Stable solution of tridiagonal systems

In this paper we present three different pivoting strategies for solving general tridiagonal systems of linear equations. The first strategy resembles the classical method of Gaussian elimination with no pivoting and is stable provided a simple and easily checkable condition is met. In the second strategy, the growth of the elements is monitored so as to ensure backward stability in most cases. Finally, the third strategy also uses the right‐hand side vector to make pivoting decisions and is proved to be unconditionally backward stable. This revised version was published online in August 2006 with corrections to the Cover Date.     

Non-linear dynamical analysis of crack surface perturbations and their dependence on velocity and direction

The fracture surfaces of single crystal [1 0 0] silicon specimens, fractured under three-point bending (3PB) and subjected to a high strain energy upon cracking, revealed exceptional surface perturbations, generated during the unstable propagation. While macroscopically the crack is propagating on the (1 1 1) low energy cleavage plane, microscopic examination revealed small angled deviations from and fluctuations along that plane. Furthermore, while the crack is propagating at a velocity of nearly 3000 m/s in the direction, its velocity in the direction is two orders of magnitude lower, with distinctive surface perturbations. The amplitude and complexity of the perturbations increase as the normal velocity vector changes its direction and magnitude. These perturbations were recorded with a profilometer and analyzed using non-linear dynamical analysis tools. This study provides an opportunity to interpret surface phenomena of one of the most general cases of fracture and to study the effect of major variables on the nature of the perturbations involved, such as the local crack tip velocity and the crystallographic orientations. It is shown that the surface perturbations are chaotic deterministic in nature and can be described by high order non-linear differential equations; the order of the equation varying with the variations of the local velocity and direction.     

Nucleophilic substitution reactions at liquid/liquid interfaces: molecular dynamics simulation of a model S(N)1 dissociation reaction at the water/carbon tetrachloride interface

The ionic dissociation step of the nucleophilic substitution reaction t-BuCl --> t-Bu(+) + Cl(-) is studied at the water/carbon tetrachloride interface using molecular dynamics computer simulations. The empirical valence bond approach is used to couple two diabatic states, covalent and ionic, in the electronically adiabatic limit. The umbrella sampling technique is used to calculate the potential of mean force along the reaction coordinate (defined as the t-Bu to Cl distance) at several interface regions of varying distances from the Gibbs dividing surface. We find a significant increase of the ionic dissociation barrier height and of the reaction free energy at the interface relative to bulk water. This is shown to be due to the reduced polarity of the interface which causes a destabilization of the pure ionic state. However, deformation to the neat interface structure in the form of water protrusions into the organic phase may provide partial stabilization of the ionic species. The importance of these structural effects is examined by repeating the calculations with an artificially smooth interface. The destabilization of the ionic state at the interface also manifests itself with a rapid (picosecond time scale) recombination dynamics of the ions to form the parent molecule followed by a slow vibrational relaxation.     

Molecular dynamics simulation of the water|nitrobenzene interface

The structure and dynamics of the neat water|nitrobenzene liquid|liquid interface are studied at 300 K using molecular dynamics computer simulations. The water is modeled using the flexible SPC potential, and the nitrobenzene is modeled using an empirically determined nitrobenzene potential energy function. Although nitrobenzene is a polar liquid with a large dielectric constant, the structure of the interface is similar to other water|non-polar organic liquid interfaces. Among the main structural features we describe are an enhancement of interfacial water hydrogen bonds, the specific orientation of water dipoles and nitrobenzene molecules, and a rough surface that is locally sharp. Surface roughness is also characterized dynamically. The dynamics of molecular reorientation are shown to be only mildly modified at the interface. The effect due to the polarizable many-body potential energy functions of both liquids is investigated and is found to affect only mildly the above results.     

Colloidal CdSe quantum dot electroluminescence: ligands and light-emitting diodes

We examine the effects of surface ligand exchange on the performance of hybrid organic/inorganic light emitting diodes (LEDs) that use colloidal nanocrystal quantum dots as emissive centers. Using a series of primary alkylamines with different alkane chain lengths, we exchange the native surface ligands on a series of CdSe/CdZnS/ZnS core/shell/shell nanocrystal quantum dots and compare the differences in photoluminescence and electroluminescence efficiency of the emissive quantum dot layer. We fabricate LEDs made with octadecylamine-, octylamine-, and butylamine-exchanged quantum dots. We find that the differences in electroluminescence efficiency of the devices are not always proportional to the photoluminescence quantum efficiency of the quantum dots. We discuss this trend both in terms of the competing needs of high photoluminescence efficiency and good charge injection and energy transfer, and also in terms of the different processability and film morphology arising from the use of nanoparticles passivated with shorter ligands. Correspondence: David S. Ginger, Department of Chemistry, University of Washington, Seattle, Washington 98195-1700, USA     

Versatility in the Brook Rearrangement for the Selective Ring-Opening of Three-Membered Rings

From a single α-silylated carbinol intermediate, easily accessible by carbometallation of cyclopropenes, various scaffolds featuring a quaternary carbon stereocenter could be obtained selectively. The selectivity towards these different products was achieved by either changing the experimental conditions or the nature of the organometallic species involved.     

