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Chapter 2 Density functional theory models of reactivity based on an energetic criterion
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Theoretical Aspects of Chemical Reactivity A. Toro-Labbé (Editor) © 2007 Published by Elsevier B.V.
Chapter 2
Density functional theory models of reactivity based on an energetic criterion Andrés Cedillo Departamento de Química, Universidad Autónoma Metropolitana-Iztapalapa San Rafael Atlixco 186, Iztapalapa DF 09340, México
Abstract Density functional theory is a very fruitful theoretical tool to understand and develop reactivity concepts. Several quantities have been defined to construct a theoretical framework of the intrinsic response of chemical species, using either intuitive or formal procedures. However, this framework is far from a final and complete form, and no systematic procedure is known to generate the appropriate quantities to describe a specific problem. This work shows that some of the most successful reactivity parameters appear from a procedure where energy stabilization takes a relevant place.
1. Introduction Density functional theory (DFT) provides an efficient method to include correlation energy in electronic structure calculations, namely the Kohn–Sham method;1 in addition, it constitutes a solid support to reactivity models.2 DFT framework has been used to formalize empirical reactivity descriptors, such as electronegativity,3 hardness4 and electrophilicity index.5 The frontier orbital theory was generalized by the introduction of Fukui function,6 and new reactivity parameters have also been proposed.78 Moreover, relationships between those parameters have been found, and general methods to relate new quantities exist.9 Usually reactivity parameters have been associated with the response of electronic properties of the system to the changes in the independent variables. Then reactivity parameters are identified with response functions, and they are represented by derivatives of electronic properties. 19
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Within the Born–Oppenheimer approximation, the number of electrons, N , and the external potential (from nonelectronic forces), r, completely determine the Hamiltonian operator, and, as a consequence, they also determine all the properties of the system. Specifically, the ground-state energy, E, is a concave function of the number of electrons and a convex functional of the external potential, E = EN The former dependence comes from experimental evidences, consecutive ionization potentials always increase, the latter was demonstrated by Lieb several years ago.10 Variations in the independent variables lead to a change in the energy which is approximated by a differential form, E
E N + rdr = N + r rdr E ≈
N r N The derivative with respect to the number of electrons,
E ≡
N is called the electronic chemical potential, or shortly the chemical potential.3 For non degenerate ground states, the functional derivative with respect to the external potential corresponds to the electron density, E r = r N Then and r represent the response of the energy to the changes on N and r, respectively. Following the same procedure, changes of other electronic properties can be analyzed. Variations of the chemical potential and electron density possess special interest,
≈ N + rdr = N + fr rdr
N r N r
r r ≈ N + r dr = fr N + r r r dr
N r N The response of the chemical potential to changes in the number of electrons is called the chemical hardness,4 2
E
= > 0 ≡
N
N 2 The positive sign of chemical hardness is a consequence of the concavity of the energy with respect to the number of electrons. The Fukui function, fr, appears in both equations since it represents the sensitivity of the chemical potential to the changes in the external potential and that of the density with respect to the number of electrons,6 E
r
fr ≡ = = r N
N
N r N
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The response of the density to the changes in the external potential, r 2 E r r ≡ = r N r r N is the density response kernel. Since and r are the first derivatives of the energy, their corresponding variations only involve the three second derivatives of the energy, namely the density response kernel, Fukui function and hardness. In addition to the canonical representation, where the independent variables are N and r, the convexity of the functional EN allows the existence of other representations.8 The grand potential,
E ≡ EN − N = EN − N
N provides an appropriate representation for open systems, the grand canonical representation. Previous equation corresponds to a Legendre transformation, which generates an alternative representation for ground states. In this case, we have the following expressions,
≈ + rdr = −N + r rdr
r
N N N ≈ + rdr = S − sr rdr
r r
r r dr = sr − r r r dr + r ≈ r
For this representation, −N and r represent the response of the grand potential to the changes of and r, respectively,
= −N = r
r As a consequence, for an open system, the number of electrons and the density depend on and r. The sensitivities of N and r correspond to the second derivatives of the grand potential, and they are the chemical softness,7 2
N
S≡ =−
2 the local softness,7 sr ≡
r
N =− r
=
r
and the softness kernel,8 r r ≡ −
r r
=−
2 r r
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Derivatives from two representations are not independent; they can be related by a reduction of derivatives scheme, analogous with the thermodynamical procedure.9 For the previous representations, one finds that the second derivatives of the grand canonical representation can be written in the following form,
N 1 1 S= = =
N
r
N
r sr ≡ = = Sfr
N −1
r
N r
r r − r r ≡ − =
r N r = Sfrfr − r r From these equations, one can easily find that sr = r r dr where we used the following properties, r r dr = 0
S=
srdr
frdr = 1
The same procedure can be applied for derivatives from other representations. Electronic properties obtained by differentiation are usually classified by its dependence on the position. Global properties have the same value everywhere, such as the chemical potential, hardness and softness. Electron density, Fukui function and local softness change throughout the molecule, and they are called local properties. Finally, kernel properties depend on two or more position vectors, like the density response and softness kernels. Global parameters describe molecular reactivity, local properties provide information on site selectivity, while kernels can be used to understand site activation. The chemical behaviour of a given species strongly depends on the nature of the other molecules involved in the interaction. For a specific type of reaction, an appropriate model is needed to simulate the chemical environment of the species of interest. In the present work, the interest is focused on the initial response of the molecule to a particular type of chemical situation, independent of the value of those parameters that characterize one specific reaction. In other words, the intrinsic capabilities of the chemical species are studied and modelled as derivatives of the electronic properties with respect to an appropriate independent variable. For example, in those processes where charge transfer is involved (such as Lewis acidity and basicity, electrophile– nucleophile interactions and coordination compounds), the number of electrons must be an independent variable; when a small molecule interacts with a very large counterpart (such as enzyme–substrate interaction and adsorption on solid surfaces), the chemical potential of the large partner will be imposed on the small molecule, and its number of electrons will not be independent. In this work, energetic criteria will be applied to support the use of derivatives, coming from some ground-state representation, to describe the intrinsic reactivity of molecules in specific types of reactions.
