Products of class sums of the symmetric group: Rules of partial elimination

Progress in the formulation of a procedure for the combinatorial evaluation of the product of a single-cycle and an arbitrary class sum in the symmetric group algebra is presented. The procedure consists of a “global conjecture” concerning the representation of the product [(p)]_n·[*]_n in terms of a set of operators referred to as reduced class sums, and of an (incomplete) set of rules for the evaluation of the (n-independent!) coefficients of these operators. Two new types of index elimination rules are suggested, and some properties of the formalism are explored. These include useful sum rules as well as a certain “detailed balance” property that sheds some light on a combinatorial aspect of the global conjecture. The present results account for several new types of reduced class coefficients and suggest some feasible further developments. Some outstanding open problems are pointed out. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 63: 961–979, 1997     

A partial recurrence relation for reduced class coefficients of the symmetric group

An expression for the product of a single-cycle class [(1)^{N - P}(p)]_N and an arbitrary class [(1)^l1(2)^l …? (N)^lN]_N of the symmetric group has recently been conjectured. This expression involves a sum over a relatively small number of reduced class sums, depending on p indices. A further conjecture is formulated and demonstrated, according to which reduced class coefficients (RCCS ) involving cycles whose length is expressed by means of a single index can be related to corresponding coefficients in the product of [(1)^{N - P+1}(p - 1)]_N with an arbitrary class sum. Consequently, the problem of evaluating the general class sum product reduces to that of obtaining a relatively small set of fundamental RCCS containing no single-index cycles. The conjectures mentioned can be used to evaluate the product [(1)^{N - p}(p)]_N · [(1)^{N - q}(q)]_N in terms of fundamental RCCS that can all be obtained from the product [(r)]_r · [(r)]_r, where r = min(p, q). For the latter product, we use a result due to Boccara.     

Eigenvalues of single-cycle class-sums in the symmetric group

Explicit expressions for the eigenvalues of the class sums [(p)(1)^n?p]_n, p = 2, 3,…,14, of the symmetric group S_n are presented. Partial results are given for the eigenvalues corresponding to arbitrary p.     

Eigenvalues of single-cycle class-sums in the symmetric group. II

Explicit expressions for the eigenvalues of the single-cycle class-sums [(p)(1)^n–p]_n of the symmetric group (S_n), with p ≤ 20, are constructed. A new algorithm is used that makes no use of representation-theoretic data. The expressions obtained consist of polynomials in the symmetric power-sums over the “contents” of the Young diagram specifying the irreducible representation, with coefficients that are polynomials in n. On the basis of the results obtained for p ≤ 20, a conjecture is proposed concerning the general form of the four leading terms in these polynomials. © 1993 John Wiley & Sons, Inc.     

Synthesis,crystal structures and xanthine oxidase inhibitory activities of two copper(II) complexes with Schiff bases

Two azido-coordinated Schiff base Cu(II) complexes with the formulae [Cu(L1)(N₃)]·MeOH and [Cu(L2)(μ_1,1-N₃)] _n , where L1 is the deprotonated form of 2-chloro-2-[(2-ethylaminoethylimino)methyl]phenol, and L2 is the deprotonated form of 2,4-dibromo-6-[(2-dimethylaminoethylimino)methyl]phenol, have been synthesized and characterized by physico-chemical and spectroscopic methods. The X-ray crystal structures of both complexes have been determined. The Cu atom in [Cu(L1)(N₃)]·MeOH is four-coordinate in a square planar geometry, while [Cu(L2)(μ_1,1-N₃)] _n is five-coordinate with a square pyramidal geometry. The molecules in [Cu(L1)(N₃)]·MeOH are linked by intermolecular O–H···O and N–H···O hydrogen bonds, forming dimers. The molecules in [Cu(L2)(μ_1,1-N₃)] _n are linked through end-on azido bridges, forming one-dimensional chains. The xanthine oxidase inhibitory activities of both complexes were evaluated.     

