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Reply to the “Comment on ‘The Concept of Protobranching and Its Many Paradigm Shifting Implications for Energy Evaluations’” Paul von Rague´ Schleyer* and W. Chad McKee Center for Computational Chemistry and Department of Chemistry, UniVersity of Georgia, Athens, Georgia, 30602 ReceiVed: October 15, 2009; ReVised Manuscript ReceiVed: January 20, 2010 Baeyer’s seminal 1885 “strain” theory of cycloalkanes considered ethylene (“dimethylene”) as the “simplest methylene ring” and was based on the deviations of regular polygon angles from the 109.47° tetrahedral value.1 Although none of the qualitative trends predicted by this model agreed with the actual energies of hydrocarbon rings, when they were determined decades later, “strain” became an guiding principle of organic chemistry and a cornerstone of conformational analysis.2 But strain energies cannot be measured directly; they are relative, rather than absolute quantities that must be defined and assessed by comparison with reference species considered by convention to be “strain free”. But the selection of such standards is arbitrary, and serious differences of opinion exist concerning the best choices of references and the justification for selecting one method of evaluation over another. Fishtik’s3 objections to Wodrich et al.’s protobranching paper4 illustrates such a contentious example, which focuses on the merits and shortcomings of two divergent estimates of the strain energy of cyclopropane (eqs 1 and 2).5 These estimates already have been debated for many years. Neither equation is ideal; in fact, a “perfect” equation does not exist. Equation 1 has historical precedence but the larger strain energy derived by eq 25 (or equivalent procedures employing group increments6 or eqs 1-15 in Fishtik’s paper)3 give the “conventional”7,8 cyclopropane strain energy most commonly reported in textbooks and the literature. Pauling provided the very first estimate of the “instability” (strain energy) of “trimethylene” in 1932.9 He employed a simple bond energy (BE) scheme based on methane, which assumed that “the H:C bond energy [100.1 kcal/mol] is the same for the higher members of the series as in methane” (the modern methane CH BE is 99.38 kcal/mol).5 On this basis, Pauling computed a “C:C bond energy” [83.0 kcal/mol] from the imprecise thermochemical data available at that time (compare the modern 78.98 kcal/mol value from ethane).5 Pauling then noted, “Similar calculations for saturated cyclic hydrocarbons show that a three-membered ring is unstable to the extent of over 1 eV.” The 24.4 kcal/mol strain energy he gave is revised to 19.49 kcal/mol, when modern experimental data5 are employed. This value, favored by Wodrich et al.,4 is identical to the isodesmic bond separation energy of Pople et al.10 (eq 1), since it also compares the CC and CH bonds of cyclopropane with those of ethane and methane energetically.
3C2H6 - 3CH4 f c-C3H6
+19.49 kcal/mol
(1)
3C3H8 - 3C2H6 f c-C3H6
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+27.56 kcal/mol
(2) 2C2H6 - CH4 f C3H8
-2.69 kcal/mol
(3)
However, such simple bond energy and isodesmic schemes do not differentiate among secondary, primary, and methane C-H bonds, which have different BE’s, or consider the variations in carbon environments and hybridization. Consequently, many more elaborate bond energy and group additivity treatments have been developed, which take such differences into account.11 These give results for the cyclopropane strain energy equivalent to those of the homodesmotic eq 2 of George et al.12 However, eq 2 also has many shortcomings. It balances the primary C-H bonds, but not the bond energies and hybridization of the CH2 groups, which are quite different in propane and cyclopropane.13-16 The torsional strain due to the eclipsed vicinal CHCH cyclopropane bonds is not taken into account. The longer range H · · · H and C · · · H and 1,3-C,C nonbonded interactions in each of the three propane reference molecules are not present in cyclopropane.4 These methylmethyl interactions, a through-space and through-bond energetic composite, can be evaluated by eq 3 (the energy of an alkane protobranch),4 which, when taken three times, is the difference between the energies of eqs 1 and 2. The large errors of HF and most DFT levels of theory in computing the energies of eq 3 and of alkane chain branching (e.g., eq 4)4,17 underscore the importance of longer range dispersion and middle range interaction effects.18
n-C4H10 f i-C4H10
-2.21 kcal/mol
(4)
