- Email: [email protected]
Intramolecular interactions along the reaction path of keto–enol tautomerism: Fukui functions as local softnesses and charges as local hardnesses
Journal of Molecular Structure (Theochem) 686 (2004) 213–218 www.elsevier.com/locate/theochem
Short communication
Intramolecular interactions along the reaction path of keto–enol tautomerism: Fukui functions as local softnesses and charges as local hardnesses Alexandre Hocqueta,*, Alejandro Toro-Labbe´b, Henry Chermettec b
a LCPM, UMR CNRS INPL 7568, B.P. 451, 54001 Nancy cedex, France QTC, Facultad de Quı´mica, Pontificia Universidad Cato´lica de Chile, Casilla 306, Correo 22, Santiago, Chile c CPT, UMR CNRS 5182, 43 Bld du 11 Novembre 1918, 69622 Villeurbanne cedex, France
Received 29 July 2004; revised 14 August 2004; accepted 16 August 2004
Abstract This paper presents an unprecedented analysis of local hard and soft descriptors of the intramolecular reactivity along the reaction path of the keto-enol tautomerism. Different kinds of atomic charges (namely Mulliken, NPA, ESP, Hirschfeld and AIM), as descriptors of local hardness, and different kinds of condensed Fukui functions, as descriptors of local softness are tested in their ability to describe a consistent picture of internal reactivity. q 2004 Elsevier B.V. All rights reserved. Keywords: Proton transfer; Tautomerism; HSAB; Fukui function; Local softness; Local hardness; Reaction path
1. Introduction In a previous publication, Perez and Toro-Labbe´ [1] investigated the reaction profile of keto–enol tautomerism on the basis of energy, chemical potential and hardness as a function of the reaction coordinate, along the reaction path. This reaction has been shown to be important in gaining insights into the popular chemical concepts associated with reactivity and selectivity. Although chemical reactivity is characterized by global reactivity parameters like electronegativity or hardness, the selectivity is usually understood in terms of local functions like the Fukui function [2,3] (f(r)) and local softness [4] (s(r)). These global and local descriptors of reactivity have been popularised within the framework of ‘conceptual density functional theory’, a field to which reviews have been dedicated recently [5,6]. According to the electronegativity equalization principle [7], ‘all the constituent atoms in a molecule have the same
* Corresponding author. Tel.: C33-3-83-17-5252; fax: 33-3-83-37-9977. E-mail address: [email protected] (A. Hocquet). 0166-1280/$ - see front matter q 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.theochem.2004.08.011
electronegativity value which is roughly equal to the geometric mean of the electronegativities of the isolated atoms’. The hard–soft acids–bases (HSAB) principle [8] states that, ‘among the potential partners of same electronegativity, hard likes hard and soft likes soft’, and the maximum hardness principle [9,10,11] (MHP) states that ‘there seems to be a rule of nature that molecules arrange themselves so as to be as hard as possible’. The analysis of such global descriptors along the reaction path had been undertaken in the case of this keto-enol tautomerism [1] and other proton transfer reactions [12] or other kinds of reaction [13]. The analysis of local descriptors, comparing various definitions of charges and Fukui functions had been done by others [14]. The condensed-to-atoms versions of these local descriptors have been defined and evaluated [15]. Recently, new descriptors have been defined within the framework of conceptual DFT [16]. But, to our knowledge, no local analysis along the reaction path has been done hereto now. Furthermore, Chattaraj recently showed that the Fukui function is highly linked to soft–soft interactions whereas they tend to be in difficulty when it comes to describe hard– hard interactions. He thus suggests that atomic charges
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should be elected as better condensed hard–hard interaction descriptors [17]. The aim of the present paper is to describe locally this reaction profile (namely, from hydroxyethylene, the enol form, to ethanal, the keto form) in terms of hard–hard or soft–soft interactions. Does the internal proton transfer (Fig. 1) from oxygen (enol form) to carbon (keto form) correspond to a hard–hard or soft–soft interaction between H7 on the one hand, and C4 and O2 on the other? The question has been asked in Ref. [1] and it is the goal of the present paper to try to answer it in comprehensive terms. For this matter, different kinds of atomic charges (namely Mulliken, NPA, ESP and AIM), as descriptors of local hardness, and different kinds of condensed Fukui functions, as descriptors of local softness are tested in their ability to describe a consistent picture of internal reactivity. This paper represents, to our knowledge, the first attempt for a local description of reactivity along a reaction path, within the framework of conceptual DFT.
