Extension of a theorem of P. Wolfe on singular integral equations

The integral operator which sends f (ζ) into $$\int_{|\xi |< 1} {\tfrac{{f(\xi )}}{{|x - \xi |^{p - 1} }}} d\xi = g(x)(|x|< 1)$$ x and ζ ∈ R^P, is shown to map the space of functions f (ζ) on |ζ|<1, square integrable with respect to the weight (1-|ξ|²)^1/2 one to one onto the space of functions g (x) on |x|<1 which possess gradients δg(x) in the sense of distribution theory, with both g(x) and δg(x) being square integrable with respect to the weight (1-|x|²)^1/2. This extends to p≥3 a result of P. Wolfe's in the case p=2.     

Anharmonicity in rotational and vibrational excitation of H2 by Li+ collisions

Integral cross sections for pure rotational and vibrational-rotational excitation of H₂(X¹Σ⁺_g) by Li⁺(¹S) impact are computed by close-coupling methods at 0.2, 0.6, and 1.2 eV in the c.m. system using vibrational functions that are numerical solutions of the one-dimensional radial Schrödinger equation for harmonic, Morse, and adiabatically corrected Kolos-Wolniewicz (KW) potential functions. Comparison of results employing KW and Morse functions shows excellent agreement for all transitions studied. Findings using harmonic oscillator functions, however, differ noticeably from KW and Morse values for vibrational (0 → 1) and very large rotational (Δj = 10) transitions, but are satisfactory for lower order (0 → 2, 4, 6, 8) rotational transitions.     

Studies in the heterocyclic series. XI. Reactions of 2-Amino-3-hydroxypyridine with 2-Chloro-3-nitropyridine

The reactions of 2-amino-3-hydroxypyridine and 2-chloro-3-nitropyridine in various solvents and in the presence of certain catalysts were studied. A dilute acid-catalysed procedure led to a 45% yield of 3-hydroxy-3-nitro-2,2-dipyridylamine which was also obtained by refluxing in alcoholic potassium hydroxide for several hours. The diarylamine was converted to 1,9-diazaphenoxazine by base-catalysed intramolecular condensation in refluxing DMSO. The same diazaphenoxazine was obtained directly by prolonged base-catalysed reaction of the starting pyridine compounds in DMSO. From a study of the infrared spectrum of the diarylamine precursor, it was found that strong NHO hydrogen bonding is responsible for the failure of the cyclization step in either aqueous or alcoholic base. This problem was overcome by the use of DMSO as the reaction solvent. Structural assignments were made by a study of their ultraviolet, infrared, nmr and mass spectra and the mechanistic pathways were also discussed.     

Thermal studies on trans-dichlorotetramminecobalt(III) complexes

Thermal studies have been carried out on trans-[Co(NH₃)₄Cl₂]X · YH₂O complexes (whereX=IO₃, BrO₃, NO₃, or NO₂ andY=0, 1, or 2) in an effort to find cases of trans to cis isomerization as occurs for the iodate. No evidence of isomerization was found for any of the other compounds. The complexes decompose in a series of steps and these reactions have been identified and kinetic parameters determined.     

Post-source decay in the analysis of polystyrene by matrix-assisted laser desorption/ionization time-of-flight mass spectrometry

Various secondary series are observed in matrix-assisted laser desorption/ionization (MALDI) time-of-flight mass spectra of polystyrene. The number and positions of the series depend on the choice of matrix and added cation. For a given treatment, series observed in linear mode are not necessarily observed in reflectron mode, and vice versa. Post-source decay analysis was used to determine that the secondary series arise at least in part from formation and decay of adducts of polystyrene with matrix species. There is some treatment-to-treatment variation, but adduct formation and decay were observed for all tested treatments. The multiplicity of secondary series makes it unclear whether post-source decay occurs for the main series (polystyrene + cation)⁺ ions under the conditions normally used for polystyrene analysis. Such ions do undergo post-source decay at laser fluences greater than normally used. Although only polystyrene was investigated in this work, other polymers may also produce adduct and decay series in MALDI analysis. Their presence can mask the presence of minor components in a sample, but at least as observed here, do not have a strong influence on molecular mass determinations.     