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2. Electronegativity and related quantities When two different atoms or fragments form a covalent bond, each one attracts the shared electron pair with different strength, giving rise to a polar bond. Pauling11 originally proposed a model to quantify this property, named electronegativity. Over the years, other electronegativity scales have been defined, using different ways to compute it.12 Electronegativity differences drive many chemical processes. Lewis acid and base model characterize acids as electron-deficient species, while bases are electron donors. Since Lewis acids show a stronger attraction for the electrons than Lewis bases, the electronegativity of the acids must be larger than that of the bases. Coordination compounds are formed by one or more Lewis acids, usually metal cations, and some ligands, Lewis bases. In organic reactions, an electrophile represents an electron-deficient species, or Lewis acid, while a nucleophile corresponds to an electron donor, or Lewis base. One molecule may have several acid and basic sites. In order to characterize the nature and strength off those sites, local properties are needed. Identification of electrophilic and nucleophilic regions of the molecule is very useful in the prediction of the initial steps of chemical reactions.
2.1. Global analysis Consider a process where a set of chemical species will receive a small portion of electron transfer, dN > 0. When the external potential remains constant for all the species, the changes of the energy are given by dEi = i dN In a species with the ability to accept charge, the energy must decrease after the electron-transfer process, and its chemical potential must be negative. The corresponding stabilization for the set of molecules becomes −dEi = − i dN > 0 Then, the negative of the chemical potential is proportional to the stabilization of a chemical species when it receives electrons. Likewise, when the same species donate electrons, dN < 0, their energies increase, dEi = − i −dN > 0 The negative of the chemical potential also measures the resistance of the chemical species to deliver electrons. From this behaviour, the negative of the chemical potential is identified with the electronegativity,3
E 1 = − = − ≈ I + A
N 2 The last approximation corresponds to the finite differences approach to the first derivative, and I and A represent the ionization potential and electron affinity of the
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species, respectively. This approximation corresponds to Mulliken’s electronegativity approach.13 In the interaction of two molecules with different chemical potentials (or electronegativities), neglecting the effects from the changes in the external potential, electron transfer from the species with higher chemical potential (lower electronegativity) to the species with lower chemical potential (higher electronegativity) is energetically preferred. Along this electron-transfer process, the chemical potential of both species also changes, d i = i dN where is the chemical hardness, which is a positive quantity, and it can be evaluated by a finite differences scheme, ≈ I − A. The chemical potential of the acceptor increases, while that of the donor decreases, until both become equal (chemical potential equalization).14 At this point, electron transfer ends. Electronegativity equalization methods have been very useful to predict atomic or fragment charges for complex systems.15 In these methods, it is shown that the maximum stabilization is obtained when the electronegativity of each fragment becomes equal. Chemical potential of a species measures the initial affinity for the electrons; however, during the electron transfer, this affinity is modified. Chemical hardness, , modulates this variation. Ignoring the external potential effects, the change in the energy, up to second order, is given by 1 E ≈ N + N2 2 Two electron acceptors (Lewis acids), with the same chemical potential (electronegativity), will show different behaviour in the charge transfer process. The species with larger hardness (harder species) easily satisfies its affinity for electrons, while the species with lower hardness (softer species) keeps its affinity for larger amounts of charge transfer, see Figure 1. Donating behaviour for bases shows that for the same change in the energy, the soft base is able to transfer a bigger amount of charge than the hard one. Then hard species are associated with small amounts of charge transfer, while for soft
Figure 1 Change in the energy for two species with the same chemical potential 1 > 2
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species with large. This accordance is consistent with Pearson’s hard and soft acids and bases principle.16 An electronegative species increases its chemical potential when it receives charge up to the time that its affinity for electrons is satisfied, that is when the chemical potential vanishes. This situation corresponds to a minimum in Figure 1. Using the second-order truncated Taylor expansion, one finds that the maximum amount of charge that a Lewis acid can accept is Nmax = − S = S This expression represents the charge capacity of the species. It is important to note that a very hard species, even when its electronegativity is large, can have a small value of Nmax , on account of the small value of its softness, see Figure 1. This situation is found on the chemical behaviour of the halogens, fluorine atom is more electronegative and harder than chlorine atom, however chloride ion is a weaker base that fluoride ion. When both atoms accept one electron, chloride ion’s chemical potential is still negative, while fluoride’s becomes positive; furthermore, stabilization of chlorine is larger than that of fluorine. The maximum stabilization of a species in a charge transfer process can be also evaluated, 1 1 ≡ − Emin = − E Nmax = 2 S = Nmax 2 2 This quantity has been identified as an electrophilicity index.5 Note that it is constructed from two important features of an electrophile, electronegativity and charge capacity, which act cooperatively. One can observe that global reactivity parameters represent the response of an electronic property to the changes of the independent variables. The physical meaning of these quantities can be rationalized by proposing specific situations where a few quantities are involved. Also specific combinations of these parameters are useful to describe important chemical trends.
