Synthesis of meso‐Pyrrole‐Substituted 22‐Oxacorroles by a “3+2” Approach

Unsymmetrical 22‐oxacorrole containing two aryl groups and one pyrrole group at the meso position was synthesized by condensing one equivalent of 16‐oxatripyrrane with one equivalent of meso aryl dipyromethane under mild acid‐catalyzed conditions followed by oxidation with 2,3‐dichloro‐5,6‐dicyano‐1,4‐benzoquinone (DDQ). This [3+2] condensation approach was expected to yield meso‐free 25‐oxasmaragdyrin but unexpectedly afforded unsymmetrical meso‐pyrrole‐substituted 22‐oxacorrole. We demonstrated the versatility of the reaction by synthesizing four new meso‐pyrrole‐substituted 22‐oxacorroles. The reactivity of α‐position of meso‐pyrrole was tested by carrying out various functionalization reactions such as bromination, formylation, and nitration and obtained the functionalized meso‐pyrrole‐substituted 22‐oxacorroles in decent yields. The X‐ray structure obtained for one of the functionalized meso‐pyrrole substituted 22‐oxacorrole revealed that the macrocycle was nearly planar and the meso‐pyrrole was in the perpendicular orientation with respect to the macrocyclic plane. The meso‐pyrrole‐substituted 22‐oxacorroles absorb strongly in 400–700 nm region with one strong Soret band and four weak Q bands. The 22‐oxacorroles are strongly fluorescent and showed emission maxima at ≈650 nm with decent quantum yields and singlet‐state lifetimes. The 22‐oxacorroles are redox‐active and exhibited three irreversible oxidations and one or two reversible reduction(s). A preliminary biological study indicated that meso‐pyrrole corroles are biocompatible.     

The complexity of approximating a nonlinear program

We consider the problem of finding the maximum of a multivariate polynomial inside a convex polytope. We show that there is no polynomial time approximation algorithm for this problem, even one with a very poor guarantee, unless P = NP. We show that even when the polynomial is quadratic (i.e. quadratic programming) there is no polynomial time approximation unless NP is contained in quasi-polynomial time.Our results rely on recent advances in the theory of interactive proof systems. They exemplify an interesting interplay of discrete and continuous mathematics—using a combinatorial argument to get a hardness result for a continuous optimization problem.     

Thermodynamic properties of gold–water nanofluids using molecular dynamics

Conflicts and discrepancies around nanoparticle (NP) size effect on the optical properties of metal NPs of sizes below the mean free path of electron can be traced to the internal damping effect of the hybrid resonance of the inner band (IB) and the conduction band (CB) electrons of the noble metals. We present a scheme to show how alternative mathematical formulation of the physics of interaction between the CB and the IB electrons of NP sizes <50?nm justifies this and resolves the conflicts. While a number of controversies exist between classical and quantum theories over the phenomenological factors to attribute to the NP size effect on the absorption bandwidth, this article shows that the bandwidth behavior can be well predicted from a different treatment of the IB damping effect, without invoking any of the controversial phenomenological factors. It finds that the IB damping effect is mainly frequency dependent and only partly size dependent and shows how its influence on the surface plasmon resonance can be modeled to show the influence of NP size on the absorption properties. Through the model, it is revealed that strong coupling of IB and CB electrons drastically alters the absorption spectra, splitting it into distinctive dipole and quadrupole modes and even introduce a behavioral switch. It finds a strong overlap between the IB and the CB absorptions for Au and Cu but not Ag, which is sensitive to the NP environment. The CB modes shift with the changing refractive index of the medium in a way that can allow their independent excitation, free of influence of the IB electrons. Through a hybrid of parameters, the model further finds that metal NP sizes can be established not only by their spectral absorption peak locations but also from a proper correlation of the peak location and the bandwidth (FWHM).     

Synthesis and characterization of stable organosols of silver nanoparticles by electrochemical dissolution of silver in DMSO

Stable organosols of silver nanoparticles (AgNPs) without any capping agents have been synthesized by an electrochemical dissolution of a sacrificing silver electrode in dimethyl sulfoxide (DMSO). The peak at 425 +/- 5 nm observed in the UV-vis spectra was attributed to the surface plasmon resonance for silver. The formation of nanoparticles of silver was confirmed by X-ray diffraction analysis (XRD). In TEM, three ranges of particle size were observed, namely, 3.75 +/- 0.8, 6.25 +/- 0.5, and 9.25 +/- 0.3 nm. The correlation among these sizes was explained by a new model based on a droplet coalition. Based on that, sizes correlation fits very well in the empirical formula 5d(1)(3)n' + (3 - n')d(2)(3) = d(3)(3) where d(1), d(2), and d(3) are three sizes of particles and n' is an integer having values 0, 1, 2, 3, .... The sols prepared in DMSO were stable against flocculation for months. An unusual solution stability without any capping agents was attributed to the formation of Ag(I)DMSO complex on particle surface, which was confirmed by FTIR, fluorescence spectroscopy, and thermal analysis.     

Effect of field quantization on Rabi oscillation of equidistant cascade four-level system

We have exactly solved a model of equidistant cascade four-level system interacting with a single-mode radiation field both semiclassically and quantum mechanically by exploiting its similarity with Jaynes-Cummings model. For the classical field, it is shown that the Rabi oscillation of the system initially in the first level (second level) is similar to that of the system when it is initially in the fourth level (third level). We then proceed to solve the quantized version of the model where the dressed state is constructed using a six-parameter four-dimensional matrix and show that the symmetry exhibited in the Rabi oscillation of the system for the semiclassical model is completely destroyed on the quantization of the cavity field. Finally, we have studied the collapse and revival of the system for the cavity field-mode in a coherent state to discuss the restoration of symmetry and its implication is discussed.      

Optical rotatory dispersion of rare earth crystal

The ORD spectrum of optically active single crystal of praseodymium oxydiacetate has been studied in the visible range at 77 K. The results have been compared with those obtained in the cases of transition metal complexes and other RE complexes with optically active ligands, and correlated with the site symmetry of the RE ion in the crystal. The optical activity has been related to the magnetic dipole strengths of the transitions.     

Molybdenum(VI)-peroxo complex catalyzed oxidation of alkylbenzenes with hydrogen peroxide

The molybdenum(VI) peroxo complex 1, synthesized from the reaction of MoO₃ with H₂O₂ and 3,5-dimethylpyrazole (dmpz), selectively oxidizes benzylic C-H bonds of alkylbenzenes to the corresponding alcohols and ketones in moderate to good yields in the presence of H₂O₂ in acetonitrile under reflux (ca. 80°C).     

A proof of Howard's conjecture in homogeneous parallel shear flows—II: Limitations of Fjortoft's necessary instability criterion

The present paper on the linear instability of nonviscous homogeneous parallel shear flows mathematically demonstrates the correctness of Howard's [4] prediction, for a class of velocity distributions specified by a monotone functionU of the altitudey and a single point of inflexion in the domain of flow, by showing not only the existence of a critical wave numberk _c>0 but also deriving an explicit expression for it, beyond which for all wave numbers the manifesting perturbations attain stability. An exciting conclusion to which the above result leads to is that the necessary instability criterion of Fjortoft has the seeds of its own destruction in the entire range of wave numbersk>k _c—a result which is not at all evident either from the criterion itself or from its derivation and has thus remained undiscovered ever since Fjortoft enunciated [3].