The generalized precursor of eq 3 dates back to Allen (1959).19 The data in Pitzer and Catalano’s 1956 analysis of hydrocarbon energies20 showed that the negative energy of eq 3, as well as that of chain branching (eq 4), “arises principally from an electron correlation effect” (see ref 20). Dewar’s insightful σ conjugation paper (1984)16 noted the consequence of the difference expressed by eq 3, “isobutane should therefore be more stable than n-butane by 2 kcal/mol” (eq 4). The relationship between the stabilization due to branching (eq 4) and the propane advantage (eq 3) also was implicit in Allen’s analysis19 and in Pitzer’s data.20 The energy of eq 3 is comparable to that of eq 4, since the branched isobutane has three protobranches, while n-butane has only two.4 Wodrich et al., also noting these energetic and the structural relationships between the “kinks” in branched alkanes and those in n-alkanes (which are not branched topologically), coined the name “protobranching” [i.e., the branching precursor or prototype] and defined it as “the net stabilizing 1,3-alkyl-alkyl interactions existing in linear, branched, and most cycloalkanes, but not in methane and ethane.” The nature of protobranching “is the same as the well known branching effect. Propane is stabilized by the net attractive21 composite of carbon and hydrogen interactions not present in methane and in ethane [eq 1].” With regard to cyclopropane, they pointed out that “no protobranching 1,3-alkyl-alkyl interactions are present since all carbons are directly bonded to one another.” Fishtik’s Criticisms. Fishtik’s1 characterization of the concept of protobranching2 as a “regrettable mistake” is based on three assertions:
10.1021/jp909910f 2010 American Chemical Society Published on Web 02/15/2010
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(1) It is a mistake to assume “that there are no protobranches in cyclopropane”. (2) “It is fundamentally wrong to employ methane as a reference species” since it “is ruled out by any group additivity method for alkanes.” (3) That the similar strain estimates of cyclopropane given by Fishtik’s (group additivity) reaction schemes reflect the “good performance of the group additivity model”. (1) How To Count Protobranches. Fishtik’s criticism results from a misinterpretation and a misuse of the concept of “protobranching”, as it was defined, in words and by usage.4 The originators of a term have the privilege to establish its meaning.22 Since Fishtik uses the term incorrectly, his criticisms are baseless. A protobranch in an alkane is like the letter “V.” It has one vertex and one associated (distal; opposite) open edge, which cannot be shared with another protobranch (see the discussion for cyclobutane and bicyclo[1.1.1]pentane below). Interacting vertices are not counted as protobranches if they are also in a 1,2 relationship to one another (e.g., as in cyclopropane). In contrast, closed edges may be shared, as in the letters “N” or “Z” (both representing n-butane), or in the letters “M” or “W” (both representing n-pentane). Each of the two vertices in n-butane or the three vertices in n-pentane has its own distal open edge. The letter “Y” can represent branching (as in isobutane). A common point joins three vertices, which along with their three associated open edges, correspond to three protobranches. Neopentane is like a three-dimensional “X”. The four C-C “edges” meet at a common point (the central carbon) and the six open edges correspond to the six protobranches. Cyclopropane has three vertices but it has no open edges and therefore no protobranches. Neither does tetrahedrane. Cyclobutane has four vertices, but only two unique open edges, since they may not be shared in our definition. Hence, we assign only two protobranches to cyclobutane, only four bicyclo[1.1.1]pentane (in which the one open edge is shared by the three fourmembered ring faces), and 12 to cubane (two for each face). Five-membered and higher rings have the same number of distal open edges as vertices. Distal open edges assignable uniquely to vertices are the essence of protobranching. It is easy to see why this is so. “Many two-, three-, four-atom, and higher terms are needed for a complete dissection of all the individual interaction energies of a molecule even as simple as propane.”16 The higher and some of the four-atom terms are not present in methane, ethane, and especially cyclopropane. This imbalance was not considered by Fishtik.3 The energy of eq 3 and the difference between the energies of eqs 1 and 2 can be attributed to imbalances in these longer range interactions.4,17,16,20 But in the simplest sense a “protobranch” is merely a structural descriptor. “Protobranching” was coined to emphasize the close relationship between structural features of topologically branched alkanes and those of propane and the higher n-alkanes. The kinked geometries of the topologically “straight chain” alkanes resemble the bent units comprising their branched counterparts. Thus, the 1,3-methyl-methyl moiety in propane corresponds to the three analogous 1,3-methyl-methyl arrangements in isobutane and to the six in neopentane. The methyl groups in these moieties are bound to a common carbon, but not to one another by conventional bonds. There are no methyl groups in cyclopropane, which is unique among the monocyclic alkanes in that all the carbons are bound to one another. As a consequence, cyclopropane has no protobranches as defined in our concept.