Copper(I) bromide and chloride complexes with urotropine and triethylenediamine: synthesis,crystal structure,and Raman characterization*

Abstract

In the present study, the oxidative dissolution of metallic copper has been explored with the intention to prepare some new complexes with urotropine (hmta) and triethylenediamine (dabco) ligands. All the compounds synthesized were characterized by single-crystal X-ray diffraction and Raman spectroscopy. Reactions performed in a DMSO/CuCl₂?2H₂O mixture resulted in [(μ-Cl)₂Cu^I(hdabco⁺)Cu^I(μ-Cl)(κS-DMSO)]_n and [Cu^ICl(hmta)₂] complexes. Their isostructural bromide analogs [(μ-Br)₂Cu^I(hdabco⁺)Cu^I(μ-Br)(κS-DMSO)]_n and [Cu^IBr(hmta)₂] were prepared by the reaction of elemental copper with respective ligands in a DMSO/CBr₄ mixture. Early interrupted reaction of the copper wire with the DMSO/CBr₄/dabco solution resulted in an appearance of crystals of the [Cu^I₂Br₂(CO)₂(dabco)]_n carbonyl complex on the copper surface. It arises with the participation of in situ formed carbon monoxide. Despite the identical stoichiometry, the crystal structure of the [Cu₂Br₂(CO)₂(dabco)]_n complex is markedly different from that of a known [Cu₂Cl₂(CO)₂(dabco)]_n analog.     

Products of class operators of the symmetric group

A combinatorial derivation of the product of the class of three cycles, [(1)^N?3(3)]_N with an arbitrary class operator of the symmetric group S_N is presented. The form of this result suggests a conjecture concerning the expression of the general class operator product in terms of a relatively small number of reduced class coefficients. The conjecture is applied to the determination of the products of [(1)^N?4(4)]_N, [(1)^N?4(2)²]_N, and [(1)^N?5(5)]_N with arbitrary class operators. General expressions for the reduced class coefficients of the simplest type are obtained.     

Synthesis,crystal structures and characterizations of two homochiral coordination polymers based on a chiral reduced Schiff base ligand

Two homochiral coordination polymers based on a chiral reduced Schiff base ligand, namely poly[(μ₅‐4‐{[(NR,1S)‐(1‐carboxylato‐2‐phenylethyl)amino]methyl}benzoato)zinc(II)], [Zn(C₁₇H₁₅NO₄)]_n, (1), and poly[(μ₅‐4‐{[(NR,1S)‐(1‐carboxylato‐2‐phenylethyl)amino]methyl}benzoato)cobalt(II)], [Co(C₁₇H₁₅NO₄)]_n, (2), have been obtained by hydrothermal methods and studied by single‐crystal X‐ray diffraction, elemental analyses, powder X‐ray diffraction, thermogravimetric analysis, IR spectroscopy and fluorescence spectroscopy. Compounds (1) and (2) are isostructural and crystallize in the P2₁2₁2₁ space group. Both display a three‐dimensional network structure with a one‐dimensional channel, with the benzyl group of the ligand directed towards the channel. An investigation of photoluminescence properties shows that compound (1) displays a strong emission in the purple region.     

Products of arbitrary class-sums in the symmetric group

An algorithm for the evaluation of products of arbitrary conjugacy class-sums in the symmetric group is conjectured. This algorithm generalizes a procedure presented sometime ago, which deals with products in which at least one of the class-sums involved consists of a single cycle (and an appropriate number of fixed points). Let the support size of a conjugacy class be the number of indices that are not fixed points. The algorithm proposed implies that the coefficient of the class-sum C in the product of the class-sums A and B is given in terms of a well-defined enumeration problem within the symmetric group S_p, where p is the smallest of the support sizes of A, B, and C. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 70: 429–440, 1998     

Static and frequency dependent polarizabilities and hyperpolarizabilities of H2Sn

The structure–polarization relationship was investigated in a series of polysulfanes, H₂S_n. The reported results demonstrate that the forms of change of the polarizability components, α_ii, and the second hyperpolarizability components, γ_iiii, as well as the average values α and γ, respectively, of H₂S_n with n are similar. This shows that polarizability components can be easily used to determine corresponding hyperpolarizability data. A remarkable change of the hyperpolarizabilities with the molecular geometry of H₂S_n was found. This result can be used for the design of nonlinear optical materials with optimum properties. The present study uses the flexible σ bonded H₂S_n and is complementary to the works that considered the effect of conformational changes of π-conjugated systems on their hyperpolarizabities. The present computations were performed using the semiempirical approaches MNDO and MNDO/d, as well as ab initio methods with STO-3G, extended with polarization and diffuse functions, and [3s2p/7s5p2d] sets for H₂S_n. At the ab initio level, the electronic and the vibrational contributions to polarizabilities and hyperpolarizabilities were both computed for several members of H₂S_n. The frequency dependence of the above contributions and the static limit were discussed. Electron correlation was taken into account for several test cases using MP2 theory. The selected methods and the variety of the approximations on which they rely allow the systematic consideration of the effect of changes of the geometry of H₂S_n on their polarizabilities and second hyperpolarizabilities. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 1698–1715, 1998     

