Universally optimal distribution of points on spheres

We study configurations of points on the unit sphere that minimize potential energy for a broad class of potential functions (viewed as functions of the squared Euclidean distance between points). Call a configuration sharp if there are distances between distinct points in it and it is a spherical -design. We prove that every sharp configuration minimizes potential energy for all completely monotonic potential functions. Examples include the minimal vectors of the and Leech lattices. We also prove the same result for the vertices of the -cell, which do not form a sharp configuration. For most known cases, we prove that they are the unique global minima for energy, as long as the potential function is strictly completely monotonic. For certain potential functions, some of these configurations were previously analyzed by Yudin, Kolushov, and Andreev; we build on their techniques. We also generalize our results to other compact two-point homogeneous spaces, and we conclude with an extension to Euclidean space.

     

Synthesis, reactions and X-ray crystal structures of metallacrown ethers with unsymmetrical bis(phosphinite) and bis(phosphite) ligands derived from 2-hydroxy-2′-(1,4-bisoxo-6-hexanol)-1,1′-biphenyl

Chlorodiphenylphosphine and 2,2′-biphenylylenephosphorochloridite react with 2-hydroxy-2′-(1,4-bisoxo-6-hexanol)-1,1′-biphenyl to yield the new α,ω-bis(phosphorus-donor)-polyether ligands, 2-Ph₂PO(CH₂CH₂O)₂–C₁₂H₈-2′-OPPh₂ (1) and 2-(2,2′-O₂C₁₂H₈)P(CH₂CH₂O)₂–C₁₂H₈-2′-P(2,2′-O₂C₁₂H₈) (2). These ligands react with Mo(CO)₄(nbd) to form the monomeric metallacrown ethers, cis-Mo(CO)₄{2-Ph₂PO(CH₂CH₂O)₂–C₁₂H₈-2′-OPPh₂} (cis-3) and cis-Mo(CO)₄{2-(2,2′-O₂C₁₂H₈)P(CH₂CH₂O)₂–C₁₂H₈-2′-P(2,2′-O₂C₁₂H₈)} (cis-4), in good yields. The X-ray crystal structures of cis-3 and cis-4 are significantly different, especially in the conformation of the metal center and the adjacent ethylene group. The very different ¹³C-NMR coordination chemical shifts of this ethylene group in cis-3 and cis-4 suggest that the solution conformations of these metallacrown ethers are also quite different. Both metallacrown ethers undergo cis–trans isomerization in the presence of HgCl₂. Although the cis–trans equilibrium constants for the isomerization reactions are nearly identical, the isomerization of cis-3 is more rapid. Phenyl lithium reacts with cis-3 to form the corresponding benzoyl complexes but does not react with either trans-3 or cis-4. Both the slower rate of cis–trans isomerization of cis-4 and its lack of reaction with PhLi are consistent with weaker interactions between the hard metal cations and the carbonyl oxygens in both trans-3 and cis-4.     

Analysis of the photo-thermal excitation in a semiconducting medium under the purview of DPL theory involving non-local effect

Meccanica - Non-local theory comprises a unique characteristics by analyzing the effects of all points of the body on a single point of the material. The present study enlightens the propagation of...     

Synthesis and Characterization of Molecules Containing Thiazole and Oxazole Moieties and Study of ESIPT Phenomenon

Two novel ESIPT molecules, 2-[4-(1,3-benzothiazol-2-yl)naphtho[1,2-d][1,3]oxazol-2-yl]phenol 9a and 4-[4-(1,3-benzothiazol-2-yl)naphtho[1,2-d][1,3]oxazol-2-yl]benzene-1,3-diol 9b were synthesized by condensing 1-amino-3-(1,3-benzothiazol-2-yl)naphthalen-2-ol with 2-hydroxybenzoic acid and 2,4-dihydroxybenzoic acid respectively. The novel compounds were characterized by FT-IR, (1)H NMR, Mass spectral and elemental analysis. Effect of polarity on photo physical properties, absorption and emission were studied. Compounds showed single absorption and dual emission due to ESIPT phenomenon. The structural changes due to ESIPT phenomenon in terms of bond angle, bond distances and geometry were investigated by using Gaussian 03 software. These two novel ESIPT molecules are thermally stable up to 200?°C.     

