a joint study based on the electron localization function and catastrophe theory of the chameleonic...

A Joint Study Based on the Electron LocalizationFunction and Catastrophe Theory of the Chameleonic and

Centauric Models for the Cope Rearrangement of1,5-Hexadiene and Its Cyano Derivatives

VICTOR POLO, JUAN ANDRESDepartament de Ciencies Experimentals, Universitat Jaume I, Apartat 224,

12080 Castello, Spain

Received 5 April 2005; Accepted 9 May 2005DOI 10.1002/jcc.20272

Published online in Wiley InterScience (www.interscience.wiley.com).

Abstract: A novel interpretation of the chameleonic and centauric models for the Cope rearrangements of 1,5-hexadiene (A) and different cyano derivatives (B: 2,5-dicyano, C: 1,3,4,6-tetracyano, and D: 1,3,5-tricyano) is presentedby using the topological analysis of the electron localization function (ELF) and Thom’s catastrophe theory (CT) on thereaction paths calculated at the B3LYP/6-31G(d,p) level. The progress of the reaction is monitorized by the changes ofthe ELF structural stability domains (SSD), each being change controlled by a turning point derived from CT. Thereaction mechanism of the parent reaction A is characterized by nine ELF SSDs. All processes occur in the vicinity of thetransition structure and corresponding to a concerted formation/breaking of C1—C6 and C3—C4 bonds, respectively, togetherwith an accumulation of charge density onto C2 and C5 atoms. Reaction B presents the same number of ELF SSDs as A, buta different order appears; the presence of 2,5-dicyano substituents favors the formation of C1—C6 bonds over the breakingof C3—C4 bond process, changing the reaction mechanism from a concerted towards a stepwise, via a cyclohexane biradicalintermediate. On the other side, reaction C presents the same type of turning points but two ELF SSD less than A or B; thereis an enhancement of the C3—C4 bond breaking process at an earlier stage of the reaction by delocalizing the electrons fromthe C3—C4 bond among the cyano groups. In the case of competitive effects of cyano subsituents on each moiety, as it is forreaction D, seven different ELF SSDs have been identified separated by eight turning points (two of them occur simulta-neously). Both processes, formation/breaking of C1—C6 and C3—C4 bonds, are slightly favored with respect to the parentreaction (A), and the TS presents mixed electronic features of both B and C. The employed methodology provides theoreticalsupport for the centauric nature (half-allyl, half-radical) for the TS of D.

© 2005 Wiley Periodicals, Inc. J Comput Chem 26: 1427–1437, 2005

Key words: electron localization function (ELF); catastrophe theory (CT); Cope rearrangement; chameleonic andcentauric mechanisms

Introduction

One of the major concerns in chemistry stands on the elucidationof the reaction mechanism for a given chemical rearrangement.Chemical reactions can be represented by the evolution of energyand charge redistribution along the channel connecting reactants toproducts via the corresponding transition structures (TSs) and/orpossible intermediates. However, the relationship between bothproperties along the reaction pathway still remains unclear. Al-though the use of standard methods is instructive and efficient inthe discussion of the reaction mechanism, theoretical analysisshould be performed to correlate with quantum mechanical con-cepts derived from first principle calculations.

Topological analysis of the electron density as the atoms inmolecules (AIM) theory of Bader1,2 or the electron localizationfunction (ELF) approach of Becke and Edgecombe3 and exten-sively developed by Silvi and Savin,4 can be considered as suitable

Correspondence to: J. Andres; e-mail: [email protected]

Contract/grant sponsor: Universitat Jaume I—Fundacio Bancaixa;contract/grant number: P1B99-02

Contract/grant sponsor: Ministerio de Cienciay Tecnologia, DGI;contract/grant number: BQU 2003-04168-C03-03

Contract/grant sponsor: Generalitat Valenciana; contract/grant number:6PUPOS02-028

© 2005 Wiley Periodicals, Inc.

tools for rationalization of reaction mechanisms because they arebased on a quantum mechanical observable (electron density).Krokidis et al.5 proposed a joint use of the ELF approach and thecatastrophe theory (CT) of Thom,6 to identify changes betweenregions of structural stability in processes of forming/breaking ofchemical bonds along the reaction path. It has been also applied toimportant chemical processes such as proton7 and electron8 trans-fers, isomerization9 and transition metal intercalation.10 Recently,we have applied the same methodology to study the molecularmechanism of organic reactions,11–14 where massive rearrange-ments of formally single, double, and triple carbon–carbon bondstake place along the reaction path. We now extend these studies toCope rearrangements, which have been object of controver-sies15–18 due to the variable electronic nature of the associatedtransition region to the presence of substituents.

