegativity of the system.

Let now return to the charge dependent second order terms that render the HMO model a non linear problem. For the ar and brs derivatives neglected in Eq (10), we have

which may be rearranged, using the chain rule, to give:

and

thereby introducing the atom-atom polarizability and the bond-bond polarizability 2,3. Note that the second

order terms introduces the inverse of both quantities. Since polarizability is related to softness 23, and using the inverse relationship between softness and hardness, we may conclude that the second order derivatives in HMO theory are hardness quantities. For instance, bond hardness in HMO theory measures the interatomic electron repulsion of the bonding electrons. The incorporation of this contribution is equivalent to the approximations made to obtain the Pariser Parr Pople (PPP) parametrization 24.

3.2 The atomic Fukui function within the HMO theory.

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 f(r), which may be expressed as:

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 (f +(r)) or decreasing (f -(r)) number of electrons. While the former is used to analyze local reactivity towards a nucleophilic attack, the latter is used to describe local reactivity towards electrophilic attacks. For radical attacks, the mean value is considered. The three point finite difference expression is however not usually well understood because the lack of a conceptual background to justify its derivation. In contrast, a single point scheme introduces several conceptual and computational advantages that we describe we will shortly. We shall illustrate the reasoning for the derivation of the nucleophilic Fukui function (f -(r)) (i.e. the Fukui function for electrophilic attack). This quantity, which is defined in the finite difference approach as 25 : f -(r) = rN(r) - rN-1(r) shows how the total electron density is modified at point r of the molecular region when the system release one electron to another interacting molecule. We use a one electron orbital representation to express the total electron density for the system with N and N-1 electrons as:


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 26. Note that within the HMO theory, the density of the HOMO is represented by the square of the HOMO coefficient, thereby providing a quick method to obtain the f -(r) quantity condensed to atom k in the molecule f -k = CHk2. By a similar argument, it may be shown that the electrophilic Fukui function f +(r) (i.e. the Fukui function for nucleophilic attack) is given by f +k = CLk2, involving this time the coefficient of the LUMO frontier orbital.

The conceptual advantage of this formalism for the Fukui function lies in the physical approximations allowing to perform the subtraction of Eqs (18a) and (18b). This is the well known frozen orbital approximation that assumes that in the charge releasing or charge withdrawing processes only the frontier orbitals HOMO and LUMO are involved, respectively, and that the remaining part of the electronic structure remains invariant. This approach is reminiscent of the valence electron theory of atomic electronic structure. The computational advantage is evident: the Fukui functions are derived within a single point calculation rather than a three point calculation involving the systems with N, N+1 and N-1 electrons 27,28.

4. Illustration of the model: reactivity and selectivity in Diels-Alder reactions.

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 29.. Diels-Alder (DA) are the largest family of cycloaddition processes. In a DA reaction one p component, called the dienophile, adds to a 1,3-diene system to afford a six membered ring product. By varying the nature of the diene and dienophile many different types of carbocyclic reactions can be obtained. The mechanism of the DA reaction has been controversial for some time. For the case of the archetypal reaction ethylene + butadiene the accepted reaction channel is a synchronous concerted mechanism along a pericyclic transition state. In such a process four atomic centers are involved and neither one of the interacting molecules is assumed to act as an electrophile-nucleophile pair. Within the present approach this means that no charge transfer takes place at the transition state, and we expect both partners to display similar electronic potential. Accordingly, the HMO model does predict that the relative electronic potential defined in Eq. 8 vanishes identically for both the diene and the dienophile, thereby predicting a pure pericyclic pathway for this process (see Figure 1).


 
Fig. 1. Relative electrophilicity values ( in b units) of some diene and dienophile reagents commonly present in Diels-Alder 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 29. The switch from pericyclic to polar stepwise mechanisms implies an increase in the charge transfer pattern at the transition state, which should be reflected in an increase of the electrophilicity of the dienophile and a decrease of electrophilicity (i.e. an enhancement of nucleophilicity) of the diene. In Figure 1 we present some examples involving chemical substitution by electron withdrawing groups (EWG) at the dienophile (i.e. the electrophile) and electron releasing groups (ERG) at the diene (the nucleophile). Substituent effects may be divided in four important contributions according to Topsom classification 30.. They are: (a) the substituent dipole leading to a field effect; (b) the electronegativity difference between the substituent and the atom directly attached to it, leading to an inductive or electronegativity effect; (c) charge transfer between suitable orbitals of the substituent and the group to which it is attached, leading to resonance or hyperconjugative effects and (d) polarizability effects. Within HMO theory however, only electronegativity effect of (b) and resonance and hyperconjugative effects of (c) are taken into account. For instance, substitution by a ­CH3 group entails the incorporation of the 2pZ empty orbital of the carbon atom to the p framework. The interaction of this empty atomic orbital with the p system leads to a stabilizing effect know as hyperconjugation. The incorporation of a ­NO2 group on the other hand introduces both, electronegativity and resonance effects. The nature of electron releasing or electron withdrawing groups effects is therefore introduced in HMO theory, in the form of a re-parametrization of the a and b integrals of heteroatoms containing information about their electronegativity. Note that the HMO scale correctly predicts that EWG substitution on the dienophile enhances its electrophilicity with respect to the reference ethylene molecule, whereas ERG substitution at the diene brings the global electrophilicity down with respect to reference diene (1,3 butadiene, see Figure 1). According to the present scale of electrophilicity, an increasing value of the relative electronic chemical potential will result in an increase in electrophilicity difference of the diene/dienophile pair, thereby favoring a polar stepwise reaction channel, in agreement with the experimental results.

Another relevant aspect of the DA cycloadditions concerns their regioselectivity. Selectivity may be handled in terms of local reactivity parameters 31,32.. The Fukui function has been proposed as a reliable descriptor of selectivity 31,32. Within HMO theory we have shown that using the finite difference approximation within a frozen core regime, simple expressions for the electrophilic and nucleophilic Fukui functions condensed to atoms may be obtained. In order to test the reliability of the HMO Fukui functions we performed a local analysis on some representative examples, for which experimental data about regioselectivity is available. The general conclusions are summarized in Figure 2. For instance, for the pure pericyclic process butadiene + ethylene, the Fukui function analysis reveals that the active sites in the diene are the 1,4 centers, whereas, as expected, ethylene shows two equivalent electrophilic sites at C1 and C2 ( i.e f +k = f -k = 0.5 ). However the most important result concerns the site activation pattern induced by chemical substitution by ERG at the diene and EWG at the dienophile. It may be seen in Figure 2 that chemical substitution by ERG at position 1 in the diene activates position 4 for an electrophilic attack. Chemical substitution at the dienophile by an EWG on the other hand activates the unsubstituted center. As a result, the reaction of 1-ERG diene with 1-EWG dienophile will afford the ortho-product, in agreement with the observed regioselectivity in the DA reactions. On the other hand, substitution by ERG at position 2 on the diene activates position 1. Therefore the reaction of this 2-substituted dienes with 1-EWG substituted ethylenes will afford the para-product, in agreement again with the observed regioselectivity.


 
Fig.2. Selectivity and site activation diagram for ethylene substituted with electron withdrawing groups (EWG) and butadiene substituted with electron releasing groups (ERG). Bold circle highlights the largest value of the nucleophilic Fukui function at the diene and the electrophilic Fukui function at the dienophile.

5. Concluding remarks.

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.

 

Acknowledgments

Work supported by UTFSM project N 130223, and Fondecyt grant N 1030548.

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Correspondencia a: e-mail:arie.aizman@qui.utfsm.cl

Received: july 15, 2003 - Acepted: September 11, 2003

 

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