2.2. Local analysis A single molecule can exhibit both acidic and basic conducts on different sites. Global properties are unable to provide this kind of description, in this case, local properties are useful. Fukui function represents, either the response of the chemical potential to variation in the external potential or the response of the density to changes in the number of electrons.6 The latter is important in charge transfer descriptions. Fukui function minimizes a hardness functional,17 Hg ≡ grr ˜ r gr dr dr where r ˜ r is the hardness kernel,8 r ˜ r ≡
2 F r r
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F is the Hohenberg–Kohn universal functional and gr represents any function that integrates to unity,
grdr = 1
By the way, the extreme value of the hardness functional is equal to the chemical hardness,17 Hf = frr ˜ r fr dr dr = Then energy and density come from the variational principle of the energy, while Fukui function and hardness are obtained from the minimization of the hardness functional. Recently, it has been shown that the previous variational principle for the Fukui function can be derived from a minimization of the energy.18 When an electronic system changes its number of electrons in dN , the variation of the density that minimizes the energy follows the Fukui function, r = frdN That is, the most favourable way to distribute (or extract) dN electrons in a chemical species is guided by the Fukui function. From Janak’s extension for noninteger occupation numbers,19 the Fukui function can be written as the sum of a frontier orbital density and relaxation terms, ⎧ N
i r − ⎪ ⎪N r + dN < 0 ⎪ ⎨
N i=1 fr = ⎪ N
i r + ⎪ ⎪ ⎩N +1 r + dN > 0
N i=1 where i is the density associated with the i-th Kohn–Sham spin orbital. In consequence, Fukui function is considered as a generalization of the frontier orbital theory,20 since it includes terms coming from the redistribution of the electronic density induced by the addition or removal of electrons. Note that discontinuity on the density, as a function of the number of electrons, leads to a couple of expressions. When the number of electrons increases, a nucleophilic attack or acidic behaviour, nucleophilic Fukui function may be related to the LUMO density, +
f r ≡
r
N
+ ≈ LUMO r
If the number of electrons decreases, an electrophilic attack or basic behaviour, electrophilic Fukui function may be related to the HOMO density,
r − − f r ≡ ≈ HOMO r
N
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Fukui function is usually computed by a finite differences procedure, f − r ≈ N r − N −1 r f + r ≈ N +1 r − N r where N represents the density of a system with N electrons. Donor or basic sites of chemical species are characterized by large positive values of the Fukui function f − , while acceptor or acidic sites correspond to large values of f + . Since Fukui function integrates to unity, it cannot be used to compare sites from different molecules. In many cases, for large systems, it is found that Fukui function distributes along the molecule. Local softness also satisfies a variational principle,18 the variation of the density that minimizes the grand potential, when the chemical potential changes in d , is given by r = srd From the relation between Fukui function and local softness, electrophilic and nucleophilic local softnesses can be computed. Donor and acceptor sites can also be identified by large values of both types of local softnesses; in addition, it can be used to compare sites of different molecules and to identify which one is softer or harder. The electrophilicity index can also be extended to a local context,21 and a comparison of the electrophilicity of sites in different species can be made.
