A set of new copper(i) complexes is synthesized and characterized using a labile PNP pincer ligand (PNP = N,N'-bis(diphenylphosphine)-2,6-diaminopyridine). A homoleptic Cu(i) complex [Cu(PNP-κP:κN)], (1), was prepared, and taking advantage of the uncoordinated phosphorus atoms in (1), reaction with a second Cu(i) atom bearing secondary ligands (PPh, phen or dmp) allows the formation of new complexes: a bimetallic helicate [Cu(PNP)(phen)], (2), a mononuclear pincer complex [Cu(PNP)(PPh)], (3), and a heteroleptic complex [Cu(PNP)(dmp)], (4). All complexes were characterized by X-ray crystallography, NMR (VT-NMR for (1) and (4)), cyclic-voltammetry, and steady-state and time-resolved luminescence spectroscopy. The fluxional behavior in (1) was studied by P VT-NMR, where an E value of 47.42 kJ mol was calculated for the intramolecular alternating coordination of -PPh moieties in PNP to the metal atom. This set of compounds reveals the versatility of the PNP ligand when added to the coordinating properties of Cu(i). The four complexes exhibit emission in solution and complexes (2)-(4) display intense luminescence in the solid state. The oscillographic traces showing the decay of the luminescence were fitted to biexponential functions with time constants: 8.0 μs > τ > 0.37 μs and 50 μs >τ > 2.2 μs for complexes (2), (3) and (4), respectively. Radiative relaxation is associated with electronic transitions in both the ligand PNP and metal-to-ligand charge transfer (MLCT).
We first establish sufficient conditions ensuring strong duality for cone constrained nonconvex optimization problems under a generalized Slater-type condition. Such conditions allow us to cover situations where recent results cannot be applied. Afterwards, we provide a new complete characterization of strong duality for a problem with a single constraint: showing, in particular, that strong duality still holds without the standard Slater condition. This yields Lagrange multipliers characterizations of global optimality in case of (not necessarily convex) quadratic homogeneous functions after applying a generalized joint-range convexity result. Furthermore, a result which reduces a constrained minimization problem into one with a single constraint under generalized convexity assumptions, is also presented.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.