A Neural Network-Based Policy Iteration Algorithm with Global $$H^2$$ -Superlinear Convergence for Stochastic Games on Domains

In this work, we propose a class of numerical schemes for solving semilinear Hamilton–Jacobi–Bellman–Isaacs (HJBI) boundary value problems which arise naturally from exit time problems of diffusion processes with controlled drift. We exploit policy iteration to reduce the semilinear problem into a sequence of linear Dirichlet problems, which are subsequently approximated by a multilayer feedforward neural network ansatz. We establish that the numerical solutions converge globally in the \(H^2\)-norm and further demonstrate that this convergence is superlinear, by interpreting the algorithm as an inexact Newton iteration for the HJBI equation. Moreover, we construct the optimal feedback controls from the numerical value functions and deduce convergence. The numerical schemes and convergence results are then extended to oblique derivative boundary conditions. Numerical experiments on the stochastic Zermelo navigation problem are presented to illustrate the theoretical results and to demonstrate the effectiveness of the method.

     

Synthesis of μ2-Oxo-Bridged Iron(III) Tetraphenylporphyrin–Spacer–Nitroxide Dimers and their Structural and Dynamics Characterization by using EPR and MD Simulations

Iron(III) porphyrins have the propensity to form μ₂-oxo-dimers, the structures of which resemble two wheels on an axle. Whereas their crystal structure is known, their solution structure and internal dynamics is not. In the present work, the structure and dynamics of such dimers were studied by means of electron paramagnetic resonance (EPR) spectroscopy and quantum chemistry based molecular dynamics (MD) simulations by using the semiempirical tight-binding method (GFN-xTB). To enable EPR investigation of the dimers, a nitroxide was attached to each of the tetraphenylporphyrin cores through a linear and a bent linker. The inter-nitroxide distance distributions within the dimers were determined by continuous-wave (cw)-EPR and pulsed electron–electron double resonance (PELDOR or DEER) experiments and, with the help of MD, interpreted in terms of the rotation of the porphyrin planes with respect to each other around the Fe–O–Fe axis. It was found that such rotation is restricted to the four registers defined by the phenyl substituents. Within the registers, the rotation angle swings between 30° and 60° in the proximal and between 125° and 145° in the distal register. With EPR, all four angles were found to be equally populated, whereas the 30° and 145° angles are strongly favored to the expense of the 60° and 125° angles in the MD simulation. In either case, the internal dynamics of these dimers thus resemble the motion of a step motor.     

Synthesis,Crystal Structure,and Selected Properties of [Au(S2CNH2)2]SCN: A Precursor for Gold Macro-Needles Consisting of Gold Nanoparticles Glued by Graphitic Carbon Nitride

A new preparation route is developed for the synthesis of needle-like crystals of [Au(S₂CNH₂)₂]SCN, which avoids disproportionation of the Au^I salt used as a starting material. In the crystal structure, the two crystallographically independent Au^III centers are in a square-planar environment of two S₂CNH₂ ligands. The Hirshfeld surface analysis reveals the presence of noncovalent intermolecular S⋅⋅⋅S interactions, which are essential for the spatial arrangement of the molecules. Density functional theory (DFT) calculations including dispersion and damping corrections result in a unit cell volume very close to the value determined experimentally. Thermal decomposition in an inert atmosphere generates black needles with lengths of up to 500 μm. X-ray powder diffraction and pair distribution function analyses demonstrate that the needles are composed of nanosized crystals with a volume-weighted average domain size of 20(1) nm. According to results of X-ray photoemission experiments, the black needles are covered by a nitrogen-rich carbon nitride with composition near (CN)₂N. ¹³C solid-state NMR investigations indicate that two different carbon species are present, with signals corresponding well to heptazine units as in melon and triazine units as in poly(triazin imide) type compounds. Scanning transmission electron microscopy tomography evidences that the needles are composed of slightly elongated nanoparticles.     

Stereoselective Cascade Cyclizations with Samarium Diiodide to Tetracyclic Indolines: Precursors of Fluorostrychnines and Brucine

A series of γ-indolylketones with fluorine, cyano or alkoxy substituents at the benzene moiety was prepared and subjected to samarium diiodide-promoted cyclization reactions. The desired dearomatizing ketyl cascade reaction forming two new rings proceeded in all cases with high diastereoselectivity, but with differing product distribution. In most cases, the desired annulated tetracyclic compounds were obtained in moderate to good yields, but as second product tetracyclic spirolactones were isolated in up to 29 % yield. The reaction rate was influenced by the substituents at the benzene moiety of the substrate as expected, with electron-accepting groups accelerating and electron-donating groups decelerating the cyclization process. In case of a difluoro-substituted γ-indolylketone a partial defluorination was observed. The intermediate samarium enolate of the tetracyclic products could be trapped by adding reactive alkylating agents as electrophiles delivering products with quarternary carbons. In the case of a dimethoxy-substituted tetracyclic cyclization product a subsequent reductive amination stereoselectively provided a pentacyclic compound that was subsequently N-protected and subjected to a regioselective elimination. The obtained functionalized pentacyclic product should be convertible into the alkaloid brucine by four well-established steps. Overall, the presented report shows that functionalized tetracyclic compounds with different substituents are rapidly available with the samarium diiodide cascade cyclization as crucial step. Hence, analogues of the landmark alkaloid strychnine, for example, with specific fluorine substitutions, should be easily accessible.     