3. Interactions with nuclei Changes in the nuclei are associated with variations in the external potential. For example, nuclear displacements are related to Hellman–Feynman forces; protonation reactions (addition of a nucleus) are used to study Lewis acidity and basicity in gas phase and solution. In a protonation reaction, a hydrogen nucleus is added to a chemical species. The proton can be placed in several positions, and the molecular energy determines stable isomers and the most stable protonated form. The addition of one proton, at the position R0 , is represented by a change in the external potential given by r = −
1 r − R0
while the number of electrons remains constant. The change in the molecular energy, Emol , includes two terms, one for the electronic part, E, and the other for the nuclei, Enn , Emol = EN + − EN + Enn The nuclear contribution is an electrostatic term, Enn =
M
Z =1 R − R0
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where M is the original number of nuclei in the species. For the electronic part, a Taylor series expansion is used,
1 r r r r dr dr + · · · 2 r 1 r r =− dr dr + · · · dr + r − R0 2 r − R0 r − R0
EN + − EN =
r rdr +
Then, the change in the molecular energy, which depends on R0 , can be approximated, up to second order, by 1 Emol R0 ≈ R0 + R0 2 where is the molecular electrostatic potential, R =
r Z − dr r − R =1 R − R M
and the last term is an integral of the density response kernel, R =
r r dr dr < 0 r − Rr − R
The sign of the last integral is a consequence of the convexity of the energy as a functional of the external potential. Stable protonated isomers are associated with minima of Emol . A first-order approximation predicts that protonation occurs at places where the molecular electrostatic potential is a minimum.22 Since, in neutral species, negative values of the electrostatic potential are usually associated with lone pairs or electron-rich regions,23 this approximation is very rational. At this level, the change in the energy, Emol R0 ≈ R0 , represents the electrostatic interaction of the proton with the nuclei and the unperturbed electron density. Relaxation of the density, induced by the presence of the proton, is taken into account by higher-order terms. The second-order term, related to the density response kernel, contributes with an additional stabilization from the initial response of the density to the new positive charge. At the present, there is no simple procedure to compute the response kernels, neither its contributions to energy, and fundamental studies on this direction are desired. Nucleophiles or Lewis bases are involved in many chemical processes, including protonation, then the characterization of the nucleophilicity should take into account the response functions associated with variations of the external potential.
4. Concluding remarks The use of an energy stabilization procedure allows the identification of some response functions as reactivity parameters. In other cases, energetic criteria were used to find insights into the physical meaning. Since the intrinsic reactivity framework is far from
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being complete, the use of this kind of procedure can be useful to develop the appropriate parameters to describe specific chemical situations. At the present, the finite differences approximation used to evaluate global and local properties seems to be reasonable. However, theoretical developments to find useful computing schemes for kernels are needed, as well as models to understand and apply kernels to specific reactivity problems, such as the site activation case. Many chemical reactivity aspects are beyond the present framework. Since this methodology studies the intrinsic or initial response, properties associated with advanced steps of the reaction, such as those related with the transition state or the reaction product, may be inaccurately described. The same can be said about situations where the Born–Oppenheimer approximation fails, like tunnelling contributions.
Acknowledgements This work received financial support from CONACYT grant 39622-F.
References 1. See for example: Parr, R. G. and Yang, W., Density Functional Theory of Atoms and Molecules, Oxford, New York (1989); Gross, E. K. U. and Dreizler, R., Density Functional Theory, Springer, Berlin (1993). 2. Sen, K. D. and Jørgensen, C. K. (Eds.), Electronegativity, Structure and Bonding 66, Springer, Berlin (1987); Sen, K. D. (Ed.), Chemical Hardness, Structure and Bonding 80, Springer, Berlin (1993); Pearson, R. G., Chemical Hardness: Applications from Molecules to Solids, Wiley-VCH, Weinheim (1997). 3. Iczkowski, R. P. and Margrave, J. L., J. Am. Chem. Soc. 83 3547 (1961); Parr, R. G., Donnelly, R. A., Levy, M. and Palke, W. E., J. Chem. Phys. 69 4431 (1978). 4. Parr, R. G. and Pearson, R. G., J. Am. Chem. Soc. 105 7512 (1983). 5. Parr, R. G. von Szentpály, L. and Liu, S., J. Am. Chem. Soc. 121 1922 (1999). 6. Parr, R. G. and Yang, W., J. Am. Chem. Soc. 106 4049 (1984). 7. Yang, W. and Parr, R. G., Proc. Natl. Acad. Sci. U.S.A 82 6723 (1985). 8. Berkowitz, M. and Parr, R. G., J. Chem. Phys. 88 2554 (1988). 9. Nalewajski, R. F. and Parr, R. G., J. Chem. Phys. 77 399 (1982); Nalewajski, R. F., J. Chem. Phys. 78 6112 (1983); Cedillo, A., Int. J. Quant. Chem. Suppl. 28 231 (1994). 10. Lieb, E. H., Int. J. Quant. Chem. 24 243 (1983). 11. Pauling, L., The Nature of the Chemical Bond, 3rd ed., Cornell, Ithaca (1960). 12. See for example: Sanderson, R. T., J. Chem. Educ. 65 112 (1988); Sanderson, R. T., J. Chem. Educ. 65 227 (1988); Huheey, J. E., Inorganic Chemistry: Principles of Structure and Reactivity, 2nd ed., pp. 159–173, Harper & Row, New York (1978). 13. Mulliken, R. S., J. Chem. Phys. 2 782 (1934). 14. Sanderson, R. T., Science 121 207 (1955). 15. See for example: Mortier, W. J., Ghosh, S. K. and Shankar, S., J. Am. Chem. Soc. 108 4315 (1986); Van Genechten, K., Mortier, W. J. and Geerlings, P., J. Chem. Phys. 86 5063 (1987); Baekelandt, B. G., Mortier, W. J., and Schoonheydt, R. A. Structure and Bonding 80 187 (1993); Nalewajski, R. F., Korchowiec, J. and Zhou, Z. Int. J. Quant. Chem. Suppl. 22 349 (1988). 16. Pearson, R. G., J. Am. Chem. Soc. 85 3533 (1963); Pearson, R. G., Hard and Soft Acids and Bases, Dowden, Hutchinson and Ross, Stroudsburg, PA (1973).