The Cope reaction19 is a thermally allowed [3,3]-sigmatropicshift involving the migration of a � bond along one or two �systems, which has attracted a great variety of both experimentaland theoretical works15–37 to investigate the molecular mechanismand the changes caused by the presence and position of radical-stabilizing substituents. The variable nature of the correspondingTS of this reaction was evidenced experimentally by secondarykinetic isotope effect experiments by Gajewski and Conrad.20 Dueto the flatness of the TS region, this reaction presents a so-calledchameleonic (term coined by Doering16) transition region. Scheme1 provides a pictorial basis to understand the TS variation; de-pending on the interallylic distance (r), then the nature of the TScan vary from diradical, at short r, to “aromatic” at intermediate r,or bis-allyl structure at long r. Therefore, the presence of thesubstituents at the central 1,5-hexadiene acting in a cooperativeway moves the position of TS along the transition region switchingthe molecular mechanism between three reaction channels. Evenrecently, new candidates for chameleonic TSs have been pro-posed38 and discarded.39

Doering and Wang,16 using thermodynamic arguments consid-ered the possibility of another kind of model, different from the

chameleonic. Competitive substituent effects acting independentlyon the two allyl units would lead to a TS where the two differentextremes of the chameleonic transition region could coexist in thesame TS, half-radical and half-allyl (called the centauric model).Hrovat et al.,18 comparing experimental results with theoreticalcalculations at the B3LYP/6-31G(d) level, demonstrated the exis-tence of cooperative/competitive rather than simply additive ef-fects of cyano and allyl substituents on 1,5-hexadiene, and recentlya simple mathematical model to estimate these effects was pro-posed by Hrovat and Borden.40 Staroverov and Davidson17 haveperformed extensive theoretical calculations maintaining the C2h

symmetry at the transition region of the parent reaction and thecorresponding cyano derivatives. By calculating the density ofeffectively unpaired electrons, they point out the importance of theinterallyl distance in determining the radical character of thetransition state region, rather than the direct electronic effect of thecyano substituents. More recently, Blavins et al.,36 employingspin-coupled (modern valence bond) theory, determined theweights of diradical/aromatic/bis-allyl contributions of the TS of1,5-hexadiene and the same cyano derivatives. Following theseworks, our analysis considers not only the stationary points of theparent 1,5-hexadiene and those cyano derivatives that might ac-count for the different possibilities, but also the full reactionpathway.

With the aim of offering a novel interpretation of these reactionmechanisms from the new point of view offered by the ELF, weobserve for each reaction the effect of the cyano substituents actingfrom the reactants to the products and the repercussion into thereaction mechanism. First, to delimit the chameleonic model, weinvestigate the parent unsubstituted reaction (A) and the twoextreme situations: 2,5-dicyano (B, via a biradical intermediate)and the 1,3,4,6-tetracyano (C, through a bis-allyl TS). Havingexplored the chameleonic model, we provide an answer, in termsof ELF and CT arguments, for the question posed by Doering etal.:16 does the 1,3,5-tricyano (D) belong to the chameleonic modelor it is beyond (Scheme 2)?

Scheme 1. General scheme of the chameleonic model of the Coperearrangement.

Scheme 2. The Cope rearrangement of 1,5-hexadiene and the threecyano substitution patterns considered in this study.

1428 Polo and Andres • Vol. 26, No. 14 • Journal of Computational Chemistry

Computational Details

The geometry optimizations and electronic structure calculationswere carried out by the Gaussian03 program.41 Structures wereoptimized using the B3LYP hybrid exchange-correlation function-al42 together with the 6-31G(d,p)43 basis set (B3LYP/6-31G(d,p)).Stationary points on the PES were characterized by harmonicanalysis using the same theoretical level as used in the optimiza-tion. Although the nature of the TS is sensitive to the choice of thecalculation level, the B3LYP activation energy is in good agree-ment with experimental data and more elaborate methods.29 Someauthors17 recommend the use of pure functionals (instead of thehybrid version) to describe this type of reaction, due to the ap-pearance of spurious stationary points for short interallyl distances,however, as we are not applying geometry constraints, we prefer toemploy the popular B3LYP method. Reactions A, C, and D wereconcerted, while B was found to react through an intermediatecalculated using BS-UB3LYP, as it was pointed out by Hrovat etal.18 for 2,5-diphenyl-1,5-hexadiene. In the present work, reactantsare connected to the TS by means of IRC calculations, using themethod proposed by Fukui44 and developed by Gonzalez andSchlegel.45

The stability of the wave function was checked at some pointsof the reaction pathway according to the procedure described byBauernschmitt and Ahlrichs.46 Those points where the restricted(RDFT) wave function presented an instability were reoptimizedusing the unrestricted broken spin symmetry (BS-UDFT) descrip-tion, to see whether the constraint �� was lifted.47 In thisway, the IRC path of B is composed of two parts: the closed-shell(RB3LYP) from the reactant to the TS, and the open-shell (BS-UB3LYP) from the TS to the intermediate. The Kohn–Shamorbitals were obtained for each point of the calculated IRC pathand the ELF analysis was carried out using a cubical grid of stepsize smaller than 0.1 Bohr, employing the TopMod48 package ofprograms. The graphical representation was obtained using theMOLEKEL program.49

The electron localization function [ELF, �(r)] provides anorbital independent description of the electron localization basedon strong physical arguments regarding the Fermi hole.50 The ELFis defined by the ratio of the excess of local kinetic energy densitydue to the Pauli exclusion principle, and the Thomas–Fermi kineticenergy density. Numerical values of the ELF are mapped on theinterval (0,1) facilitating its analysis. The topological analysis ofthe ELF gradient field, ��(r), provides a mathematical modelenabling the partition of the molecular position space in basins ofattractors, which present, in principle, a one-to-one correspon-dence with chemical local objects such as bonds and lone pairs.These basins are either core basins labeled C(A) or valence basinsV(A, . . .) belonging to the outermost shell and characterized bytheir coordination number with core basins, which is called thesynaptic order. This method has been well documented in series ofarticles presenting its theoretical foundations.4 Recently, Silvi51

has demonstrated that the ELF is a good approximation to the moregeneral size-independent spin-pair composition function, whichhas a clear meaning as a local indicator of chemical bonding.