Comments The difference between the heat of formation of ethane and propane is essentially the same as that between propane and n-butane (as are all further stepwise enlargements of the n-alkane series). This allows the simplifying assumption that differentiation between methyl and methylene groups is not necessary in the protobranching sense. Thus, n-butane can be considered to have two protobranches, n-pentane three, and so on. Even methine, HC(C)3, and C(C)4 groups can be considered formally in the same way. Hence, 2-methylbutane has four protobranches, 2,2-dimethylbutane seven, and cyclohexane six. The key to understanding is that a protobranch, as we have defined it, does not describe arrangements in which 1,3-disposed groups are bound to one another by conventional chemical bonds. Cyclopropane has no protobranches as we have defined the term. (2) Is It “Fundamentally Wrong To Employ Methane as a Reference Species in Evaluating the Strain Energy of Cycloalkanes”? Although this is exactly what Pauling did in the first evaluation of the strain energy of cyclopropane in 1932,9 Fishtik states, “methane is ruled out by any group additivity method for alkanes.” But there is no inherent superiority of group additivity methods, either in concept or performance. Cohen and Benson list 47 hydrocarbon group increments, as well as 39 corrections for use in different environments. Bond energy and other schemes can be adjusted to give results of the same quality with fewer parameters (see below). While the average bond dissociation energy of methane is the measurable C-H bond energy reference value (BE’s are different from bond dissociation energies), refined bond energy schemes for predicting hydrocarbon energies employ various derived CH bond energy values for different environments, primary, secondary, and tertiary alkyl, arenyl, vinyl, ethynyl, etc. Excluding methane from evaluations of hydrocarbon energies is like counting objects starting from the number two, rather than the number one. Fishtik’s reasoning forbids H2, as well as diatomic and almost all simple molecules from being employed in energy evaluations. In contradiction, methane, as well as molecules like NH3 and OH2, are the basis of Pople’s isodesmic bond separation energy (BSE) scheme,10 which has long been used not only for error reduction in theoretical energy evaluations but also for interpretive purposes. Pople’s BSE of cyclopropane was 19.6 kcal/mol (compare eq 1).5 George et al.12 gave the same value in their comparison of isodesmic ring strain energies with those employing homodesmotic schemes. Earlier uses of methane as a reference are too numerous to review. Allen’s,19 Cox’s,11 and Dewar’s16 papers are notable examples. Pauling’s use of methane to estimate the strain energy of cyclopropane9 is described above. Methane is not included in Benson’s and related group additivity schemes only because it is a unique entity, rather than being a part of a larger neutral hydrocarbon. However, as Benson pointed out, “The earliest attempts at an enthalpy additivity scheme were prompted by the recognition that if one listed the enthalpies of straight-chain saturated hydrocarbonssmethane, ethane, propane, butane, etc.sthe difference between two successive enthalpies was very nearly a constant value of approximately 5.0 kcal/mol.”6 Indeed, the methane enthalpy fits into Benson’s alkyl group increment series beautifully, when the necessary allowance is made for protobranching.4 A simple extension of the 1993 Cohen-Benson alkane group increments6 by adding methane gives (in kcal/mol) -17.83 C-(H)4,5 -10.00 C-(C)(H)3, -5.00 C-(C)2(H)2, -2.40 C-(C)3(H), and -0.10 C-(C)4. If the 7.83 kcal/mol difference between the first two values continued linearly along the series, C-(C)2(H)2 ) -2.17 (instead of -5.00
Comments