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Chattaraj, P. K., Cedillo, A. and Parr, R. G., J. Chem. Phys. 103 7645 (1995). Ayers, P. W. and Parr, R. G., J. Am. Chem. Soc. 122 2010 (2000). Janak, J. F., Phys. Rev. B 18 7165 (1978). Fukui, K., Theory of Orientation and Stereoselection, Springer, Berlin (1973); Fukui, K., Science 218 747 (1982). 21. Fuentealba, P. and Contreras, R., in Reviews of Modern Quantum Chemistry, Sen, K. D. (Ed.), pp. 1013–1052, World Scientific, Singapore (2002); Chattaraj, P. K., Maiti, B. and Sarkar, U., J. Phys. Chem. A 107 4973 (2003); Cedillo, A. and Contreras, R. (to be published). 22. Politzer, P. and Daiker, K. C., J. Chem. Phys. 68 5289 (1978). 23. See for example: Murray, J.S. and Sen, K.D. (Eds.), Molecular Electrostatic Potentials: Concepts and Applications, Elsevier, Amsterdam (1996).
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Theoretical Aspects of Chemical Reactivity A. Toro-Labbé (Editor) © 2007 Published by Elsevier B.V.
Chapter 2
Density functional theory models of reactivity based on an energetic criterion Andrés Cedillo Departamento de Química, Universidad Autónoma Metropolitana-Iztapalapa San Rafael Atlixco 186, Iztapalapa DF 09340, México
Abstract Density functional theory is a very fruitful theoretical tool to understand and develop reactivity concepts. Several quantities have been defined to construct a theoretical framework of the intrinsic response of chemical species, using either intuitive or formal procedures. However, this framework is far from a final and complete form, and no systematic procedure is known to generate the appropriate quantities to describe a specific problem. This work shows that some of the most successful reactivity parameters appear from a procedure where energy stabilization takes a relevant place.
1. Introduction Density functional theory (DFT) provides an efficient method to include correlation energy in electronic structure calculations, namely the Kohn–Sham method;1 in addition, it constitutes a solid support to reactivity models.2 DFT framework has been used to formalize empirical reactivity descriptors, such as electronegativity,3 hardness4 and electrophilicity index.5 The frontier orbital theory was generalized by the introduction of Fukui function,6 and new reactivity parameters have also been proposed.78 Moreover, relationships between those parameters have been found, and general methods to relate new quantities exist.9 Usually reactivity parameters have been associated with the response of electronic properties of the system to the changes in the independent variables. Then reactivity parameters are identified with response functions, and they are represented by derivatives of electronic properties. 19
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Within the Born–Oppenheimer approximation, the number of electrons, N , and the external potential (from nonelectronic forces), r, completely determine the Hamiltonian operator, and, as a consequence, they also determine all the properties of the system. Specifically, the ground-state energy, E, is a concave function of the number of electrons and a convex functional of the external potential, E = EN The former dependence comes from experimental evidences, consecutive ionization potentials always increase, the latter was demonstrated by Lieb several years ago.10 Variations in the independent variables lead to a change in the energy which is approximated by a differential form, E
E N + rdr = N + r rdr E ≈
N r N The derivative with respect to the number of electrons,
E ≡
N is called the electronic chemical potential, or shortly the chemical potential.3 For non degenerate ground states, the functional derivative with respect to the external potential corresponds to the electron density, E r = r N Then and r represent the response of the energy to the changes on N and r, respectively. Following the same procedure, changes of other electronic properties can be analyzed. Variations of the chemical potential and electron density possess special interest,
≈ N + rdr = N + fr rdr
N r N r
r r ≈ N + r dr = fr N + r r r dr
N r N The response of the chemical potential to changes in the number of electrons is called the chemical hardness,4 2
E
= > 0 ≡
N
N 2 The positive sign of chemical hardness is a consequence of the concavity of the energy with respect to the number of electrons. The Fukui function, fr, appears in both equations since it represents the sensitivity of the chemical potential to the changes in the external potential and that of the density with respect to the number of electrons,6 E
r
fr ≡ = = r N
N
N r N
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The response of the density to the changes in the external potential, r 2 E r r ≡ = r N r r N is the density response kernel. Since and r are the first derivatives of the energy, their corresponding variations only involve the three second derivatives of the energy, namely the density response kernel, Fukui function and hardness. In addition to the canonical representation, where the independent variables are N and r, the convexity of the functional EN allows the existence of other representations.8 The grand potential,