As previously pointed out, in the framework established by theELF analysis, changes in the control parameters defining thereaction pathway (such as the nuclear coordinates and the elec-

tronic state) can lead to different topologies of the ELF. Accordingto the theory of dynamical systems, a system is said structurallystable if a small perturbation of the vector field does not change theindexes of its critical points. Thus, each ELF topological config-uration is only possible for values of the control parameters com-prised into well-defined ranges or, in other words, to subsets calledELF structural stability domains (SSD). Two points of the controlspace belonging to the same SSD possess the same number ofcritical points of each type in the ELF gradient field. Within a SSD,the critical points on the ELF scalar field (i.e., points at which��(r) � 0.0) are said to be hyperbolic, that is, all eigenvalues ofthe Hessian matrix are not zero. The turning points between twoconsecutive SSDs, present at least one critical exponent equal tozero. Along the reaction path the chemical system goes from agiven ELF SDD to another by means of bifurcation catastrophesoccurring at the turning points. Each catastrophe transforms theoverall topology in such a way that the Poincare–Hopf relation isfulfilled. Three types of bifurcation catastrophes have been foundin the study of chemical reactivity: (1) the fold catastrophe, cor-responding to the creation or to the annihilation of two criticalpoints of different parity; for example, a wandering point gives riseto an attractor (index 0), and a saddle point of index 1. (2) The cuspcatastrophe, which transforms one critical point into three (andvice versa) such as in the formation or the breaking of a covalentbond. (3) The elliptic umbilic in which the index of a critical pointchanges by two.

Results and Discussion

Lewis Structures of the Cyano Substituted Allyl-Radical

Prior to the presentation of results for reactions A–D, the effect ofcyano substituents in the allyl radical is analyzed by calculation ofthe Lewis resonance structures employing the ELF basin popula-tions. Three possible substitution patterns of the allyl radical, areshown in Scheme 3. As shown in previous articles,12 a basinpopulation larger than 2.00e for a C—H bond suggests that canhide a monosynaptic basin V(C), corresponding to an unpairedelectron on the C atom, overlapped by the hydrogenated basin. Forthis reason, we consider the presence of unpaired electrons oncarbons bonded to hydrogen (but not on C—CN). To reduceoverlapping of small V(C) basins by large V(C,C), we havecalculated the populations for � and � electrons separately.52 Theallyl radical can be reasonably described by the superposition ofthe following three distinct resonance forms shown in the Scheme4 (5 including degenerate forms of I-A and III-A):

Structure I-A has a double bond and one unpaired electronlocated on the other terminal C atom. In II-A, there is one unpaired

Scheme 3. Cyano substitution patterns of the allyl radical.

Electron Localization Function and Catastrophe Theory 1429

electron on each C atom, while III accounts for the ionic form. Thecalculated weights of each form are: wI-A � 0.59, wII-A � 0.11 andwIII-A � 0.30. As expected, the form I-A is dominant, althoughIII-A has an important contribution.

Cyano substituents offer a multitude of possible resonanceforms. Those with a negative charge on the N atom can bereasonable, but according to this analysis, their contribution isfound to be very small to the total superposition structure; there-fore, they are not included here. For 2-cyano-allyl radical, II-B candelocalize the unpaired electron on the C—CN bond (Scheme 5).

The calculated weights are: wI-B � 0.59, wII-B � 0.08 andwIII-B � 0.33. Although the difference with regard to the unsub-stituted allyl is small, there is a noticeable increment of wIII-B.

The cyano substitution in terminal carbons (1,3-dicyano-allylradical), can be represented by the analogous Lewis resonancestructures; the unpaired electron is localized on the nitrogen atomin III-C (Scheme 6): The calculated weights are: wI-C � 0.63,wII-C � 0.13, wIII-C � 0.24; form III-C contributes less, and thereis a considerable increment of I-C and II-C weights.

Hence, different effects are observed depending on the positionof substituents, 2-cyano increases the weights of form III-B, wherenonbonded electrons localized on terminal carbons, whereas the1,3-dicyano-allyl radical is composed by forms where the unpairedelectron prefers to be more localized on the cyano groups, thecentral carbon or the bond C1—C2—C3.

