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Understanding Reaction Mechanisms in Organic Chemistry from Catastrophe Theory: Ozone Addition on Benzene Ibrahim Mbouombouo Ndassa, Bernard Silvi,* and Franc¸ois Volatron Laboratoire de Chimie The´orique (UMR-CNRS 7616), UniVersite´ Pierre et Marie Curie, 3 rue Galile´e 94200-IVry sur Seine, France ReceiVed: June 25, 2010; ReVised Manuscript ReceiVed: August 4, 2010
The potential energy profiles of the endo and exo additions of ozone on benzene have been theoretically investigated within the framework provided by the electron localization function (ELF). This has been done by carrying out hybrid Hartree-Fock DFT B3LYP calculation followed by a bonding evolution theory (BET) analysis. For both approaches, the reaction is exothermic by ∼98 kJ mol-1. However, the activation energy is calculated to 10 kJ mol-1 lower in the endo channel than in the exo one; therefore the formation of the endo C6H6O3 adduct is kinetically favored. Six structural stability domains are identified along both reaction pathways as well as the bifurcation catastrophes responsible for the changes in the topology of the system. This provides a chemical description of the reaction mechanism in terms of heterolytic synchronous bond formation. SCHEME 1
Introduction Quantum chemistry has been very useful and successful for the theoretical analysis of chemical reactions and chemical reactivity. The frontier orbitals theory1 and the orbital symmetry rules of Woodward and Hoffman2 are paradigmatic examples of the possibilities of quantum chemistry within the MO theory. The conceptual density functional theory pioneered by R. G. Parr3 has been at the origin of very useful reactivity descriptors,4 whereas a general model for transition states has been proposed by Shaik,5 which has been applied successfully to many areas of chemical reactivity.6 The attempts made to extract the flow and electron-transfer processes along the reaction pathway associated with a chemical reaction from quantum chemical calculations are based either on wave-function-based and orbital-based methods or on the topology of scalar fields associated with the electron density distributions. The former techniques rely, by construction, on the approximations made to calculate the approximate wave function, whereas the topological approaches are, in their principles, free of arbitrariness. The localized orbital centroid evolution technique of Leroy et al.7 and the valence bond (VB) approaches used by Karadakov8 and by Harcourt9 belong to the former group. The molecular electrostatic potential (MESP) topography approach of Balanarayan et al.10 is very attractive, although the correspondences between the evolution of the MESP and the charge density transfers are stated rather than rigorously established. The catastrophe theory has been used to study the evolution along a reaction path of the topologies of the electron density,11 the laplacian of the electron density12 and of the electron localization function (ELF),13 without investigating so much the electronic transfer aspects of the reactions. The analysis of the ELF function topology evolution along a reaction pathway is often referred to as bonding evolution theory (BET) after Krokidis et al.13 In this paper, we are interested in describing the so-called endo effect within this method. From the frontier molecular * To whom correspondence should be addressed. E-mail: silvi@ lct.jussieu.fr.
orbital (FMO) theory,1 this endo effect arises from secondary molecular orbital interactions (SOI). However, following Garcia et al., there is no evidence of the involvement of such SOI in cycloaddition reactions.14 In many cases, they show that the endo selectivity may be rationalized by well-known mechanisms such as solvent effects, steric interactions, hydrogen bonds, and electrostatic forces. On the opposite, Wannere et al.15 showed that the endo preferences in a series of Diels-Alder cycloadditions may come from SOI on the basis of magnetic properties of the transition structure (TS). Clearly, the role of the SOI is therefore still under debate. The reaction under study is the benzene ozonolyzis, which has been studied experimentally.16 According to the Criegee mechanism17 (which has been recently confirmed in the cyclopentadiene18 and cis-2-butene19 cases), the first step of this reaction is a [3s + 2s] cycloaddition, which leads to a primary ozonide. The activation barrier of this first step has been experimentally found to be 61.0 kJ mol-1. This reaction has also been studied by theoretical means20 at the CCSD(T)//B3LYP/6-31G** level. In the cycloaddition step, the endo TS has been found to be lower in energy than the exo one in both benzene and phenol ozonolysis. The computed activation barrier (66.3 kJ mol-1) is close to that experimentally determined. In FMO analysis, two interactions are to be considered in this cycloaddition (Scheme 1). The first involves the LUMO of the benzene and the HOMO of ozone (Scheme 1a). Because the coefficient of the central O atom on ozone is 0, the endo and exo approaches are expected to be equivalent with respect to this interaction. The second FMO interaction involves the HOMO of benzene with the LUMO of ozone (Scheme 1b). In this case, the coefficient of the central atom is different from 0, and a secondary orbital interaction takes place. Because it
