Simulation of scanning tunneling microscope images of 1,3-cyclohexadiene bound to a silicon surface

Scanning tunneling microscope (STM) images of 1,3-cyclohexadiene bound to silicon are interpreted using a nonequilibrium Green's function method. The resolution of the carbon-carbon double bond for positive bias voltages but not for negative bias voltages is explained using a quasiprobability density analysis. The asymmetry in the images arises from the system's voltage dependent electronic structure. A pi* orbital is found to be responsible for the empty state STM images of the carbon-carbon double bond, which is observed experimentally. The pi orbital relevant for the opposite bias does not produce an STM image sharply localized in the bond region because the molecule induces a Si-surface dipole dependent on the bias. The dipole voltage dependence arises from molecular charging. This result emphasizes the importance of simulating the molecule as an element in an open quantum system.     

Nonlinear waves in zonostrophic turbulence

The Charney-Hasegawa-Mima equation applies to a broad variety of hydrodynamic systems ranging from the large-scale planetary circulations to small-scale processes in magnetically confined plasma. This equation harbors flow regimes that have not yet been fully understood. One of those is the recently discovered regime of zonostrophic turbulence emerging in the case of small-scale forced, barotropic two-dimensional turbulence on the surface of a rotating sphere or in its beta-plane approximation. The commingling of strong nonlinearity, strong anisotropy and Rossby waves underlying this regime is highlighted by the emergence of stable systems of alternating zonal jets and a new class of nonlinear waves, or zonons. This Letter elucidates the physics of the zonons and their relation to the large-scale coherent structures.     

Optimal Secant-Type Methods for Operator Equations

We prove an existence and uniqueness theorem for operator equations in Banach spaces with (generally non-differentiable) operators whose divided differences are Lipschitz continuous on operator's domain. The theorem makes possible to apply the concept of entropy optimality of iterative methods introduced in our earlier work to the class of secant-type methods. Using this concept, we show that it is feasible to get a method that needs the same information (one value of the operator) per iteration but exhibits a faster convergence than the secant and secant-update methods.     

Billiards with Pockets: A Separation Principle and Bound for the Number of Orbit Types

 We introduce and prove a Separation Principle, similar in form to the familiar Uncertainty Principle of quantum mechanics, which separates the position and direction of any two phase points on distinct unfoldings of (non-parallel) trajectories on a polygonal billiard table with pockets. Applying this principle, we demonstrate that the number of orbit types (that is, classes of trajectories, up to parallelism) on a polygonal billiard table with area A and pockets of area a is strictly bounded above by . More generally, the same bound applies to any compact polyhedral surface with pockets at its vertices. If the boundary is empty (so that billiard trajectories are just geodesics), the bound is reduced by a factor of two to . We believe the Separation Principle will also have fundamental applications to other problems in the theory of billiards and related dynamical systems. Received: 28 December 2001 / Accepted: 9 April 2002 Published online: 4 September 2002     

Reactions of 3а,6a-diaza-1,4-diphosphapentalene with activated acetylenes

The reaction of cyclohexane-annulated 3 a,6a-diaza-1,4-diphosphapentalene (DDP) with di-tert-butyl acetylenedicarboxylate (DBAD) affords the 1,3-dipolar cycloaddition product of the acetylene moiety to the phosphorus and sp²-carbon atoms of DDP in 90% yield. No individual products were isolated in the reaction of DDP with dimethyl acetylenedicarboxylate (DMAD). In the three-component DDP–DMAD–carbazole system (1: 2: 1), a product was generated in 73% yield via successive reactions, including the 1(P),3(C)-dipolar cycloaddition of DMAD to DDP, the addition of the second equivalent of DMAD to the three-coordinate phosphorus atom of the intermediate, and the NH addition of carbazole at the unsaturated C=C bond of the second DMAD moiety. The structures of the reaction products were established by X-ray diffraction.