The most important local descriptor of site selectivity is the Fukui function [2,3,18], which is defined as the partial derivative of the electronic density r(r) with respect to the total number of electrons N, at constant external potential v(r) and also as the partial derivative of the chemical potential m with respect to external potential v(r), at constant number of electrons N (Eq. 3). vm vrðrÞ Z f ðrÞ Z (3) vvðrÞ N vN vðrÞ The Fukui function is a function defined in each point of space, but it is useful to define a condensed Fukui function [15]. In this case, Fukui function are condensed to an atom, as defined by a partition scheme, just like the one defined for charges. Three types of Fukui functions can be defined [2,3] on the basis of the discontinuity of the f(r) versus the number of electrons N curve [19] namely: fCZq(NC1)Kq(N), for a nucleophilic attack fKZq(N)Kq(NK1), for a electrophilic attack f 8Z1/2(q(NC1)Kq(NK1)), for a homolytic attack.
2. Theoretical background We consider the keto–enol tautomerism as a chemical reaction of the type R/TS/P, where R represents the ground state of the enol form (reactant), P, represents the ground state of the keto tautomer (product) and TS is the corresponding transition state (Fig. 1). A reduced coordinate can be defined through a scaling procedure on the internal reaction coordinate (IRC) procedure obtained from ab initio calculations [1]. An IRC calculation gives the reaction path leading down to reactants and products from the TS. At each step it optimises the geometry of the system. The chemical potential m (Eq. 1) and the hardness h (Eq. 2) are the response of the system when the total number of electrons N changes and the external potential v(r) remains constant. vE mZ (1) vN vðrÞ hZ
vE vN
(2)
The local softness s(r) can also be defined [4,20] by multiplying f(r) by global softness [21] S which is the inverse of global hardness h. All these definitions are comprehensively reviewed in Ref. [6]. Thus, for a given molecule where global softness is constant, the Fukui function is a descriptor of local softness, and can be used to describe soft–soft interactions, between zones (or atoms) of maximal Fukui functions. Hard–hard interactions ought to correspond to minimal Fukui functions but the behaviour of hard–hard interactions is much more difficult to evaluate with Fukui functions in various examples [22] as commented by Chattaraj [17]. One of the goals of this paper is to assess the validity of minimal Fukui functions as a descriptor of hard–hard interaction, compared to other descriptors like atomic charges.
3. Computational details
vðrÞ
Fig. 1. Representation of the keto, transition state, and enol structures involved in the keto-enol tautomerism (with numbering of the atoms).
For the sake of comparison of results, all calculations have been performed exactly like in the former publication [1], at the HF/6-311G** level of theory using the GAUSSIAN 98 suite of programs [23]. It is obviously not the highest theoretical level feasible for such systems nowadays, but the idea is to compare the local reactivity analysis of this paper, to the global reactivity analysis of Ref. [1], in the same conditions. The molecular structures along the IRC were fully optimized at the same level of theory. The Mulliken [24], NPA [25] and ESP [26] population analyses have been performed with the GAUSSIAN 98 package.
A. Hocquet et al. / Journal of Molecular Structure (Theochem) 686 (2004) 213–218
The Hirshfeld analysis has been performed with the ADF code [27], at the BLYP/DZP Level of theory as no HF calculation and no Gaussian-type orbitals are available within that code. It must then be remembered when comparing results that the Hirshfeld population analysis is done in a slightly different calculation scheme. Comparative BLYP/6-311G** calculations have been done to check the reliability of comparisons of Hirshfeld charges to other population analyses. Qualitative trends of results are preserved. The AIM [28] analysis has been performed with the AIM 2000 code [29], with all default options. Integration of atomic properties over the atomic basins have been performed in natural coordinates, with a tolerance of 10K4 per integration steps. The radius of the beta sphere used for
215
integration of atomic properties (default value 0.5 a.u.) had sometimes to be set to 0.4 a.u., when the bond critical point (BCP) was too close from the nucleus.
4. Results and discussion 4.1. Analysis of charges First, we compare the Mulliken, ESP, NPA, Hirschfeld and AIM population analysis, leading to the definition of atomic charges along the reaction profile. These profiles are gathered in Fig. 2. From now on, we shall concentrate on the profiles of the three atoms that are primordially involved in the proton transfer mechanism. Hydrogen atom H7
Fig. 2. Reaction profiles of charges of H7 (circle), C4 (square) and 02 (black diamond). The Y-axis units are atomic units and the X-axis represents the IRC from enol to keto as in Ref. [1].
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(the transferred proton itself), the C4 carbon atom (to which it is linked in the enol reactant form) and the O2 oxygen atom (to which H7 is linked in the keto product form). The profiles all show the same qualitative pattern, as was the one obtained in Ref. [1], which corresponded to a Mulliken scheme. There is thus consistency with respect to the population scheme regarding the variation of the charges along the reaction path. The profiles all show the same qualitative pattern, as was the one obtained in Ref. [1], which corresponded to a Mulliken scheme. The only exception is the Hirshfeld scheme which shows too little a variation to fit into a well defined pattern.
