How a Quantum Chemical Topology Analysis Enables Prediction of Electron Density Transfers in Chemical Reactions

Letter

pubs.acs.org/JPCL

How a Quantum Chemical Topology Analysis Enables Prediction of Electron Density Transfers in Chemical Reactions. The Degenerated Cope Rearrangement of Semibullvalene Patricio Gonzalez-Navarrete,́ † Juan Andres,́*,† and Slawomir Berski‡

† Departamento de Química Física y Analítica, Universitat Jaume I, 12071 Castellóde la Plana, Spain ‡ Faculty of Chemistry, University of Wroclaw, F. Joliot-Curie 14, 50-383 Wroclaw, Poland

*S Supporting Information

ABSTRACT: Recent works on the reaction mechanism for the degenerated Cope rearrangement (DCR) of semibullvalene (SBV) in the ground state prompted us to investigate this complex rearrangement in order to assign experimentally observed contrast features in the simulated electron distribution. We present a joint use of the electron localization function (ELF) and Thom's catastrophe theory (CT) as a powerful tool to analyze the electron density transfers along the DCR. The progress of the reaction is monitored by the structural stability domains of the topology of ELF, while the change between them is controlled by turning points derived from CT. The ELF topological analysis shows that the DCR of SBV corresponds to asynchronous electron density rearrangement taking place in three consecutive stages. We show how the pictures anticipated by drawing Lewis structures of the rearrangement correlate with the experimental data and time-dependent quantum description of the process. SECTION: Molecular Structure, Quantum Chemistry, and General Theory

o a large extent, chemistry can be viewed as the art of allow a Lewis type of arrow representation in terms of T making and breaking bonds, and understanding the basic electronic fluxes between neighboring bonds, revealing their science behind this has been one of the main challenges of asynchronous nature. theoretical and computational chemistry. A reaction mechanism The DCR of SBV corresponds to a complex and coupled represents a sequence of elementary steps by which overall forming and breaking of chemical bonds, and it has been widely studied in order to provide relevant insights into those chemical change occurs, describing in detail what takes place at − each stage of a chemical transformation, that is, chemical bonds properties that are related to pericyclic reactivity.10 14 As are breaking/forming processes, electron pair rearrangements, shown in Scheme 1, the net structural outcome is the breaking/ transformation of formally double to simple bonds or vice versa, and so forth. Measurements visualizing the progress of Scheme 1 chemical reactions on their natural time scale can be considered the holy grail of chemical physics as it is now done for the resolution of molecular structure by using ultrafast electron diffraction1 or X-ray diffraction.2 Furthermore, recent advances in the emerging area of attosecond physics3 (providing real- time access to the motion of electrons on atomic and subatomic − scales) and real-time vibrational spectroscopy4 6 by a few femtosecond pulse laser (enabling the observation of dynamic forming of the C2−C8/C4−C6 bonds and the transformations behavior of molecular vibrations during chemical reactions) of formal single C2−C3 and C7−C8 bonds to double ones, have been recently achieved. while an opposite behavior takes place in C3−C4 and C6−C7 The motivation of our investigation essentially arises from bonds. The delineation of factors that control the pair electron the results of three recent works concerning electronic reorganization on this rearrangement is complicated. Although mechanistic aspects of pericyclic reactions. Bredtmann, Manz, a large number of experimental and computational studies have et al.,7,8 by means of selective laser pulses, monitored electronic mainly been focused on the energy barriers (with special fluxes that accompany the breaking and making of covalent attention in the transition states, TSs) and the rearrangement of bonds during the degenerated Cope rearrangement (DCR) of semibullvalene (SBV). In addition, these authors9 have carried Received: July 18, 2012 out time-dependent quantum simulations in order to analyze Accepted: August 20, 2012 the electronic fluxes as the reaction proceeds, and their results

Figure 1. Energy proﬁles for the DCR of SBV calculated by means of the IRC method. Below the graph, a schematic representation of the reaction mechanism for each ﬁgure is depicted from the perspective of the ELF analysis (full lines and ellipses represent disynaptic and monosynaptic basins, respectively; dotted lines indicate a large basin population).

