Stochastic search of the quantum conformational space of small lithium and bimetallic lithium-sodium clusters

In this paper we report the results obtained by an implementation and application of the simulated annealing optimization procedure to the exploration of the conformational space of small neutral and charged lithium clusters (Li(n)(q), n = 5, 6, 7; q = 0, +/-1) and of the bimetallic lithium/sodium clusters (Li5Na) in their lowest spin states. Our methodology eliminates the structure guessing procedure in the process of generating cluster configurations. We evaluate the quantum energy, typically with the Hartree-Fock Hamiltonian, of randomly generated points in the conformational space and use a modified Metropolis test in the annealing algorithm to generate candidate structures for atomic clusters. The structures are further optimized by analytical methods (gradient following) at the M?ller-Plesset second order perturbation theory level (MP2), in conjunction with basis sets including polarization functions with and without diffuse functions. High accuracy ab initio energies at the coupled clusters level, with single, double, and triple substitutions from the Hartree-Fock determinant (CCSD(T)), on the MP2 geometries were calculated and used to establish the relative stability of the isomers within each potential energy surface. Various cluster properties were computed and compared to existing values in order to validate our methods. Our results show excellent agreement with previous experimental and theoretical reports. Even at these small sizes, evidence for 10 new structures never reported before for the lithium clusters and four new structures for the bimetallic clusters is presented.     

Thick clusters for the radially symmetric nonlinear Schrödinger equation

This article is devoted to the study of radially symmetric solutions to the nonlinear Schrödinger equation

$\varepsilon^2 \Delta u - V(r)u + |u|^{p-1}u = 0\, {\rm in} B,\quad \frac{\partial u}{\partial n} = 0\, {\rm on}\,{\partial}B,$

where B is a ball in \({\mathbb{R}}^N\) , 1 <  p <  (N +  2)/(N ? 2), N ≥ 3 and the potential V is radially symmetric. We construct positive clustering solutions in an annulus having O(1/?) critical points, as well as sign changing solutions with O(1/?) zeroes concentrating near zero.

     

A simple analysis of the influence of the solvent-induced electronic polarization on the <Superscript>15</Superscript>N magnetic shielding of pyridine in water

Electronic polarization induced by the interaction of a reference molecule with a liquid environment is expected to affect the magnetic shielding constants. Understanding this effect using realistic theoretical models is important for proper use of nuclear magnetic resonance in molecular characterization. In this work, we consider the pyridine molecule in water as a model system to briefly investigate this aspect. Thus, Monte Carlo simulations and quantum mechanics calculations based on the B3LYP/6-311++G (d,p) are used to analyze different aspects of the solvent effects on the ¹⁵N magnetic shielding constant of pyridine in water. This includes in special the geometry relaxation and the electronic polarization of the solute by the solvent. The polarization effect is found to be very important, but, as expected for pyridine, the geometry relaxation contribution is essentially negligible. Using an average electrostatic model of the solvent, the magnetic shielding constant is calculated as −58.7 ppm, in good agreement with the experimental value of −56.3 ppm. The explicit inclusion of hydrogen-bonded water molecules embedded in the electrostatic field of the remaining solvent molecules gives the value of −61.8 ppm.     

Exhaustive oxidation of a nickel dithiolate complex: some mechanistic insights en route to sulfate formation

A study of the step-wise oxidation of a Ni(II) diaminodithiolate complex through the formation of sulfate, the ultimate sulfur oxygenate, is reported. Controlled oxygenations or peroxidations of a neutral, planar, tetracoordinate, low-spin Ni(II) complex of a N(2)S(2)-donor ligand, (N,N'-dimethyl-N-N'-bis(2-mecaptoethyl)-1,3-propanediaminato) nickel(ii) (1), led to a series of sulfur oxygenates that have been isolated and characterized by ESI-MS and single-crystal X-ray diffraction. A monosulfenate complex (2) was detected by ESI-MS as a product of oxidation with one equivalent of H(2)O(2). However, this complex proved too unstable to isolate. Reaction of the dithiolate (1) with two equivalents of H(2)O(2) or one O(2) molecule leads to the formation of a monosulfinate complex (3), which was isolated and fully characterized by crystallography. The oxidation product of the monosulfinate (3) produced with either O(2) or H(2)O(2) is an interesting dimeric complex containing both sulfonate and thiolate ligands (4), this complex was fully characterized by crystallography, details of which were reported earlier by us. A disulfonate complex (7) is produced by reaction of 1 in the presence of O(2) or by reaction with exactly six equivalents of H(2)O(2). This complex was isolated and also fully characterized by crystallography. Possible intermediates in the conversion of the monosulfinate complex (3) to the disulfonate complex (7) include complexes with mixed sulfonate/sulfenate (5) or sulfonate/sulfinate (6) ligands. Complex 5, a four-oxygen adduct of 1, was not detected, but the sulfonate/sulfinate complex (6) was isolated and characterized. The oxidation chemistry of 1 is very different from that reported for other planar cis-N(2)S(2) Ni(ii) complexes including N,N'-dimethyl-N-N'-bis(2-mecaptoethyl)-1,3-ethylenediaminato) nickel(II), (8), and N,N'-bis(mercaptoethyl)-1,5-diazacyclooctane nickel(II). To address the structural aspects of the reactivity differences, the crystal structure of 8 was also determined. A comparison of the structures of planar Ni(II) complexes containing cis-dithiolate ligands, strongly suggests that the differences in reactivity are determined in part by the degree of flexibility that is allowed by the NN' chelate ring.     

Students’ Conceptions of Congruency Through the Use of Dynamic Geometry Software

This paper describes students’ interactions with dynamic diagrams in the context of an American geometry class. Students used the dragging tool and the measuring tool in Cabri Geometry to make mathematical conjectures. The analysis, using the cK￠ model of conceptions, suggests that incorporating technology in mathematics classrooms enabled a measure-preserving conception of congruency with which students’ could shift focus from shapes to properties. Students also interacted with dynamic diagrams in a novel way, which we call the functional mode of interaction with diagrams, relating outputs and inputs that result when dragging a figure. Students’ participation in classroom interactions through discourse and through actions on diagrams provided evidence of learning using tools within dynamic geometry software.     

Rational inversion of the Laplace transform

This paper studies new inversion methods for the Laplace transform of vector-valued functions arising from a combination of A-stable rational approximation schemes to the exponential and the shift operator semigroup. Each inversion method is provided in the form of a (finite) linear combination of the Laplace transform of the function and a finite amount of its derivatives. Seven explicit methods arising from A-stable schemes are provided, such as the Backward Euler, RadauIIA, Crank-Nicolson, and Calahan scheme. The main result shows that, if a function has an analytic extension to a sector containing the nonnegative real line, then the error estimate for each method is uniform in time.     

Free‐decomposability in varieties of semi‐Heyting algebras

In this paper we prove that the free algebras in a subvariety $\mathcal V$ of the variety $\mathcal {SH}$ of semi‐Heyting algebras are directly decomposable if and only if $\mathcal V$ satisfies the Stone identity.