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Aromaticity on the Fly: Cyclic Transition State Stabilization at Finite Temperature Tama´s Rozgonyi,† Albert Barto´k-Pa´rtay,‡ and Andra´s Stirling*,† Chemical Research Center of the Hungarian Academy of Sciences, Pusztaszeri ut 59-67, Budapest, 1025 Hungary, and CaVendish Laboratory, UniVersity of Cambridge, 19 JJ Thomson AVenue, Cambridge CB3 0HE, United Kingdom ReceiVed: October 20, 2009
We study the transition state of pericyclic reactions at elevated temperature with unbiased ab initio molecular dynamics. We find that the transition state of the intramolecular rearrangements for barbaralane and bullvalene remains aromatic at high temperature despite the significant thermal atomic motions. Structural, magnetic, and electronic properties of the dynamical transition state show the concertedness and aromatic character. Free-energy calculations also support the validity of the transition state theory for the present rearrangement reactions. The calculations demonstrate that cyclic delocalization represents a strong force to synchronize the thermal atomic motions even at high temperatures. SCHEME 1a
Introduction The characteristic features of cyclic, concerted reactions can be very efficiently explained by invoking the concept of aromaticity.1-4 The aromaticity of the transition state (TS) in thermally allowed pericyclic reactions (the so-called DewarEvans-Zimmermann concept) can be derived from the Woodward-Hoffmann rules.1,5-7 The aromaticity is not a physical observable; therefore, it has no exact definition. Various criteria have been developed and applied for aromaticity,8 including structural,9 energetic,10,11 magnetic,4 or electronic12 measures. Applying the aromaticity concept in the theoretical interpretation of a pericyclic TS is usually done by inspecting the electronic structure of the TS identified on the underlying potential energy surface (PES). This implies 0 K for the reaction course. However, it has already been shown that the shape of the PES alone may not be sufficient to determine the actual reaction mechanism because the PES only partially captures the effects of thermal motions.13,14 For example, nonpericyclic reactions predicted to be concerted from the PES topology have been proven to be stepwise at finite temperature.15 Clearly, elevated temperature introduces noises to the concerted motion of the atoms and destroys local symmetries, and the thermal fluctuations may prevent the formation of the aromatic state. It is therefore an important question of whether finite temperature alters concerted mechanisms featuring aromatic TS. Alternative routes can be radical mechanisms or charge separation along the reaction coordinate. Indeed, similar questions have already been addressed concerning the PES of various molecules predisposed for intramolecular rearrangements, such as retro Diels-Alder reactions.16 In this computational study, we show representative examples for pericyclic reactions where aromaticity remains the organizing force even at high temperature and the ground-state reaction paths continue to proceed through aromatic configurations. For our study, we have selected barbaralane (1) and bullvalene (2) as their low activation barrier on the intramolecular rearrangement path allows the efficient sampling of the corresponding TS-s at elevated temperature without other, competing * Corresponding author. † Chemical Research Center of the Hungarian Academy of Sciences. ‡ University of Cambridge.
a (A) Barbaralane (1) rearrangement. The breaking bonds are labelled by b1 and b2. (B) bullvalene (2) rearrangement.