A redox series of gallium(III) complexes: ligand-based two-electron oxidation affords a gallium-thiolate complex

We have prepared a series of gallium(III) complexes of the redox active iminopyridine ligand (IP). Reaction of GaCl(3) with iminopyridine ligand (IP) in the presence of either two or four equivalents of sodium metal resulted in the formation of deep green (IP(-))(2)GaCl (1), or deep purple [(DME)(3)Na][(IP(2-))(2)Ga] (2a), respectively. Complex 1 is paramagnetic with a room temperature magnetic moment of 2.3 μ(B) which falls to 0.5 μ(B) at 5 K. These observations indicate that two ligand radicals comprise a triplet at room temperature which becomes a singlet due to antiferromagnetic coupling at low temperature. Complex 2 is diamagnetic. Cyclic voltammograms recorded on 0.3 M Bu(4)NPF(6) THF solutions of [Na(THF)(6)][(IP(2-))(2)Ga](-) (2b) indicate that oxidation of 2b occurs in two two-electron steps at -1.31 V and -0.54 V vs. SCE. The observation of two-electron redox events indicates that electronic coupling through the gallium(III) center is minimal and that the two IP ligand on 2b are oxidized concurrently. Oxidation of 2 with one equivalent of MeS-SMe afforded the two-electron oxidized product (IP(-))(2)Ga(SMe) (3). This complex has an electronic structure analogous to 1. Accordingly, both 1 and 3 are deep green in color and magnetic susceptibility measurements performed on 3 confirm the triplet character of the complex at room temperature. Electron paramagnetic resonance experiments on 1 and 3 display a quartet signal at g = 2.0 which confirmed the triplet nature of the compounds, and a half field signal consistent with the integer spin state.     

Non-catalytic approach to the synthesis of partially reduced 'S' shaped dioxathia- and oxadithiahelicenes through base induced inter- and intramolecular C-C bond formation

An efficient and convenient route for the construction of helical 'S' shaped dioxathia- and oxadithiahelicenes with oxygen and sulfur atoms located in the middle of the outer helix has been developed through base induced inter- and intramolecular C-C bond formation from the reaction of 4-sec-amino-2-oxo-2,5-dihydrothiochromeno[4,3-b]pyran-3-carbonitriles with 3,4-dihydro-2H-benzo[b]oxepin-5(2H)-ones, 3,4-dihydrobenzo[b]thiepin-5(2H)-one and thiochroman-4-ones separately. Quantum chemical calculations have also been carried out to explore the geometries and electronic structures of newly synthesized compounds to envisage the pathway for interconversion of both atropisomers. The determination of helicity parameters and configurational stability demonstrate that the energy barrier is strongly dependent on the nature of hetero-atoms present.     

Synthesis,Structure and Light‐Harvesting Properties of Some New Transition‐Metal Dithiocarbamates Involving Ferrocene

Nine new transition‐metal dithiocarbamates involving ferrocene (Fc), namely, [M(FcCH₂Bzdtc)₂] (M=Ni^II ( 1 ), Cu^II ( 2 ), Cd^II ( 3 ), Hg^II ( 4 ), Pd^II ( 5 ), Pt^II ( 6 ) and Pb^II ( 7 ); Bzdtc=N‐benzyl dithiocarbamate) and [M(FcCH₂Bzdtc)₃] (M=Co^II ( 8 ) and UO₂^VI ( 9 )), have been synthesised and characterised by micro analyses, IR spectroscopy, ¹H and ¹³C NMR spectroscopy, and in three cases by single‐crystal X‐ray analysis. The peak broadening in the ¹H spectrum of the copper complex indicates the paramagnetic behaviour of this compound. A square‐planar geometry around the nickel and copper complexes and distorted linear geometry around the mercury complex have been found. The latter geometry is attributed to the bulkiness of the methylferrocenyl and benzyl groups. The observed single quasi‐reversible cyclic voltammograms for complexes 2 , 8 and 9 indicate the stabilisation of a metal centre other than Fe in their characteristic oxidation state. These complexes have been used as a photosensitiser in dye‐sensitised solar cells.