− the geometrical parameters,12,15 21 the electronic mechanism the reactants as the reaction takes place, providing a nice guide concerning DCR of SBV is still not well understood.22 to elucidate the mechanism of chemical reactions and further One step further is based on the idea that it is reasonable to understanding of the chemical reactivity. This methodology think that an adequate representation of these chemical events proposed by Krokidis et al. is known as bonding evolution should be given by a physical observable defined in coordinate theory (BET)30 (more details concerning theoretical aspects of space. The electron density, ρ(r), is the best choice because it is the Thom’s CT and BET are available in the Supporting a local function defined within the exact many body theory, and Information). Thus, questions such as how could the electronic it is also an experimentally accessible scalar field. Therefore, in reorganization proceed along the reaction path, is the electronic the deeper study of chemical reactivity, we want to identify how density flowing synchronously, in which direction, and do the electron density transfers occur as a function of reaction bond forming/breaking processes take place at the TS may be progress, which constitutes the motivation of the present work. answered. This combined method that we use herein has been − In doing so, we can provide a connection between the ρ(r) described in much detail previously32 41 distribution and the chemical reactivity. The importance of In particular, in the present work, we want to assess the ρ(r), as a fundamental property of an electronic system usefulness of this theoretical protocol by comparing with containing all information of physical relevance, is highlighted previous results obtained by means of selective laser pulses − by the Hohenberg−Kohn theorem.23 ρ(r) of a molecule experiments and time-dependent quantum simulations.7 9 contains information not only on the atomic structure and Complementarily, from the ELF topological and CT analysis, electronic properties but also on the nature of the chemical it is also possible to symbolize the electronic transfer in bonds that lead ultimately to chemical reactivity. Very recently, pericyclic reactions suggesting a graphical representation of Stalke24 has provided an introduction to the basics of ρ(r) curved arrows in Lewis structures as the reaction proceeds. It investigations from a theoretical point of view. will also be a challenge to extend the present investigations to Herein, we present an alternative representation of the complex reactions of other systems. electron density transfers during the DCR of SBV in the ground In order to analyze the electron density transfer, we have state in the domain of quantum chemical topology (QCT), a traced the intrinsic reaction coordinate (IRC)42,43 pathway subarea of quantum mechanics,25 in which different methods from reactant to product. B3LYP/6-311+G(2d,p) calculations based on the seminal work of Bader are included.26,27 In doing have been performed using the Gaussian 0944 code in order to so, we show how the DCR of SBV can be analyzed. To this end, localize the structures involved in the chemical rearrangement. we have developed the joint use of an electronic localization For each point obtained on the IRC pathway, we have carried function (ELF)28,29 and Thom's catastrophe theory (CT).30,31 out the topological analysis of the ELF field by means of the In this framework, the mechanism of chemical reactions can be TopMod package45 considering a cubical grid of step size rationalized in terms of chemical events (bond-forming or smaller than 0.1 bohr.46 The ELF function is a convenient tool -breaking processes, creation and annihilation of electron pairs) for the analysis of chemical bonding as it reveals regions in that drive the chemical rearrangement. This analysis allows us molecular space where the probability of finding an electron to understand the electronic structure and related properties of pair is high; thus, numerical values of the ELF are mapped on

