Electronic Structure of [Ta(5)(NH)(4)Cl(17)](6-): A Cluster with a Distorted Square Pyramidal Ta(5) Core

We report approximate molecular orbital calculations on the [Ta(5)(NH)(4)Cl(17)](6-) cluster synthesized by Simon and Meyer. The cluster is based on a "flattened" square pyramid of tantalum atoms, basal bridging imides, and terminal chlorides. This cluster was of interest to us due to the unusual presence of imide ligands, the distorted nature of the metal core, and the possible resemblance to B(5)H(9). Our calculations indicate that metal-metal bonding is limited to Ta(apical)-Ta(basal) bonding, with no significant bonding between the basal metal atoms. The imide ligands, which bridge the base of the pyramid, were found to have a significant amount of capping character. The metal-metal bonding orbitals have some unusual features due to the pyramid's distortion. Additionally, the flattened nature of the pyramid leads to an interesting energy ordering of the metal-metal bonding orbitals.     

Ditungsten Siloxide Hydrides, [(silox)(2)WH(n)()](2) (n = 1, 2; silox = (t)BuSiO), and Related Complexes

The addition of 4.0 equiv of Na(silox) to Na[W(2)Cl(7)(THF)(5)] afforded (silox)(2)ClW&tbd1;WCl(silox)(2) (1, 65%). Treatment of 1 with 2.0 equiv of MeMgBr in Et(2)O provided (silox)(2)MeW&tbd1;WMe(silox)(2) (2, 81%). In the presence of 1 atm of H(2), reduction of 1 with 2.0 equiv of Na/Hg in DME provided (silox)(2)HW&tbd1;WH(silox)(2) (3, 70%), characterized by a hydride resonance at delta 19.69 (J(WH) = 325 Hz, (1)H NMR). Exposure of 2 to 1 atm of H(2) yielded 3 and CH(4) via (silox)(2)HW&tbd1;WMe(silox)(2) (4); use of D(2) led to [(silox)(2)WD](2) (3-d(2)). Exposure of 3 to ethylene ( approximately 1 atm, 25 degrees C) in hexanes generated (silox)(2)EtW&tbd1;WEt(silox)(2) (5), but solutions of 5 reverted to 3 and free C(2)H(4) upon standing. NMR spectral data are consistent with a sterically locked, gauche, C(2) symmetry for 1-5. Thermolysis of 3 at 100 degrees C (4 h) resulted in partial conversion to (silox)(2)HW&tbd1;W(OSi(t)Bu(2)CMe(2)CH(2))(silox) (6a, approximately 60%) and free H(2), while extended thermolysis with degassing (5 d, 70 degrees C) produced a second cyclometalated rotational isomer, 6b (6a:6b approximately 3:1). When left at 25 degrees C (4 h) in sealed NMR tubes, 6 and free H(2) regenerated 3. Reduction of 1 with 2.0 equiv of Na/Hg in DME also afforded 6a (25%). When 3 was exposed to approximately 3 atm of H(2), equilibrium amounts of [(silox)(2)WH(2)](2) (7) were observed by (1)H NMR spectroscopy (3 + H(2) right harpoon over left harpoon 7; 25.9-88.7 degrees C, DeltaH = -9.6(4) kcal/mol, DeltaS = -21(2) eu). Benzene solutions of 3 and 1-3 atm of D(2) revealed incorporation of deuterium into the silox ligands, presumably via intermediate 6. In sealed tubes containing [(silox)(2)WCl](2) (1) and dihydrogen (1-3 atm), (1)H NMR spectral evidence for [(silox)(2)WCl](2)(&mgr;-H)(2) (8) was obtained, suggesting that formation of 3 from 1 proceeded via reduction of 8. Alternatively, 3 may be formed from direct reduction of 1 to give [(silox)(2)W](2) (9), followed by H(2) addition. Hydride chemical shifts for 7 are temperature dependent, varying from delta 1.39 (-70 degrees C, toluene-d(8)), to delta 3.68 (90 degrees C). (29)Si{(1)H} NMR spectra revealed a similar temperature dependence of the silox (delta 12.43, -60 degrees C, to delta 13.64, 45 degrees C) resonances. These effects may arise from thermal population of a low-lying, deltadelta, paramagnetic excited state of D(2)(d)() [(silox)(2)W](2)(&mgr;-H)(4) (DeltaE approximately 2.1 kcal/mol, chi(7a) approximately 0.03), an explanation favored over thermal equilibration with an energetically similar but structurally distinct isomer (e.g., [(silox)(2)WH(2)](2)(&mgr;-H)(2), DeltaG degrees approximately 0.69 kcal/mol, chi(7b) approximately 0.25) on the basis of spectral arguments. Extended Hückel and ab initio molecular orbital calculations on model complexes [(H(3)SiO)(2)W](2)(&mgr;-H)(4) (staggered bridged 7a', EHMO), [(H(3)SiO)(2)WH(2)](2) (all-terminal 7b', EHMO), [(H(3)SiO)(2)W](2) (9', EHMO), (HO)(4)W(2)(H(4)) (staggered-bridged 7", ab initio), and (HO)(4)W(2)(H(4)) (bent-terminal 7, ab initio) generally support the explanation of a thermally accessible excited state and assign 7 a geometry intermediate between the all-terminal and staggered-bridged forms.     