reactions occurring on the simulation time scale (Scheme 1).17 1 and 2 are fluxional molecules; that is, they easily rearrange themselves into chemically identical but configurationally different structures.18 Their fascinating intramolecular transformations are Cope rearrangements and have attracted wide interest on both experimental19-21 and theoretical21-23 fronts. The TS identified on the theoretical PES is of aromatic nature in both cases,19-23 but the effects of thermally induced deviations from the 0 K reaction path on the aromaticity of their TS have not yet been studied. Computational Section For the calculations, we used the CP2K24 and the Gaussian 0325 program suites. We used the BLYP exchange-correlation functional26 for both electronic structure methods. We have checked that the two quantum chemical programs (CP2K and Gaussian 03) give essentially the same results. It is known that the inclusion of exact exchange into the functional would yield more accurate activation energies, but it also increases the CPU time of the MD simulations with orders of magnitude.27 The better performance of the hybrid functionals for aromatic states is due to the improved treatment of the self interaction in the exchange-correlation terms compared with the local or semilocal functionals.28 The extent of the bias for delocalized states shows quite large variation in the set of the most frequently used functionals.29 Stepwise intramolecular transformations of 1 and 2 also feature delocalized states. Numerical tests have showed
10.1021/jp910042r 2010 American Chemical Society Published on Web 12/11/2009
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that the tendency of BLYP to stabilize delocalized states are of the same magnitude or higher for other electronic states (e.g., radical) for both 1 and 2.30 Therefore, we expect that the reaction barriers are underestimated for all possible transformation routes, and thus the BLYP functional is suitable to identify the reaction mechanism in the present study, despite its well-known tendency to stabilize delocalized structures. We carried out the ab initio Born-Oppenheimer molecular dynamics (MD) calculations by using the CP2K-Quickstep software. We have employed the double-ζ basis sets augmented with a set of polarization functions in conjunction with the Goedecker-Teter-Hutter31 pseudopotentials. In Quickstep, the electronic charge density is expanded by an auxiliary planewave basis set. The cutoff used to define the larger grid for the expansion was 300 Ry. We have employed a cubic nonperiodic box of 11 Å, and the Martyna-Tuckerman Poisson’s equation solver32 has been used. We have carried out the simulations by employing canonical (NVT) conditions. To ensure proper thermal equilibrium, we used a separate Nose´-Hoover thermostat for all degrees of freedom. Note that canonical simulations yield Helmholtz free energy, whereas available experimental data provide Gibbs free energies. Because the reactions studied here are intramolecular rearrangements, the two energy values are very close. For barbaralane, we used a 0.5 fs time step to integrate out the Verlet equations. For bullvalene, we used a larger, 0.8 fs time step for the 800 K simulations and 0.4 fs timesteps for the simulations at 1000, 1200, 1500, and 1800 K. We have substituted the hydrogen atoms with deuterium in all simulations. Separate NVE calculations with the selected time steps showed that they are sufficiently small to integrate the equations of motions correctly, and the energy drifts are within 0.5 K/atom/ps. The equilibration periods were 4 ps under the selected NVT conditions for every simulations. The Gaussian 03 program package has been used for calculating the intrinsic reaction coordinate (IRC) paths, the NICS values along the IRC paths, and for the dynamical trajectories. The structure calculations have been performed at the BLYP/6-31+G* level. The NICS values have been obtained from single-point GIAOBLYP/6-31+G* calculations. Results and Discussion We have performed ab initio molecular dynamics simulations in a wide temperature range (600-1250 and 800-1800 K for 1 and 2, respectively). We have followed the simulations up to 80 ps at each selected temperature.33 The simulations yielded trajectories of successive intramolecular rearrangements.34 The trajectories were stable against spin polarization, indicating nonradical mechanism.35 In this study, we make use of the fact that elevated temperature boosts chemical reactions, albeit we do not have any control on which particular reaction paths will be active. There are various methods for biasing a particular set of degrees of freedom and observing chemical reactions as a function of these variables (see, e.g., ref 36 and references therein). In the present case, however, we preferred to observe the chemical reactions in a completely unbiased manner; therefore, we chose the finite temperature technics. The reaction course in terms of structural parameters during an arbitrarily chosen interval of 6 ps is shown for 1 in Figure 1. Along the trajectories, we can follow the rearrangements by introducing a reaction coordinate. A simple definition is based on a cutoff distance (d0). This defines the reactant and transition states: if the lengths (d1, d2) of bonds b1 and b2 are larger than d0, then the configuration is a TS, otherwise a reactant state.37 This definition still permits large structural variations in both
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Figure 1. Structural evolution of 1 at 1000 K during 6 ps. (A) Top: Variations of the CC bond lengths during the simulations. Red and blue curves correspond to b1 and b2. (See Scheme 1.) Bottom: Variation of the reaction coordinate defined in the text (rounded window averages over an interval of 0.25 ps). Values 1 and 1′ represent the reactant states. TS corresponds to the transition state. (B) Detailed view of the bond evolutions within a picosecond interval.
