Quantal Two-Centre Coulomb Problem treated by means of the Phase-Integral Method II. Quantization Conditions in the Symmetric Case Expressed in Terms of Complete Elliptic Integrals. Numerical Illustration

The contour integrals, occurring in the arbitrary-order phase-integral quantization conditions given in a previous paper, are in the first- and third-order approximations expressed in terms of complete elliptic integrals in the case that the charges of the Coulomb centres are equal. The evaluation of the integrals is facilitated by the knowledge of quasiclassical dynamics. The resulting quantization conditions involving complete elliptic integrals are solved numerically to obtain the energy eigenvalues and the separation constants of the $1s\sigma$ and $2p\sigma$ states of the hydrogen molecule ion for various values of the internuclear distance. The accuracy of the formulas obtained is illustrated by comparison with available numerically exact results.Comment: 19 pages, RevTeX 4, 4 EPS figures, submitted to J. Math. Phy

Quantal Two-Centre Coulomb Problem treated by means of the Phase-Integral Method I. General Theory

The present paper concerns the derivation of phase-integral quantization conditions for the two-centre Coulomb problem under the assumption that the two Coulomb centres are fixed. With this restriction we treat the general two-centre Coulomb problem according to the phase-integral method, in which one uses an {\it a priori} unspecified {\it base function}. We consider base functions containing three unspecified parameters $C, \tilde C$ and $\Lambda$. When the absolute value of the magnetic quantum number $m$ is not too small, it is most appropriate to choose $\Lambda=|m|\ne 0$. When, on the other hand, $|m|$ is sufficiently small, it is most appropriate to choose $\Lambda = 0$. Arbitrary-order phase-integral quantization conditions are obtained for these choices of $\Lambda$. The parameters $C$ and $\tilde C$ are determined from the requirement that the results of the first and the third order of the phase-integral approximation coincide, which makes the first-order approximation as good as possible. In order to make the paper to some extent self-contained, a short review of the phase-integral method is given in the Appendix.Comment: 23 pages, RevTeX, 4 EPS figures, submitted to J. Math. Phy

Parametric triggering of vortices in toroidally trapped rotating Bose-Einstein condensates

We study the creation of vortices by triggering the rotating Bose-Einstein condensates in a toroidal trap with trap parameters such as laser beam waist and Gaussian potential depth. By numerically solving the time-dependent Gross-Pitaevskii equation in two dimensions, we observe a change in vortex structure and a considerable increase in the number of vortices when the waist of the irradiated laser beam is in consonance with the area of the condensate as we vary the Gaussian potential depth. By computing the root mean square radius of the condensate, we confirm the variation in the number of vortices generated as a function of the ratio between the root-mean-square radius of the condensate and the laser beam waist. In particular, the number of hidden vortices reaches the maximum value when the above ratio is close to the value 0.7. We find the variation in the number of vortices is rapid for deeper Gaussian potentials, and we conclude that the larger beam waist and deeper Gaussian potentials generate more vortices. Further, we calculate the number of vortices using the Feynman rule with Thomas Fermi approximation and compare them with the numerical results. We also observe that the critical rotation frequency decreases with an increase in depth of Gaussian potential.Comment: 14 pages, 11 figure

Direct Evidence for Competitive C-H Activation by a Well-Defined Silver XPhos Complex in Palladium-Catalyzed C-H Functionalization

Increasing evidence indicates that silver salts can play a role in the C-H activation step of palladium-catalyzed C-H functionalization. Here we isolate a silver(I) complex by C-H bond activation and demonstrate its catalytic competence for C-H functionalization. We demonstrate how silver carbonate, a common but highly insoluble additive, reacts with pentafluorobenzene in the presence of a bulky phosphine, XPhos, to form the C-H bond activation product Ag(C6F5)(XPhos). By isolating and fully characterizing this complex and the related carbonate and iodide complexes, [Ag(XPhos)]2(μ-κ2,κ2-CO3) and [AgI(XPhos)]2, we show how well-defined Ag(I) complexes can operate in conjunction with palladium complexes to achieve C-H functionalization even at ambient temperature. Reactions are tested against the standard cross-coupling of C6F5H with 4-iodotoluene, catalyzed by palladium acetate at 60 °C in the presence of silver carbonate and Xphos. Key observations are that (a) PdI(C6H5)(XPhos) reacts stoichiometrically with Ag(C6F5)(XPhos) to form Ph-C6F5 instantly at room temperature; (b) catalytic cross coupling can be achieved using 5% Ag(C6F5)(XPhos) as the sole silver source; and (c) palladium acetate (typical precatalyst) can be replaced for catalytic cross coupling by the expected oxidative addition compound PdI(C6H5)(XPhos). These investigations lead to a catalytic cycle in which Ag(I) plays the C-H bond activation role and palladium plays the coupling role. Moreover, we show how the phosphine can be exchanged between silver complexes, ensuring that it is recycled even though silver carbonate is consumed during catalytic cross-coupling

Opening a Pandora’s Flask on a Prototype Catalytic Direct Arylation Reaction of Pentafluorobenzene : The Ag2CO3/Pd(OAc)2/PPh3 System

