Phase Space Prediction of Product Branching Ratios: Canonical Competitive Nonstatistical Model Jingjing Zheng, Ewa Papajak, and Donald G. Truhlar* Department of Chemistry and Supercomputing Institute, UniVersity of Minnesota, Minneapolis, Minnesota 55455-0431 Received May 31, 2009; E-mail: truhlar@umn.edu Abstract: We present a new model for predicting branching ratios of chemical reactions when a branching of the reaction path occurs after the dynamical bottleneck, including the case where it occurs after an intermediate. The model is based on combining nonstatistical phase space theory for the direct component of a reaction with variational transition-state theory for an indirect component of reaction. The competition between direct and indirect processes is treated by an extension of the unified statistical model. This new method provides a way to understand the factors that control this kind of chemical reaction and to perform calculations using high-level electronic structure methods for complex systems. The model is based on quantized energy levels of transition states and products, and it involves the same information as required for calculating transition-state rate constants and equilibrium constants plus a phenomenological relaxation time, which was taken from previous work. For the textbook reaction of the hydroboration of propene by BH 3 it has recently been inferred that the selectivity can only be understood by consideration of dynamical trajectories. However, the calculated branching fraction of this prototype reaction increases from 2%-3% when calculated under the inappropriate assumption of complete equilibration of the intermediate to from 8%-9% when calculated with the new theory, which requires only limited information about the system and does not involve running trajectories. The calculated result is in reasonable agreement with experiment (∼10%). Introduction The addition of HX to an alkene has been studied for a long time, and it is conventional to label the products as Markovni- kov 1 (M) or anti-Markovnikov 2 (anti-M). This was originally understood in terms of an acid mechanism, where H + adds first, or a radical mechanism, where X • adds first. In the former case the more stable carbocation is obtained if H + adds to the least substituted end of the double bond, and in the latter case the more stable radical is obtained if X • adds first to the first least substituted end of the double bond. 3 For the case where X is BH 2 (Scheme 1), Brown and Zweifel 4 found 6-7% M, and they explained the result by a four-center transition state where H δ- BH 2 δ+ interacts most favorably with the alkene when BH 2 is partially bonded to the least substituted end of the double bond because that end is more electronegative. Lipscomb and co-workers 5 studied the addition of BH 3 (hydroboration) com- putationally and found M and anti-M π complexes followed by M and anti-M transition states. They concluded that: “The observed regioselectivity of the hydroboration reaction is reflected in the differences in the stability of the transition states”. However, their transition states were lower in energy than the reactant, and they did not compute free energies to see whether the association reaction or the post-complex transition states provided the overall dynamical bottleneck. Oyola and Singleton 6 recently performed much more quantitatively reliable calculations that included estimates of the free energies, and they found for the hydroboration of propene with BH 3 that the association bottleneck is overall rate determining. Rate processes when a branching of the reaction path occurs after the dynamical bottleneck are an important challenge for theory because in such a case transition-state theory only predicts the sum of the rate constants for the two branches. The hydroboration reaction was studied by Oyola and Singleton 6 using transition-state theory and trajectory calculations. They concluded the following: “In this most ‘textbook’ of reactions, transition state theory fails and the selectivity can only be understood by consideration of dynamics trajectories”. 6 “For reactions exhibiting dynamic effects, chemistry must develop new qualitative ideas to account for reactivity and selectivity.” 7 In the present article we will present a qualitative theory not involving dynamical trajectories that accounts for the experi- (1) Markovnikov, V. Annalen 1870, 153, 256. (2) Markovnikov, V. Comptes Rendu 1875, 82, 868. Kharasch, M.; Mayo, F. J. Am. Chem. Soc. 1933, 55, 2468. (3) Isenberg, N.; Grdinic, M. J. Chem. Educ. 1969, 46, 601. (4) Brown, H. C.; Zweifel, G. J. Am. Chem. Soc. 1960, 82, 4708. (5) Graham, G. D.; Froeilich, S. C.; Lipscomb, W. N. J. Am. Chem. Soc. 1981, 103, 2546. See also: Dewar, M. J. S.; McKee, M. Inorg. Chem. 1978, 17, 1075. Wang, X.; Li, Y.; Wu, Y.-D.; Paddon-Row, M. N.; Rondan, N. G.; Houk, K. N. J. Org. Chem. 1990, 55, 2601. (6) Oyola, Y.; Singleton, D. A. J. Am. Chem. Soc. 2009, 131, 3130. (7) Thomas, J. B.; Waas, J. R.; Harmata, M.; Singleton, D. A. J. Am. Chem. Soc. 2006, 130, 14544. Scheme 1 Published on Web 10/07/2009 10.1021/ja904405v CCC: $40.75  2009 American Chemical Society 15754 9 J. AM. CHEM. SOC. 2009, 131, 15754–15760