The highest contrast in charge is obtained for the ESP scheme while the lowest is obtained with the Hirschfeld scheme. The Hirschfeld values, and above all the Hirschfeld charges variations are very small, thus rendering the analysis along the reaction path very difficult. The major discrepancy is encountered within the AIM charges. If the profile of charges show the same qualitative pattern as the other schemes, charges values are quite different from all the other charge definitions, and above all, they do not show the same sign on all circumstances. This is a major problem. As a hard interaction descriptor, the local hardness concept imbedded in the charge is understood as
Fig. 3. Reaction profiles of Fukui functions of H7 (circle), C4 (square) and 02 (black diamond). The Y-axis units are atomic units and the X-axis represents the IRC from enol to keto as in Ref. [1].
A. Hocquet et al. / Journal of Molecular Structure (Theochem) 686 (2004) 213–218
an absolute value: be it positive or negative, the farther from zero means the harder. The AIM scheme provides a C4 charge that is translated towards positive values (with respect to the other schemes), which means that its value along the reaction path oscillates between positive and negative values, thus implying its uselessness as a hardness descriptor. Apart from this, the major discrepancy between all schemes is the profile of the H7 atom: In two occurrences, the ESP and the Hirshfeld scheme, the H7 profile does not show any peak at the transition state. 4.2. Analysis of Fukui functions On the basis of these various charges definitions, we took advantage of the definition of condensed Fukui functions to analyse them along the reaction path, and compare the various population schemes. The profiles of the Fukui functions are presented in Fig. 3. For C4 and O2, fK is calculated, as the interaction between them and H7 corresponds to a electrophilic attack on them. For H7, on the contrary, fC is calculated, as H7 undergoes a nucleophilic attack. The profiles of fK(C4) and fK(O2) all show the same qualitative pattern. Feeble contrast in the charge definitions result in feeble contrast in the Fukui function definition, which is the case for the Hirschfeld scheme. Likewise, the problem with the C4 charge values within the AIM scheme result in a feeble contrast for the C4 Fukui function, too. For all schemes, the definition of fC(H7), the electrophilic Fukui function of H7 led to inaccurate results and the profile is not very consistent from one scheme to the other, due to weak contrast. Fukui functions are supposed to be always positive [18]. The Mulliken and above all the ESP scheme do not strictly obey to this law, in the same post transition state region, while other schemes do not show this problem. 4.3. Comparison of Fukui functions and charges The analysis of intramolecular reactivity along the reaction profile must then be consistent with the conclusions provided by the analysis of global descriptors, such as chemical potential and hardness, as described in Ref. [1]. The principle of maximum hardness states that, along a reaction, hardness goes down from a local maximum at the reactant to a minimum, at the transition state, and then goes up again to reach another local maximum at the product. This is indeed the case for this reaction as calculated in Ref. [1]. Consistently, the negative charges of C4 present a minimum at the transition state and the positive charges of H7 present a maximum at this point (except for the ESP and Hirschfeld scheme, see above). The O2 charge does not seem to vary much around the transition state. This in turn can be interpreted in terms of maximal hard–hard
217
interactions (between C4 and H7) at the transition state, thus favouring the formation of the keto form (where C4 and H7 are actually linked) over the enol form [1]. In the case of Fukui functions profiles, a maximum occurs at the transition state for fK(C4) but not for fC(H7). Instead, the fC(H7) profiles show no clear maximum along the reaction path, and fK(O2) abruptly changes around the transition state but shows no extremum either. This is in concordance with Chattaraj’s statement that charges are better descriptors of hard–hard interactions and Fukui functions better descritptors of soft–soft interactions [17]. The intramolecular keto enol tautomerism can thus be interpreted as a hard–hard interaction between the H7 hydrogen atom and the C4 and O2 atoms: at the transition state, H7 favours C4 and the keto form is actually the most stable. The thermodynamics of this equilibrium and the electronics of its reaction path are thus in concordance.
5. Conclusions This work thus presents an analysis of different definitions of charges, as local hardness descriptors, on the one hand, and a comparison of these charges with their respective Fukui functions, based on the same population analysis schemes. It is concluded that for this kind of reaction path involving the transfer of a proton, the intramolecular hard–hard interactions are better described by charges as local hardnesses than by Fukui functions. As a matter of fact, the charge profile along the reaction path is consistent with the global hardness profile as described in Ref. [1], whereas the Fukui function profile is not. This example supports also Chattaraj’s comments on the weakness of Fukui function in the case of hard–hard interactions [17]. It is also demonstrated that the Hirschfeld scheme fails to describe the variation of charges along the reaction path for its weak contrast. The AIM population analysis poses problem of scale, and the ESP profiles are not fully consistent with the others. The Mulliken and the NPA population analysis schemes are thus the more consistent, though the Mulliken Fukui function definition leads to occasional negative values. As the Mulliken population scheme is subject to basis set fluctuations [16], the NPA scheme seems the more appropriate.
Acknowledgements Friedrich Biegler-Ko¨nig and the University of Bielefeld for making available the AIM2000 program, IDRIS and CINES for providing time and space for calculations, FONDECYT, project number 1020534, are kindly acknowledged.