states. However, as Figure 1 shows, the atomic motions remain correlated along the trajectory. Clearly, there is no 1 f TS f 1′ or 1′ f TS f 1 transition in unconcerted manner; that is, the CC breaking and formation always occur simultaneously at successful transitions. Sampling the trajectories has showed that this is true throughout all simulations at any selected temperature. In addition, we have found that the only chemical reactions taking place during the simulations are the 1 h 1′ (2 h 2′ for bullvalene) rearrangements. Other chemical transformations do not take place,30 although the fluctuations of the CC bond lengths can be quite large, as seen in Figure 1. To reveal aromaticity along the rearrangement paths, we have chosen a magnetic criterion, the nucleus independent chemical shift (NICS) value.4,38 The NICS value corresponds to the negative of the magnetic shielding computed at a given point. Large negative values indicate the presence of induced diatropic ring currents, that is, aromaticity. We have calculated this index at a single point below the center of the ring formed by the carbon atoms participating in the Cope reaction.39 On the PES (0 K), the NICS is -17.5 and -16.8 ppm at the TS, whereas it is -7.8 and -2.8 ppm at the reactant state for 1 and 2, respectively. In Figure 2, we show the temporal variation of this quantity for 1 along with the evolution of the |d1 - d2| difference. We have selected |d1 - d2| because the correlated motion of b1 and b2 allows the use of a single variable as reaction coordinate. The variation of the NICS nicely correlates with the variation of |d1 - d2|. The absolute value of the NICS dramatically increases around the TS (|d1 - d2| ≈ 0) and thus clearly indicates the presence of aromaticity in the TS region.40 In particular, we see in Figure 2 that each attempt to cross the TS region is accompanied by a sharp peak in the NICS variation.
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Figure 4. Correlated motion of seven Wannier centers representing the breaking and forming bonds of 1 at the TS at 1000 K. Blue and red balls indicate the positions of the Wannier centers (A) at the beginning and (C) at the end of the 10 fs period. (B) Superposition of 21 frames spanning 10 fs. Light-blue tubes indicate the superpositioned frameworks. The number of WFCs around a bond indicates its bond order. For simplicity, we omitted the H atoms and the other WFCs. The Figure was made using the VMD program.43
Figure 2. Evolution of the |d1 - d2| reaction coordinate and the NICS value representative of the aromaticity of the six participating carbon atoms of 1 within a short time period of 2 ps. The NICS axis is reversed for better visualization.
Figure 5. Activation free energy (∆F) versus T. The lines fitted to the points represent the ∆F ) ∆U - T∆S equation. The error bars indicate the standard deviations of the calculated values.44
Figure 3. NICS versus |d1 - d2| collected from a 6 ps interval of a simulation for 1 at 1000 K.
The strong correlation between the variation of the aromaticity and the |d1 - d2| values is shown in Figure 3. Despite the thermal fluctuations, the diagonal localization of the signals at Figure 3 represents a clear trend: approaching the TS the system increases its aromatic character. The formation of the aromatic ring is therefore an essential motif of the TS. This is indeed remarkable because the six atoms participating in the transformation are never in the same plane during the rearrangement, and the thermal fluctuations may interfere with concerted motions. The trends emerging from Figures 2 and 3 show that the energy gain provided by the cyclic delocalization of six electrons plays an essential role in properly phasing the motion of the reacting six C atoms, even at elevated temperature. We note that the same conclusion can be drawn from the results of the simulations at all other selected temperatures for both 1 and 2. We present additional support for concertedness from an electronic point of view: the correlated motion of the Wannier function centers41 (WFCs) of the participating valence electrons spectacularly traces out the pattern of an electrocyclic reaction.