E ≡ EN − N = EN − N
N provides an appropriate representation for open systems, the grand canonical representation. Previous equation corresponds to a Legendre transformation, which generates an alternative representation for ground states. In this case, we have the following expressions,
≈ + rdr = −N + r rdr
r
N N N ≈ + rdr = S − sr rdr
r r
r r dr = sr − r r r dr + r ≈ r
For this representation, −N and r represent the response of the grand potential to the changes of and r, respectively,
= −N = r
r As a consequence, for an open system, the number of electrons and the density depend on and r. The sensitivities of N and r correspond to the second derivatives of the grand potential, and they are the chemical softness,7 2
N
S≡ =−
2 the local softness,7 sr ≡
r
N =− r
=
r
and the softness kernel,8 r r ≡ −
r r
=−
2 r r
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Derivatives from two representations are not independent; they can be related by a reduction of derivatives scheme, analogous with the thermodynamical procedure.9 For the previous representations, one finds that the second derivatives of the grand canonical representation can be written in the following form,
N 1 1 S= = =
N
r
N
r sr ≡ = = Sfr
N −1
r
N r
r r − r r ≡ − =
r N r = Sfrfr − r r From these equations, one can easily find that sr = r r dr where we used the following properties, r r dr = 0
S=
srdr
frdr = 1
The same procedure can be applied for derivatives from other representations. Electronic properties obtained by differentiation are usually classified by its dependence on the position. Global properties have the same value everywhere, such as the chemical potential, hardness and softness. Electron density, Fukui function and local softness change throughout the molecule, and they are called local properties. Finally, kernel properties depend on two or more position vectors, like the density response and softness kernels. Global parameters describe molecular reactivity, local properties provide information on site selectivity, while kernels can be used to understand site activation. The chemical behaviour of a given species strongly depends on the nature of the other molecules involved in the interaction. For a specific type of reaction, an appropriate model is needed to simulate the chemical environment of the species of interest. In the present work, the interest is focused on the initial response of the molecule to a particular type of chemical situation, independent of the value of those parameters that characterize one specific reaction. In other words, the intrinsic capabilities of the chemical species are studied and modelled as derivatives of the electronic properties with respect to an appropriate independent variable. For example, in those processes where charge transfer is involved (such as Lewis acidity and basicity, electrophile– nucleophile interactions and coordination compounds), the number of electrons must be an independent variable; when a small molecule interacts with a very large counterpart (such as enzyme–substrate interaction and adsorption on solid surfaces), the chemical potential of the large partner will be imposed on the small molecule, and its number of electrons will not be independent. In this work, energetic criteria will be applied to support the use of derivatives, coming from some ground-state representation, to describe the intrinsic reactivity of molecules in specific types of reactions.
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2. Electronegativity and related quantities When two different atoms or fragments form a covalent bond, each one attracts the shared electron pair with different strength, giving rise to a polar bond. Pauling11 originally proposed a model to quantify this property, named electronegativity. Over the years, other electronegativity scales have been defined, using different ways to compute it.12 Electronegativity differences drive many chemical processes. Lewis acid and base model characterize acids as electron-deficient species, while bases are electron donors. Since Lewis acids show a stronger attraction for the electrons than Lewis bases, the electronegativity of the acids must be larger than that of the bases. Coordination compounds are formed by one or more Lewis acids, usually metal cations, and some ligands, Lewis bases. In organic reactions, an electrophile represents an electron-deficient species, or Lewis acid, while a nucleophile corresponds to an electron donor, or Lewis base. One molecule may have several acid and basic sites. In order to characterize the nature and strength off those sites, local properties are needed. Identification of electrophilic and nucleophilic regions of the molecule is very useful in the prediction of the initial steps of chemical reactions.