Syntheses,Structures, and Catalytic Reactions of Palladium Adducts of Chiral Diphosphines That Contain a Rhenium Stereocenter in the Backbone

Reactions of the title diphosphines [(η⁵‐C₅H₄PPh₂)Re(NO)(PPh₃)((CH₂)_nPPh₂)] (n=0, (R)‐ 1 ; n=1, racemic or (S)‐ 2 ) with [PdCl₂(PhCN)₂] give the palladium/rhenium chelate complexes [(η⁵‐C₅H₄PPh₂)Re(NO)(PPh₃)((μ‐CH₂)_nPPh₂)PdCl₂] (n=0, (S)‐ 5 ; n=1, racemic or (S)‐ 6 ) in 75–92% yield. The crystal structure of racemic 6 shows a twisted‐boat conformation of the chelate ring, giving a chiral pocket very different from that in a related rhodium chelate. However, NOE experiments suggest a similar ensemble of conformations in solution. Catalysts are generated from various combinations of a) Pd(OAc)₂ and (R)‐ 1 or (S)‐ 2 (1 : 2), b) (S)‐ 5 or (S)‐ 6 and (R)‐ 1 or (S)‐ 2 (1 : 2), or c) (i‐Bu)₂AlH with the preceding recipes. These factors effect the Heck arylation of 2,3‐dihydrofuran with phenyl trifluoromethylsulfonate. In contrast to analogous reactions with (R)‐binap (=(R)‐2,2′‐bis(diphenylphosphanyl)‐1,1′‐binaphthalene), the major product 2‐phenyl‐2,3‐dihydrofuran is nearly racemic (≤12% ee).     

Comparative Study of Phosphine and N‐Heterocyclic Carbene Stabilized Group 13 Adducts [L(EH3)] and [L2(E2Hn)]

Quantum chemical calculations using density functional theory at the BP86/TZ2P level have been carried out to determine the geometries and stabilities of Group 13 adducts [(PMe₃)(EH₃)] and [(PMe₃)₂(E₂H_n)] (E=B–In; n=4, 2, 0). The optimized geometries exhibit, in most cases, similar features to those of related adducts [(NHC^Me)(EH₃)] and [(NHC^Me)₂(E₂H_n)] with a few exceptions that can be explained by the different donor strengths of the ligands. The calculations show that the carbene ligand L=NHC^Me (:C(NMeCH)₂) is a significantly stronger donor than L=PMe₃. The equilibrium geometries of [L(EH₃)] possess, in all cases, a pyramidal structure, whereas the complexes [L₂(E₂H₄)] always have an antiperiplanar arrangement of the ligands L. The phosphine ligands in [(PMe₃)₂(B₂H₂)], which has C_s symmetry, are in the same plane as the B₂H₂ moiety, whereas the heavier homologues [(PMe₃)₂(E₂H₂)] (E=Al, Ga, In) have C_i symmetry in which the ligands bind side‐on to the E₂H₂ acceptor. This is in contrast to the [(NHC^Me)₂(E₂H₂)] adducts for which the NHC^Me donor always binds in the same plane as E₂H₂ except for the indium complex [(NHC^Me)₂(In₂H₂)], which exhibits side‐on bonding. The boron complexes [L₂(B₂)] (L=PMe₃ and NHC^Me) possess a linear arrangement of the LBBL moiety, which has a B?B triple bond. The heavier homologues [L₂(E₂)] have antiperiplanar arrangements of the LEEL moieties, except for [(PMe₃)₂(In₂)], which has a twisted structure in which the PInInP torsion angle is 123.0°. The structural features of the complexes [L(EH₃)] and [L₂(E₂H_n)] can be explained in terms of donor–acceptor interactions between the donors L and the acceptors EH₃ and E₂H_n, which have been analyzed quantitatively by using the energy decomposition analysis (EDA) method. The calculations predict that the hydrogenation reaction of the dimeric magnesium(I) compound L′MgMgL′ with the complexes [L(EH₃)] is energetically more favorable for L=PMe₃ than for NHC^Me.     