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kcal/mol) would be expected. The 2.83 kcal/mol deviation is the protobranching stabilization. Similarly, the linearly extrapolated C-(C)3(H) increment, +5.66, deviates from the -2.40 Benson value due to the stabilization of the three protobranches (as in i-butane) by 8.06 kcal/mol (2.69 per protobranch, nearly the same value). Finally, the C-(C)4 deviation on the same basis, 13.59 kcal/mol, gives only a modestly attenuated value, 2.27, for each of the six protobranches involved (as in neopentane). We also have shown how methane, along with only the methane-ethane enthalpy difference, protobranching corrections, and an attenuation term, can serve as the basis of an additivity scheme. This employs the smallest number of parameters (four) but still reproduces the heats of formation of conventionally unstrained alkanes accurately.23 Other unstrained hydrocarbons, alkenes, alkynes, alkylated cycloalkanes, and alkyl radicals are predicted equally well by the same method, when methane is replaced by the parent compound of each type as the base. A newly proposed method for evaluating hydrocarbon enthalpies of formation based on hydrogen atom counting (rather than group increments) gives a 24.3 kcal/mol cyclopropane strain energy.24 Pople’s Energies of Complete Hydrogenation (ECR),10 is another conceptually different method of evaluating data, having nothing to do with group increments. Methane is the key, as it is the only product formed from all hydrocarbons! For example, the strain energy of cyclopropane is evaluated by comparing the ECR of cyclopropane (eq 5), i.e., the energy required to reduce the three ring C-C bonds, with the energy of hydrogenolysis of ethane. This is taken three times to give the reference value for reducing three unstrained C-C bonds (eq 6). The difference between eqs 5 and 6, 19.49 kcal/mol, is identical with the isodesmic cyclopropane strain energy (eq 1). The protobranching effect can be included in the strain assessment by employing the heat of hydrogenolysis of only one of the C-C bonds of propane similarly, as in eq 7. The difference between eqs 5 and 7, 27.56 kcal/mol, is identical with the conventional cyclopropane strain energy (eq 2).
(CH2)3 + 3H2 f 3CH4
3[CH3CH3 + H2 f 2CH4]
∆H ) -66.23 kcal/mol, expt5 (5) ∆H ) -46.74 kcal/mol, expt5
(6) 3[CH3CH2CH3 + H2 f C2H6 + CH4] ∆H ) -38.67 kcal/mol, expt5
(7)
These evaluations of cyclopropane strain involving methane explicitly are conceptually very different from those based on group increments, but both give the same results. Other cycloalkanes can be evaluated similarly. It is not “fundamentally wrong to employ methane as a reference species in evaluating the strain energy of cycloalkanes.” (3) Do the Similar Strain Estimates of Eqs 1-15 Employed by Fishtik Prove “the Group Additivity Model Perform(s) Remarkably Well”? Fishtik’s 15 equations give essentially the same, conventional 27.7 kcal/mol strain energy of cyclopropane because they all reduce to the same basic homodesmotic comparison (eq 2, or the difference between eqs 5 and 7). As Fishtik stressed, this can be expressed as a simple Benson group enthalpy evaluation (eq 7), employing the -5.00 kcal/mol
C-(C)2(H)2 value6 and the cyclopropane heat of formation, 12.74 kcal/mol:
3C-(C)2(H)2 f cyclopropane
+27.74 kcal/mol
(8) As in eq 8, the three CH2 groups of cyclopropane in eq 2 are balanced by the three CH2’s on the right that remain after subtraction of the two methyl group increments of ethane from those of propane. All Fishtik’s equations, which employ “conventionally strain free” reference molecules, reduce to eqs 2 and 8. The insignificant differences in the evaluations and the trivial variations in the exact strain energies arise from slight variations (or errors) in the experimental data of the reference alkanes. The Benson CH2 group increment (C-(C)2(H)2)6 is an average value, but it differs little from alkane to alkane. The numerical success of group increment or other methods mentioned above in reproducing experimental enthalpies does not ensure their conceptual appropriateness for evaluating virtual quantities. The numerically similar strain estimates given by Fishtik’s reactions only reflect only their equivalence within the group