10.1021/jp105874j 2010 American Chemical Society Published on Web 11/16/2010
Mechanisms in Organic Chemistry and Catastrophe Theory increases the overlap between the interacting orbitals, this secondary interaction is expected to favor the endo approach. Methodology Sketch of the Topological Analysis of the Electron Localization Function. ELF was originally designed by Becke and Edgecombe to identify “localized electronic groups in atomic and molecular systems”.21 It relies, through its kernel, on the laplacian of the conditional same-spin pair probability scaled by the homogeneous electron gas kinetic energy density
χσ(r) )
Dσ(r) Dσ0 (r)
ELF ) η(r) )
1 1 + χ(r)2
(1)
in which
Dσ(r) ) tσ(r) -
2 1 (Fσ(r)) 4 Fσ(r)
(2)
is the difference of the actual definite positive kinetic energy tσ(r) density and the von Weizsa¨cker kinetic energy density functional,22 whereas
3 Dσ0 (r) ) (6π2)2/3Fσ5/3(r) 5
(3)
is the kinetic energy density of the homogeneous electron gas. This formulation led Savin to propose an interpretation of ELF in terms of the local excess kinetic energy because the Pauli repulsion enabled its calculation from Kohn-Sham orbitals.23 Orbital-based interpretations of ELF have been proposed by Burdett24 and more recently by Nalewajski et al.,25 who considered the nonadditive interorbital Fisher information. Another route pioneered by Dobson26 explicitly considers the pair functions. It has been independently developed by Kohout et al.27 and by one of us,28 allowing the extension of ELF to correlated wave functions.29 The topological analysis of the ELF gradient field is achieved by applying the dynamical system theory.30 It provides a mathematical model enabling the partition of the molecular position space in basins of attractors which present, in principle, a one-to-one correspondence with chemical local objects such as bonds and lone pairs. Therefore, the Lewis picture of bonding and the electronic domains of the valence shell electron pair repulsion (VSEPR) approach are recovered. Moreover, it has been recently shown that the electrostatic repulsions between the ELF basins provide a justification of the VSEPR rules.31 The core basins surround nuclei with atomic number Z > 2 and are labeled C(A), where A is the atomic symbol of the element. The union of the valence basins encompassing a given core C(A) constitutes the valence shell of atom A. A valence basin may be shared by several valence shells; this is a generalization of Lewis’s fourth postulate: “Two atomic shells are mutually interpenetrable”.32 The valence basins are characterized by the number of atomic valence shells to which they participate or, in other words, by the number of core basins with which they share a boundary. This number is called the synaptic order. Thus, there are monosynaptic, disynaptic, trisynaptic basins, and so on. Monosynaptic basins, labelled V(A), correspond to the lone pairs of the Lewis model, and polysynaptic basins correspond to the shared pairs of the Lewis
J. Phys. Chem. A, Vol. 114, No. 49, 2010 12901 model. In particular, disynaptic basins, labelled V(A,X) correspond to two-center bonds, trisynaptic basins, labelled V(A,X,Y) correspond to three-center bonds, and so on.33 ELF along a Reaction Pathway: The BET Analysis. Within the framework provided by the ELF analysis, a chemical reaction is viewed as a series of topological changes occurring along the reaction path. The parameters defining the reaction pathway (such as the nuclear coordinates and the electronic state) constitute the control space. The evolution of the bonding along the reaction path is modeled by the changes in the number and synaptic orders of the valence basins. Each structure is only possible for values of the control parameters belonging to definite ranges, in other words, to subsets called structural stability domains. For any two points of the control space belonging to a given structural stability domain (SSD), there is the same number of critical points of each type in the ELF gradient field. Along the reaction path, the chemical systems goes from a structural stability domain to another by means of bifurcation catastrophes occurring at the turning points. The changes are ruled by the Poincare´-Hopf theorem, which states that
∑ (-1)I
P
)1
(4)
P
introducing accordingly a very strong constraint due to the structure of the geometrical space. Along a reaction pathway which links the chemical structures and therefore the topologies of the ELF gradient fields of the reactants with those of the product, the system experiences a series of structural stability domains within which all of the critical points are hyperbolic, separated by catastrophic points at which at least one critical point is nonhyperbolic. The bifurcation catastrophes occurring at these turning points are identified according to Thom’s classification,34 which gives access to their unfolding, and a compact polynomial expression which contains all of the information about how ELF may change as the control parameters change. In this way, a chemical reaction is viewed as a sequence