Matroid intersection algorithms

LetM ₁ = (E, 9₁),M ₂ = (E, 9₂) be two matroids over the same set of elementsE, and with families of independent sets 9₁, 9₂. A setI 9₁ 9₂ is said to be anintersection of the matroidsM ₁,M ₂. An important problem of combinatorial optimization is that of finding an optimal intersection ofM ₁,M ₂. In this paper three matroid intersection algorithms are presented. One algorithm computes an intersection containing a maximum number of elements. The other two algorithms compute intersections which are of maximum total weight, for a given weighting of the elements inE. One of these algorithms is primal-dual, being based on duality considerations of linear programming, and the other is primal. All three algorithms are based on the computation of an augmenting sequence of elements, a generalization of the notion of an augmenting path from network flow theory and matching theory. The running time of each algorithm is polynomial inm, the number of elements inE, and in the running times of subroutines for independence testing inM ₁,M ₂. The algorithms provide constructive proofs of various important theorems of matroid theory, such as the Matroid Intersection Duality Theorem and Edmonds' Matroid Polyhedral Intersection Theorem.Research sponsored by the Air Force Office of Scientific Research Grant 71-2076.     

The Brownian loop soup

We define a natural conformally invariant measure on unrooted Brownian loops in the plane and study some of its properties. We relate this measure to a measure on loops rooted at a boundary point of a domain and show how this relation gives a way to chronologically add Brownian loops to simple curves in the plane.Cornell University; Research supported in part by the National Science FoundationUniversité Paris-Sud and IUFMathematics Subject Classification (2000):60J65, 81T40     

Erratum: The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization

Journal of the Operational Research Society -     

Raman spectrographic system for quantitative analysis of isotopic hydrogen mixtures for muon catalysis experiments

We show that Raman spectral lines from H₂, D₂, T₂, HD, HT and DT are readily resolved, permitting an effective means to analyze isotopic hydrogen mixtures used in muon-catalyzed fusion experiments. We propose a Raman spectrographic system to allow for real-time analysis of targets involving all three isotopes of hydrogen.     

Anomalous crosslinking in acrylic acid and acrylic acid-polyethylene systems by γ-irradiation

Studies have been carried out on irradiated systems of acrylic acid in aqueous solution and acrylic acid-polyethylene systems in order to discover the cause of the enhanced crosslinking mentioned in previous literature from this laboratory. Theories which have been investigated include: free radical mechanisms, anhydride mechanisms, ionic mechanism, entanglements, hydrogen bonding. It seems that hydrogen bonding is the most plausible.