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References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
[14]
P. Perez, A. Toro-Labbe´, J. Phys. Chem. A 104 (2000) 1557–1562. R.G. Parr, J. Am. Chem. Soc. 106 (1984) 4049. K. Fukui, Science 218 (1987) 747. W. Yang, R.G. Parr, Proc. Natl Acad. Sci. USA 82 (1985) 6723. H. Chermette, J. Comput. Chem. 20 (1999) 129. P. Geerlings, F. De Proft, W. Langenaeker, Chem. Rev. 103 (2003) 1793–1874. R.T. Sanderson, Science 121 (1955) 207. R.G. Parr, R.G. Pearson, J. Am. Chem. Soc. 105 (1983) 7512. R.G. Pearson, Acc. Chem. Res. 26 (1993) 250. P.K. Chattaraj, Proc. Indian Natl Sci. Acad. Part A 62 (1996) 513. P.W. Ayers, R.G. Parr, J. Am. Chem. Soc. 122 (2000) 2010. E. Chamorro, A. Toro-Labbe´, P. Fuentealba, J. Phys. Chem. A 106 (2002) 3891–3898. P.K. Chattaraj, P. Perez, J. Zevallos, A. Toro-Labbe´, J. Phys. Chem. A 105 (2001) 4272; R. Parthasarathi, J. Padmanabhan, V. Subramanian, B. Maiti, P.K. Chattaraj, J. Phys. Chem. A 107 (2003) 10346–10352; B. Herrera, A. Toro-Labbe´, J. Phys. Chem. A 108 (2004) 1830–1836; E. Chamorro, P.K. Chattaraj, P. Fuentealba, J. Phys. Chem. A 107 (2003) 7068–7072. P. Bultinck, W. Langenaeker, P. Lahorte, F. De Proft, P. Geerlings, M. Waroquier, J.P. Tollenaere, J. Phys. Chem. A 106 (2002) 7887–7894; P. Bultinck, W. Langenaeker, P. Lahorte, F. De Proft, P. Geerlings, M. Waroquier, J.P. Tollenaere, J. Phys. Chem. A 106 (2002) 7895–7901; P. Thanikaivelan, J. Padmanabhan, V. Subramanian, T. Ramasami, Theor. Chem. Acc. 107 (2002) 326; J. Olah, C. Van Alsenoy, A.B. Sannigrahi, J. Phys. Chem. A 106 (2002) 3885–3890; P. Mondal, K. Hazarika, R. Deka, Phys. Chem. Commun. 6 (2003) 24–27.
[15] P. Ayers, R. Parr, J. Am. Chem. 122 (2000) 2010–2018; P. Ayers, R.C. Morrison, R. Roy, J. Chem. Phys. 116 (2002) 8731–8744. [16] P.K. Chattaraj, B. Maiti, U. Sarkar, J. Phys. Chem. A 107 (2003) 4973–4975; J. Padmanabhan, U. Sarkar, P.K. Chattaraj, R. Parthasarathi, V. Subramanian, Chem. Phys. Lett. 383 (2004) 122–128; L. Meneses, W. Tiznado, R. Contreras, P. Fuentealba, Chem. Phys. Lett. 383 (2004) 181–187. [17] P.K. Chattaraj, J. Phys. Chem. A 105 (2001) 511; J. Melin, F. Aparicio, V. Subramanian, M. Galvan, P.K. Chattaraj, J. Phys. Chem. A 108 (2004) 2487–2491. [18] P. Ayers, M. Levy, Theor. Chem. Acc. 103 (2000) 53–360. [19] J.P. Perdew, R.G. Parr, M. Levy, J.M. Balduz, Phys. Rev. Lett. 49 (1982) 1691. [20] W. Yang, C. Lee, S.K. Ghosh, J. Phys. Chem. 89 (1985) 5413. [21] M. Berkowitz, R.G. Parr, J. Chem. Phys. 88 (1988) 2554. [22] A.K. Chandra, M.T. Nguyen, Int. J. Mol. Sci. 3 (2002) 310–323. [23] M.J. Frisch et al., Gaussian, Inc., Pittsburgh PA, 1998. [24] P. Politzer, R.S. Mulliken, J. Chem. Phys. 55 (1971) 1833. [25] J.E. Carpenter, F. Weinhold, J. Mol. Struct. (Theochem) 169 (1988) 41. [26] B.H. Besler, K.M. Merz Jr., P.A. Kollman, J. Comput. Chem. 11 (1990) 431. [27] E.J. Baerends, et al., ADF1999; SCM, Theoretical Chemistry, Vrije Universiteit, Amsterdam, The Netherlands, 1999. [28] R.F.W. Bader, Atoms in Molecules. A Quantum Theory, Clarendon Press, Oxford, 1990. [29] F. Biegler-Ko¨nig, J. Scho¨nbohm, D. Bayles, J. Comput. Chem. 22 (2001) 545–559.