of elementary chemical processes characterized by a catastrophe. Thus, chemical processes are classified according to the variations in the number of basins µ and/or in the synaptic order σ of at least one basin. There are accordingly three types of chemical processes which correspond to ∆µ > 0, ∆µ < 0 and ∆µ ) 0, ∆σ * 0. Only three elementary catastrophes have been recognized so far in the chemical reactions, the fold, cusp, and elliptic umbilic catastrophe. The fold catastrophe transforms a wandering point (i.e., a point which is not a critical one) into two critical points of different parity. Its unfolding is x3 + ux; x is the direction of the eigenvector corresponding to the eigenvalue of the Hessian matrix which changes sign, and u is the control space parameter which governs the discontinuity. For u > 0, the first derivative is positive for all x; the catastrophe takes place at u ) 0, for which both first and second derivatives are zero. For u < 0, there are two critical points at x ) ((-u/3)1/2. The cusp catastrophe transforms a critical point of a given parity into two critical points of the same parity and one of the opposite parity. Finally, the elliptic umbilic catastrophe changes the index of one critical point by 2. The sequence of turning points occurring along the reaction pathway is represented by the general formula N1-N2FCSHEBP-N3 introduced by Berski et al.35 In this notation, N1 is the ordinal number of an analyzed sequence that can be omitted when only one reaction is considered (i.e., N1 ) 1), N2
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TABLE 1: Structural Parameters of Reactants, Molecular Complexes, Transition States, and Productsa complex R (O1,O2) ∠O2O1O3 R(C1,C2) R(C1,C3) R(C3,C5) R(C5,C6) R(C1,O2) ∠C6C1H ∠C6C1O2 dihedral (C3C1C2O3) dihedral (C1O2O3O1) a
transition state
product
reactants
endo
exo
endo
exo
endo
exo
1.264 117.9 1.396 1.396 1.396 1.396
1.269 117.2 1.402 1.398 1.394 1.400 2.947 178.4 106.3 104.7 115.6
1.268 117.4 1.400 1.398 1.394 1.399 3.041 179.2 106.4 106.0 247.1
1.320 109.5 1.453 1.427 1.370 1.427 2.022 163.3 108.9 108.8 122.4
1.324 109.4 1.451 1.429 1.368 1.430 2.024 161.7 109.4 108.8 240.0
1.441 101.1 1.569 1.502 1.340 1.464 1.447 131.5 124.4 122.2 128.8
1.443 101.6 1.570 1.500 1.341 1.463 1.444 130.0 120.4 117.4 211.0
Distances are in Å, and angles are in degree.
is the number of observed steps associated with the SSDs usually greater than the number of catastrophes, FCSHEBP are the symbols of the catastrophes taken from the first letters in the original Thom’s classification, that is, F ) fold, C ) cusp, S ) swallow tail, H ) hyperbolic umbilic, E ) elliptic umbilic, B ) butterfly, and P ) parabolic umbilic, and N3 indicates the end of the sequence. Turning points of the same type occurring simultaneously are indicated by [A]n, where n is the multiplicity of the catastrophe labeled by A. Moreover, bold symbols are used to emphasize the first bond formation, whereas the superscript is used for those catastrophes that increase either the number of basins or the synaptic order. For example, C corresponds to a cusp catastrophe in which an attractor gives rise to two new attractors and a saddle point of index 1. In this way, a chemical reaction can be decomposed in a well-defined sequence of electron pair topologies which can be identified with chemical concepts commonly used. This method has been applied to characterize different chemical processes such as isomerization,36 electron transfers,37 proton transfers,38 hydrogen transfers,39 cylclizations or cycloadditions,40 substitutions,41 and two-state reactions.42 Technically, the ELF analysis is performed for a series of structures calculated along the reaction path by the IRC method; the turning points between structural stability domains are then located; and the catastrophe is identified when the two successive domains belong to the same Born-Oppenheimer energy surface. Computational Methods The calculations of the wave function have been carried out at the hybrid Hartree-Fock density functional B3LYP level43 with the Gaussian 03 software.44 Stationary points on the potential energy surface were confirmed by calculation of harmonic vibrational frequencies, all positive for a minimum and one imaginary for a TS. The intrinsic reaction coordinate (IRC) method of Fukui,45 developed by Gonzalez and Schlegel,46 was employed to follow the reaction path in mass-weighted coordinates in the forward and reverse directions starting at the TS. The calculated reaction path comprises a total of 98/95 points in the endo/exo approaches with a step size of 0.1 amu1/2 Bohr. The topological analysis of ELF was carried out with the TopMod47 suite. The ELF function was calculated over a rectangular box by using a cubic grid of step size smaller than 0.1 Bohr. The accuracy of the integrated densities is on the order of 0.02 e. The ELF isosurfaces have been visualized with the Amira 3.0 software.48 Results and Discussion The geometrical parameters of the reactants, molecular complexes, transition states, and products are listed in Table 1,
Figure 1. Ball and stick representation of the endo (left) and exo (right) isomers of C6H6O3, indicating the labeling of the atoms.