Alkali Blues: Blue-Emissive Alkali Metal Pyrrolates

2-Iminopyrroles [H^tBuL, 4-tert-butyl phenyl(pyrrol-2-ylmethylene)amine] are non-fluorescent π systems. However, they display blue fluorescence after deprotonation with alkali metal bases in the solid state and in solution at room temperature. In the solid state, the alkali metal 2-imino pyrrolates, M(^tBuL), aggregate to dimers, [M(^tBuL)(NCR)]₂ (M=Li, R=CH₃, CH(CH₃)CNH₂), or polymers, [M(^tBuL)]_n (M=Na, K). In solution (solv=CH₃CN, DMSO, THF, and toluene), solvated, uncharged monomeric species M(^tBuL)(solv)_m with N,N′-chelated alkali metal ions are present. Due to the electron-rich pyrrolate and the electron-poor arylimino moiety, the M(^tBuL) chromophore possesses a low-energy intraligand charge-transfer (ILCT) excited state. The chelated alkali cations rigidify the chromophore, restricting intramolecular motions (RIM) by the chelation-enhanced fluorescence (CHEF) effect in solution and, consequently, switch-on a blue fluorescence emission.     

Iodine(III)‐Catalyzed Rearrangements of Imides: A Versatile Route to α,α‐Dialkylated α‐Hydroxy Carboxylamides

A tertiary hydroxy group α to a carboxyl moiety comprises a key structural motif in many bioactive substances. With the herein presented metal‐free rearrangement of imides triggered by hypervalent λ³‐iodane, an easy and selective way to gain access to such a compound class, namely α,α‐disubstituted‐α‐hydroxy carboxylamides, was established. Their additional methylene bromide side chain constitutes a useful handle for rapid diversification, as demonstrated by a series of further functionalizations. Moreover, the in situ formation of an iodine(III) species under the reaction conditions was proven. Our findings clearly corroborate that hypervalent λ³‐benziodoxolones are involved in these organocatalytic reactions.     

The Effect of the Spacer of Bis(biurea) Ligands on the Structure of A2L3‐type (A=anion) Phosphate Complexes

By tuning the length and rigidity of the spacer of bis(biurea) ligands L, three structural motifs of the A₂L₃ complexes (A represents anion, here orthophosphate PO₄^3?), namely helicate, mesocate, and mono‐bridged motif, have been assembled by coordination of the ligand to phosphate anion. Crystal structure analysis indicated that in the three complexes, each of the phosphate ions is coordinated by twelve hydrogen bonds from six surrounding urea groups. The anion coordination properties in solution have also been studied. The results further demonstrate the coordination behavior of phosphate ion, which shows strong tendency for coordination saturation and geometrical preference, thus allowing for the assembly of novel anion coordination‐based structures as in transition‐metal complexes.     

Association and Dissociation of Grignard Reagents RMgCl and Their Turbo Variant RMgCl⋅LiCl

Grignard reagents RMgCl and their so‐called turbo variant, the highly reactive RMgCl?LiCl, are of exceptional synthetic utility. Nevertheless, it is still not fully understood which species these compounds form in solution and, in particular, in which way LiCl exerts its reactivity‐enhancing effect. A combination of electrospray‐ionization mass spectrometry, electrical conductivity measurements, NMR spectroscopy (including diffusion‐ordered spectroscopy), and quantum chemical calculations is used to analyze solutions of RMgCl (R=Me, Et, Bu, Hex, Oct, Dec, iPr, tBu, Ph) in tetrahydrofuran and other ethereal solvents in the absence and presence of stoichiometric amounts of LiCl. In tetrahydrofuran, RMgCl forms mononuclear species, which are converted into trinuclear anions as a result of the concentration increase experienced during the electrospray process. These trinuclear anions are theoretically predicted to adopt open cubic geometries, which remarkably resemble structural motifs previously found in the solid state. The molecular constituents of RMgCl and RMgCl?LiCl are interrelated via Schlenk equilibria and fast intermolecular exchange processes. A small portion of the Grignard reagent also forms anionic ate complexes in solution. The abundance of these more electron‐rich and hence supposedly more nucleophilic ate complexes strongly increases upon the addition of LiCl, thus rationalizing its beneficial effect on the reactivity of Grignard reagents.