The Effect of Cyano Substituents on Cope Rearrangements

The reaction paths calculated using the IRC method for reactionsA–D are shown in Figure 1. Below each potential energy profile aschematic representation of the change of topology of ELF basinsis presented, indicating the type of turning point between SSDs:nine for A and B, and seven for C and D. The populations forvalence basins at the first and last points of each SSD are presentedin Tables 1–4, respectively. Geometrical parameters, such as dis-tances rC1—C6 and rC3—C4 are also included in the table tocorrelate electronic and geometrical changes. The population ofcore basins remains constant along the IRC path at 12.54, 20.90,29.24, and 25.06 electrons for reactions A–D, respectively. Asreactant and products are identical, we will discuss only the part of

the IRC pathway going from the reactant to the TS (or intermediatefor B). In the same way, for reactions A–C, both moieties formedby C1,6—C2,5—C3,4 units are equivalent by symmetry; thus, aturning point on one moiety presents another degenerate point onthe other unit.

Reaction A

The ELF analysis of 1,5 hexadiene shows (Fig. 2) disynapticbasins V1,2(C1,5,C2,6), V(C2,4,C3,5), and V(C3,C4) correspondingto double bonds between C1,5—C2,6 atoms with an integratedpopulation of 3.54e and single bonds between C2,4—C3,5 andC3—C4 atoms, whose electronic population integrates to 2.00e and1.81e, respectively. The first ELF SSD is the most costly in termsof energy, there is an increment of 28.0 kcal/mol with respect tothe reactant. Looking at the integrated electron populations ofTable 1, there is an electronic charge process of 0.44e from theV(C3,C4) basin to V(C2,4,C3,5) ones, which is accompanied by theelongation of the C3—C4 bond (1.553 to 1.793 Å).

At Rx � �0.998 amu1/2bohr, the first turning point occurs. Itbelongs to the cusp type (we label it as C1), and degeneratedisynaptic basins V1,2(C1,5,C2,6) evolve into V(C1,5,C2,6) basinspointing out the participation of double bonds in the reaction. Theenergy in this SSD increases by 4.8 kcal/mol. The third ELF SSDcommences when the disynaptic V(C3,C4) is divided in two mono-synaptic basins V(C3) and V(C4) by a cusp type catastrophe (C2)characteristic of a covalent bond breaking process5 and the ELFanalysis reveals that the bond C3—C4 is broken at an interatomicdistance rC3—C4 of 1.923 Å. Figure 2 shows the ELF at this SSDof the reaction path. It is interesting to compare the deformedshape of V(C1,5,C2,6) basins in Figure 2 at the C2 turning pointwith the typical degenerate double bond as shown for the reactant.

One point later on the IRC path, a new turning point of foldtype occurs (F1). Two new electron pairs are formed on the C1 andC6 atoms with an integrated electron population of 0.34e each,coming from the V(C1,5, C2,6) basins. At the next point on the IRCpath, the TS is reached and a new turning point of fold type happens(F2). Again, two new monosynaptic ELF basins are created but on C2

and C5 atoms, with a population of 0.29e each, the electron density ofthese basins also comes from V(C1,5,C2,6) basins.

The analysis of the TS by the ELF reveals some interestingfeatures about the distribution of electronic pairs that explain thechameleonic properties of this reaction mechanism. In Figure 3,the tree reduction diagram of ELF localization domains shows thehierarchy of ELF basin domains. All monosynaptic basins areconnected by saddle points of high ELF value (�0.74), pointingout the very large delocalization between valence electron pairs atthe TS. According to the ELF terminology, the TS presents two“proto-covalent” bonds between the moieties.53 Summarizing, theuphill part of the IRC pathway of the Cope rearrangement of

Scheme 6. Proposed resonance forms for the 1,3-dicyano-allyl radi-cal.Scheme 4. Proposed resonance forms for the allyl radical.

Scheme 5. Proposed resonance forms for the 2-cyano-allyl radical.


1,5-hexadiene can be described as the stretching and depopulation ofthe C3—C4 bond, and the movement of electron density to adjacentELF basins. All bond breaking/forming processes occur suddenly inthe vicinity of the TS, revealing the highly concerted nature of theseprocesses. The populations of hydrogenated basins V(H,C) changeslightly their populations along the IRC path (see Table 1).

Reaction B

The perturbation introduced by the 2,5-dicyano-substituents mod-ifies vaguely the ELF populations of the 1,5-hexadiene main unit

at the reactant. The cyano group is represented by a lone pair onthe nitrogen atom, V(N), counting with a population of 3.30e, thebond N—C by two degenerate disynaptic basins V1,2(C7,N1) withtotal population of 4.41e while the C2 and C7 atoms are connectedby a V(C7,C2) basin, which integrates to 2.27e, more than a typicalsingle C—C bond (around 1.86e).