whereas Figure 1 displays the geometry of the two isomers of the adduct. The optimized geometries of the reactants are in rather good agreement with experimental data, the largest deviation occurring for the O-O internuclear distance calculated to be 1.264 Å instead of 1.278 Å experimentally.49 Prior to the addition reaction itself, benzene and ozone form a weakly bound molecular complex, the stabilization energy of which is 7.5 kJ mol-1 for the endo conformer and 3.3 kJ mol-1 for the exo one in vacuum. The main expected contributions to the complex stabilization energies are, on the one hand, the ozone dipolebenzene quadrupole interaction and, on the other hand, the ozone dipole-benzene induced dipole interaction. The former tends to put the ozone dipole moment perpendicular to the benzene ring plane, whereas the latter has its maxima when the permanent and induced dipoles are either aligned or parallel. In the endo complex, the dipole moment direction forms an angle of 40° with the benzenic ring plane, and therefore, its stabilization energy is ruled by the competition of the purely electrostatic contribution and of the induction energy, whereas in the exo one, the induction energy is the driving factor because the molecular planes of the C6H6 and O3 moieties are almost parallel. However, the QTAIM50 partition of the electron density reveals a noticeable electron density transfer toward the O3 fragment, of an amount of 0.11 e and 0.12 e for the endo and exo conformers, respectively, which is consistent with the large global electron affinity difference between ozone and benzene (∼3.2 eV.).51 In the molecular complexes, the geometry of the isolated species is almost conserved. Except for the formation of the C1-O2 and C2-O3 bonds, which correspond to the translation of the ozone group toward the benzene, the most important structural changes occurring during the reaction are the out-of-plane displacements of the hydrogen atoms linked to C1 and C2, the lengthening of the O1-O2, O1-O3, C1-C2, C1-C3, C2-C4, and C5-C6 bonds,
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Figure 2. IRC profiles for the two channels of the addition of ozone on benzene, indicating the structural stability domains. Left: endo; right: exo.
TABLE 2: Activation Enthalpy and Reaction Enthalpy of the Addition of Ozone on Benzene ∆H† (kJ mol-1) ∆H (kJ mol-1)
endo
exo
30.1 -97.4
41.0 -97.8
ZPE-Corrected Values ∆H† (kJ mol-1) ∆H (kJ mol-1)
35.4 -84.0
44.8 -86.1
and the shortening of the C3-C5 and C4-C6 bonds, which become double bonds. These effects are already noticeable in the transition states. At the transition state, the benzene ring has lost its D6h symmetry, and therefore, according to usual criteria, its aromaticity has decreased with respect to that of the isolated benzene molecule. The deformations are larger for the exo TS than those for the endo one. The electronic energy profiles along the IRC path in the gas phase are shown in Figure 2, where the zero of the energy corresponds to the reactants at infinite separation. The activation enthalpy of the endo channel is calculated to be lower than that of the exo one by about 10 kJ mol-1 (see Table 2). The addition reaction is exothermic for both channels, and the enthalpies of reaction are -84.0 and -86.1 kJ mol-1 for the endo and exo isomers, respectively. Therefore, the formation of the endo isomer is kinetically favored. The magnitudes of the deformations of the internal coordinates of the fragments in the endo and exo transition states with respect to the isolated reactants provide a geometrical explanation of the difference of the activation enthalpies. In other words, the endo TS is found to be earlier than the exo one. Consequently, the former is lower in energy, as expected in an exothermic reaction. The ELF population analysis of the reactants, molecular complexes, transition states, and products is presented in Table 3. Isolated benzene and ozone molecules have 12 and 7 valence basins, respectively. The ELF description of the bonding in benzene is well-known;52 the aromatic ring gives rise to six disynaptic basins V(C,C), whose populations are equal and close to 3 e, accounting for the delocalization. The valence basin populations of isolated ozone are typical of charge-shift bonding,53 in which Sanderson’s lone pair bond-weakening effect54 is responsible for the low population of the V(O1,O2) and V(O1,O3) basins. This result and the absence of a V(O2,O3) basin accounting for the cyclic structure seems to be in contradiction with the resonance structures55 deduced from the NRT analysis.56 The dominant valence bond (VB) structure of a charge shift bond is often built with hybrid atomic orbitals
outwardly directed. Although this structure is covalent, it does not contribute to the population of the disynaptic basin but rather to that of monosynaptic ones located on the bonded atoms. Another ELF signature of charge shift bonds is the large magnitude of the covariance matrix elements between monosynaptic basins belonging to different atoms. In O, 〈cov[V(O1),V(O2)]〉 ) -0.36 and 〈cov[V(O2),V(O3)]〉 ) -0.26. The latter value accounts for a strong interaction between O2 and O3, which is represented by the cyclic structure in the resonance description. The formation of the molecular complexes gives rise to a reorganization of the electron density among the valence basins in order to account for the density transfer and to polarization effects. The populations of the V(C,C) basins involving C1 and C2 decrease as well as V(C5,C6), whereas those of V(C3,C5) and V(C4,C6) increase. On the ozone side, an electron density transfer toward V(O1) takes place at the expense of the V(O1,O2) and V(O1,O3) basins. The ELF topology of both transition states is the sum of the topologies of the reactants; there is the same number of valence basins whose synaptic orders have not changed. As the addition of ozone on benzene is exothermic, this result appears to be a consequence of the Hammond’s postulate.57 With respect to the free reactants, the basin populations have noticeably changed, amplifying the trends observed for the molecular complexes. The overall difference with respect to the corresponding molecular complex is larger for the exo TS than that for the endo one, and the V(O,O) basin populations are smaller in the exo TS. The larger relaxation of the electron density implied by the exo channel might be a cause of its higher calculated activation enthalpy. In addition, the increase of the V(O,O) basin population in the endo channel with respect to the exo one is consistent with a secondary orbital interaction which predicts a
Figure 3. ELF ) 0.7 localization domains of the endo (left) and exo (right) addition products of ozone on benzene. Color code: brick red, monosynaptic; light blue, protonated disynaptic; green, disynaptic; magenta, core.