2501 dx.doi.org/10.1021/jz300974v | J. Phys. Chem. Lett. 2012, 3, 2500−2505 The Journal of Physical Chemistry Letters Letter the interval (0,1) facilitating its analysis. The topological partition of the ELF gradient field29 yields basins of attractors that can be thought of as corresponding to atomic cores, bonds, and lone pairs. In molecules, two types of basins are found, (i) core basins surrounding nuclei and labeled C(A) (where A is the atomic symbol of the element) and (ii) valence basins that are characterized by the number of core basins with which they share a boundary. This number is called the synaptic order.47 Thus, there are monosynaptic, disynaptic, trisynaptic basins, and so on. Monosynaptic basins, labeled V(A), correspond to the lone pairs of the Lewis model, and polysynaptic basins correspond to the shared pairs of the Lewis model. In particular, disynaptic basins, labeled V(A,X), correspond to two-center bonds, trisynaptic basins, labeled V(A,X,Y), to three-center bonds, and so on. The valence shell of a molecule is the union of its valence basins. As hydrogen nuclei are located within the valence shell, they are counted as a formal core in the synaptic order because hydrogen atoms have a valence shell. Therefore, they are called protonated disynaptic. In addition, the electronic population of the bonding basin, Figure 2. Snapshots of the ELF localization domains for (a) SBV, (b) obtained by integration of the electronic density, defines the turning point-I (TP-I), and (c) the TS. Color code: blue for core basins, red for monosynaptic basins, white for protonated disynaptic number of electrons shared in a bond. Accordingly, by using the basins, and green for disynaptic basins. molecular structure defined through the topology of ELF, the knowledge of the electronic density transfers is a valuable and C6C7 bonds are not reflected by a pair of disynaptic completion of the structural evolution, which describes the basins Vi=1,2(C3,C4) and Vi=1,2(C6,C7), as sometimes ob- change in structure of a system, that is, the connectivity among served,49 but only single V(C3,C4) and V(C6,C7) disynaptic atoms, along a chemical reaction, and serves as a basis for a basins were found. The V(C3,C4) and V(C6,C7) disynaptic better understanding of such process, as well as to undertake a basins integrate the electronic charge to 3.42 e. meaningful assessment of the physical origins of potential A quantitative analysis is further achieved by integrating the energy barriers. electron density over the volume of the basin yielding basin According to the theory of dynamics systems, it can be populations in order to understand how the electron density considered structurally stable if a small perturbation is only transfers are proceeding during the chemical rearrangement. fi possible for values of the control parameters in well-de ned Figure 3 shows the evolution of the basin population along the ranges, namely, structural stability domains (SSDs), where all of the critical points are hyperbolic and separated by catastrophic points at which at least one critical point is nonhyperbolic. Along the reaction pathway, the chemical system goes from a given SSD to another by means of bifurcation catastrophes occurring at the turning points (TPs). The bifurcation catastrophes occurring at these TPs are identified according to Thom’s classification.31 The energy profile along the reaction coordinate is reported in Figure 1 together with the SSDs representing the different ELF topologies along the reaction coordinate. The electronic rearrangement reveals five different SSDs, which can be viewed as a sequence of chemical events. The SSDs are separated by respective TPs derived from CT that are responsible for the topological changes of the system. For each point obtained on the IRC pathway, we have evaluated the basin populations of some specific attractors with the aim of following their ff respective evolutions in the di erent SSDs. Our calculations Figure 3. Basin populations along the IRC path of the DCR. Basin predict an energy barrier of 4.45 kcal/mol, in good agreement population in electrons. with previous studies carried out by Bader and co-workers48 at the B3PW91 level. The ELF topology of SBV presents eight core basins of IRC pathway of the five SDDs, while the basin populations are carbon C(Ci=1,8) (blue button), which characterize the electron summarized in Table1. The integration of the electronic charge density of core regions (see Figure 2a) with basin populations over the ELF basins along SSD-I reveals the following (see of 2.09 e. In addition, the formal single C−C bonds of SBV are Figure 3): (1) A pronounced decrease in the V(C2,C8) basin represented by only single disynaptic basins V(C,C) (green population is observed (0.29 e). (2) A very small increment of button) between the respective core carbon basins C(C). The the V(C2,C3) basin population takes place (0.09 e) while the V(C2,C3) and V(C7,C8) disynaptic basin populations were V(C3,C4) basin population decreases (0.05 e). (3) The calculated to be to 2.14 e, slightly higher than that expected for V(C1,C5) basin population practically remains constant, formal single bonds. Conversely, the formal double C3C4 indicating that electronic charge is not mainly transferred

2502 dx.doi.org/10.1021/jz300974v | J. Phys. Chem. Lett. 2012, 3, 2500−2505 The Journal of Physical Chemistry Letters Letter

Table 1. Integrated Electron Populations of the ELF Basins in Diﬀerent Structural Stability Domains (SSDs) of the DCR of SBV a Calculated for the Initial and Final Points of Each SSD