Along the first ELF SSD, there is a less efficient transfer ofelectronic charge from the V(C3,C4) basin to the V(C2,4,C3,5) ones(0.25e), hindering the C3—C4 bond breaking process. Similar toA, the second ELF SSD starts when the two degenerate disynaptic

Figure 1. Energy profiles (a)–(d) for the Cope rearrangement of reactions A–D calculated by means of the IRC method, with a step size of 0.1[amu1/2bohr]. Below the graph a schematic representation of the reaction mechanism for each figure is depicted from the perspective of the ELFanalysis (full lines and ellipses representing disynaptic and monosynaptic basins, respectively, dotted lines indicates a large basin population), thetype of turning point is noted as in the text.


basins V1,2(C1,5,C2,6) are transformed into one basin V(C1,5,C2,6)by a cusp type turning point (C1). Compared with A, the popula-tion of V(C3,C4) at the point where C1 takes place is 1.62e, 0.26e

more populated in B; the distance rC3—C4 is 1.622 Å, 0.185 Åshorter, and the increment of energy to reach this point is of 19.4kcal/mol, which is 8.6 kcal/mol less than for A.

Table 1. Rx (in amu1/2bohr), Increments of Energy within Each SSD in kcal/mol (�E � Elast � Einitial),Geometrical Parameters (in Å), and Integrated Electron Populations for Valence ELF Basins of Reaction Aon the C1—C2—C3 Moiety, Calculated for the Initial and Final Points of Each SSD Identified on the EnergyReaction Profile Using the ELF Analysis.

I II II IV V

Rx Reactive �1.098 �0.998 �0.299 �0.199 �0.0998 0.0 (TS)�E �28.0 �4.8 �0.3 �0.3 �0.4rC3—C4 1.553 1.793 1.807 1.909 1.923 1.938 1.986rC1—C6 4.563 2.188 2.173 2.065 2.050 2.035 1.986

V1(C1,C2) 1.77 1.843.42 3.35 3.36 3.03 2.55V2(C1,C2) 1.77 1.60

V(C2,C3) 2.00 2.18 2.21 2.37 2.41 2.41 2.54V(C3,C4) 1.81 1.37 1.36 1.14 — — —V(C1) — — — — — 0.34 0.45V(C2) — — — — — — 0.29V(C3) — — — — 0.55 0.53 0.45

V(H1,C1) 2.08 2.11 2.10 2.10 2.09 2.09 2.07V(H2,C1) 2.09 2.11 2.11 2.11 2.09 2.09 2.09V(H3,C2) 2.10 2.10 2.10 2.11 2.10 2.10 2.12V(H4,C3) 2.02 2.04 2.04 2.06 2.06 2.07 2.07V(H5,C3) 2.00 2.07 2.06 2.08 2.08 2.08 2.09

First column is the reactant and the last column is the TS.

Table 2. Rx (in amu1/2bohr), Increments of Energy within Each SSD in kcal/mol (�E � Elast � Efirst),Geometrical Parameters (in Å), and Integrated Electron Populations for Valence ELF Basins of Reaction Bon the C1—C2—C3 Moiety (V(C2—C7N1), Accounts for All Valence Basins Belonging to the Cyano Group,Including the C2—C7 Bond), Calculated for the Initial and Final Points of Each SSD Identified on theEnergy Reaction Profile Using the ELF Analysis.

I II III IV V

Rx Reactive �1.788 �1.688 �0.789 �0.689 �0.195 �0.096 0.0 (TS) Int.�E �19.4 �4.6 �1.4 �1.2 �4.3rC3—C4 1.545 1.619 1.622 1.657 1.662 1.686 1.690 1.684 1.574rC1—C6 4.419 2.219 2.198 2.013 1.992 1.891 1.871 1.830 1.574

V1(C1,C2) 1.72 1.803.39 3.34 3.05 2.91 2.89 2.41 2.13V2(C1,C2) 1.75 1.58

V(C2,C3) 1.99 2.05 2.06 2.13 2.13 2.17 2.18 2.18 2.11V(C3,C4) 1.80 1.62 1.62 1.53 1.51 1.44 1.43 1.43 1.66V(C1) — — — — 0.31 0.46 — — —V(C1,C6) — — — — — — 0.96 1.07 1.66V(C2) — — — — — — — 0.42 0.30

V(C2—C7N1) 9.98 10.02 10.01 10.05 10.05 10.06 10.07 10.08 10.28V(H1,C1) 2.09 2.12 2.12 2.10 2.10 2.08 2.08 2.07 2.03V(H2,C1) 2.10 2.12 2.12 2.10 2.10 2.08 2.08 2.07 2.03V(H3,C3) 2.00 2.03 2.03 2.03 2.04 2.04 2.04 2.04 2.03V(H4,C3) 2.00 2.03 2.03 2.04 2.04 2.04 2.04 2.04 2.01

First column is the reactant, the last column is the intermediate.


