Structural manifestation of the delocalization error of density functional approximations: C(4N+2) rings and C(20) bowl, cage, and ring isomers.

The ground state structure of C(4N+2) rings is believed to exhibit a geometric transition from angle alternation (N < or = 2) to bond alternation (N > 2). All previous density functional theory (DFT) studies on these molecules have failed to reproduce this behavior by predicting either that the transition occurs at too large a ring size, or that the transition leads to a higher symmetry cumulene. Employing the recently proposed perspective of delocalization error within DFT we rationalize this failure of common density functional approximations (DFAs) and present calculations with the rCAM-B3LYP exchange-correlation functional that show an angle-to-bond-alternation transition between C(10) and C(14). The behavior exemplified here manifests itself more generally as the well known tendency of DFAs to bias toward delocalized electron distributions as favored by Huckel aromaticity, of which the C(4N+2) rings provide a quintessential example. Additional examples are the relative energies of the C(20) bowl, cage, and ring isomers; we show that the results from functionals with minimal delocalization error are in good agreement with CCSD(T) results, in contrast to other commonly used DFAs. An unbiased DFT treatment of electron delocalization is a key for reliable prediction of relative stability and hence the structures of complex molecules where many structure stabilization mechanisms exist.

Nonholonomically Constrained Dynamics and Optimization of Rolling Isolation Systems

Rolling Isolation Systems provide a simple and effective means for protecting components from horizontal floor vibrations. In these systems a platform rolls on four steel balls which, in turn, rest within shallow bowls. The trajectories of the balls is uniquely determined by the horizontal and rotational velocity components of the rolling platform, and thus provides nonholonomic constraints. In general, the bowls are not parabolic, so the potential energy function of this system is not quadratic. This thesis presents the application of Gauss's Principle of Least Constraint to the modeling of rolling isolation platforms. The equations of motion are described in terms of a redundant set of constrained coordinates. Coordinate accelerations are uniquely determined at any point in time via Gauss's Principle by solving a linearly constrained quadratic minimization. In the absence of any modeled damping, the equations of motion conserve energy. This mathematical model is then used to find the bowl profile that minimizes response acceleration subject to displacement constraint.