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TABLE 3: Basin Population of the Skeleton Atoms for the Isolated Reactants, The Transition States, And the Aduct Isomers complex
transition state
product
reactants
endo
exo
endo
exo
endo
exo
V(C1,C2) V(C1,C3) V(C3,C5)
2.80 2.80 2.80
2.71 2.72 2.89
2.72 2.72 2.88
2.50 2.56 3.12
2.49 2.49 3.18
V(C5,C6) V(O1)
2.80 3.82
2.71 4.00
2.68 4.06
2.45 4.66
2.39 4.78
V(O2)
2.85 2.85
2.89 2.86
2.89 2.86
2.89 2.94
2.88 2.94
1.96 2.14 1.77 1.71 2.17 2.65 2.70 2.54 2.66 0.26 0.26 0.22
1.97 2.06 1.80 1.67 2.18 2.70 2.73 2.66 2.66 0.26 0.26 0.21
1.19
1.12
1.09
0.84
0.79 1.22
1.23
V(O1) V(O2) V(O1,O2) V(C1,O2)
larger electron transfer from the C6H6 fragment toward the O3 one in the endo TS. Figure 3 displays the localization domains of the C6H6O3 isomers. With respect to the reactants, the number of valence basins has increased by 7. The two V(O,O) basins have disappeared, leaving a place for four monosynaptic basins, accounting for the protocovalent58 character of the O1O2 and O1O3 interactions, the V(O1) basins have been split; two (C,O) basins have appeared, corresponding to the formation of the CO ring as well as two new V(C,C) basins due to the evolution of the C3C5 and C4C6 bonds from aromatic to double bonds. Figure 4 shows the evolution of the basin population along the section of the IRC path scanning the six SDDs. The ELF topology evolution along the reaction path involves six structural stability domains for both channels; however, the order and the nature of the changes occurring at the turning points show some differences. The first SDD corresponds to the sum of the reactant topologies. It corresponds to the longest step on the reaction path, running over 59 and 52 steps for the endo and exo routes, respectively. The first catastrophe along the endo pathway is of the cusp type; it corresponds to the division of the V(O1) basin, whose population had reached 4.80 e. Two active control space parameters are responsible for this catastrophe; they are identified as the ∠O2O1O3 and the symmetric OO stretching. It is worth noting that ∠O2O1O3 ) 108.5° at this point, and therefore, the splitting of V(O1) is consistent with the AX2E2 geometry of the VSEPR model. The second turning point occurs two steps later; it corresponds to the formation through a dual fold catastrophe of two monosynaptic basins on top of the C1
and C2 carbons. Each catastrophe is driven by a single control space parameter, which appears to be the pyramidalization around the carbon atom which acquires a partial sp3 character. The step discontinuity of the [V(C1,C3)] clearly indicates that the electron density transfers toward V(C1) and V(C2) are from V(C1,C3) and V(C2,C4). In the exo channel, these two turning points belonging to the same step have not been distinguished. It is worth noting that the evolution of the ELF topology, corresponding to steps II and III, yields arrangements of the basins around O1, O2, and C2, consistent with the VSEPR equilibrium geometry prediction. Such a relaxation is therefore expected to be stabilizing, which is verified on the energy profiles of Figure 2. The third SDD (second for exo) runs over 13 steps for both channels. The transition with the next SDD involves a dual cusp catastrophe, which splits the V(C3,C5) and V(C4,C6) basins into basins symmetrically disposed on each side of the benzenic plane. It occurs for both routes when the population of the V(C,C) basin has reached 3.35 e. The next turning point in the exo path corresponds to the formation of the V(O2) and V(O3) basins, in front of the V(C1) and V(C2) basins, which is due to a dual fold catastrophe. The fourth turning point gives rise to the formation of the CO bonds. For the endo channel, this is achieved by a dual elliptic umbilic catastrophe, which increases the synaptic order of the V(C) by 1. This catastrophe is accompanied by a density transfer of ∼0.33 e from the oxygen lone pairs toward the new V(C,O) basins. In the exo path, a dual cusp catastrophe merges the V(C) and V(O) monosynaptic basins into V(C,O) disynaptic ones. The last step is characterized
Figure 4. Basin populations along the IRC path of the addition of ozone on benzene. Left: endo; right: exo. Equivalent monosynaptic and disynaptic basins have been merged in V(O1), V(O2), and V(C3,C5).