WFCs are the centers of the density of the Wannier functions. The centers are derived from the maximally localized Wannier functions, which are analogous to the localized molecular orbitals and derived from the set of the occupied periodic Kohn-Sham orbitals.41 They are very useful for interpreting electronic change induced by structural or electronic effects. For instance, WFCs have been used to analyze Diels-Alder reactions.42 In Figure 4, we show a short 1000 K trajectory of seven WFCs localized around the six reacting C atoms of 1. The snapshots cover a 10 fs interval at the TS when a large electron redistribution can be observed. Three WFCs represent the electrons participating in the concerted bond rupture and formation processes. The other four correspond to the flexible, but stable σ frame of the six C atoms. The pattern drawn by the moving WFCs is a strong indication of the concerted nature of the structural and electronic reorganization at the TS. Clearly, the three electron pairs move in a strictly concerted fashion. Their movement defines a rotation direction that is determined by the bond length fluctuations at the beginning of the rotation. The circular motion induced by the asymmetry of the atomic movements indicates the inherent asynchronicity of the pericyclic reaction at finite temperature. Extensive sampling of the trajectories of both 1 and 2 shows that this pattern always accompanies the rearrangements at the TS regions. Free-energy calculations also support our observation that aromaticity remains the governing force at the TS, even at high temperature. Figure 5 displays the activation free energy of 1 and 2 as a function of T. The free-energy barriers have been calculated by sampling the trajectories. In general, the free energy as a function of a given reaction coordinate (s) can be given (up to an additive constant) by eq 1
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F(s) ) -kBT ln P(s)
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(1)
where P(s) is the probability of finding the system in a configuration with the reaction coordinate s, kB is the Boltzmann constant, and T is the temperature. In the present study, s is discretized and represents only reactant (R) and transition (T) states.44 From eq 1, it follows that the activation energy is calculated as ∆F ) F(T) - F(R) ) -kBT ln((P(T))/(P(R))). As Figure 5 shows, in both cases, the ∆F values vary linearly with T. The extrapolation to 0 K gives 2.6 and 6.3 kcal/mol activation energies for 1 and 2, respectively, which is in good agreement with the values obtained from static DFT calculations: 3.4 and 8.7 kcal/mol (experimental values:45,46 7.6 and 13.8 kcal/ mol, respectively). The deviations can be attributed to the statistical errors of the simulations. For both cases, the linear dependence of ∆F versus T indicates the validity of the ∆F ) ∆U - T∆S equation; that is, the mechanism does not change in the given temperature range. The concerted mechanism featuring aromatic TS at 0 K remains valid at higher temperatures, albeit with more apparent entropy contributions to the free energy. Conclusions In summary, we have shown in several ways that during electrocyclic rearrangements of 1 and 2, the aromatic character of their TS is conserved, even at high temperature. Despite the random thermal fluctuations of the various internal coordinates, the aromaticity represents a governing force strong enough to lead the reactions through the concerted reaction channels. We note that 1 and 2 are molecules featuring structures that are well suited for concerted rearrangements. However, our results strongly suggest that many other concerted reactions with aromatic TS character predicted from their electronic structure at 0 K may also preserve their TS aromaticity at elevated temperature. Indeed, in all pericyclic reactions, the bond breaking and formation processes take place in close vicinity; therefore, the structural conditions are easily fulfilled for the aromatic ring formations. The current finding significantly extends the validity of the concept of aromaticity and indicates that cyclic delocalization on dynamical pathways can resist extreme conditions. Acknowledgment. We thank T. A. Rokob and A. Laio for fruitful discussions. This work has been supported by OTKA grant K68360. Supporting Information Available: Details of the calculations on the structures at the potential energy minima of 1 and 2: comparison of theory and experiment, TDDFT, IRC, and NICS results. Evolution of all CC bonds within 6 ps at 1000 K for 1. NICS on the dynamical pathways for 2. Details of the selection of the reaction coordinates and the error estimation for the free energy calculations. This information is available free of charge via the Internet at http://pubs.acs.org/. References and Notes (1) Zimmermann, H. E. Acc. Chem. Res. 1971, 4, 272–280. (2) Wiest, O.; Montiel, D. C.; Houk, K. N. J. Phys. Chem. A 1997, 101, 8378–8388. (3) Jiao, H.; von Rague´ Schleyer, P. J. Phys. Org. Chem. 1998, 11, 655–662. (4) Chen, Z.; Wannere, C. S.; Corminboeuf, C.; Puchta, R.; von Rague´ Schleyer, P. Chem. ReV. 2005, 105, 3842–3888. (5) Colin Day, A. J. Am. Chem. Soc. 1975, 97, 2431–2438. (6) Woodward, R. B.; Hoffmann, R. The ConserVation of Orbital Symmetry; Verlag Chemie: Weinheim, Germany, 1970. (7) Dewar, M. J. S. Angew. Chem., Int. Ed. Engl. 1971, 11, 761–776.
