Calculation of SN 1 + N 2 ⊃ SN1 ⊗ sn 2 and U(n1 + n2) ⊃ u (n1) ⊗ u (n2) subduction coefficients by using spin graph

By use of the graphical method of spin algebra, simple, and closed expressions for S_N1+_N2 ? S_N1 ? S_N2 and U(n₁ + n₂) ? U(n₁) ? U(n₂), subduction coefficients are derived.     

Unitary group approach to general system partitioning. I. Calculation of U(n = n1 + n2): U(n1) × u(n2) reduced matrix elements and reduced wigner coefficients

This paper is the first in a series of two directed toward a unitary calculus for group-function-type approaches to the many-electron correlation problem. In this paper we present a complete derivation of the matrix elements of the U(n = n₁ + n₂) generators, for the representations approapriate to many-electron systems, in a basis symmetry adapted to the subgroup U(n₁) × U(n₂). Explicit formulae for the fundamental U(n):U(n₁) × U(n₂) reduced Wigner coefficients, which are needed for the general multishell problem, are also obtained. The symmetry properties of the reduced Wigner coefficients and reduced matrix elements are investigated, and a suitable phase convention is given.     

The evaluation of matrix elements for non-canonical Weyl tableau basis states adapted toU(n 1+n 2)⊃U(n 1)×U(n 2)

Summary This is the first paper of a series of two, which enables the evaluation ofU(n) generator matrix elements in the non-canonical Weyl tableau basis adapted to subgroupU(n ₁)×U(n ₂). In this paper the explicit closed formulae for subduction coefficients are presented. These formulae will become useful through an inductive method to be presented in the second paper.     

The Electron Transfer Series [MoIII(bpy)3]n (n=3+, 2+, 1+, 0, 1−), and the Dinuclear Species [{MoIIICl(Mebpy)2}2(μ2‐O)]Cl2 and [{MoIV(tpy.)2}2(μ2‐MoO4)](PF6)2⋅4 MeCN

The electronic structures of the five members of the electron transfer series [Mo(bpy)₃]ⁿ (n=3+, 2+, 1+, 0, 1?) are determined through a combination of techniques: electro‐ and magnetochemistry, UV/Vis and EPR spectroscopies, and X‐ray crystallography. The mono‐ and dication are prepared and isolated as PF₆ salts for the first time. It is shown that all species contain a central Mo^III ion (4d³). The successive one‐electron reductions/oxidations within the series are all ligand‐based, involving neutral (bpy⁰), the π‐radical anion (bpy^.)^1?, and the diamagnetic dianion (bpy^2?)^2?: [Mo^III(bpy⁰)₃]³⁺ (S=3/2), [Mo^III(bpy^.)(bpy⁰)₂]²⁺ (S=1), [Mo^III(bpy^.)₂(bpy⁰)]¹⁺ (S=1/2), [Mo^III(bpy^.)₃] (S=0), and [Mo^III(bpy^.)₂(bpy^2?)]^1? (S=1/2). The previously described diamagnetic dication “[Mo^II(bpy⁰)₃](BF₄)₂” is proposed to be a diamagnetic dinuclear species [{Mo(bpy)₃}₂(μ₂‐O)](BF₄)₄. Two new polynuclear complexes are prepared and structurally characterized: [{Mo^IIICl(^Mebpy⁰)₂}₂(μ₂‐O)]Cl₂ and [{Mo^IV(tpy^.)₂}₂(μ₂‐Mo^VIO₄)](PF₆)₂?4 MeCN.     

S1 (n,π*), T1 (n,π*) and S2(n,π*) emissions in 3,6-diphenyl-s-tetrazine

Simultaneous emissions from S₁(n,π^*) and S₂(n,π^*) states in 3,6-diphenyl-s-tetrazinc (DPT) have been observed along with weak luminescence from T₁ (n,π^*). The occurrence of the S₂(n,π^*) fluorescence has been justified on the basis of the slow S₂ XXX S₁ internal conversion resulting from the large energy gap between the two states. This is the first case of dual fluorescence where both the emitting states are of (n,π^*) nature.     

Unitary group approach to the many-electron problem. II. Adjoint tensor operators for U(n)

In this paper we present a derivation of the U(n) adjoint coupling coefficients for the representations appropriate to many-electron systems. Since the states of a many-fermion system are to comprise the totally antisymmetric Nth rank tensor representation of U(2n), the work of this paper enables the matrix elements of the U(2n) generators to be evaluated directly in the U(n) × U(2) (i.e., spin orbit) basis using their transformation properties as adjoint tensor operators. A connection between the adjoint coupling coefficients, as derived in this paper, and the matrix elements of certain (spin independent) two-body operators is also presented. This indicates that in CI calculations, one may obtain the matrix elements of spin-dependent operators from the known matrix elements of certain spin-independent two-body operators. In particular this implies a segment-level formula for the matrix elements of the U(2n) generators in the spin-orbit basis.     