Tetrahydropyridoazepines and tetrahydropyridoazepinones from the corresponding dihydroquinolones

The p-toluenesulfonate of 7,8-dihydro-5(6H)quinoloneoxime( 3 ) was subjected of a Beckmann rearràngement. The resulting 2,3,4,5-tetrahydro-1H-pyrido[3,2-b]azepin-2-one ( 4 ) was reduced with lithium aluminum hydride affording 2,3,4,5-tetrahydro-1H-pyrido[3,2-b] azepine ( 5 ). 5,6-l)ihydro-8(7H)quinolone ( 7 ), obtained by oxidation of 5,6,7,8-tetrahydro-8-quinolinol ( 6 ), was converted into the p-toluenesulfonate of 5,6-dihydro-8(7H)quinolone oxitne ( 9 ). Similarly the latter compound could be rearranged into 2,3,4,5-letrahydro-1H-pyrido [2,3-b] azepin-2-one ( 10 ) which on reduction produced 2,3,4,5-tetrahydro-1H-pyrido [2,3-b] azepine ( 11 ).     

The structure of the H3O+ (hydronium) ion

Near Hartree-Fock level ab initio molecular orbital calculations on H₃O⁺ and a minimum energy structure with θ(HOH) = 112.5° and r(OH) = 0.963 Å and an inversion barrier of 1.9 kcal/mole. By comparing these results to calculations on NH₃ and H₂O, where precise experimental geometries are known, we estimate the “true” geometry of isolated H₃O⁺ to have a structure with θ(HOH) = 110-112°, r(OH) = 0.97–0.98 Å and an inversion barrier of 2–3 kcal/mole. Our prediction for the proton affinity of water is ≈ 170 kcal/mole, which is somewhat smaller than the currently accepted value.     

The role of kinetic energy in chemical binding

The wavefunctions and various partitions of the energy are examined for a variety of small molecules (H₂, H₃, H₄, HeH, HeH₂, He₂, LiH, and BH) in order to isolate the factors crucial for bond formation. We find that a natural partition of the energy leads to the conclusion that the crucial factor is the exchange, or nonclassical, part of the kinetic energy, T ^x. The change in T ^xupon pushing the atoms towards one another is the dominant term in the binding energy; it is negative when the resulting molecule is stable and positive when it is unstable. We show that T ^x is related to the interference kinetic energy considered by Ruedenberg.

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Zusammenfassung Die Wellenfunktionen und verschiedene Zerlegungen der Energie werden für eine Reihe kleiner Moleküle untersucht (H₂, H₃, H₄, HeH, HeH₂, He₂, LiH und BH), um die Faktoren zu finden, die für die Bindungsbildung ausschlaggebend sind. Die natürliche Zerlegung der Energie läßt die Folgerung zu, daß der bestimmende Faktor der Austauschanteil T ^x(oder nichtklassische Anteil) der kinetischen Energie ist. Die Änderung von T ^xbeim Zusammenführen der Atome ist der dominierende Term für die Bindungsenergie; er ist negativ, wenn das resultierende Molekül stabil ist, und positiv, falls es instabil ist. Es wird gezeigt, daß T ^x im Zusammenhang zum Wechselwirkungsanteil der kinetischen Energie nach Ruedenberg steht.

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Partially supported by a grant (GP-15423) from the National Science Foundation. This paper is based on a portion of the PhD thesis (California Institute of Technology, 1970) by CWW.

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National Science Foundation Predoctoral Trainee.

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Alfred P. Sloan Foundation Research Fellow.

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Contribution No. 3917.     

Some positive cotes numbers for the chebyshev weight function

We show that, for the Chebyshev weight function (1–x ²)^–1/2, the Cotes numbers for the quadrature rule with nodes at the zeros of the ultraspherical polynomialP _n ^/() are nonnegative if and only if –1/2<1.     

An ab initio study of some structural features and the rotational barriers in performic acid,formic acid and hydrogen peroxide

Calculations on performic acid at the 4-31G level, with and without bond functions with complete geometry optimization, and at the (9, 5) level, with and without polarization functions and rigid rotation, all give no sign of a well in the potential energy curve for rotation about the O/O bond axis in the region of 50° – 90° ; and all but the unaugmented 4-31G basis set find the cis-cis planar conformer to be the most stable form. Calculations at the (9,5) level with rigid rotation find the energies of the other planar conformers, relative to the cis-cis conformer, to be 0.94, 1.50 and 14.80 kcal mol^?1 for the trans-trans, cis-trans and trans-cis structures respectively. These energies and also that for the barrier separating the cis-cis and cis-trans conformers, 1–2 kcal mol^?1, are discussed in relation to corresponding data for formic acid, hydrogen peroxide and several planar four heavy-atom molecules. Dipole moment calculations using the same basis sets would seem to favor a skew conformation as the most stable for performic acid, but comparisons between calculated and experimental values for formic acid and for hydrogen peroxide cast doubt on the validity of such results.