additivity formalism, but not any inherent superiority over other divergent estimates. Many group-increment and bondadditivity schemes deduced from experimental data based on Boltzmann distributions of alkane conformers were shown in 1970 to be flawed for cyclic systems as they result in unexpected “strain energies” of 1.35 and 6.5 kcal/mol for the punitively “strain free” diamond-lattice hydrocarbons, cyclohexane and adamantane.25 Wodrich et al. argued that the commonly used group increments also hide contaminating effects like protobranching stabilization, which should be considered and compensated in more refined treatments of the subjective and vague concept of “strain.”4 Concluding Viewpoint: Strain Definitions. We emphasize that “strain” is a virtual concept, whose definition and quantitative evaluation are not absolute but depend on the context, the arbitrary choice of “strain-free” reference models, and methods of evaluation. Thus, by convention, ethane, ethene, ethyne, benzene, and graphite are all considered to be “strain-free”, despite their large energy differences. For example, the energy of benzene is 143.2 kcal/mol lower than that of three acetylenes even though the conventional strain energies of C2H2 and of C6H6 are both zero. Ideal “strain-free” reference models should resemble the target molecule as closely as possible in all essential aspects but must be free from all perturbing interactions (“contaminations”), both stabilizing and destabilizing, not present in the target. In short, the “strain-free” models should be just the same as the target molecule but should lack all strain features and unbalanced stabilizing interactions. Since this ideal is impossible to achieve in any experimental comparison, strain estimates, like those for cyclopropane, are arbitrary; several very different values may be valid. When strain energies are quoted, their basis should be made clear. Equation 1 gives the isodesmic strain energy of cyclopropane (19.5 kcal/mol), while homodesmotic eq 2 gives the “conventional” value (27.6 kcal/mol). Other evaluations also can be justified. Thus, even a “negative” strain energy of cyclopropane also can be deduced based on Baeyer’s original consideration of ethylene (“dimethylene”) “as the simplest methylene ring”,1 the well-known “olefinic character” of cyclopropane, and Walsh’s sp2 model of its bonding.13-16 This entirely different analysis also can employ Fishtik’s favored group increment approach. However, ethylene (or Benson’s olefin Cd(H)2 group increment)6 is chosen as the conventional strain-free standard for evaluation.
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Comments
Cyclopropane (trimethylene) is 6.07 kcal/mol more stable than ethene (dimethylene) on this basis (eq 9)!
3/2 C2H4 f c-C3H6
-6.07 kcal/mol
(9)
Much besides the ethene-like chemical behavior and reactivity of cyclopropane speaks in favor of the eq 8 method for its “strain” estimation, rather than eqs 1 and 2. Ethylene models the greater C(sp2)-H bond energies, the widened HCH bond angles, and the vicinal CH-CH eclipsing interactions of cyclopropane;14 alkanes do not. The “conventional” strain energy of ethylene is taken to be zero, even though alkenes are less stable than alkanes. But if we follow Baeyer in considering ethylene as a two-membered dimethylene ring (presciently anticipating Pauling’s “bent-bond” description)26 and as the origin of the (CH2)n series with n ) 2,1 its isodesmic strain energy would be 17.0 kcal/mol (based on methane and ethane, cf. eq 1) or 22.5 kcal/mol (based on Benson’s alkane CH2 group increment, cf. eq 2). Should we really think of cyclopropane as a cyclic alkane when all its structural features have so little in common with, e.g., cyclohexane? Or is better to compare (CH2)3 with (CH2)2, as a kind of “triangular olefin” (trimethylene) analog? Or is cyclopropane somewhere in between an alkane and an alkene? How does one decide the best model (or select the best group increment) on which to base its “strain energy”? Regard molecules in many