Mechanisms in Organic Chemistry and Catastrophe Theory SCHEME 2
by a dual cusp catastrophe, which splits the V(O1,O2) and V(O1,O3) disynaptic basins into monosynaptic basins, two V(O1), one V(O2), and one V(O3), which are typical protocovalent bonds. The driving force responsible for this change in the bonding is the symmetric lengthening of the OO distances. The sequence of turning points corresponds to 6-C†[F†]2[C†]2[U]2[C†]2-0 for the exo approach and to 6-C†[F†]2[C†]2 [F†]2[C]2[C†]20 for the exo one. The analysis of the basin populations along the reaction path suggests the following mechanism (Scheme 2), in which the curly arrows stand for partial electron transfers. In this reaction, as well as in other cycloadditions which have been previously investigated,40 all of the bifurcation catastrophes occur after the transition state. In the first step of the reaction, from the molecular complex to the TS structures, there is no significant electronic rearrangement, and the geometry changes, essentially the shortening of the inter-reactant fragment distance, yield an increase of the energy due to the strain. The second part of the reaction path can be viewed as a relaxion process in which the reorganization of the valence molecular shell accompanies the geometry changes. In Figure 2, it is interesting to note that the energetic distances between the groups of catastrophes are on the order of 50 kJ mol-1. Conclusion The ELF and catastrophe theory provide a powerful technique for characterizing the different steps of a chemical reaction in order to shed light onto its electronic mechanism. The reaction is viewed as a transfer of electron density from the benzene fragment toward the ozone fragment and the two basins corresponding to the CO bonds. This study clearly shows how well the Hammond’s postulate works in the case of the present reaction. The first step of the reaction is characterized by a reorganization of the electron density, which affects the basin population but not their topology. A possible effect of this topological constraint is the increase of the repulsive energy between basins, which explains the activation energy. In the following steps, the number and the arrangement of the basins fits the nuclear potential, yielding a lowering of the energy of the system. References and Notes (1) Fukui, K.; Yonezawa, T.; Shinghu, H. J. Chem. Phys. 1952, 20, 722. (2) Woodward, R. B.; Hoffmann, R. Angew. Chem., Int. Ed. Engl. 1969, 8, 781–932. (3) (a) Parr, R. G.; Donnelly, R. A.; Levy, M.; Palke, W. E. J. Chem. Phys. 1978, 68, 3801–3807. (b) Parr, R. G.; Pearson, R. G. J. Am. Chem. Soc. 1983, 105, 7512–7516. (4) Geerlings, P.; De Proft, F.; Langenaeker, W. Chem. ReV. 2003, 103, 1793–1873. (5) Shaik, S. S. J. Am. Chem. Soc. 1981, 103, 3692–3701. (6) Shaik, S. S.; Shurki, A. Angew. Chem., Int. Ed. 1999, 38, 586– 625. (7) Leroy, G.; Sana, M.; Burke, L. A.; Nguyen, M. T. In Theory of Chemical Reactions; Daudel, R., Diner, S., Malrieu, J. P., Eds.; Reidel: Dordrecht, The Netherlands, 1979; pp 395-448. (8) Karadakov, P. B.; Cooper, D. L.; Gerratt, J. Theor. Chem. Acc. 1998, 100, 222–229.
