A hypothetical five-step catalytic cycle for Brønsted-mediated fission of an all-trans n-alkane was examined using density functional theory. Optimized geometries and transition states were determined for catalystreactant complexes, using three different monodentate catalyst ions (NH 4 + , H 3 O + , and H 2 F + ). Despite the wide variety of catalyst acidities, protonated hexane appears as an intermediate (not a transition state) in each case. The protonated cyclopropane structure is the most likely initial form of the dissociated product ion. The predicted intermediates were seen to vary with catalyst acidity. The complete energy profiles of this model catalytic cycle are provided and fitted to a cosine expansion, which allows for generation of the energy profile for any Brønsted catalyst and any n-alkane only on the basis of the proton affinities of the n-alkane and the conjugate base of the catalyst. Remarks on the applicability to zeolites and ionic liquid catalysts are given.
Ab initio calculations at the MP2/6-31G(d) level of theory have been performed to determine the geometries and relative energies of many isomers of protonated octane (also known as octonium or octanium ions). Five of the 18 structural isomers of octane were considered for protonation, as they provided C-C bonds containing all possible combinations of carbon substitution types (quaternary-tertiary, quaternary-secondary, etc.). All resulting isomers of C 8 H 19 + feature either a CHC or a CHH 3-center-2-electron (3c2e) bond, although barrierless dissociation into an ion-molecule complex was very common. Octonium ion properties such as relative energies, 3c2e bond geometries, Mulliken partial charges, and the frequency of the most intense infrared absorption, have also been calculated. Each property is correlated to the level of substitution of C atoms in the 3c2e bond. The proton affinities of individual bonds in octane range from 154 to 187 kcal mol -1 for C-C bonds and 139 to 150 kcal mol -1 for C-H bonds. Alkanium (carbonium) ions of greater than four carbons have never before been studied in this depth.
The vibrational frequencies of several linear polyynes HCNH, HCNI and ICJ, for even-valued N up to 60, have been obtained using computational quantum chemistry. The bending normal modes have the appearance of classical transverse normal modes of a vibrating string fixed at both ends, for which the frequencies vary with the first power of the harmonic. Our calculations, however, reveal that in the limit of infinite chain length and infinite mass on the ends of the 'molecular string', the bending frequencies vary with the square of the harmonic, at all levels of theory. A derivation is presented to explain the discrepancy.
The geometry and relative energies of torsional conformers of centrally protonated C 4 H 11 ϩ were studied with ab initio methods, to ͑a͒ obtain the most accurate geometry of the three-centertwo-electron CHC bond to date, ͑b͒ evaluate the performance of lower levels of approximation upon this challenging structure, and ͑c͒ gain an understanding of the torsional dynamics of C 4 H 11 ϩ . Twenty-nine combined levels of theory were used to optimize the geometry of the C 2 -symmetry minimum for trans-C 4 H 11 ϩ , and the most accurate one ͓CCSD͑T͒/cc-pVTZ͔ gave the following CHC bond geometry: CHC ϭ122.4°, R CC ϭ2.177 Å, R CH ϭ1.2424 Å. Molecular-orbital-based methods generally perform better than density functional methods for describing the three-centertwo-electron bond. A smaller subset of levels of theory was used to optimize other torsional conformers of centrally protonated C 4 H 11 ϩ , varying the CCCC dihedral ͑trans, gauche, cis͒ and the dihedral for the bridging proton ͑various eclipsed and staggered positions͒. The results show that all conformers lie within a 4 kJ mol Ϫ1 range, with the lowest-energy conformer being either trans or gauche with a staggered dihedral for the bridging proton. The effect of core-valence correlation was also investigated. Finally, the potential energy surface as a function of the CCCC and bridging-proton dihedral angles was qualitatively estimated and drawn, based on our computed data, to aid in understanding the fluxional character of C 4 H 11 ϩ .
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