Short communication
Intramolecular interactions along the reaction path of keto–enol tautomerism: Fukui functions as local softnesses and charges as local hardnesses Alexandre Hocqueta,*, Alejandro Toro-Labbe´b, Henry Chermettec b
a LCPM, UMR CNRS INPL 7568, B.P. 451, 54001 Nancy cedex, France QTC, Facultad de Quı´mica, Pontificia Universidad Cato´lica de Chile, Casilla 306, Correo 22, Santiago, Chile c CPT, UMR CNRS 5182, 43 Bld du 11 Novembre 1918, 69622 Villeurbanne cedex, France
Received 29 July 2004; revised 14 August 2004; accepted 16 August 2004
Abstract This paper presents an unprecedented analysis of local hard and soft descriptors of the intramolecular reactivity along the reaction path of the keto-enol tautomerism. Different kinds of atomic charges (namely Mulliken, NPA, ESP, Hirschfeld and AIM), as descriptors of local hardness, and different kinds of condensed Fukui functions, as descriptors of local softness are tested in their ability to describe a consistent picture of internal reactivity. q 2004 Elsevier B.V. All rights reserved. Keywords: Proton transfer; Tautomerism; HSAB; Fukui function; Local softness; Local hardness; Reaction path
1. Introduction In a previous publication, Perez and Toro-Labbe´ [1] investigated the reaction profile of keto–enol tautomerism on the basis of energy, chemical potential and hardness as a function of the reaction coordinate, along the reaction path. This reaction has been shown to be important in gaining insights into the popular chemical concepts associated with reactivity and selectivity. Although chemical reactivity is characterized by global reactivity parameters like electronegativity or hardness, the selectivity is usually understood in terms of local functions like the Fukui function [2,3] (f(r)) and local softness [4] (s(r)). These global and local descriptors of reactivity have been popularised within the framework of ‘conceptual density functional theory’, a field to which reviews have been dedicated recently [5,6]. According to the electronegativity equalization principle [7], ‘all the constituent atoms in a molecule have the same
* Corresponding author. Tel.: C33-3-83-17-5252; fax: 33-3-83-37-9977. E-mail address: [email protected] (A. Hocquet). 0166-1280/$ - see front matter q 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.theochem.2004.08.011
electronegativity value which is roughly equal to the geometric mean of the electronegativities of the isolated atoms’. The hard–soft acids–bases (HSAB) principle [8] states that, ‘among the potential partners of same electronegativity, hard likes hard and soft likes soft’, and the maximum hardness principle [9,10,11] (MHP) states that ‘there seems to be a rule of nature that molecules arrange themselves so as to be as hard as possible’. The analysis of such global descriptors along the reaction path had been undertaken in the case of this keto-enol tautomerism [1] and other proton transfer reactions [12] or other kinds of reaction [13]. The analysis of local descriptors, comparing various definitions of charges and Fukui functions had been done by others [14]. The condensed-to-atoms versions of these local descriptors have been defined and evaluated [15]. Recently, new descriptors have been defined within the framework of conceptual DFT [16]. But, to our knowledge, no local analysis along the reaction path has been done hereto now. Furthermore, Chattaraj recently showed that the Fukui function is highly linked to soft–soft interactions whereas they tend to be in difficulty when it comes to describe hard– hard interactions. He thus suggests that atomic charges
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should be elected as better condensed hard–hard interaction descriptors [17]. The aim of the present paper is to describe locally this reaction profile (namely, from hydroxyethylene, the enol form, to ethanal, the keto form) in terms of hard–hard or soft–soft interactions. Does the internal proton transfer (Fig. 1) from oxygen (enol form) to carbon (keto form) correspond to a hard–hard or soft–soft interaction between H7 on the one hand, and C4 and O2 on the other? The question has been asked in Ref. [1] and it is the goal of the present paper to try to answer it in comprehensive terms. For this matter, different kinds of atomic charges (namely Mulliken, NPA, ESP and AIM), as descriptors of local hardness, and different kinds of condensed Fukui functions, as descriptors of local softness are tested in their ability to describe a consistent picture of internal reactivity. This paper represents, to our knowledge, the first attempt for a local description of reactivity along a reaction path, within the framework of conceptual DFT.
The most important local descriptor of site selectivity is the Fukui function [2,3,18], which is defined as the partial derivative of the electronic density r(r) with respect to the total number of electrons N, at constant external potential v(r) and also as the partial derivative of the chemical potential m with respect to external potential v(r), at constant number of electrons N (Eq. 3). vm vrðrÞ Z f ðrÞ Z (3) vvðrÞ N vN vðrÞ The Fukui function is a function defined in each point of space, but it is useful to define a condensed Fukui function [15]. In this case, Fukui function are condensed to an atom, as defined by a partition scheme, just like the one defined for charges. Three types of Fukui functions can be defined [2,3] on the basis of the discontinuity of the f(r) versus the number of electrons N curve [19] namely: fCZq(NC1)Kq(N), for a nucleophilic attack fKZq(N)Kq(NK1), for a electrophilic attack f 8Z1/2(q(NC1)Kq(NK1)), for a homolytic attack.