The third ELF SSD commences 10 points later on the calcu-lated IRC path; two new basins are formed on C1 and C6 atoms bymeans of a fold type turning point, which is of the same type as F1and opposite direction than in A but acting on C1 and C6 atoms.The population of V(C2,5—C7,8N1,2) has increased to 10.05e,revealing the active role of the cyano groups on the electronicrearrangements. Creation of V(C1,6) basins before breaking ofC3—C4 bond causes an alteration of the order of appearance ofELF SSDs with regard to A, which can be rationalized by theanalysis of the cyano substitution in the allyl radical. The incre-ment of resonance form III-B, with more nonbonded electrons onterminal carbons than forms I-B or II-B, explains the rapid appear-ance of V(C1,6) basins and the stability of C3—C4 bond withrespect to A. The fourth ELF SSD starts by the formation of theC1—C6 covalent bond from V(C1) and V(C6) basins by a cusptype turning point, which corresponds to the inverse of C2 in A.Looking at Table 2, two movements of electron charge can bequantified, because the beginning of the rearrangement up to theend of fourth SSD: (1) V(C1,5,C2,6) basins population have low-ered in 1.16e (in total), the destinations of this electronic chargehave been the formation of C1—C6 bond (0.96e) and an incrementof population of cyano groups (V(C2,5—C7,8N1,2)) (�0.10e), and(2) V(C3,C4) has transferred 0.37e to V(C2,4,C3,5) basins.

In this region, the open-shell singlet biradical found usingBS-UB3LYP methodology is lower in energy than the RB3LYP,which becomes unstable, and the method of calculation is switchedto the BS-U version. The next point along the IRC path reaches theTS and a new turning point (F2) takes place, two new basins arecreated at C2 and C5, whose electronic density is coming fromV(C1,5,C2,6) basins. Contrary to A, the downhill IRC pathwayfrom the TS does not drive to the product, but to a stationary point

on the PES 4.3 kcal/mol below the TS having all real vibrationalfrequencies. This intermediate has low-populated monosynapticbasins V(C2,5) (0.30e) accounting for the unpaired electrons, andthe population of valence basins of the cyano group (including theC—C bond) adds 10.28e, suggesting that a considerable part of thebiradical is overlapped by the C—CN bond. The intermediate hasdistances rC1—C6 and rC3—C4 equal to 1.574 Å, and V(C1,3,C4,6)basins having increased their populations to 1.66e. The system issimilar to a cyclohexane ring in the chair conformation (see Fig. 2,intermediate B). Along the fifth ELF SSD, electron density charge(0.56e) flows from V(C1,5,C2,6) basins to the new bond C1—C6.Table 2 reveals that the process of transference of electroniccharge from V(C3,C4) to V(C2,4,C3,5) basins is stopped and in-versed, also distance rC3—C4 suffers no further elongation untilthe intermediate is found.

In summary, the perturbation on the Cope parent rearrangementintroduced by cyano substituents on C2,5 atoms correlates with theeffect observed in the 2-cyano-allyl radical: the forms with non-bonded electrons on terminal atoms (C1,C3, C4, and C6) are fa-vored and the intermoiety bonds are stabilized. Although theelectronic effect is very subtle on the reactant, its continuous effectbrings a different rearrangement of the flow of electron densitywithin the molecule along the IRC pathway, causing the dramaticalteration of the reaction mechanism from concerted to stepwisevia a stable biradical intermediate.

Reaction C

Again, the four cyano substituents do not change significatively theintegrated populations of ELF basins of the 1,5-hexadiene unit.However, looking at the population of the V(C1,6—C7,10N1,4)

Table 3. Rx (in amu1/2bohr), Increments of Energy within Each SSD in kcal/mol (�E � Elast � Efirst),Geometrical Parameters (in Å), and Integrated Electron Populations for Valence ELF Basins of Reaction Con the C1—C2—C3 Moiety (V(C1,3—C7,8N1,2) Accounts for All Valence Basins Belonging to the CyanoGroup, Including the C1,3—C7,8 Bonds), Calculated for the Initial and Final Points of Each SSD Identifiedon the Energy Reaction Profile Using the ELF Analysis.

I II III IV

Rx Reactive �2.656 �2.556 �1.557 �1.457 �0.866 �0.768 0.0 (TS)�E �18.7 �5.0 �1.7 �0.9rC3—C4 1.588 1.967 1.986 2.176 2.195 2.300 2.317 2.473rC1—C6 5.384 2.730 2.724 2.660 2.652 2.601 2.592 2.473

V1(C1,C2) 1.74 1.66 1.67 1.733.24 3.16 3.16 3.03V2(C1,C2) 1.76 1.70 1.68 1.51

V(C2,C3) 2.00 2.16 2.18 2.37 2.37 2.51 2.88 2.97V(C3,C4) 1.75 1.32 — — — — — —V(C3) — — 0.65 0.50 0.48 0.37 — —

V(C1—C7N1) 9.95 9.95 9.98 9.98 9.96 9.98 10.00 10.00V(C3—C8N2) 9.88 9.95 9.94 9.96 9.96 9.96 9.96 10.00V(H1,C1) 2.11 2.12 2.12 2.13 2.12 2.13 2.13 2.14V(H2,C2) 2.08 2.11 2.11 2.13 2.12 2.13 2.13 2.13V(H3,C3) 2.02 2.07 2.07 2.11 2.11 2.13 2.13 2.14



(9.95e) and V(C3,4—C8,9N2,3) (9.88e) basins, it is shown that theyare playing an active role. Considering the 1,3-dicyano-allyl rad-ical, the delocalization of the unpaired electron on the cyanogroups and the higher weight of form I-C favors breaking of theC3—C4 bond and localization of the electronic charge comingfrom the C3—C4 bond on each 1,3-dicyano-allyl structure.