Stereoselective tandem iridium-catalyzed alkene isomerization-cope rearrangement of ω-diene epoxides: efficient access to acyclic 1,6-dicarbonyl compounds

The Cope rearrangement of 2,3-divinyloxiranes, a rare example of epoxide C–C bond cleavage, results in 4,5-dihydrooxepines which are amenable to hydrolysis, furnishing 1,6-dicarbonyl compounds containing two contiguous stereocenters at the 3- and 4-positions. We employ an Ir-based alkene isomerization catalyst to form the reactive 2,3-divinyloxirane in situ with complete regio- and stereocontrol, which translates into excellent control over the stereochemistry of the resulting oxepines and ultimately to an attractive strategy towards 1,6-dicarbonyl compounds.

Iridium catalyzed alkene isomerization-cope rearrangement of ω-diene epoxide furnishes 3,4-dihydrooxepines. These oxepines are hydrolyzed to diastereomerically pure 1,6-dicarbonyl compound containing two contiguous stereocenters within acyclic system.

1,6-Dicarbonyl compounds are widespread as targets and intermediates in organic synthesis.¹ Due to the “dissonant” polarizing effect induced by the two carbonyl groups,² these motifs are challenging to retrosynthetically disconnect into classical synthons. Unsurprisingly, many approaches toward 1,6-dicarbonyls rely on dimerization of α,β-unsaturated carbonyl compounds (Scheme 1a)³ or oxidative cleavage of substituted cyclohexene derivatives⁴ which significantly limits the range of possible products. Alternative strategies, such as the ring-opening of donor–acceptor cyclopropanes with enolate nucleophiles, efficiently form the 1,6-dicarbonyl skeleton, albeit with limited substrate scope (Scheme 1b).⁵ The Cope rearrangement of 1,5-dienes, featuring oxygen functionality in the 3- and 4-positions,⁶ represents a promising strategy towards 1,6-dicarbonyl compounds but suffers from lack of stereocontrol over the diene substrates, resulting in diastereomeric mixtures of products (Scheme 1c).Open in a separate windowScheme 1Selected approaches towards the formation of 1,6-dicarbonyl compounds and our proposed approach.A conceptually related approach towards the preparation of 1,6-dicarbonyl compounds is through the hydrolysis of 3,4-dihydrooxepines (Scheme 1d), which are in turn generated through the Cope rearrangement of 2,3-divinyloxiranes.⁷ Such a sigmatropic rearrangement is also noteworthy as a rare example where an epoxide C–C bond is selectively cleaved over the usually more reactive C–O bond. This intriguing rearrangement has been studied but its use in synthesis is scarce, presumably due to difficulties in the stereoselective synthesis and handling of the key divinyl epoxides.In line with our interest in the strategic application of alkene isomerization to generate reactive synthetic intermediates in stereodefined form,⁸ we posited to form the reactive 2,3-divinyloxiranes in situ, through alkene isomerization^9,10 of the simpler allyl epoxides, which are accessible in enantiomerically enriched form.¹¹ Such a strategy might greatly facilitate access to these intermediates and therefore uncover a synthetically attractive route toward 1,6-dicarbonyl compounds featuring two contiguous stereocenters.With this idea in mind, we first explored the isomerization and subsequent Cope rearrangement of allyl-vinyl epoxides 1 (Scheme 2). To induce isomerization, we employed a cationic iridium-based catalytic system,¹² which is known to reliably isomerize alkenes with high degrees of regio- and stereocontrol.¹³Open in a separate windowScheme 2Substrate scope for the tandem iridium-catalyzed alkene isomerization-Cope rearrangement of allyl-vinyl epoxides.In line with our expectations, our model substrate 1a (R² = R³ = H, R⁴ = Me, R⁵ = CO₂Et) was smoothly isomerized at 65 °C in the presence of 1.5 mol% of Ir dimer to obtain the corresponding divinyl epoxide with a complete E-selectivity. With suitable conditions for alkene isomerization in hand, we exposed substrate 1a to the Ir-based catalytic system at 120 °C and were equally pleased to observe the 4,5-dihydrooxepine product 2a, resulting from the tandem isomerization-Cope rearrangement as a single diastereoisomer in 81% yield. We proceeded to test the generality of our protocol with respect to different alkene and epoxide substitution patterns. Pleasingly, product 2b was generated with complete stereoselectivity, showcasing the compatibility of the reaction conditions with potentially labile tertiary stereocenters α to the ester group. We then wondered whether the anti-diastereomer could be accessed starting from the corresponding cis allyl-vinyl epoxide. Indeed, in line with the known stereospecific behavior of the Cope rearrangement, we obtained the complementary diastereomer 2c. Turning our attention to more highly substituted epoxides, we were pleased to observe the formation of dihydrooxepines 2d and 2e, which correspond to 1,6-keto-aldehyde and diketone products, respectively. Substrate 1f (R² = R⁴ = R⁵ = H, R³ = Ph), which features an unactivated vinyl group, also underwent the rearrangement, demonstrating that an activated alkenyl group is not required for a successful outcome. Similarly, product 2g featuring two alkyl groups is also generated, with high diastereoselectivity albeit in moderate yield. Products featuring ethyl and methyl ester 2h, 2i could also be obtained in good yields and diastereoselectivity. We next tested substrate 1j (R² = Me, R³ = Ph, R⁴ = CH₂CH₂Ph, R⁵ = H), as a geometric-mixture of the double bond (E : Z = 1.1 : 1) and in accordance with the stereospecificity of the process, the oxepine 2j was obtained as a mixture of two diastereomers with the same ratio. Disappointingly, substrate 1k did not undergo isomerization, presumably due to the Lewis basic nature of the ketone, likely poisoning the Ir-catalyst.During our study, we noticed that allyl-vinyl epoxides bearing electron donating groups on the vinyl moiety tend to decompose during purification by column chromatography on silica gel. This obstacle further motivated us to explore diallyl epoxides 3 as substrates, where the reactive divinyl epoxide would be generated by isomerization of both allyl fragments. Notably, these diallyl epoxides are much more stable compared to their vinyl counterparts and can be readily prepared in two steps from simple alkynes.¹⁴ To our delight, diallyl epoxide 3a (R = CH₂OMe) smoothly underwent the double isomerization-Cope rearrangement cascade at 140 °C, furnishing oxepine 2l with impressive yield and diastereoselectivity (Scheme 3). The use of alkene isomerization to form the reactive divinyl epoxide in situ avoids the isolation of the unstable divinyl epoxide, while controlling the stereochemistry of both double bonds, particularly not trivial to achieve using classical olefination reactions. Products 2m and 2n feature ester and silyl groups, highlighting the functional group tolerance of the catalytic system.Open in a separate windowScheme 3Substrate scope for tandem iridium-catalyzed double alkene isomerization-Cope rearrangement of diallyl epoxides.Our next objective was to hydrolyze the diastereomerically pure oxepines obtained through the rearrangement in a stereoretentive fashion, revealing the acyclic 1,6-dicarbonyl motif. Pleasingly, diversely substituted oxepines 2 underwent smooth hydrolysis either using 5 mol% of Pd(MeCN)₂Cl₂¹⁵ at 50 °C or an acidic aqueous solution to form 1,6-dicarbonyls 4 in diastereomerically pure form (Scheme 4).¹⁶ Dicarbonyl products featuring labile tertiary centers 4a and 4b are formed under these conditions with excellent diastereoselectivities and yields. Without surprise, oxepine 2f (R² = R⁴ = R⁵ = H, R³ = Ph) furnished the keto-substituted product 4c in good yield. The relative stereochemistry of 4b was unambiguously confirmed by single crystal X-ray diffraction analysis of the corresponding carboxylic acid 7 (Scheme 4b).¹⁷ The reaction is scalable to ½ gram of substrate and could be performed in a single-pot operation without isolation of the intermediate oxepine (Scheme 4b). By using this approach, 1h provides 4b in 61% yield as a single diastereomer, underlining the synthetic potential and efficiency of this method.Open in a separate windowScheme 4Hydrolysis of oxepines and one-pot sequence.     

Molecular dynamics study of the vibrational relaxation of OCl and OCl in the bulk and the surface of water and acetonitrile

The vibrational relaxation of OCl and OCl⁻ in the bulk and the liquid/vapor interface of water and acetonitrile is studied by molecular dynamics computer simulations. Both equilibrium calculations of the vibrational friction and non-equilibrium simulation of the energy relaxation are used to elucidate the factors that influence the rate of energy relaxation in systems that represent polar ionic and non-ionic solutes in polar protic and non-protic solvents. We find that, in agreement with previous experimental and theoretical studies, the relaxation of the ionic solute is much faster than that of the non-ionic solute in both the solvents. However, while the relaxation is slowed down considerably when the non-ionic solute is transferred from the bulk to the interface, no such surface effect is found in the case of the ionic solute. This behavior can be explained by noting that the ionic solute is able to keep its first solvation shell intact upon transfer to the interface and that the main contribution to the friction is due to the Lennard-Jones part of the intermolecular potential.     

Hats,auctions and derandomization

We investigate derandomizations of digital good randomized auctions. We propose a general derandomization method which can be used to show that for every random auction there exists a deterministic auction having asymptotically the same revenue. In addition, we construct an explicit optimal deterministic auction for bi‐valued auctions. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 478–493, 2015