Influence of the Coordination of Donor Solvent Molecules to 1,4,7,10-Tetramethyl-1,4,7,10- tetraazacyclododecanenickel(II) and Analogous Nickel(II) Complexes on Their Steric Factors

Coordination equilibrium constants (K _NiS) of some donor solvent molecules to 1,4,7,10-tetramethyl-1,4,7,10-tetraazacyclododecanenickel(II) ([Ni(Me₄[12]aneN₄)]²⁺) were determined in nitrobenzene (a noncoordinating bulk solvent). The first (K _NiS1) and second stepwise coordination equilibrium constants (K _NiS2) for 1,4,7,10-tetraazacyclododecanenickel(II) ([Ni([12]aneN₄)]²⁺), 1,4,8,11-tetraazac yclotetradecane- nickel(II) ([Ni([14] aneN₄)]²⁺), 1,4,8,11-tetrathiacyclotetra-decanenickel(II) ([Ni([14]aneS₄)]²⁺) were also reinvestigated. The K _NiS values for [Ni(Me₄[12]aneN₄)]²⁺ were compared to those of [Ni([12]aneN₄)]²⁺, (1R,4S, 8R,11S)-1,4,8,11-tetramethyl-1,4,8,11-tetraazacyclotetradecanenickel(II) (R,S,R,S-[Ni(Me₄[14]aneN₄)]²⁺), R,R,S,S-[Ni(Me₄[14]aneN₄)]²⁺, [Ni([14]aneN₄)]²⁺, and [Ni([14]aneS₄)]²⁺. Coordination of pyridine (Py), N,N,N′,N′-tetramethylurea (TMU), and N,N-dimethylacetamide (DMA) to [Ni(Me₄[12]aneN₄)]²⁺ was observed, although these donor solvent molecules did not coordinate to R,S,R,S-[Ni(Me₄[14]aneN₄)]²⁺. The K _NiS values for Py, TMU, and DMA are 7.9, 2.8, and 9.0 dm³⋅mol⁻¹, respectively. Some hydrogen-bonding waters were coordinated to R,S,R,S-[Ni(Me₄[14]aneN₄)]²⁺, but such waters did not coordinate to [Ni(Me₄[12] aneN₄)]²⁺. Also, the K _NiS2 values were larger than the corresponding K _NiS1 values for [Ni([14]aneS₄)]²⁺. Furthermore, the K _NiS1 values for [Ni([12]aneN₄)]²⁺ were the largest among these nickel(II) complex cations. The K _NiS, K _NiS1, and K _NiS2 values are discussed in terms of properties of the donor solvents and steric strains of these nickel(II) complex cations.     

Syntheses and Characterization of Organotin(IV) Complexes Derived from 2‐Mercapto‐1,3,4‐thiadiazole

Four organotin complexes of the types [(Ph₃Sn)(C₂HN₂S₂)] ( 1 ), [(CH₃)₃Sn(C₂HN₂S₂)]_n ( 2 ), [(Bu₂Sn)(C₂HN₂S₂)₂] ( 3 ), and [(Me₂Sn)₄(C₂HN₂S₂)₂(µ ₃‐O)₂(C₂H₅O)₂] ( 4 ) have been obtained by 2‐mercapto‐1,3,4‐thiadiazole and triorganotin chloride or diorganotin dichloride. All the complexes were characterized by elemental analysis, IR and NMR spectroscopy, and X‐ray diffraction analyses, which revealed that complexes 1 and 3 are mononuclear structures, complex 1 can further form a one‐dimensional (1D) helical chain, and complex 3 can further form a 22‐membered macrocycle through the intermolecular C–H·N hydrogen bond; complex 2 is a 1D infinite chain linked by intermolecular N→Sn and S→Sn bonding interactions; complex 4 is a typical ladder structure. © 2012 Wiley Periodicals, Inc. Heteroatom Chem 00:1–8, 2012; View this article online at wileyonlinelibrary.com . DOI 10.1002/hc.21046     

Über [Ag(S9)]−, ein symmetrisches zehngliedriges Ringsystem Darstellung,Struktur und spektroskopische Charakterisierung der schwefelreichen Verbindung [(PPh3)2N][Ag(S9)] · S8

About [Ag(S₉)]^?, a Symmetric Ten-Membered Ring System; Preparation, Structure, and Spectroscopic Characterization of the Sulfur Rich Compound [(PPh₃)₂N][Ag(S₉)] · S₈ Orange [(PPh₃)₂N][Ag(S₉)] · S₈ ( 1 ) could be obtained by reaction of a definite S_x^2?-solution with AgNO₃ and characterized by vibrational spectra (IR/Raman) and X-ray structure analysis. The anion [Ag(S₉)]^? shows a symmetric conformation of a ten-membered ring system. 1 crystallizes in the triclinic space group P1 (a = 1383.8(4), b = 1429.5(4), c = 1540.5(5) pm, α 62.38(2), β 68.05(2), γ 65.86(2)°, V = 2399.1 · 10⁶ pm³, Z = 2; R = 0.077 for 5433 independent reflections (F₀ > 3.92 σ(F₀))).     

