Symbolic dynamics for the renormalization map of a quasiperiodic Schrödinger equation

A rigorous analysis is given of the dynamics of the renormalization map associated to a discrete Schrödinger operatorH onl ²(), defined byH(n)=(n+1)+(n–1)+Vf(n)(n), whereV is a real parameter,f is a certain discontinuous period-1 function, and is the golden mean. The renormalization map forH is a diffeomorphism,T, of ³, preserving a cubic surfaceS _V . ForV8 we prove that the non-wandering set of the restriction ofT toS _v is a hyperbolic set, on whichT is conjugate to a subshift on six symbols. It follows from results in dynamical systems theory that the optimally approximating periodic operators toH have spectra which obey a global scaling law. We also define a set which we call the pseudospectrum of the operatorH. We prove it to be a Cantor set of measure zero, and obtain bounds on its Hausdorff dimension. It is an open question whether the pseudospectrum coincides with the spectrum ofH.     

Donor—acceptor interactions of substituted benzenes with molecular chlorine and carbon disulfide

CNDO molecular orbital calculations have been performed to analyze donor—acceptor interactions between molecular chlorine and benzene, toluene, mesitylene and hexamethylbenzene and the, as yet, unreported chlorine—hexafluorobenzene and carbon disulfide—benzene pairs. The stabilization energy and the dipole moment and its derivative (?p/?R_CICI) calculated for the benzene—chlorine complex are in good agreement with the estimated experimental values. The trends in the experimental stabilization energies and the Cl-Cl vibrational frequencies with increasing methyl substitution appear to be well reproduced by the calculations. The charge transferred from the benzene donor is polarized toward the outer chlorine atom or sulfur atom. For hexafluorobenzene-chlorine the direction of electronic charge polarization is reversed from that of the benzene and methylbenzene complexes. The calculated results are discussed within the framework of Muliiken's simplified resonance theory for complexes.     

Analytical applications of infra-red spectroscopy : The behaviour of the oxinates of niobium,tantalum and associated elements

The behaviour of the oxinates of niobium, tantalum and associated metals in the infra-red region was studied and a method developed for the determination of niobium and tantalum. Vanadium caused no interference, but other heavy metals, such as molybdenum, manganese and cobalt, which interfered were removed by preliminary treatment when the method was applied to the determination of niobium and tantalum in steels.     

The direct determination of Brueckner orbitals with application to the H2 molecule

An iterative procedure is presented which allows for the direct determination of approximate Brueckner orbitals for small atomic and molecular systems. Starting from the Hartree-Fock determinant one first determines pair natural orbitals (PNOs) of independent electron pairs in the HF-field of the remaining electrons. The use of the generalized Brillouin-theorem then leads to an approximate Brueckner orbital for each electron pair. This procedure must be repeated up to self-consistency which is reached generally after 4–5 macroiterations. Applications to the ground state of H₂ show how important the use of Brueckner orbitals is to get good expectation values of one-electron operators and the correct asymptotic behaviour of the potential energy curve for large internuclear distances.

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Zusammenfassung Es wird eine Iterationsmethode beschrieben, die eine direkte Bestimmung genäherter Bruecknerorbitale für kleine atomare und molekulare Systeme gestattet. Ausgehend von der Hartree-Fock-Determinante bestimmt man zunächst Paar-NOs (PNOs) der unabhängigen Elektronenpaare im HF-Feld der übrigen Elektronen des Systems. Mit Hilfe des verallgemeinerten Brillouin-Theorems erhält man dann ein genähertes Bruecknerorbital für jedes Elektronenpaar. Dies Verfahren muß bis zur Selbstkonsistenz wiederholt werden, die man im allgemeinen nach 4–5 solcher Makroiterationen erreicht. Anwendung auf den Grundzustand des H₂-Moleküls zeigt, wie wichtig die Benutzung von Bruecknerorbitalen ist, wenn man gute Erwartungswerte von Einelektronenoperatoren und das richtige asymptotische Verhalten der Potentialkurve bei großen Abständen erhalten will.

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Résumé On propose un schéma itératif qui permet le calcul direct des orbitales de Brueckner approchées pour les atomes et les petites molécules. Partant du déterminant Hartree-Fock on calcule d'abord les PNO (pair natural orbitals) des paires indépendantes d'électrons dans le champ effectif des autres électrons dans l'approximation Hartree-Fock. Ensuite le théorème de Brillouin généralisé permet de calculer les orbitales de Brueckner approchées pour chaque paire d'electrons. On recommence le calcul des PNO, la self-consistance de cette méthode itérative s'établit en général au bout de 4 ou 5 macroitérations. Les applications à la molécule H₂ démontrent que l'utilisation d'orbitales de Brueckner s'impose si l'on s'intéresse à des bonnes valeurs moyennes d'opérateurs monoélectroniques ou au comportement asymptotique correct des courbes de potentiel aux grandes distances.

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Dedicated to the memory of Prof. K. H. Hansen.     

Indirect electrochemical methods of nitration: Novel nitroacetamidation of dienes

Solutions of nitronium tetrafluoroborate in acetonitrile prepared by anodic oxidation of nitrogen dioxide, efficiently nitrate aromatics, enolsilyethers, alkenes and conjugated dienes to afford respectively nitroaromatics, α-nitroketones, vicinal nitroamides and mixtures of nitroacetamides by 1,2- and 1,4-addition.     

Mass spectral fragmentation patterns of heterocycles. VI. Carbazole and 1,8-dideuteriocarbazole

The mass spectral fragmentation of carbazole was reinvestigated using metastable ion studies, exact mass measurements and 1,8-dideuteriocarbazole. 1,8-Dideuteriocarbazole was prepared by successive metalations and deuterations of carbazole. The ¹H- and ¹³C-nmr spectra of 1,8-dideuteriocarbazole confirmed earlier assignments for carbazole.     

The preparation of hot-pressed chalcogenide spinels. II. Stoichiometry and optical transparency in CdCr2S4 and CoCr2S4

The compounds CdCr₂S₄ and CoCr₂S₄ have been hot pressed into disks that are highly transparent in the infrared. Stoichiometry has been altered by varying the $\text{Cr}{}^{\mathrm{3+}}\text{M}{}^{\mathrm{2+}}$ ratio, where M²⁺ is Cd²⁺ or Co²⁺. The effects of nonstoichiometry on optical transmission were determined. Excess M²⁺ attenuates the transmission much more than excess Cr³⁺.     

Nouveaux exemples d'acétonides d'aldéhydosucres furanniques

Further examples of O-isopropylidenaldehydosugars in the furanose series Formyl-bearing furanose derivatives 1–3 (α-D -xylo or α-D -erythro configurations) and 5–8 (configurations α-D -ribo, α-D -lyxo, β-L -threo and D -arabino) have been prepared by classical synthetic steps and their properties mainly spectroscopic, reported. The coupling constant between the formyl and the vicinal proton is always small. Like other members of the series, these new aldehydosugars constitute very useful and flexible synthetic intermediates.     

Potassium addition to Fe/MgO, Ni/MgO and Ni/SiO2: Reducibility, dispersity and stabilization of the promoter by the support

The influence of potassium addition on the morphological properties of Fe/MgO and Ni/MgO has been investigated: KNO₃ addition to precursors results in a decrease of the reducibility and in a smaller dispersity of the metallic phase. A loss of potassium is observed at reduction temperatures in excess of 773 K. Silica-supported Ni behaves differently: silica inhibits the potassium volatilization and nickel reducibility is enhanced. This illustrates how promoter effects can be influenced by the nature of support.

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Fe/MgO Ni/MgO: KNO₃ . , 773 K. , : . .