Following the IRC path from the reactant, the first turning pointcorresponds to the C2 type as defined for the parent reaction. It isfound at Rx � �2.556 amu1/2bohr and determines the transition tothe second SSD. The V(C3,C4) basin associated to the C3—C4

covalent bond is broken into two monosynaptic basins V(C3) andV(C4) at a distance rC3—C4 of 1.986 Å. Note that the incrementof the energy of the system to reach the breaking of the C3—C4

bond is only 18.7 kcal/mol compared with 32.8 kcal/mol for A.The third ELF SSD is introduced by a cusp type turning point(C1): formal double bonds C1,5—C2,6 represented by two degen-erate basins V1,2(C1,5,C2,6) evolve to one basin V(C1,5,C2,6). AtRx � �0.768 amu1/2bohr, the V(C3) and V(C4) basins are mergedinto V(C2,4,C3,5) by means of a fold type turning point (F1)forming a bis-allyl complex. Along this SSD, the distance rC3—C4

is enlarged up to 2.473 Å, when the TS is reached. The attractiveinteractions between the two allyl groups are due only to mutualpolarization forces between fragments; there are neither proto-covalent (as in A) nor covalent bonds (as in B) between themoieties. The structure of the molecule at the TS is a bis-allylcomplex with no monosynaptic basins on the allyl moieties; theelectron density from the initial C3—C4 bond has been completelyredistributed through other ELF basins, mainly on the cyanogroups and V(C,C) basins as the resonance forms of the 1,3-dicyano-allyl radical pointed out.

As it was observed for reactions A and B, the driving process ofC is the stretching of C3—C4 bond and depopulation of the associatedV(C3,C4) ELF basin, which is helped by the steric repulsions betweenthe cyano substituents on atoms C3 and C4. As distinct from the parentreaction, the cyano groups on atoms C1, C3, C4, and C6 can easilyaccept electronic charge arising from the elongation of the C3—C4

bond, greatly favoring the bond-breaking process and the formation ofa bis-allyl complex at the TS. Contrary to the previous reactions, thekey turning points involving bond breaking/forming are not takingplace in the vicinity of the TS.

Table 4. Rx (in amu1/2bohr), Increments of Energy within Each SSD in kcal/mol (�E � Elast � Efirst),Geometrical Parameters (in Å), and Integrated Electron Populations for Valence ELF Basins of Reaction DV(C1,3,5—C7,8,9N1,2,3) Accounts for All Valence Basins Belonging to the Cyano Group, Including theC1,3,5—C7,8,9 Bonds), Calculated for the Initial and Final Points of Each SSD Identified on the EnergyReaction Profile Using the ELF Analysis.

I II III IV

Rx Reactive �1.298 �1.198 �0.498 �0.398 �0.099 0.0 (TS)�E �24.0 �4.9 �1.3 �0.3rC3—C4 1.561 1.877 1.893 2.007 2.023 2.070 2.119rC1—C6 4.311 2.329 2.316 2.223 2.209 2.167 2.119

V1(C1,C2) 1.74 1.653.27 3.22 3.21 3.18 2.67V2(C1,C2) 1.73 1.64

V(C2,C3) 1.99 2.2 2.22 2.36 2.38 2.48 2.67V(C3,C4) 1.81 1.28 1.25 1.00V(C4,C5) 1.96 2.21 2.23 2.38 2.41 2.51 2.66V1(C5,C6) 1.75 1.69

3.27 3.21 3.2 3.16 2.66V2(C5,C6) 1.73 1.6V(C3) — — — — 0.59 0.50 0.40V(C4) — — — — 0.37 0.31 0.24V(C1) — — — — — — 0.40V(C6) — — — — — — 0.24V(C5) — — — — — — 0.20

V(H1—C7N1) 9.95 9.97 9.97 9.97 9.96 9.96 9.95V(C3—C8N2) 9.84 9.92 9.92 9.95 9.95 9.95 9.97V(C5—C9N3) 9.96 9.98 9.98 9.98 9.98 9.99 9.98V(H1,C1) 2.12 2.15 2.16 2.15 2.15 2.15 2.15V(H2,C2) 2.08 2.13 2.14 2.13 2.13 2.13 2.12V(H3,C3) 2.02 2.07 2.07 2.09 2.09 2.11 2.12V(H4,C4) 2.02 2.08 2.08 2.11 2.11 2.11 2.13V(H5,C4) 2.02 2.08 2.09 2.11 2.12 2.13 2.14V(H6,C6) 2.09 2.14 2.15 2.14 2.14 2.14 2.14V(H7,C6) 2.11 2.13 2.14 2.13 2.13 2.13 2.13



Reaction D

Now that the chameleonic model has been characterized and itsdifferent reaction mechanism pathways have been interpreted bythe ELF analysis, we focus on the possible existence of a hybrid orcentauric reaction mechanism. Similar to the first ELF SSD of thereactions A–C, the stretching of the C3—C4 bond leads to aconsiderable depopulation of the V(C3,C4) basin (it lowers in0.53e), the electronic charge is mainly transferred to V(C2,4,C3,5)basins (0.21 and 0.26e). In the same way as A and B, the second