Transfer Hydrogenation of Ketones Catalyzed by Surface‐Active Ruthenium and Rhodium Complexes in Water

An improved, high‐yield, one‐pot synthetic procedure for water‐soluble ligands functionalized with trialkyl ammonium side groups H₂N(CH₂)₂NHSO₂‐p‐C₆H₄CH₂[NMe₂(C_nH_2n+1)]⁺ ( [HL ⁿ ]⁺ ; n=8, 16) was developed. The corresponding new surface‐active complexes [(p‐cymene)RuCl( L ⁿ )] and [Cp*RhCl( L ⁿ )] (Cp*=η⁵‐C₅Me₅) were prepared and characterized. For n=16 micelles are formed in water at concentrations as low as 0.6 mM , as demonstrated by surface‐tension measurements. The complexes were used for catalytic transfer hydrogenation of ketones with formate in water. Highly active catalyst systems were obtained in the case of complexes bearing C₁₆ tails due to their ability to be adsorbed at the water/substrate interface. The scope of these catalyst systems in aqueous solutions was extended from partially water soluble aryl alkyl ketones (acetophenone, butyrophenone) to hydrophobic dialkyl ketones (2‐dodecanone).     

Theoretical Study on Molecular Structures of Methylaluminoxane Nanotubes

Density functional theory was used to optimize structures of different methylaluminoxane nanotubes with general formula [(AlOMe)₂]_n, [(AlOMe)₃]_n and [(AlOMe)₄]_n cycle unit, where n ranges from 1 to 10. To explore the stability of nanotubes, the binding energies and total energies are calculated. The results indicate that [(AlOMe)₃]_n and [(AlOMe)₄]_n have the stable structure of nanotubes. When n is 3, they have the most stable structure in all systems. Moreover, [(Al₅O₅)]_n and [(Al₇O₇)]_n are also considered, but their dimers have irregular and distorted structures. So [(Al₅O₅)]_n and [(Al₇O₇)]_n nanotubes are impossible to exist.     

Synthesis and Reactivity of New Functionalized Perfluoroalkylfluorophosphates

A new efficient synthesis of functionalized perfluoroalkyl fluorophosphates by oxidative addition of Me₂NCH₂F to the electron‐deficient phosphanes (C₂F₅)_nPF_3?n (n=0–3) is reported. The initially formed zwitterionic, hexacoordinated phosphates [(C₂F₅)_nF_5?nP(CH₂NMe₂?CH₂NMe₂)] are converted into the corresponding phosphonium salts [(Me₃PCH₂NMe₂]⁺[(C₂F₅)_nF_5?nP(CH₂NMe₂)]^? by treatment with PMe₃. In addition [(C₂F₅)₃F₂P(CH₂NMe₂?CH₂NMe₂)] can undergo a 1,3‐methyl shift from the internal to the terminal nitrogen—a structural characterization was achieved from the CF₃ analogue. Reaction of [(C₂F₅)₃F₂P(CH₂NMe?CH₂NMe₃)] and PMe₃ gave rise to the formation of the zwitterionic phosphonium phosphate [(C₂F₅)₃F₂P(CH₂NMe?CH₂PMe₃)], which was fully characterized by X‐ray diffraction analysis. Moreover, an efficient one‐pot synthesis of Cs⁺[(C₂F₅)₃F₂P(CH₂NMe₂)]^? was pursued. This salt turned out to be a useful nucleophile in several alkylation reactions.     

Synthesen und Kristallstrukturen von Cu‐ und Ag‐Komplexen mit Dithiophosphinat‐ und Trithiophosphonat‐Liganden

Syntheses and Crystal Structures of Copper and Silver Complexes containing Dithiophosphinato and Trithiophosphonato Ligands The reactions of Cu^I and Ag^I salts with diphenyldithiophosphinic acid trimethylsilylester in the presence of tertiary phosphines yield the complexes [Cu(μ‐S)SPPh₂(PR₃)]₂ (R = Me 1a , ⁱPr 1b ), [Ag(μ‐S)SPPh₂(PⁿPr₃)]₂ ( 2 ), [Ag(S₂PPh₂)(PEt₃)]₂ ( 3 ), and [Cu₈(μ₈‐S)(S₂PPh₂)₆] ( 4 ). The cage complex [(PhPS₃)₂Cu₄(PMe₃)₅] ( 5 ) is obtained by the reaction of phenyltrithiophosphonic acid trimethylester. All compounds were structurally characterised by X‐ray crystallography.