different ways. Each provides instructive insights. The protobranching concept4 enriches understanding.27 Acknowledgment. We thank NSF Grant CHE-0716718 for financial support and Professor C. Kemnitz for information regarding his related work and suggestions, and M. Randic for illuminating comments. References and Notes (1) Baeyer, A. Ber. Dtsch. Chem. Ges. 1885, 18, 2269. (2) Eliel, E. L.; N L. Allinger, N. L.; Angyal, S. J.; Morrison, G. A. Conformational Analysis; Wiley: New York, 1965. (3) Fishtik, I. J. Phys. Chem. A, DOI: 10.1021/jp908894q. (4) Wodrich, M. D.; Wannere, C. S.; Mo, Y.; Jarowski, P. D.; Houk, K. N.; Schleyer, P. v. R. Chem.sEur. J. 2007, 13, 7731. (5) All experimental data employed here are based on the experimental enthalpies of formation at 298 K given in the updated Sept. 2006 NIST
CCCBDB Database (http://cccbdb.nist.gov/) and may differ slightly from earlier recommendations. (6) Cohen, N.; Benson, S. W. Chem. ReV. 1993, 93, 2419. Note the extensive historical survey of the development of enthalpy estimation methods. (7) Cox, J. D.; Pilcher, G. Thermochemistry of Organic and Organometallic Compounds; Academic Press: London, 1970. Cox and Pitcher introduced “conventional strain energies” noting that these may include stabilizing as well as destabilizing effects. (8) Lewis, L. L.; Turner, L. L.; Salter, E. A.; Magers, D. H. J. Mol. Struct. 2002, 592, 161; see also other cited literature. (9) Pauling, L. J. Am. Chem. Soc. 1932, 54, 3570. (10) Hehre, W. J.; Ditchfield, R.; Radom, L.; Pople, J. A. J. Am. Chem. Soc. 1970, 78, 4796. Hehre, W. J.; Radom, L.; Schleyer, P. v. R.; Pople, J. A. Ab Initio Molecular Orbital Theory; Wiley: New York, 1986. (11) Cox, J. D. Tetrahedron 1962, 18, 1337–1350. 1963, 19, 1175. (12) George, P.; Trachtman, M.; Bock, C. W.; Brett, A. M. Tetrahedron 1976, 32, 317. For discussion of a hierarchy of homodesmotic reactions for thermochemical evaluations see: Wheeler, S. E.; Houk, K. N.; Schleyer, P. v. R.; Allen, W. D. J. Am. Chem. Soc. 2009, 131, 2547. (13) Walsh, A. D. Trans. Faraday Soc. 1949, 45, 179. (14) For an extensive discussion of factors influencing the strain of cyclopropane see: Bachrach, S. M. Computational Organic Chemistry; Wiley-Interscience: New York, 2007; pp 70ff. (15) For the latest study of σ-aromaticity (see ref 16) and other energetic contributions to cyclopropane see:Wu, W.; Ma, B.; Wu, J. I.; Schleyer, P. v. R.; Mo, Y. Chem. Eur. J., in press. (16) Dewar, M. J. S. J. Am. Chem. Soc. 1984, 106, 669. Dewar, M. J. S.; Pettit, R. J. Chem. Soc. 1954, 1625. (17) Wodrich, M. D.; Corminboeuf, C.; Schleyer, P. v. R. Org. Lett. 2006, 8, 3631. (18) Grimme, S. Angew. Chem., Int. Ed., 2006, 45, 4460. Korth, M.; Grimme, S. J. Chem. Theory Comput. 2009, 5, 993; see also other cited literature. (19) Allen, T. L. J. Chem. Phys. 1959, 31, 1039. Cignitti, M.; Allen, T. L. J. Chem. Phys. 1965, 43, 4472. Zahn, C. T. J. Chem. Phys. 1934, 2, 671. (20) Pitzer, K. S.; Catalano, E. J. Am. Chem. Soc. 1956, 78, 4844. (21) Laidig, K. E. J. Phys. Chem. 1991, 95, 7709. (22) See: Carroll, L. Through the Looking Glass; 1871; “‘When I use a word,’ Humpty Dumpty said in a rather scornful tone, ‘it means just what I choose it to mean - neither more nor less.’” (23) Wodrich, M. D.; Schleyer, P. v. R. Org. Lett. 2006, 8, 2135. (24) Zavitsas, A. A.; Matsunaga, N.; Rogers, D. W. J. Phys. Chem. A 2008, 112, 5734. (25) Schleyer, P. v. R.; Williams, J. E.; Blanchard, K. R. J. Am. Chem. Soc. 1972, 92, 2337. (26) Pauling, L. J. Am. Chem. Soc. 1931, 53, 1367. (27) A rebuttal of other criticisms of protobranching: Gronert, S. Chem.sEur. J. 2009, 15, 5372. Also see: Gronert, S. J. Org. Chem. 2006, 71, 1209. Gronert, S. Org. Lett. 2007, 9, 2211. For generally supporting views, see: Poutsma, M. L. J. Org. Chem. 2008, 73, 8921. Estrada, E. Chem. Phys. Lett. 2008, 463, 422. Kemnitz, C. B.; Mackey, J. L.; Loewen, M. J.; Hargrove, J. L.; Lewis, J. L.; Hawkins, W. E.; Nielsen, A. F. Chem.sEur. J. 2009, DOI: 10.1002/chem.200.
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