SSD starts by a C1 turning point which transforms C1,5—C2,6

double bonds into singles. The basin V(C3,C4) continues beingdepopulated until a C2 turning point breaks the bond in a hetero-lytic way at Rx � �0.398 amu1/2bohr. After the bond breaking,more electron density goes to the V(C3) (0.60e) than to V(C4)(0.39e) basin due to the electron withdrawing effect of the cyanogroup on C3 atom. The fourth ELF SSD comprises only the TS,and it starts by the simultaneous formation of V(C1,6) and V(C5)by means of the F1 and F2 turning points, respectively, but now

Figure 2. Snapshots of ELF localization domains (� � 0.75 isosurface, except 0.80 for C and reactive D) for: (a) reactive (b) C—C bond breaking(A, C, and D)/forming (B) turning point, (c) TS (A, C, and D) or intermediate (B). Basin color legend: purple for core, orange for monosynaptic,and green for disynaptic. Hydrogenated and basins attached to nitrogen are not displayed for clarity.


the F2 turning point is acting only on the C4—C5—C6 moiety.Considering the interallyl distance at the TS (rC1—C6 � 2.119 Å)the situation resembles A (rC1—C6 � 1.986 Å) by the presence of“proto-covalent” bonding between moieties, while B (rC1—C6 �1.574 Å) presents a covalent bond and C (rC1—C6 � 2.473 Å)merely an electrostatic binding. Looking at Figure 2 (all ELFisosurfaces), the TS of the parent reaction can be described as anintermediate situation of the two extremes cases B and C, but thedifferences between moieties of TS of reaction D indicate thecoexistence of both structures. In contrast with previous studies,36

the ELF analysis provides support for some kind of centaur TS byrevealing an allyl- and radical-like character for C1—C2—C3 andC4—C5—C6 moieties, respectively.

Conclusions

The present work is based on the analysis of the evolution ofelectron pairs along the reaction pathway, determining the timingof changes in the number and type of ELF structural domains bymeans of turning points and relating them to chemical events, suchas bond-forming/breaking processes and other electronic rear-rangements. This procedure is in agreement with physical laws andquantum theoretical insights, and it can be considered as a new toolto tackle chemical reactivity with a wide range of possible appli-cations, and the universal behavior that it predicts. Thus, newinsights about the role of cyano substituents and their influence onthe reaction mechanism of the Cope reactions for four cases:1,5-hexadiene (A) and cyano derivatives (B: 2,5-dicyano, C:1,3,4,6-tetracyano, and D: 1,3,5-tricyano) are gained. A better

mechanistic understanding of bond-breaking/forming processesand charge redistribution along the reaction pathways has beenobtained. Thus, the specific flows of electron density that occurnaturally in the unsubstituted Cope rearrangement can be advancedor retarded by the number and placement of cyano groups.

The main conclusions of the present work can be summarizedas follows:

1. The simple case of cyano substitution on the allyl radicalillustrates how the cyano group alters the weights of the pro-posed resonance Lewis structures, favoring the location ofnonbonded electrons on terminal carbons or among the cyanogroups and C—C bonds.

2. Although the perturbation exerted by cyano substituents on theelectronic structure of the 1,5-hexadiene is rather small, they playa very important role in the context of the multiple electronicreorganizations occurring along the intrinsic reaction path.

3. The presence of a cyano on C2,5 atoms (B) stabilizes theresonance structures with nonbonding electrons on terminalcarbons, favoring the formation of the C1—C6 bond and penal-izing the breaking of the C3—C4 bond process, forming a stablecyclohexane biradical intermediate.

4. The highly delocalized nature of the TS parent reaction (A) ismanifested in the ELF description by the presence of fourdisynaptic and six monosynaptic basins (one per carbon atom)enclosed by an isocontour surface of high ELF(r) value (0.74).

5. Both reactions B and C reduce the activation energy barrier inalmost the same amount by different changes on the reactionmechanism: cyano substituents in C2,5 atoms (B) stabilize in-termoiety bonding, while in C1,3,4,6 atoms (C) favored theintramoiety delocalization.

6. Substitution on C1,3,4,6 atoms (C) allows the stabilization of theelectrons coming from the breaking of C3—C4 bond by delo-calization onto cyano groups.

7. ELF analysis provides support for the centauric model of the Coperearrangement of 1,3,5-tricyano-1,5-hexadiene (D), whose TS pre-sents both features of the chameleonic model. The radical moietypresents the characteristic monosynaptic basin on the C5 carbonatom, while in the allyl moiety the electronic charge from thebroken C3—C4 bond is partially delocalized on the cyano groups.

8. Turning points connecting the ELF SSD for the four reactions(A, B, C, and D) are of the same type according to CTclassification, but the order of appearance is altered, leading todifferent sequence of SSDs and reaction mechanisms.

Acknowledgments

The authors thank Dr. Luis Domingo and Prof. Ian Williams forstimulating discussions about substituent effects and Lewis reso-nance forms. The authors acknowledge the Servei d’Informatica,Universitat Jaume I for generous allotment of computer time.

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