J. Phys. Chem. A, Vol. 114, No. 49, 2010 12905 (9) Harcourt, R. D.; Schulz, A. J. Phys. Chem. A 2000, 104, 6510– 6516. (10) Balanarayan, P.; Gadre, S. R. J. Chem. Phys. 2003, 119, 5037– 5043. (11) Tal, Y.; Bader, R. F. W.; Nguyen-Dang, T. T.; Ojha, M.; Anderson, S. G. J. Chem. Phys. 1981, 74, 5162–5167. (12) Malcolm, N. O. J.; Popelier, P. L. A. J. Phys. Chem. A 2001, 105, 7638–7645. (13) Krokidis, X.; Noury, S.; Silvi, B. J. Phys. Chem. A 1997, 101, 7277– 7282. (14) Jose´ I. Garcı´a, J. A. M.; Salvatella, L. Acc. Chem. Res. 2000, 33, 658–664. (15) Wannere, C. S.; Paul, A.; Herges, R.; Houk, K. N.; Schaefer, H. F., III; Schleyer, P. V. R. J. Comput. Chem. 2007, 28, 344–361. (16) Atkinson, R. J. Phys. Chem. Ref. Data, Monogr. 1994, 2, R1–R261. (17) Kuczkowski, R. L. Acc. Chem. Res. 1983, 16, 42–47. (18) Hoops, M. D.; Ault, B. S. J. Am. Chem. Soc. 2009, 131, 2853– 2863. (19) Clay, M.; Ault, B. S. J. Phys. Chem. A 2010, 114, 2799–2805. (20) Hendrickx, M. F. A.; Vinckier, C. J. Phys. Chem. A 2003, 107, 7574–7580. (21) Becke, A. D.; Edgecombe, K. E. J. Chem. Phys. 1990, 92, 5397– 5403. (22) von Weizsa¨cker, C. F. Z. Phys. 1935, 96, 431–458. (23) (a) Savin, A.; Becke, A. D.; Flad, J.; Nesper, R.; Preuss, H.; von Schnering, H. G. Angew. Chem., Int. Ed. Engl. 1991, 30, 409. (b) Savin, A.; Jepsen, O.; Flad, J.; Andersen, O. K.; Preuss, H.; von Schnering, H. G. Angew. Chem., Int. Ed. Engl. 1992, 31, 187–190. (c) Savin, A.; Nesper, R.; Wengert, S.; Fa¨ssler, T. F. Angew. Chem., Int. Ed. Engl. 1997, 36, 1809– 1832. (24) Burdett, J. K.; McCormick, T. A. J. Phys. Chem. A 1998, 102, 6366–6372. (25) Nalewajski, R. F.; Koster, A. M.; Escalante, S. J. Phys. Chem. A 2005, 109, 10038–10043. (26) Dobson, J. F. J. Chem. Phys. 1991, 94, 4328–4333. (27) (a) Kohout, M.; Pernal, K.; Wagner, F. R.; Grin, Y. Theor. Chem. Acc. 2004, 112, 453–459. (b) Kohout, M.; Pernal, K.; Wagner, F. R.; Grin, Y. Theor. Chem. Acc. 2005, 113, 287–293. (28) Silvi, B. J. Phys. Chem. A 2003, 107, 3081–3085. (29) Matito, E.; Silvi, B.; Duran, M.; Sola, M. J. Chem. Phys. 2006, 125, 024301. (30) Abraham, R. H.; Marsden, J. E. Foundations of Mechanics; Addison Wesley: Redwood City, CA, 1994. (31) Martn Penda´s, A.; Francisco, E.; Blanco, M. Chem. Phys. Lett. 2008, 454, 396–403. (32) Lewis, G. N. J. Am. Chem. Soc. 1916, 38, 762–786. (33) Silvi, B. J. Mol. Struct. 2002, 614, 3–10. (34) Thom, R. Stabilite´ Structurelle et Morphoge´nese; Intereditions: Paris, 1972. (35) Berski, S.; Andre´s, J.; Silvi, B.; Domingo, L. R. J. Phys. Chem. A 2006, 110, 13939–13947. (36) (a) Krokidis, X.; Silvi, B.; Alikhani, M. E. Chem. Phys. Lett. 1998, 292, 35–45. (b) Polo, V.; Andre´s, J. J. Comput. Chem. 2005, 26, 1427– 1437. (37) Krokidis, X.; Silvi, B.; Dezarnaud-Dandine, C.; Sevin, A. New J. Chem. 1998, 22, 1341–1350. (38) (a) Krokidis, X.; Goncalves, V.; Savin, A.; Silvi, B. J. Phys. Chem. A 1998, 102, 5065–5073. (b) Krokidis, X.; Vuilleumier, R.; Borgis, D.; Silvi, B. Mol. Phys. 1999, 96, 265–273. (c) Alikhani, M. E.; Silvi, B. J. Mol. Struct. 2004, 706, 3–6. (39) Berski, S.; Latajka, Z. Chem. Phys. Lett. 2006, 426, 273–279. (40) (a) Berski, S.; Andre´s, J.; Silvi, B.; Domingo, L. J. Phys. Chem. A 2003, 107, 6014–6024. (b) Polo, V.; Andres, J.; Castillo, R.; Berski, S.; Silvi, B. Chem.sEur. J. 2004, 10, 5165–5172. (c) Santos, J. C.; Polo, V.; Andre´s, J. Chem. Phys. Lett. 2005, 406, 393–397. (d) Santos, J. C.; Andre´s, J.; Aizman, A.; Fuentealba, P.; Polo, V. J. Phys. Chem. A 2005, 109, 3687– 3693. (e) Polo, V.; Andres, J.; Berski, S.; Domingo, L. R.; Silvi, B. J. Phys. Chem. A 2008, 112, 7128–7136. (41) Polo, V.; Gonzalez-Navarrete, P.; Silvi, B.; Andre´s, J. Theor. Chem. Acc. 2008, 120, 341–349. (42) (a) Michelini, M. C.; Sicilia, E.; Russo, N.; Alikhani, M. E.; Silvi, B. J. Phys. Chem. A 