Limiting Lagrangians: A primal approach

In this paper, we consider a convex program with either a finite or an infinite number of constraints and its formal Lagrangian dual. We show that either the primal program satisfies a general condition which implies there is no duality gap or that there is a nonzero vectord with the following properties: First, wheneverd is added to the objective function, where is a positive number not greater than one, the resulting program satisfies the general sufficient condition cited above for no duality gap. Second, the optimal value of this perturbed program is attained and tends to the optimal value of the original program as tends to zero. Third, the optimal solutions of the perturbed programs form a minimizing sequence of the original program. As a consequence of the above, we derive the limiting Lagrangian theory of Borwein, Duffin, and Jeroslow.The authors are indebted to an unknown referee who suggested the very short and elegant proofs of Lemma 2.3 and Theorem 2.3.This work was completed while the first author was a member of the College of Management, Georgia Institute of Technology, Atlanta, Georgia.     

Möbius aromaticity in [12]annulene: cis-trans isomerization via twist-coupled bond shifting

Density functional and coupled cluster calculations show that facile thermal configuration change in [12]annulene occurs via a twist-coupled bond-shifting mechanism. The transition state for this process is highly aromatic with M?bius topology. At the CCSD(T)/cc-pVDZ//BH&HLYP/6-311+G** level, the isomerization of tri-trans-[12]annulene 1a (CTCTCT) to its di-trans isomer 2 (CCCTCT) via such a mechanism has a barrier of 18.0 kcal/mol, in good agreement with earlier experiments. Two other aromatic M?bius bond-shifting transition states were located that result in configuration change for other [12]annulene conformers. This mechanism contrasts sharply with diradical configuration change for acyclic polyenes and with planar bond-shifting mechanisms generally assumed for annulenes. This constitutes evidence that neutral M?bius aromatic annulenes play a role in the dynamic processes of neutral [4n]annulenes.     

Competing isomerizations of [12]annulenes: Diels-Alder reaction versus electrocyclization

Experimentally, tri-trans-[12]annulene and tris(cyclohexeno)[12]annulene exhibit differing reactivities. Whereas the former, after isomerizing to its di-trans isomer, undergoes sequential electrocyclizations, the latter follows a Diels-Alder pathway after initial electrocyclization. B3PW91/6-31+G*//B3LYP/6-31G* calculations indicate that cyclohexenofusion simultaneously hinders the second electrocyclization and facilitates Diels-Alder reaction, primarily by inducing greater puckering in the intermediate eight-membered ring.     

Investigation of a putative mobius aromatic hydrocarbon. The effect of benzannelation on mobius [4n]annulene aromaticity

The first experimental example of a [4n]annulene derivative with one Mobius twist, 1, was synthesized recently [Ajami, D.; Oeckler, O.; Simon, A.; Herges, R. Nature 2003, 426, 819] and was purported to possess aromatic character. However, critical analysis of the published crystallographic data indicates that the Mobius [16]annulene core of 1 shows large bond alternation (Deltar up to 0.157 A). Delocalization in this core is inhibited by large dihedral angles, which hinders effective pi overlap. This conclusion is supported by computational results (B3LYP/6-311+G) on 1 and several less benzannelated derivatives, based on geometric (Deltar, Deltar(m), Julg A, HOMA) and magnetic (NICS, magnetic susceptibility exaltation) criteria of aromaticity. That benzannelation results in bond localization in the [16]annulene core is shown by additional computations on benzannelated derivatives of other Mobius aromatic species. Additionally, the aromatic stabilization energy (ASE) of 1 has been reinvestigated using two different procedures. Evaluation of uncorrected ISE(II) values of just the polyene bridge portion of 1 and its Huckel counterpart suggests that stabilization of 1 relative to its Huckel isomer is confined to the polyene bridge and is not due to a delocalized pi circuit. Furthermore, application of s-cis/s-trans corrections lowers the ISE(II) value of 1 from 4.0 kcal/mol to 0.6 kcal/mol, suggesting that 1 is nonaromatic.     

Singlet-triplet energy separation of cyclobutylidene

Ab initio (MP2, CCSD(T)) and density functional theory (BLYP, B3LYP) calculations provide insight concerning novel aspects of structure and bonding in cyclobutylidene (1). Singlet cyclobutylidene ((1)1) adopts a bicyclobutane-like structure (C(s) symmetry) that includes a weak, transannular bonding interaction between the carbene carbon and the opposing CH(2) group. Conformational ring inversion in (1)1 occurs through a transition state of C(2)(v)() symmetry (TS(1)1) with an enthalpy barrier of approximately 3 kcal/mol. Stabilization afforded the singlet state by the transannular interaction appears to be largely offset by a loss of hyperconjugative stabilization from the adjacent C-H bonds. Triplet cyclobutylidene ((3)1) exhibits a C(2)(v)() structure and conventional bonding. The triplet state lies 5.9 kcal/mol above the singlet ground state at the CCSD(T)/TZP//CCSD(T)/DZP level of theory. The singlet-triplet energy gap of cyclobutylidene (-5.9 kcal/mol) lies between that of an acyclic analogue, dimethylcarbene (-1.6 kcal/mol), and a highly strained analogue, cyclopropylidene (-13.8 kcal/mol). The magnitude of the energy gap suggests that triplet cyclobutylidene ((3)1) will be thermally accessible under a variety of experimental conditions.