SDD SSD-I SSD-I SSD-II SSD-III species semibullvalene TS Rx reactive −3.06b −2.29c −2.15b −1.34c −1.21b 0.00 d(C2−C8) 1.610 1.614 1.735 1.759 1.902 1.926 2.120 V(C1,C5) 1.90 1.90 1.90 1.89 1.88 1.89 1.89 V(C1,C2) 1.83 1.83 1.89 1.91 2.00 2.01 2.06 V(C2,C3) 2.14 2.15 2.24 2.26 2.47 2.83 3.02 V(C3,C4) 3.42 3.42 3.37 3.35 3.24 3.22 2.93 V(C2,C8) 1.61 1.59 1.32 V(C4,C5) 1.97 1.97 1.98 1.99 2.00 2.01 2.06 V(C2) 0.63 0.38 V(C8) 0.63 0.38 ΔE 1.64 2.13 0.68d a Δ Δ − Δ 1/2 Increments of energy within each SSD in kcal/mol ( E = Elast Einitial); the reaction coordinate (Rx) is in amu bohr, and the distances are in angstroms. bInitial point of the SSD. cLast point of the SSD. dCalculated energy between the first point of the SSD-III and the TS. toward the C1−C5 region. Additionally, the V(C1,C2) basin Figure 2c). While the change of the energy between the TP-II population increases slightly (0.06 e) as a consequence of the and TS is calculated to be 0.68 kcal/mol, the SDD-III is found internal electronic redistribution. Note that the SSD-I entails an to be the longest SDD along the chemical process. energetic cost of 1.64 kcal/mol and eight points along the IRC Subsequently, after reaching the TS, the basin population pathway. The first turning point (TP-I) indicates a cusp (C†) analysis of the SSD-IV and SSD-V can be interpreted as SSD-II type catastrophe (connecting SSD-I and SSD-II). It takes place and SDD-I, respectively. at −2.15 amu1/2 bohr and d(C2,C8) = 1.759 Å. The valence Using the notation previously defined by Berski et al.36 (see disynaptic basin V(C2,C8) is transformed into two mono- more details in Supporting Information), the sequence of TPs † TS † synaptic basins, V(C2) and V(C8) (see Figure 2b), which have can be represented as 5-C [F]2 [F ]2C-0. According to the been localized in the region of C2 and C8 carbon atoms, above findings, the reaction mechanism can be illustrated as it is respectively. depicted in Scheme 2, in which the curly arrows stand for At the beginning of the SSD-II, both V(C2) and V(C8) monosynaptic basin populations are calculated to be 0.63 e and Scheme 2 electron density is highly delocalized. From a strictly topological point of view, one may state that the C2−C8 bond has been broken. Along SSD-II, it is possible to observe a continuous increase of the V(C2,C3) basin population while V(C2) and V(C8) basins populations decrease progressively until their annihilation. This fact indicates that the electron density from V(C2,C8) basin and the respective V(C2) and V(C8) monosynaptic basins along SSD-I and SSD-II, respectively, may be transferred toward the C2−C3 region and its symmetric analogue (C7−C8 region). The gradual decrease of the V(C3,C4) basin population electron density transfers accompanying the breaking of along both SSD-I and SSD-II is even more pronounced along chemical bonds and the forming of new chemical bonds or SSD-III, pointing out that the electron density is principally the rearrangements of electron pairs, together with associated transferred toward the C4 region (and its symmetric analogue transitions from single to double bonds or vice versa. From this the V(C7,C6) basin, toward the C6 region.). It is worth noting type of analysis, we can conclude that this reaction can be that SSD-II is the most energetic SDD that entails an energetic dissected in three consecutive stages; the first stage yields to the cost of 2.13 kcal/mol and seven points along the IRC pathway. C2−C8 bond-breaking process with no significant electronic Subsequently, the TP-II is found in the region between SSD-II rearrangement. The second part of the reaction path can be and SSD-III at −1.21 amu1/2 bohr and d(C2−C8) = 1.926 Å. viewed as a reorganization of the valence molecular shells with fi Two simultaneous fold ([F]2) catastrophes are identi ed in the concomitant reorganization of C3−C4 and C6−C7 bonds, region of C2 and C8 atoms. Both the V(C2) and V(C8) from double to single, while an opposite behavior appears in attractors are annihilated. The nonbonding V(C2) and V(C8) the C2−C3 and C7−C8 bonds. In the third stage, the C4−C8 monosynaptic basins are not observed, and from a chemical bond-forming process takes place. This result shows that the point of view, there are no signs of the former C2−C8. The reaction DCR of SBV corresponds to asynchronous electron electron density is distributed on the ring formed by the cyclic density rearrangement. carbon skeleton C1−C2−C3−C4−C5-C6−C7−C8. Note that Although most current research is focused on the accurate the V(C1,C5) basin population does not undergo significant prediction of energy barriers and reaction energies of chemical changes even at the TS, while the V(C1,C2) basin population reactions, an understanding of their origin based on the reaches its maximum value at this point; see Table 1 and Figure electron density is desirable. This approximation can adequately 3. Interestingly, the breaking/forming processes of C2−C8 and describe the thermal DCR of SBV, showing the order, C4−C6 bonds are not taking place at the TS structure (see direction, and asynchronicity of the electron flux to be in

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M.; Fernandes, E.; Heubes, M.; Quast, H. The The authors declare no competing financial interest. Effects of Substituents on the Degenerate Cope Rearrangement of Semibullvalenes and Barbaralanes. Eur. J. Org. Chem. 1998, 1998, ■ ACKNOWLEDGMENTS 2209−2227. (18) Williams, R. V. Semibullvalenes and Related Molecules: Ever This work is supported by Generalitat Valenciana for Closer Approaches to Neutral Homoaromaticity. Eur. J. Org. Chem. Prometeo/2009/053 project, Spanish Ministry Ministerio de − ́ 2001, 2001, 227 235. Economia y Competitividad for Project CTQ2009-14541-C02, (19) Hrovat, D. A.; Williams, R. V.; Goren, A. C.; Borden, W. T. as well as to Fundacioń Bancaixa-Universitat Jaume I (UJI) for B3LYP Calculations on Bishomoaromaticity in Substituted Semi- financial support during S.B.'s research stay at UJI. P.G.-N. bullvalenes. J. Comput. Chem. 2001, 22, 1565−1573. gratefully acknowledges the Postdoctoral grant provided by (20) Brown, E. C.; Henze, D. K.; Borden, W. T. 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