2. Theoretical background We consider the keto–enol tautomerism as a chemical reaction of the type R/TS/P, where R represents the ground state of the enol form (reactant), P, represents the ground state of the keto tautomer (product) and TS is the corresponding transition state (Fig. 1). A reduced coordinate can be defined through a scaling procedure on the internal reaction coordinate (IRC) procedure obtained from ab initio calculations [1]. An IRC calculation gives the reaction path leading down to reactants and products from the TS. At each step it optimises the geometry of the system. The chemical potential m (Eq. 1) and the hardness h (Eq. 2) are the response of the system when the total number of electrons N changes and the external potential v(r) remains constant. vE mZ (1) vN vðrÞ hZ
vE vN
(2)
The local softness s(r) can also be defined [4,20] by multiplying f(r) by global softness [21] S which is the inverse of global hardness h. All these definitions are comprehensively reviewed in Ref. [6]. Thus, for a given molecule where global softness is constant, the Fukui function is a descriptor of local softness, and can be used to describe soft–soft interactions, between zones (or atoms) of maximal Fukui functions. Hard–hard interactions ought to correspond to minimal Fukui functions but the behaviour of hard–hard interactions is much more difficult to evaluate with Fukui functions in various examples [22] as commented by Chattaraj [17]. One of the goals of this paper is to assess the validity of minimal Fukui functions as a descriptor of hard–hard interaction, compared to other descriptors like atomic charges.
3. Computational details
vðrÞ
Fig. 1. Representation of the keto, transition state, and enol structures involved in the keto-enol tautomerism (with numbering of the atoms).
For the sake of comparison of results, all calculations have been performed exactly like in the former publication [1], at the HF/6-311G** level of theory using the GAUSSIAN 98 suite of programs [23]. It is obviously not the highest theoretical level feasible for such systems nowadays, but the idea is to compare the local reactivity analysis of this paper, to the global reactivity analysis of Ref. [1], in the same conditions. The molecular structures along the IRC were fully optimized at the same level of theory. The Mulliken [24], NPA [25] and ESP [26] population analyses have been performed with the GAUSSIAN 98 package.
A. Hocquet et al. / Journal of Molecular Structure (Theochem) 686 (2004) 213–218
The Hirshfeld analysis has been performed with the ADF code [27], at the BLYP/DZP Level of theory as no HF calculation and no Gaussian-type orbitals are available within that code. It must then be remembered when comparing results that the Hirshfeld population analysis is done in a slightly different calculation scheme. Comparative BLYP/6-311G** calculations have been done to check the reliability of comparisons of Hirshfeld charges to other population analyses. Qualitative trends of results are preserved. The AIM [28] analysis has been performed with the AIM 2000 code [29], with all default options. Integration of atomic properties over the atomic basins have been performed in natural coordinates, with a tolerance of 10K4 per integration steps. The radius of the beta sphere used for
215
integration of atomic properties (default value 0.5 a.u.) had sometimes to be set to 0.4 a.u., when the bond critical point (BCP) was too close from the nucleus.
4. Results and discussion 4.1. Analysis of charges First, we compare the Mulliken, ESP, NPA, Hirschfeld and AIM population analysis, leading to the definition of atomic charges along the reaction profile. These profiles are gathered in Fig. 2. From now on, we shall concentrate on the profiles of the three atoms that are primordially involved in the proton transfer mechanism. Hydrogen atom H7
Fig. 2. Reaction profiles of charges of H7 (circle), C4 (square) and 02 (black diamond). The Y-axis units are atomic units and the X-axis represents the IRC from enol to keto as in Ref. [1].
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(the transferred proton itself), the C4 carbon atom (to which it is linked in the enol reactant form) and the O2 oxygen atom (to which H7 is linked in the keto product form). The profiles all show the same qualitative pattern, as was the one obtained in Ref. [1], which corresponded to a Mulliken scheme. There is thus consistency with respect to the population scheme regarding the variation of the charges along the reaction path. The profiles all show the same qualitative pattern, as was the one obtained in Ref. [1], which corresponded to a Mulliken scheme. The only exception is the Hirshfeld scheme which shows too little a variation to fit into a well defined pattern.
