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Three sources of quantitative data for methyl-substituted allyl cations are available experimentally: Gas phase heats of formation, rotational barriers in solution, and rates of solvolysis. N M R chemical shifts also provide an index of charge distribution. This paper draws all these lines together in comparison with results of STO-3G ab initio molecular orbital calculations performed on planar and perpendicular methyl-substituted allyl cations. Sequential substitution of the terminal positions by one, two, three, and four methyl groups stabilizes allyl cations electronically by 17, 15, 13, and I 1 kcal/mol, respectively; a methyl group on the central carbon has a much smaller stabilizing effect (-5 kcal/mol). The steric strain for the first endomethyl group was determined to be 3 kcal/mol, whereas a second endo-methyl experiences a somewhat greater strain of 5 kcal/mol. Allyl solvolysis rates correlate well with gas phase allyl cation stability data, but there is a reduction in magnitude due to solvation. The calculated rotational barriers also are 4-1 2 kcal/mol higher than the activation free enthalpies determined i n superacid solution; the difference provides further evidence for differential solvation effects-the more highly delocalized planar forms are solvated to a lesser extent than their rotational transition states. The rotational barrier of the parent allyl cation is predicted to be 34 kcal/mol in the gas phase but to decrease to 23.7 f 2 kcal/mol in superacid solution. Allyl cations, the prototype H delocalized carbenium ions, have been studied experimentally for many decades.' The solvolysis studies, which first proved the existence of allyl cations as short-lived reactive intermediates, preceded generation and direct observation of stable allyl cations in superacid media.' In recent years, many thermodynamic data on allyl cations in the gas phase have become available, particularly by means of ion cyclotron resonance measurements.2 Previous theoretical studies have concentrated on the parent allyl With few exception^,^^ methyl-substituted allyl cations have only been investigated by semiempirical methods"-" or by a b initio calculations without geometry optimization.'* Since these calculations give considerable deviations with experimentally determined energy differences, we undertook a comprehensive a b initio MO study of methyl-substituted allyl cations with geometry optimization in order to assess and to augment the experimental data. Methods All a b initio MO calculations on planar and perpendicular allyl cations were performed by utilizing the STO-3G minimal basis setI3 and the Gaussian 70 series of p r o g r a m ~ . ' ~ For the planar allyl cations, the heavy atom framework was fully optimized, until the total energy was constant to better than au, bond lengths to 0.001 A, and angles to 0. I degrees; standard bond lengths and angles were used for the CH bonds (Table I ) . The energy values thus obtained were only 0002-7863/79/1501-6032$01 .OO/O slightly higher than those of completely optimized structures, e g , 1 ( A E = 0.5 kcal/mol), 4 (AE = 0.7 kcal/mol), and 6 ( A E = 1.0 kcal/mol).Is The fully optimized structures are included in Table I . The geometries of the perpendicular allyl cations were based on the completely optimized structure of the parent system.' This geometry proved to be satisfactory also for substituted cations, since complete optimization of perpendicular 5 and 8 (rotation of CH2 group) resulted in energies only 0.5 and I .8 kcal/mol, respectively, below those given in Table IV. Since the errors introduced by our less than complete optimization appear to be small and more or less constant, they tend to cancel in many of the energy comparisons. In section A2, it will be shown that the relative energies of allyl cations, thus calculated, coincide with experimental gas phase data, which justifies the approximations employed. Results and Discussion A. Planar Allyl Cations. 1. Geometry. Heavy-Atom Framework. The results of STO-3G calculations on the planar allyl cations are summarized in Table I . All allyl cations were assumed to have planar heavy-atom ground states. This was justified on the basis of the planar preference of a member of the set more prone to twisting: An energy increase was calculated when the 3,3-dimethylallyl cation (8), which has a relatively low rotational barrier (see below), was twisted by only 15' around the C-2--C-3 bond. In contrast, INDO predicted

Methyl-substituted allyl cations. A comparison of experimental stability, rotational barrier, and solvolysis data with ab initio calculations