2003, 107, 4862–4868. (b) Michelini, M. C.; Russo, N.; Alikhani, M. E.; Silvi, B. J. Comput. Chem. 2004, 25, 1647–1655. (c) Michelini, M. C.; Russo, N.; Alikhani, M. E.; Silvi, B. J. Comput. Chem. 2005, 26, 1284–1293. (d) Alikhani, M. E.; Michelini, M. d. C.; Russo, N.; Silvi, B. J. Phys. Chem. A 2008, 112, 12966–12974. (43) (a) Becke, A. D. J. Chem. Phys. 1993, 98, 5648–5652. (b) Becke, A. D. Phys. ReV. 1988, A38, 3098–3100. (c) Lee, C.; Yang, Y.; Parr, R. G. Phys. ReV. 1988, B37, 785. (d) Miehlich, B.; Savin, A.; Stoll, H.; Preuss, H. Chem. Phys. Lett. 1989, 157, 200. (44) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Montgomery, J. A., Jr.; Vreven, T.; Kudin, K. N.; Burant, J. C.; Millam, J. M.; Iyengar, S. S.; Tomasi, J.; Barone, V.;
12906
J. Phys. Chem. A, Vol. 114, No. 49, 2010
Mennucci, B.; Cossi, M.; Scalmani, G.; Rega, N.; Petersson, G. A.; Nakatsuji, H.; Hada, M.; Ehara, M.; Toyota, K.; Fukuda, R.; Hasegawa, J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.; Nakai, H.; Klene, M.; Li, X.; Knox, J. E.; Hratchian, H. P.; Cross, J. B.; Bakken, V.; Adamo, C.; Jaramillo, J.; Gomperts, R.; Stratmann, R. E.; Yazyev, O.; Austin, A. J.; Cammi, R.; Pomelli, C.; Ochterski, J. W.; Ayala, P. Y.; Morokuma, K.; Voth, G. A.; Salvador, P.; Dannenberg, J. J.; Zakrzewski, V. G.; Dapprich, S.; Daniels, A. D.; Strain, M. C.; Farkas, O.; Malick, D. K.; Rabuck, A. D.; Raghavachari, K.; Foresman, J. B.; Ortiz, J. V.; Cui, Q.; Baboul, A. G.; Clifford, S.; Cioslowski, J.; Stefanov, B. B.; Liu, G.; Liashenko, A.; Piskorz, P.; Komaromi, I.; Martin, R. L.; Fox, D. J.; Keith, T.; Al-Laham, M. A.; Peng, C. Y.; Nanayakkara, A.; Challacombe, M.; Gill, P. M. W.; Johnson, B.; Chen, W.; Wong, M. W.; Gonzalez, C.; Pople, J. A. Gaussian 03, revision A.1; Gaussian, Inc.: Wallingford, CT, 2003. (45) (a) Fukui, K. J. Phys. Chem. 1970, 74, 4161–4163. (b) Fukui, K. Acc. Chem. Res. 1981, 12, 363–368. (46) Gonzalez, C.; Schlegel, H. B. J. Phys. Chem. 1990, 94, 5523. (47) (a) Noury, S.; Krokidis, X.; Fuster, F.; Silvi, B. TopMoD package; Université Pierre et Marie Curie: Paris, 1997. (b) Noury, S.; Krokidis, X.; Fuster, F.; Silvi, B. Comput. Chem. 1999, 23, 597–604. (48) Amira 3.0; TGS, Template Graphics Sofware, Inc.: San Diego, CA, 2002.
Ndassa et al. (49) Greenwood, N. N.; Earnshaw, A. Chemistry of the Elements; Pergamon Press: Oxford, U.K., 1984. (50) Bader, R. F. W. Atoms in Molecules: A Quantum Theory; Oxford Univ. Press: Oxford, U.K., 1990. (51) Rienstra-Kiracofe, J. C.; Tschumper, G. S.; Schaefer, H. F., III; Nandi, S.; Ellison, G. B. Chem. ReV. 2002, 102, 231–282. (52) Savin, A.; Silvi, B.; Colonna, F. Can. J. Chem. 1996, 74, 1088– 1096. (53) (a) Shaik, S. S.; Maitre, P.; Sini, G.; Hiberty, P. C. J. Am. Chem. Soc. 1992, 114, 7861–7866. (b) Shaik, S. S.; Danovich, D.; Silvi, B.; Lauvergnat, D.; Hiberty, P. Chem.sEur. J. 2005, 21, 6358–6371. (c) Shaik, S. S.; Danovich, D.; Wu, W.; Hiberty, P. Nature Chem. 2009, 1, 443–449. (54) Sanderson, R. T. Polar CoValence; Academic Press: New York, 1983. (55) Pakiari, A. H.; Nazari, F. J. Mol. Struct.: THEOCHEM 2003, 640, 109–115. (56) Glendening, E. D.; Weinhold, F. J. Comput. Chem. 1998, 19, 593– 609. (57) Hammond, G. S. J. Am. Chem. Soc. 1953, 77, 334–338. (58) Llusar, R.; Beltra´n, A.; Andre´s, J.; Noury, S.; Silvi, B. J. Comput. Chem. 1999, 20, 1517–1526.
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