The highest contrast in charge is obtained for the ESP scheme while the lowest is obtained with the Hirschfeld scheme. The Hirschfeld values, and above all the Hirschfeld charges variations are very small, thus rendering the analysis along the reaction path very difficult. The major discrepancy is encountered within the AIM charges. If the profile of charges show the same qualitative pattern as the other schemes, charges values are quite different from all the other charge definitions, and above all, they do not show the same sign on all circumstances. This is a major problem. As a hard interaction descriptor, the local hardness concept imbedded in the charge is understood as
Fig. 3. Reaction profiles of Fukui functions of H7 (circle), C4 (square) and 02 (black diamond). The Y-axis units are atomic units and the X-axis represents the IRC from enol to keto as in Ref. [1].
A. Hocquet et al. / Journal of Molecular Structure (Theochem) 686 (2004) 213–218
an absolute value: be it positive or negative, the farther from zero means the harder. The AIM scheme provides a C4 charge that is translated towards positive values (with respect to the other schemes), which means that its value along the reaction path oscillates between positive and negative values, thus implying its uselessness as a hardness descriptor. Apart from this, the major discrepancy between all schemes is the profile of the H7 atom: In two occurrences, the ESP and the Hirshfeld scheme, the H7 profile does not show any peak at the transition state. 4.2. Analysis of Fukui functions On the basis of these various charges definitions, we took advantage of the definition of condensed Fukui functions to analyse them along the reaction path, and compare the various population schemes. The profiles of the Fukui functions are presented in Fig. 3. For C4 and O2, fK is calculated, as the interaction between them and H7 corresponds to a electrophilic attack on them. For H7, on the contrary, fC is calculated, as H7 undergoes a nucleophilic attack. The profiles of fK(C4) and fK(O2) all show the same qualitative pattern. Feeble contrast in the charge definitions result in feeble contrast in the Fukui function definition, which is the case for the Hirschfeld scheme. Likewise, the problem with the C4 charge values within the AIM scheme result in a feeble contrast for the C4 Fukui function, too. For all schemes, the definition of fC(H7), the electrophilic Fukui function of H7 led to inaccurate results and the profile is not very consistent from one scheme to the other, due to weak contrast. Fukui functions are supposed to be always positive [18]. The Mulliken and above all the ESP scheme do not strictly obey to this law, in the same post transition state region, while other schemes do not show this problem. 4.3. Comparison of Fukui functions and charges The analysis of intramolecular reactivity along the reaction profile must then be consistent with the conclusions provided by the analysis of global descriptors, such as chemical potential and hardness, as described in Ref. [1]. The principle of maximum hardness states that, along a reaction, hardness goes down from a local maximum at the reactant to a minimum, at the transition state, and then goes up again to reach another local maximum at the product. This is indeed the case for this reaction as calculated in Ref. [1]. Consistently, the negative charges of C4 present a minimum at the transition state and the positive charges of H7 present a maximum at this point (except for the ESP and Hirschfeld scheme, see above). The O2 charge does not seem to vary much around the transition state. This in turn can be interpreted in terms of maximal hard–hard
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interactions (between C4 and H7) at the transition state, thus favouring the formation of the keto form (where C4 and H7 are actually linked) over the enol form [1]. In the case of Fukui functions profiles, a maximum occurs at the transition state for fK(C4) but not for fC(H7). Instead, the fC(H7) profiles show no clear maximum along the reaction path, and fK(O2) abruptly changes around the transition state but shows no extremum either. This is in concordance with Chattaraj’s statement that charges are better descriptors of hard–hard interactions and Fukui functions better descritptors of soft–soft interactions [17]. The intramolecular keto enol tautomerism can thus be interpreted as a hard–hard interaction between the H7 hydrogen atom and the C4 and O2 atoms: at the transition state, H7 favours C4 and the keto form is actually the most stable. The thermodynamics of this equilibrium and the electronics of its reaction path are thus in concordance.
5. Conclusions This work thus presents an analysis of different definitions of charges, as local hardness descriptors, on the one hand, and a comparison of these charges with their respective Fukui functions, based on the same population analysis schemes. It is concluded that for this kind of reaction path involving the transfer of a proton, the intramolecular hard–hard interactions are better described by charges as local hardnesses than by Fukui functions. As a matter of fact, the charge profile along the reaction path is consistent with the global hardness profile as described in Ref. [1], whereas the Fukui function profile is not. This example supports also Chattaraj’s comments on the weakness of Fukui function in the case of hard–hard interactions [17]. It is also demonstrated that the Hirschfeld scheme fails to describe the variation of charges along the reaction path for its weak contrast. The AIM population analysis poses problem of scale, and the ESP profiles are not fully consistent with the others. The Mulliken and the NPA population analysis schemes are thus the more consistent, though the Mulliken Fukui function definition leads to occasional negative values. As the Mulliken population scheme is subject to basis set fluctuations [16], the NPA scheme seems the more appropriate.
Acknowledgements Friedrich Biegler-Ko¨nig and the University of Bielefeld for making available the AIM2000 program, IDRIS and CINES for providing time and space for calculations, FONDECYT, project number 1020534, are kindly acknowledged.
