Theoretical modeling of electrochemical interactions in bimetallic molybdenum nitrosyl complexes incorporating saturated bridges

Hybrid B3LYP and non-hybrid OLYP DFT formalism has been applied to neutral and reduced forms of bimetallic hydrotris(3-methylpyrazol-1-yl)borato (Tp^3-Me) molybdenum nitrosyl complexes incorporating ethane-1,2-diolate bridges. Direct evidence for localization of an extra electron in mixed-valence compounds {16e:17e}⁻ is based on the analysis of electron density, energetic stabilization of asymmetric structures with an electron trapped on one Mo and the splitting of both calculated and experimental ν_NO stretching frequencies. Differences in the first and second electron affinities calculated in PCM solvent model have been successfully related to cyclic voltammetry measurements. Electronic interactions through saturated ethanediolato bridges are evidenced by the extent of spin density delocalization towards the second Mo center.     

An isoperimetric inequality in the universal cover of the punctured plane

We find the largest ? (approximately 1.71579) for which any simple closed path α in the universal cover of R²?Z², equipped with the natural lifted metric from the Euclidean two-dimensional plane, satisfies L(α)≥?A(α), where L(α) is the length of α and A(α) is the area enclosed by α. This generalizes a result of Schnell and Segura Gomis, and provides an alternative proof for the same isoperimetric inequality in R²?Z².     

Convergence of third order correlation energy in atoms and molecules

We have investigated the convergence of third order correlation energy within the hierarchies of correlation consistent basis sets for helium, neon, and water, and for three stationary points of hydrogen peroxide. This analysis confirms that singlet pair energies converge much slower than triplet pair energies. In addition, singlet pair energies with (aug)-cc-pVDZ and (aug)-cc-pVTZ basis sets do not follow a converging trend and energies with three basis sets larger than aug-cc-pVTZ are generally required for reliable extrapolations of third order correlation energies, making so the explicitly correlated R12 calculations preferable.     

[Ni(bpy)(mal)(H2O)3]·H2O and [Ni(4,4′-dmbpy)(mal)(H2O)3]·1.5H2O: syntheses,crystal structures,magnetic properties,and computational study of stacking interactions

[Ni(bpy)(mal)(H₂O)₃]·H₂O and [Ni(4,4′-dmbpy)(mal)(H₂O)₃]·1.5H₂O (mal = maleato; bpy = 2,2′-bipyridine; 4,4′-dmbpy = 4,4′-dimethyl-2,2′-bipyridine) exhibit molecular crystal structures. The Ni(II) central ions in both complexes are six-coordinate by one chelate bonded L–L ligand, three aqua ligands, and one position is occupied by a maleato oxygen donor atom. Hydrogen bonded ribbon-like supramolecular structural motifs are present in both studied complexes; these are linked by weaker C–H?O hydrogen bonds in [Ni(bpy)(mal)(H₂O)₃]·H₂O, whereas in [Ni(4,4′-dmbpy)(mal)(H₂O)₃]·1,5H₂O the hydrogen bonded ribbons are linked by O?H-O-H?O hydrogen bonds with the participation of the additional water solvate molecule positioned on the twofold axis. In both structures, π–π stacking interactions with different patterns in respective structures were found. The role of dispersion energy and many-body effects in the stabilization of bpy and 4,4′-dmbpy stacking interactions were investigated using methods of computational chemistry. Those confirm the dispersion-dominated stabilization of the 4,4′-dmbpy supramolecular chain-like structure, with only marginal impact of cooperativity effects. Thermal decompositions of both complexes start with dehydration. Magnetic susceptibility studies performed from 2 to 300 K revealed a dominant effect of the zero-field splitting of the Ni(II) ion, governing the low-temperature magnetic properties of both compounds.     

Privileged users in zero-error transmission over a noisy channel

The k-th power of a graph G is the graph whose vertex set is V(G) ^k , where two distinct k-tuples are adjacent iff they are equal or adjacent in G in each coordinate. The Shannon capacity of G, c(G), is lim _k→∞ α(G ^k )^1/k , where α(G) denotes the independence number of G. When G is the characteristic graph of a channel C, c(G) measures the effective alphabet size of C in a zero-error protocol. A sum of channels, C = Σ _i C _i , describes a setting when there are t ≥ 2 senders, each with his own channel C _i , and each letter in a word can be selected from any of the channels. This corresponds to a disjoint union of the characteristic graphs, G = Σ _i G _i . It is well known that c(G) ≥ Σ _i c(G _i ), and in [1] it is shown that in fact c(G) can be larger than any fixed power of the above sum. We extend the ideas of [1] and show that for every F, a family of subsets of [t], it is possible to assign a channel C _i to each sender i ∈ [t], such that the capacity of a group of senders X ⊂ [t] is high iff X contains some F ∈ F. This corresponds to a case where only privileged subsets of senders are allowed to transmit in a high rate. For instance, as an analogue to secret sharing, it is possible to ensure that whenever at least k senders combine their channels, they obtain a high capacity, however every group of k − 1 senders has a low capacity (and yet is not totally denied of service). In the process, we obtain an explicit Ramsey construction of an edge-coloring of the complete graph on n vertices by t colors, where every induced subgraph on exp vertices contains all t colors. Research supported in part by a USA-Israeli BSF grant, by the Israel Science Foundation and by the Hermann Minkowski Minerva Center for Geometry at Tel Aviv University. Research partially supported by a Charles Clore Foundation Fellowship.     

An explicitly correlated Mukherjee's state specific coupled cluster method: development and pilot applications

This paper reports development of the explicitly correlated variant of Mukherjee's state specific multireference coupled cluster method (MkCC-F12). The current implementation is restricted to conventional single and double excitations and to pseudo-double excitations related to the Slater Type Geminal (STG) correlation factor using the SP ansatz. The performance of the MkCCSD-F12 was tested on calculations of singlet methylene, dissociation curve of the fluorine molecule, and the BeH(2) insertion pathway. As expected, the results of the newly developed method reconfirm the significantly faster convergence with respect to the basis set limit compared to the traditional expansion in Slater determinants. Results prove that treating the correlation factor separately for each reference is appropriate.     

Sums and products along sparse graphs

In their seminal paper from 1983, Erdős and Szemerédi showed that any n distinct integers induce either n ^1+ɛ distinct sums of pairs or that many distinct products, and conjectured a lower bound of n ^2−o(1). They further proposed a generalization of this problem, in which the sums and products are taken along the edges of a given graph G on n labeled vertices. They conjectured a version of the sum-product theorem for general graphs that have at least n ^1+ɛ edges.     

Counting contours on trees

We calculate the exact number of contours of size n containing a fixed vertex in d-ary trees and provide sharp estimates for this number for more general trees. We also obtain a characterization of the locally finite trees with infinitely many contours of the same size containing a fixed vertex.