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G.; Dappich, S.; Daniels, A. D.; Strain, M. C.; Farkas, O.; Malick, D. K.; Rabuck, A. D.; Raghavachari, K.; Forseman, 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.; Nanykkara, A.; Callacombe, M.; Gill, P. M. W.; Johnson, B.; Chen, W.; Wong, M. W.; Gonzalez, C.; Pople, J. A. Gaussian 03; Gaussian, Inc.: Wallingford, CT, 2004. (26) (a) Becke, A. Phys. ReV. A 1988, 38, 3098–3100. (b) Lee, C.; Yang, W.; Parr, R. Phys. ReV. B 1988, 37, 785–789. (27) For a comparison of theory and experiment, see the Supporting Information. (28) Parr, R. G.; Yang, W. Density-Functional Theory of Atoms and Molecules, Oxford University Press: New York1989. (29) (a) Guner, V.; Khuong, K. S.; Leach, A. G.; Lee, P. S.; Bartberger, M. D.; Houk, K. N. J. Phys. Chem. A 2003, 107, 11445–11459. (b) Pieniazek, S. N.; Clemente, F. R.; Houk, K. N. Angew. Chem., Int. Ed. 2008, 47, 7746–7749. (c) Johnson, E. R.; Mori-Sa´nchez, P.; Cohen, A. J.; Yang, W. J. Chem. Phys. 2008, 129, 204112. (30) For further details, see the Supporting Information. (31) Goedecker, S.; Teter, M.; Hutter, J. Phys. ReV. B 1996, 54, 1703–1710. (32) Martyna, G. J.; Tuckermann, M. E. J. Chem. Phys. 1999, 110, 2810. (33) For 1, we have selected four temperatures from the 600-1250 K range, whereas for 2, five values have been chosen from the 800-1800 K temperature range to calculate the temperature dependence of the activation free energy. The minimum size of the intervals is 200 K, which is sufficiently large to observe the effect of the temperature. (34) Depending on the system and temperature, we have observed 6251227 reactant state-TS transitions along the trajectories. (35) TDDFT calculations also showed that the first excited states for both 1 and 2 are >50 kcal/mol higher than the ground state at the TS at 0 K. For further details, see the Supporting Information. (36) Free Energy Calculations: Theory and Applications in Chemisty and Biology; Chipot, Ch., Pohorille, A., Eds.; Springer Series in Chemical Physics 86; Springer: Berlin, 2007.
Cyclic TS Stabilization at Finite Temperature (37) For 1, d0 ) 1.75 Å. The selected cutoff distance is obtained from the inflexion point of the IRC curve of the rearrangement paths. For the details, see the Supporting Information. (38) von Rague´ Schleyer, P.; Maerker, C.; Dransfeld, A.; Jiao, H.; van Eikema Hommes, N. J. R. J. Am. Chem. Soc. 1996, 118, 6317–6318. (39) We have calculated the NICS in the point obtained by shifting the center of mass of the six carbon atoms participating in the Cope reactions perpendicularly to their least-square plane by 0.5 Å outward from both molecules 1 and 2. 0 K calculations showed that the variation of the NICS in this point nicely reflects the formation of the TS aromaticity. See the Supporting Information for further details. (40) The same behavior has been obtained for other temperatures and for bullvalene as well. See the Supporting Information for further details.
J. Phys. Chem. A, Vol. 114, No. 2, 2010 1211 (41) (a) Marzari, N.; Vanderbilt, D. Phys. ReV. B 1997, 56, 12847– 12865. (b) Berghold, G.; Mundy, C. J.; Romero, A. H.; Hutter, J.; Parrinello, M. Phys. ReV. B 2000, 61, 10040–10048. (42) Aktah, D.; Passerone, D.; Parrinello, M. J. Phys. Chem. A 2004, 108, 848–854. (43) Humprey, W.; Dalke, A.; Schulten, K. J. Mol. Graphics 1996, 14, 33–38. (44) See the Supporting Information for further details. (45) Jackman, L. M.; Fernandes, E.; Heubes, M.; Quast, H. Eur. J. Org. Chem. 1998, 22099–2227. (46) Moreno, P. O.; Suarez, C.; Tafazzoli, M.; True, N. S.; LeMaster, C. B. J. Phys. Chem. 1992, 96, 10206–10212.
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