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References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
[14]
P. Perez, A. Toro-Labbe´, J. Phys. Chem. A 104 (2000) 1557–1562. R.G. Parr, J. Am. Chem. Soc. 106 (1984) 4049. K. Fukui, Science 218 (1987) 747. W. Yang, R.G. Parr, Proc. Natl Acad. Sci. USA 82 (1985) 6723. H. Chermette, J. Comput. Chem. 20 (1999) 129. P. Geerlings, F. De Proft, W. Langenaeker, Chem. Rev. 103 (2003) 1793–1874. R.T. Sanderson, Science 121 (1955) 207. R.G. Parr, R.G. Pearson, J. Am. Chem. Soc. 105 (1983) 7512. R.G. Pearson, Acc. Chem. Res. 26 (1993) 250. P.K. Chattaraj, Proc. Indian Natl Sci. Acad. Part A 62 (1996) 513. P.W. Ayers, R.G. Parr, J. Am. Chem. Soc. 122 (2000) 2010. E. Chamorro, A. Toro-Labbe´, P. Fuentealba, J. Phys. Chem. A 106 (2002) 3891–3898. P.K. Chattaraj, P. Perez, J. Zevallos, A. Toro-Labbe´, J. Phys. Chem. A 105 (2001) 4272; R. Parthasarathi, J. Padmanabhan, V. Subramanian, B. Maiti, P.K. Chattaraj, J. Phys. Chem. A 107 (2003) 10346–10352; B. Herrera, A. Toro-Labbe´, J. Phys. Chem. A 108 (2004) 1830–1836; E. Chamorro, P.K. Chattaraj, P. Fuentealba, J. Phys. Chem. A 107 (2003) 7068–7072. P. Bultinck, W. Langenaeker, P. Lahorte, F. De Proft, P. Geerlings, M. Waroquier, J.P. Tollenaere, J. Phys. Chem. A 106 (2002) 7887–7894; P. Bultinck, W. Langenaeker, P. Lahorte, F. De Proft, P. Geerlings, M. Waroquier, J.P. Tollenaere, J. Phys. Chem. A 106 (2002) 7895–7901; P. Thanikaivelan, J. Padmanabhan, V. Subramanian, T. Ramasami, Theor. Chem. Acc. 107 (2002) 326; J. Olah, C. Van Alsenoy, A.B. Sannigrahi, J. Phys. Chem. A 106 (2002) 3885–3890; P. Mondal, K. Hazarika, R. Deka, Phys. Chem. Commun. 6 (2003) 24–27.
[15] P. Ayers, R. Parr, J. Am. Chem. 122 (2000) 2010–2018; P. Ayers, R.C. Morrison, R. Roy, J. Chem. Phys. 116 (2002) 8731–8744. [16] P.K. Chattaraj, B. Maiti, U. Sarkar, J. Phys. Chem. A 107 (2003) 4973–4975; J. Padmanabhan, U. Sarkar, P.K. Chattaraj, R. Parthasarathi, V. Subramanian, Chem. Phys. Lett. 383 (2004) 122–128; L. Meneses, W. Tiznado, R. Contreras, P. Fuentealba, Chem. Phys. Lett. 383 (2004) 181–187. [17] P.K. Chattaraj, J. Phys. Chem. A 105 (2001) 511; J. Melin, F. Aparicio, V. Subramanian, M. Galvan, P.K. Chattaraj, J. Phys. Chem. A 108 (2004) 2487–2491. [18] P. Ayers, M. Levy, Theor. Chem. Acc. 103 (2000) 53–360. [19] J.P. Perdew, R.G. Parr, M. Levy, J.M. Balduz, Phys. Rev. Lett. 49 (1982) 1691. [20] W. Yang, C. Lee, S.K. Ghosh, J. Phys. Chem. 89 (1985) 5413. [21] M. Berkowitz, R.G. Parr, J. Chem. Phys. 88 (1988) 2554. [22] A.K. Chandra, M.T. Nguyen, Int. J. Mol. Sci. 3 (2002) 310–323. [23] M.J. Frisch et al., Gaussian, Inc., Pittsburgh PA, 1998. [24] P. Politzer, R.S. Mulliken, J. Chem. Phys. 55 (1971) 1833. [25] J.E. Carpenter, F. Weinhold, J. Mol. Struct. (Theochem) 169 (1988) 41. [26] B.H. Besler, K.M. Merz Jr., P.A. Kollman, J. Comput. Chem. 11 (1990) 431. [27] E.J. Baerends, et al., ADF1999; SCM, Theoretical Chemistry, Vrije Universiteit, Amsterdam, The Netherlands, 1999. [28] R.F.W. Bader, Atoms in Molecules. A Quantum Theory, Clarendon Press, Oxford, 1990. [29] F. Biegler-Ko¨nig, J. Scho¨nbohm, D. Bayles, J. Comput. Chem. 22 (2001) 545–559.