Influence of the chelate effect on the electronic structure of one-electron oxidized group 10 metal(II)-(disalicylidene)diamine complexes

The neutral and one-electron oxidized group 10 metal, Ni(II), Pd(II) and Pt(II), six-membered chelate Salpn (Salpn = N,N'-bis(3,5-di-tert-butylsalicylidene)-1,3-propanediamine) complexes have been investigated and compared to the five-membered chelate Salen (N,N'-bis(3,5-di-tert-butylsalicylidene)-1,2-ethanediamine) and Salcn (N,N'-bis(3,5-di-tert-butylsalicylidene)-1,2-cyclohexane-(1R,2R)-diamine) complexes. Reaction of the Salpn complexes with 1 equivalent of AgSbF(6) affords the oxidized complexes which exist as ligand radical species in solution and in the solid state. The solid state structures of the oxidized complexes have been determined by X-ray crystal structure analysis. While the Ni and Pt analogues exhibit an essentially symmetric coordination sphere contraction upon oxidation, the oxidized Pd derivative exhibits an asymmetric metal binding environment demonstrating at least partial ligand radical localization. In comparison to the oxidized Salen and Salcn complexes, the propyl backbone of the Salpn complexes leads to a larger deviation from a planar geometry in the solid state. The electronic structure of the oxidized Salpn complexes was further probed by UV-vis-NIR measurements, electrochemistry, EPR spectroscopy, and theoretical calculations. The intense NIR band for the one-electron oxidized Salpn complexes shifts to lower energy in comparison to the 5-membered chelate analogues, which is attributed to lower metal d(xz) character in the β-LUMO for the Salpn series. The reactivity of the one-electron oxidized Salpn complexes with exogenous ligands was also studied. In the presence of pyridine, the oxidized Ni analogue exhibits a shift in the locus of oxidation to a Ni(III) species. The oxidized PtSalpn complex rapidly decomposes in the presence of pyridine, even at low temperature. Interestingly, electronic and EPR spectroscopy suggests that the addition of pyridine to the oxidized Pd analogue results in initial dissociation of the phenoxyl radical ligand, likely due to the increased flexibility of the propyl backbone.     

Class III Delocalization and Exciton Coupling in a Bimetallic Bis‐ligand Radical Complex

The geometric and electronic structure of an oxidized bimetallic Ni complex incorporating two redox‐active Schiff‐base ligands connected via a 1,2‐phenylene linker has been investigated and compared to a monomeric analogue. Information from UV/Vis/NIR spectroscopy, electron paramagnetic resonance (EPR) spectroscopy, electrochemistry, and density functional theory (DFT) calculations provides important information on the locus of oxidation for the bimetallic complex. The neutral bimetallic complex is conformationally dynamic at room temperature, which complicates characterization of the oxidized forms. Comparison to an oxidized monomer analogue 1 provides critical insight into the electronic structure of the oxidized bimetallic complex 2 . Oxidation of 1 provides [ 1 ^.]⁺, which is characterized as a fully delocalized ligand radical complex; the spectroscopic signature of this derivative includes an intense NIR band at 4500 cm^?1. Oxidation of 2 to the bis‐oxidized form affords a bis‐ligand radical species [ 2 ^..]²⁺. Variable temperature EPR spectroscopy of [ 2 ^..]²⁺ shows no evidence of coupling, and the triplet and broken symmetry solutions afforded by theoretical calculations are essentially isoenergetic. [ 2 ^..]²⁺ is thus best described as incorporating two non‐interacting ligand radicals. Interestingly, the intense NIR intervalence charge transfer band observed for the delocalized ligand‐radical [ 1 ^.]⁺ exhibits exciton splitting in [ 2 ^..]²⁺, due to coupling of the monomer transition dipoles in the enforced oblique dimer geometry. Evaluating the splitting of the intense intervalence charge transfer band can thus provide significant geometric and electronic information in less rigid bis‐ligand radical systems. Addition of excess pyridine to [ 2 ^..]²⁺ results in a shift in the oxidation locus from a bis‐ligand radical species to the Ni^III/Ni^III derivative [ 2 (py)₄]^{2 +} , demonstrating that the ligand system can incorporate significant bulk in the axial positions.     

On the existence of conically self-similar free-vortex solutions to the Navier-Stokes equations

In this paper a proof of existence and non-existence of theconically self-similar free-vortex solutions to the Navier-Stokesequations, originally found by Yih et al. (1982, Phys. Fluids.25, 2147-2158), is presented. This proof clearly establishesthat these solutions do not have any kind of singularity atthe symmetry axis. This analysis gives considerably improvedexistence and non-existence bounds and it is shown that thesebounds are close to optimal in the low-swirling limit. Thisapproach links the questions of existence and non-existencefor the swirling case and for the non-swirling case. The proof,which is an extension of techniques developed by Serrin (1972,Phil. Trans. R. Soc. Lond. 271, 325-360), is based on Schauder'sFixed Point Theorem and is, therefore, non-constructive. Therefore,the paper ends with a brief discussion of the question of howto compute the conically self-similar free-vortex solutionsto the Navier-Stokes equations.     

Pricing vulnerable European options with stochastic default barriers

CF Lo and KC Ku Institute of Theoretical Physics and Department of Physics, The Chinese University of Hong Kong, Shatin, Hong Kong, China Email: cho-hoi_hui{at}hkma.gov.hk Received on 31 July 2006. Accepted on 15 March 2007. This paper develops a valuation model of European options incorporatinga stochastic default barrier, which extends a constant defaultbarrier proposed in the Hull–White model. The defaultbarrier is considered as an option writer's liability. Closed-formsolutions of vulnerable European option values based on themodel are derived to study the impact of the stochastic defaultbarriers on option values. The numerical results show that negativecorrelation between the firm values and the stochastic defaultbarriers of option writers gives material reductions in optionvalues where the options are written by firms with leverageratios corresponding to BBB or BB ratings.