“Brown-Ring”-Related Coordination Polymers of the Quartet-{FeNO}7 Chromophore

A conspicuous detail of the so-called brown-ring test (the analytical test on nitrate) is the reddish color of the bottom layer of concentrated sulfuric acid, which develops upon the bleeding of the brown layer into the acid. Crystals of the same color form from a solution of ferrous sulfate in concentrated sulfuric acid on saturation with gaseous nitric oxide. The structure of this H₃O[{Fe(NO)(μ₄-SO₄)(μ₂-SO₄)_0.5}_n/n] ( 1a ) is made up from infinite chessboard-type layers with sulfur on the field junctions and Fe(NO) moieties below the black and above the white fields. An Fe–N–O angle of about 160° causes disorder in the tetragonal space group I4/mmm. A similar crystal pathology was found in the related [{Fe(MeOH)(NO)(μ₄-SO₄)}_n/n] ( 1b ) in the same crystal class. A one-dimensional coordination polymer is formed in crystals of a third compound that comprises the Fe(NO)O₅ coordination pattern, namely the brown oxalato species [{Fe(H₂O)(NO)(μ₂-ox)}_n/n · H₂O] ( 2 ). A still larger NO tilt of about 156° is not obscured by disorder in the triclinic crystals of 2 .     

Short interval asymptotics for a class of arithmetic functions

Summary We provide a general asymptotic formula which permits applications to sums like <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"3"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"4"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"5"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"6"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"7"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"8"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"9"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation> \sum_{x< n\le x+y} \big(d(n)\big)^2, \quad \sum_{x< n\le x+y} d(n^3),\quad \sum_{x< n\le x+y}\big(r(n)\big)^2, \quad \sum_{x< n\le x+y}r(n^3), $$ where $d(n)$ and $r(n)$ are the usual arithmetic functions (number of divisors, sums of two squares), and $y$ is small compared to~$x$.     

Lattice points in a circle and divisors in arithmetic progressions

Let A(x) denote the number of lattice points in the circle u²+v²x and define as the infimum of all reals for which . The objective of this paper is to show that 35/108 which improves upon all previously known results. This estimate is an immediate consequence of a surprisingly easy generalization of KOLESNIK's work on Dirichlet's divisor problem to divisor functions with respect to arithmetic progressions.     

Higher order derivative tests for exponential sums incorporating the Discrete Hardy–Littlewood method

This paper provides estimates for exponential sums, combining classic tools of Van der Corput type with a deep result from the modern “discrete Hardy–Littlewood method”. As an application, an improved bound for the lattice point discrepancy of a large ellipsoid of rotation is deduced.     

On the Glivenko-Cantelli problem in stochastic programming: Mixed-integer linear recourse

Expected recourse functions in linear two-stage stochastic programs with mixed-integer second stage are approximated by estimating the underlying probability distribution via empirical measures. Under mild conditions, almost sure uniform convergence of the empirical means to the original expected recourse function is established.     

Thermal decomposition of trimethylgallium Ga(CH3)3: a shock-tube study and first-principles calculations

The thermal decomposition of Ga(CH3)3 has been studied both experimentally in shock-heated gases and theoretically within an ab-initio framework. Experiments for pressures ranging from 0.3 to 4 bar were performed in a shock tube equipped with atomic resonance absorption spectroscopy (ARAS) for Ga atoms at 403.3 nm. Time-resolved measurements of Ga atom concentrations were conducted behind incident waves as well as behind reflected shock waves at temperatures between 1210 and 1630 K. The temporal variation in Ga-atom concentration was described by a reaction mechanism involving the successive abstraction of methyl radicals from Ga(CH3)3 (R1), Ga(CH3)2 (R2), and GaCH3 (R3), respectively, where the last reaction is the rate-limiting step leading to Ga-atom formation. The rate constant of this reaction (R3) was deduced from a simulation of the measured Ga-atom concentration profiles using thermochemical data from ab-initio calculations for the reactions R1 and R2 as input. The Rice-Ramsperger-Kassel-Marcus (RRKM) method including variational transition state theory was applied for reaction R3 assuming a loose transition state. Structural parameters and vibrational frequencies of the reactant and transition state required for the RRKM calculations were obtained from first-principles simulations. The energy barrier E3(0) of reaction R3, which is the most sensitive parameter in the calculation, was adjusted until the RRKM rate constant matched the experimental one and was found to be E(0) = 288 kJ/mol. This value is in a good agreement with the corresponding ab-initio value of 266 kJ/mol. The rate constant of reaction R3 was found to be k 3/(cm(3) mol(-1)s(-1)) = 2.34 x 10(11) exp[-23330(K/ T)].     

Synthesis of amino acid derived enaminones via Wolff rearrangement using vinylogous amides as carbon nucleophiles

Cyclic enaminones were synthesized in high yields from amino acids in two steps via Wolff rearrangement. The cyclization represents a rare 6-exo-dig cyclization involving a ketene as an electrophile. No racemization was observed during this reaction.     

On the equivalence of ring-coupled cluster and adiabatic connection fluctuation-dissipation theorem random phase approximation correlation energy expressions

The correlation energy in the direct random phase approximation (dRPA) can be written, among other possibilities, either in terms of the interaction strength averaged correlation density matrix, or in terms of the coupled cluster doubles amplitudes obtained in the direct ring approximation (drCCD). Although the corresponding dRPA correlation density matrix on the one hand, and the drCCD amplitude matrix on the other hand, differ significantly, they yield identical energies. Similarly, the analogous RPA and rCCD correlation energies calculated from antisymmetrized two-electron integrals are identical to each other despite very different underlying working equations. In the present communication, a direct correspondence between amplitudes and densities is established and investigated with perturbation theory arguments. Our analysis also sheds some light on the properties of recently proposed RPA/rCCD variants which use antisymmetrized integrals in part of the equations and nonantisymmetrized integrals in others.