2.1. Global analysis Consider a process where a set of chemical species will receive a small portion of electron transfer, dN > 0. When the external potential remains constant for all the species, the changes of the energy are given by dEi = i dN In a species with the ability to accept charge, the energy must decrease after the electron-transfer process, and its chemical potential must be negative. The corresponding stabilization for the set of molecules becomes −dEi = − i dN > 0 Then, the negative of the chemical potential is proportional to the stabilization of a chemical species when it receives electrons. Likewise, when the same species donate electrons, dN < 0, their energies increase, dEi = − i −dN > 0 The negative of the chemical potential also measures the resistance of the chemical species to deliver electrons. From this behaviour, the negative of the chemical potential is identified with the electronegativity,3
E 1 = − = − ≈ I + A
N 2 The last approximation corresponds to the finite differences approach to the first derivative, and I and A represent the ionization potential and electron affinity of the
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species, respectively. This approximation corresponds to Mulliken’s electronegativity approach.13 In the interaction of two molecules with different chemical potentials (or electronegativities), neglecting the effects from the changes in the external potential, electron transfer from the species with higher chemical potential (lower electronegativity) to the species with lower chemical potential (higher electronegativity) is energetically preferred. Along this electron-transfer process, the chemical potential of both species also changes, d i = i dN where is the chemical hardness, which is a positive quantity, and it can be evaluated by a finite differences scheme, ≈ I − A. The chemical potential of the acceptor increases, while that of the donor decreases, until both become equal (chemical potential equalization).14 At this point, electron transfer ends. Electronegativity equalization methods have been very useful to predict atomic or fragment charges for complex systems.15 In these methods, it is shown that the maximum stabilization is obtained when the electronegativity of each fragment becomes equal. Chemical potential of a species measures the initial affinity for the electrons; however, during the electron transfer, this affinity is modified. Chemical hardness, , modulates this variation. Ignoring the external potential effects, the change in the energy, up to second order, is given by 1 E ≈ N + N2 2 Two electron acceptors (Lewis acids), with the same chemical potential (electronegativity), will show different behaviour in the charge transfer process. The species with larger hardness (harder species) easily satisfies its affinity for electrons, while the species with lower hardness (softer species) keeps its affinity for larger amounts of charge transfer, see Figure 1. Donating behaviour for bases shows that for the same change in the energy, the soft base is able to transfer a bigger amount of charge than the hard one. Then hard species are associated with small amounts of charge transfer, while for soft
Figure 1 Change in the energy for two species with the same chemical potential 1 > 2
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species with large. This accordance is consistent with Pearson’s hard and soft acids and bases principle.16 An electronegative species increases its chemical potential when it receives charge up to the time that its affinity for electrons is satisfied, that is when the chemical potential vanishes. This situation corresponds to a minimum in Figure 1. Using the second-order truncated Taylor expansion, one finds that the maximum amount of charge that a Lewis acid can accept is Nmax = − S = S This expression represents the charge capacity of the species. It is important to note that a very hard species, even when its electronegativity is large, can have a small value of Nmax , on account of the small value of its softness, see Figure 1. This situation is found on the chemical behaviour of the halogens, fluorine atom is more electronegative and harder than chlorine atom, however chloride ion is a weaker base that fluoride ion. When both atoms accept one electron, chloride ion’s chemical potential is still negative, while fluoride’s becomes positive; furthermore, stabilization of chlorine is larger than that of fluorine. The maximum stabilization of a species in a charge transfer process can be also evaluated, 1 1 ≡ − Emin = − E Nmax = 2 S = Nmax 2 2 This quantity has been identified as an electrophilicity index.5 Note that it is constructed from two important features of an electrophile, electronegativity and charge capacity, which act cooperatively. One can observe that global reactivity parameters represent the response of an electronic property to the changes of the independent variables. The physical meaning of these quantities can be rationalized by proposing specific situations where a few quantities are involved. Also specific combinations of these parameters are useful to describe important chemical trends.
2.2. Local analysis A single molecule can exhibit both acidic and basic conducts on different sites. Global properties are unable to provide this kind of description, in this case, local properties are useful. Fukui function represents, either the response of the chemical potential to variation in the external potential or the response of the density to changes in the number of electrons.6 The latter is important in charge transfer descriptions. Fukui function minimizes a hardness functional,17 Hg ≡ grr ˜ r gr dr dr where r ˜ r is the hardness kernel,8 r ˜ r ≡
2 F r r
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F is the Hohenberg–Kohn universal functional and gr represents any function that integrates to unity,
grdr = 1
By the way, the extreme value of the hardness functional is equal to the chemical hardness,17 Hf = frr ˜ r fr dr dr = Then energy and density come from the variational principle of the energy, while Fukui function and hardness are obtained from the minimization of the hardness functional. Recently, it has been shown that the previous variational principle for the Fukui function can be derived from a minimization of the energy.18 When an electronic system changes its number of electrons in dN , the variation of the density that minimizes the energy follows the Fukui function, r = frdN That is, the most favourable way to distribute (or extract) dN electrons in a chemical species is guided by the Fukui function. From Janak’s extension for noninteger occupation numbers,19 the Fukui function can be written as the sum of a frontier orbital density and relaxation terms, ⎧ N
i r − ⎪ ⎪N r + dN < 0 ⎪ ⎨
N i=1 fr = ⎪ N
i r + ⎪ ⎪ ⎩N +1 r + dN > 0
N i=1 where i is the density associated with the i-th Kohn–Sham spin orbital. In consequence, Fukui function is considered as a generalization of the frontier orbital theory,20 since it includes terms coming from the redistribution of the electronic density induced by the addition or removal of electrons. Note that discontinuity on the density, as a function of the number of electrons, leads to a couple of expressions. When the number of electrons increases, a nucleophilic attack or acidic behaviour, nucleophilic Fukui function may be related to the LUMO density, +
f r ≡
r
N
+ ≈ LUMO r
If the number of electrons decreases, an electrophilic attack or basic behaviour, electrophilic Fukui function may be related to the HOMO density,
r − − f r ≡ ≈ HOMO r
N
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Fukui function is usually computed by a finite differences procedure, f − r ≈ N r − N −1 r f + r ≈ N +1 r − N r where N represents the density of a system with N electrons. Donor or basic sites of chemical species are characterized by large positive values of the Fukui function f − , while acceptor or acidic sites correspond to large values of f + . Since Fukui function integrates to unity, it cannot be used to compare sites from different molecules. In many cases, for large systems, it is found that Fukui function distributes along the molecule. Local softness also satisfies a variational principle,18 the variation of the density that minimizes the grand potential, when the chemical potential changes in d , is given by r = srd From the relation between Fukui function and local softness, electrophilic and nucleophilic local softnesses can be computed. Donor and acceptor sites can also be identified by large values of both types of local softnesses; in addition, it can be used to compare sites of different molecules and to identify which one is softer or harder. The electrophilicity index can also be extended to a local context,21 and a comparison of the electrophilicity of sites in different species can be made.