Direct C-H functionalization reactions have opened new avenues in catalysis, removing the need for prefunctionalization of at least one of the substrates. Although C-H functionalization catalyzed by palladium complexes in the presence of a base is generally considered to proceed by the CMD/AMLA-6 mechanism, recent research has shown that silver(I) salts, frequently used as bases, can function as C-H bond activators instead of (or in addition to) palladium(II). In this study, we examine the coupling of pentafluorobenzene 1 to 4-iodotoluene 2a (and its analogues) to form 4-(pentafluorophenyl)toluene 3a catalyzed by palladium(II) acetate with the commonplace PPh3 ligand, silver carbonate as base, and DMF as solvent. By studying the reaction of 1 with Ag2CO3/PPh3 and with isolated silver (triphenylphosphine) carbonate complexes, we show the formation of C-H activation products containing the Ag(C6F5)(PPh3)n unit. However, analysis is complicated by the lability of the Ag-PPh3 bond and the presence of multiple species in the solution. The speciation of palladium(II) is investigated by high-resolution-MAS NMR (chosen for its suitability for suspensions) with a substoichiometric catalyst, demonstrating the formation of an equilibrium mixture of Pd(Ar)(κ1-OAc)(PPh3)2 and [Pd(Ar)(μ-OAc)(PPh3)]2 as resting states (Ar = Ph, 4-tolyl). These two complexes react stoichiometrically with 1 to form coupling products. The catalytic reaction kinetics is investigated by in situ IR spectroscopy revealing a two-term rate law and dependence on [Pdtot/nPPh3]0.5 consistent with the dissociation of an off-cycle palladium dimer. The first term is independent of [1], whereas the second term is first order in [1]. The observed rates are very similar with Pd(PPh3)4, Pd(Ph)(κ1-OAc)(PPh3)2, and [Pd(Ph)(μ-OAc)(PPh3)]2 catalysts. The kinetic isotope effect varied significantly according to conditions. The multiple speciation of both AgI and PdII acts as a warning against specifying the catalytic cycles in detail. Moreover, the rapid dynamic interconversion of AgI species creates a level of complexity that has not been appreciated previously

Chaos–order–chaos-type transition in hydrogen molecular ion placed in external magnetic field

The occurrence of Hamiltonian chaos in the system of hydrogen molecular ion placed in external magnetic field has been investigated by using electronic trajectory plot, Poincare surface of section and maximal Lyapunov exponent. Our analysis reveals that the underlying dynamics of this system is very much sensitive to the magnetic field. The numerical calculations show that the regular dynamics in the field free case begins to deviate, with the addition of the external magnetic field, and shows chaotic character for certain field strengths. Moreover, a gradual increase in the strength of the magnetic field makes the system to posses chaos–order–chaos type of regions. When the magnetic field is very large, we have observed that the entire phase space has been occupied by chaotic trajectories

Design, synthesis and biological evaluation of 2-(4-phenylthiazol-2-yl) isoindoline-1,3-dione derivatives as anti-prostate cancer agents

The structural modification and molecular docking-based screening approaches on thiazole-based isoindolinediones were imposed to find the novel 2-(4-phenylthiazol-2-yl) isoindoline-1,3-dione derivatives. The best fit compounds (6a-n) were synthesized and evaluated their antiproliferative activities on the prostate cancer cell lines (PC-3 & LNCaP). Among them, the compound, 6m exhibited good activity, particularly on LNCaP (IC50=5.96±1.6μM), moderately active against PC-3 cell lines as compared to bicalutamide. The compound, 6m decreased the androgen-mediated transcription of ARE-mRNA in PSA, TMPRSS2, c-myc and cyclin D1 than R-bicalutamide. The compounds, 6e and 6f were reconfirmed through single crystal XRD analysis. The ADME profiling of the test compounds was evaluated to find the drug-likeness and pharmacokinetic parameters. These findings may provide vital information for the development of anti-prostate cancer agents

Design, synthesis and biological evaluation of 2-(4-phenylthiazol-2-yl) isoindoline-1,3-dione derivatives as anti-prostate cancer agents

The structural modification and molecular docking-based screening approaches on thiazole-based isoindolinediones were imposed to find the novel 2-(4-phenylthiazol-2-yl) isoindoline-1,3-dione derivatives. The best fit compounds (6a-n) were synthesized and evaluated their antiproliferative activities on the prostate cancer cell lines (PC-3 & LNCaP). Among them, the compound, 6m exhibited good activity, particularly on LNCaP (IC50=5.96±1.6μM), moderately active against PC-3 cell lines as compared to bicalutamide. The compound, 6m decreased the androgen-mediated transcription of ARE-mRNA in PSA, TMPRSS2, c-myc and cyclin D1 than R-bicalutamide. The compounds, 6e and 6f were reconfirmed through single crystal XRD analysis. The ADME profiling of the test compounds was evaluated to find the drug-likeness and pharmacokinetic parameters. These findings may provide vital information for the development of anti-prostate cancer agents