Qualitative properties of modified equations

Suppose that a consistent one-step numerical method of orderr is applied to a smooth system of ordinary differential equations.Given any integer m 1, the method may be shown to be of orderr + m as an approximation to a certain modified equation. Ifthe method and the system have a particular qualitative propertythen it is important to determine whether the modified equationsinherit this property. In this article, a technique is introducedfor proving that the modified equations inherit qualitativeproperties from the method and the underlying system. The techniqueuses a straightforward contradiction argument applicable toarbitrary one-step methods and does not rely on the detailedstructure of associated power series expansions. Hence the conclusionsapply, but are not restricted, to the case of Runge-Kutte methods.The new approach unifies and extends results of this type thathave been derived by other means: results are presented forintegral preservation, reversibility, inheritance of fixed points.Hamiltonian problems and volume preservation. The techniquealso applies when the system has an integral that the methodpreserves not exactly, but to order greater than r. Finally,a negative result is obtained by considering a gradient systemand gradient numerical method possessing a global property thatis not shared by the associated modified equations.     

Chemical bonding in the hydrogen molecule

The chemical bond in the hydrogen molecule is examined using the electron density and the generalized overlap amplitudes. Logarithmic derivatives of the electron density provide a clear picture of its behavior in the bonding region as well as in the outer region. The GOA expansion of the density is used to examine the dependence of the rate of decay of the density on the GOA ionization potentials. The increase in the electron density at the nuclei and in the bonding region coincides with the higher ionization potential of H₂ over the H atom. The density in the bonding region along the internuclear axis does not decay exponentially, but its shape is very nearly an inverted Gaussian. © 1996 John Wiley & Sons, Inc.     

Effects of rotational symmetry order on the solid state dynamics of phenylene and diamantane rotators

The rotational dynamics of phenylene and diamantane rotators in crystals of 1,9-bis(4-[3,3,3-triphenylpropynyl]phenyl)diamantane were analyzed independently within the same crystal structure. The dynamics of phenylene rotation were established by dynamic line shape analysis using 13C CPMAS NMR. The phenylene signals were selectively highlighted by deuteration of the aromatic trityls and the use of short contact times for cross polarization. The dynamics of the diamantane group were established by 1H spin-lattice relaxation under conditions where dipolar relaxation was shown to be the dominant mechanism between 250 and 425 K. A factor of 20 000 between the rates of rotation of the faster diamantane and the slower phenylene at 300 K supports expectations that higher symmetry rotors should have significantly faster dynamics.     

Synthetic Routes to the Cyclo[n]carbons

Cyclo[n]carbons (cyclo-C_n) are n-membered monocyclic rings of sp-hybridized C-atoms with unique electronic structures resulting from two perpendicular systems of conjugated π-orbitals, one in-plane and one out-of-plane. Several synthetic approaches to generate cyclo-C₁₈ from stable precursors were investigated. In a six-step sequence from anthracene, tris(anthraceno)hexadehydro[18]annulene 5 was prepared and shown by laser-desorption time-of-flight mass spectrometry to generate C₁₈ in three successive retro-Diels-Alder reactions, together with anthracene as the by-product. The attempted preparation of C₁₈ by flash-vacuum pyrolysis of 5 using solvent-assisted sublimation only afforded anthracene next to polymers. The reaction of 1,6-bis(triisopropylsilyl)hexa-1,3,5-triyne with [Co₂(CO)₈] followed by exchange of two CO groups for a bridging bis(diphenylphosphino)methane (dppm) ligand gave Co complex 22 which, after removal of the silyl groups, was oxidatively cyclized to afford the very stable trimeric and tetrameric macrocycles 6 and 7 , complexes with cyclo-C₁₈ and cyclo-C₂₄, respectively. An X-ray crystal structure established the identity of 6 and showed that the butadiyne units within the C₁₈ core are considerably bent. Attempts to free cyclo-C₁₈ from the coordinating metal atoms in 6 did not succeed, presumably due to the steric shielding of the Co-atoms by the dppm ligands. Low-temperature matrix isolation studies using IR and UV/VIS spectroscopy showed that irradiation of carbon oxide 2 (C₂₄O₆) leads to ketene intermediates and, by subsequent loss of six CO molecules, presumably to cyclo-C₁₈.