3. Interactions with nuclei Changes in the nuclei are associated with variations in the external potential. For example, nuclear displacements are related to Hellman–Feynman forces; protonation reactions (addition of a nucleus) are used to study Lewis acidity and basicity in gas phase and solution. In a protonation reaction, a hydrogen nucleus is added to a chemical species. The proton can be placed in several positions, and the molecular energy determines stable isomers and the most stable protonated form. The addition of one proton, at the position R0 , is represented by a change in the external potential given by r = −
1 r − R0
while the number of electrons remains constant. The change in the molecular energy, Emol , includes two terms, one for the electronic part, E, and the other for the nuclei, Enn , Emol = EN + − EN + Enn The nuclear contribution is an electrostatic term, Enn =
M
Z =1 R − R0
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where M is the original number of nuclei in the species. For the electronic part, a Taylor series expansion is used,
1 r r r r dr dr + · · · 2 r 1 r r =− dr dr + · · · dr + r − R0 2 r − R0 r − R0
EN + − EN =
r rdr +
Then, the change in the molecular energy, which depends on R0 , can be approximated, up to second order, by 1 Emol R0 ≈ R0 + R0 2 where is the molecular electrostatic potential, R =
r Z − dr r − R =1 R − R M
and the last term is an integral of the density response kernel, R =
r r dr dr < 0 r − Rr − R
The sign of the last integral is a consequence of the convexity of the energy as a functional of the external potential. Stable protonated isomers are associated with minima of Emol . A first-order approximation predicts that protonation occurs at places where the molecular electrostatic potential is a minimum.22 Since, in neutral species, negative values of the electrostatic potential are usually associated with lone pairs or electron-rich regions,23 this approximation is very rational. At this level, the change in the energy, Emol R0 ≈ R0 , represents the electrostatic interaction of the proton with the nuclei and the unperturbed electron density. Relaxation of the density, induced by the presence of the proton, is taken into account by higher-order terms. The second-order term, related to the density response kernel, contributes with an additional stabilization from the initial response of the density to the new positive charge. At the present, there is no simple procedure to compute the response kernels, neither its contributions to energy, and fundamental studies on this direction are desired. Nucleophiles or Lewis bases are involved in many chemical processes, including protonation, then the characterization of the nucleophilicity should take into account the response functions associated with variations of the external potential.
4. Concluding remarks The use of an energy stabilization procedure allows the identification of some response functions as reactivity parameters. In other cases, energetic criteria were used to find insights into the physical meaning. Since the intrinsic reactivity framework is far from
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being complete, the use of this kind of procedure can be useful to develop the appropriate parameters to describe specific chemical situations. At the present, the finite differences approximation used to evaluate global and local properties seems to be reasonable. However, theoretical developments to find useful computing schemes for kernels are needed, as well as models to understand and apply kernels to specific reactivity problems, such as the site activation case. Many chemical reactivity aspects are beyond the present framework. Since this methodology studies the intrinsic or initial response, properties associated with advanced steps of the reaction, such as those related with the transition state or the reaction product, may be inaccurately described. The same can be said about situations where the Born–Oppenheimer approximation fails, like tunnelling contributions.
Acknowledgements This work received financial support from CONACYT grant 39622-F.
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Chattaraj, P. K., Cedillo, A. and Parr, R. G., J. Chem. Phys. 103 7645 (1995). Ayers, P. W. and Parr, R. G., J. Am. Chem. Soc. 122 2010 (2000). Janak, J. F., Phys. Rev. B 18 7165 (1978). Fukui, K., Theory of Orientation and Stereoselection, Springer, Berlin (1973); Fukui, K., Science 218 747 (1982). 21. Fuentealba, P. and Contreras, R., in Reviews of Modern Quantum Chemistry, Sen, K. D. (Ed.), pp. 1013–1052, World Scientific, Singapore (2002); Chattaraj, P. K., Maiti, B. and Sarkar, U., J. Phys. Chem. A 107 4973 (2003); Cedillo, A. and Contreras, R. (to be published). 22. Politzer, P. and Daiker, K. C., J. Chem. Phys. 68 5289 (1978). 23. See for example: Murray, J.S. and Sen, K.D. (Eds.), Molecular Electrostatic Potentials: Concepts and Applications, Elsevier, Amsterdam (1996).