Photochemical reactions of Re(CO)5Br with Ph2P(CH2) n PPh2 (n = 1, dppm; 2, dppe; 3, dppp)

The new complexes fac-[Re(CO)₃Br{Ph₂P(CH₂) _n PPh₂}] (1a–3a) [(1a), n = 1; (2a), n = 2; (3a), n = 3] and [Re₂(CO)₈Br₂{-Ph₂P(CH₂) _n PPh₂}] (1b–3b) [(1b), n = 1; (2b), n = 2; (3b), n = 3] have been prepared by the photochemical reaction of Re(CO)₅Br with Ph₂P(CH₂) _n PPh₂ (n = 1, dppm; 2, dppe; 3, dppp). The complexes have been characterized by elemental analysis, mass spectroscopy, f.t.-i.r. and ³¹P-[¹H]-n.m.r. spectrometry. The spectroscopic studies suggest cis-chelate bidentate coordination of the ligand in fac-[Re(CO)₃Br{Ph₂P(CH₂) _n PPh₂}] (1a–3a) and cis-bridging bidentate coordination of the ligand between two metals in [Re₂(CO)₈Br₂{cis--Ph₂P(CH₂) _n PPh₂}] (1a–3a).     

Density Functional Theory Study on (Cl2GaN3)n(n=1–4) Clusters

The molecular geometries, vibrational properties, and thermodynamic properties of the clusters (Cl₂GaN₃)_n(n=1–4) have been predicted at the B3LYP/6‐311+G* level. The optimized clusters (Cl₂GaN₃)_n (n=2–4) all possess cyclic structures containing Ga N_α Ga linkages. The relationships between geometrical parameters and oligomerization degree n are discussed. The gas‐phase structures of the trimers prefer to exist in boat‐twisting conformation. As for the tetramer, the S₄ symmetry structure is the most stable. The infrared spectra are obtained and assigned by vibrational analysis. Thermodynamic properties derived from the infrared spectra on the basis of statistical thermodynamic principles are linearly correlated with the oligomerization degree n as well as the temperature. Meanwhile, thermodynamic analysis of the gas‐phase reaction suggests that the oligomerization is exothermic and favorable under high temperature.     

Synthesis and X-Ray Crystal Structures of Fe2Ru(CO)12−n (CNBu t ) n and FeRu2(CO)12−n (CNBu t ) n (n=1, 2)

The clusters Fe₂Ru(CO)_12–n (CNBu ^t ) _n (3, n=1; 4, n=2), FeRu₂(CO)_12–n (CNBu ^t ) _n (5, n=1, 6, n=2) and FeRu₂(CO)₁₁(CNCy) (5a) have been prepared by direct substitution from the parent carbonyl precursors Fe₂Ru(CO)₁₂ (1) and FeRu₂(CO)₁₂ (2). All compounds have been characterized spectroscopically and clusters 3, 4, 5, and 6 by single crystal X-ray determinations. In all cases, the isonitrile ligands adopt axial or pseudo-axial positions on a ruthenium atom. The structures of 3–5 are very similar to their parent clusters, but the extent of metal framework disorder is significantly less. Cluster 6 adopts the same C _2v Fe₃(CO)₁₂ type structure as 4, and thus differs markedly from the parent compound 2, which has a D ₃ structure .     

Matrix elements of the U(2n) generators in the spin-orbit basis

An algorithm for the evaluation of the matrix elements of the U(2n) generators in the spin-orbit basis [induced by the U(n) ? U(2) subgroup] is proposed. This algorithm can be used for calculations with spin-dependent Hamiltonians for many-electron problems.     

Four‐Center Oxidation State Combinations and Near‐Infrared Absorption in [Ru(pap)(Q)2]n (Q=3,5‐Di‐tert‐butyl‐N‐aryl‐1,2‐benzoquinonemonoimine,pap=2‐Phenylazopyridine)

The complex series [Ru(pap)(Q)₂]ⁿ ([ 1 ]ⁿ–[ 4 ]ⁿ; n=+2, +1, 0, ?1, ?2) contains four redox non‐innocent entities: one ruthenium ion, 2‐phenylazopyridine (pap), and two o‐iminoquinone moieties, Q=3,5‐di‐tert‐butyl‐N‐aryl‐1,2‐benzoquinonemonoimine (aryl=C₆H₅ ( 1⁺ ); m‐(Cl)₂C₆H₃ ( 2⁺ ); m‐(OCH₃)₂C₆H₃ ( 3⁺ ); m‐(tBu)₂C₆H₃ ( 4 ⁺)). A crystal structure determination of the representative compound, [ 1 ]ClO₄, established the crystallization of the ctt‐isomeric form, that is, cis and trans with respect to the mutual orientations of O and N donors of two Q ligands, and the coordinating azo N atom trans to the O donor of Q. The sensitive C? O (average: 1.299(3) Å), C? N (average: 1.346(4) Å) and intra‐ring C? C (meta; average: 1.373(4) Å) bond lengths of the coordinated iminoquinone moieties in corroboration with the N?N length (1.292(3) Å) of pap in 1 ⁺ establish [Ru^III(pap⁰)(Q^.?)₂]⁺ as the most appropriate electronic structural form. The coupling of three spins from one low‐spin ruthenium(III) (t_2g⁵) and two Q^.? radicals in 1 ⁺– 4 ⁺ gives a ground state with one unpaired electron on Q^.?, as evident from g=1.995 radical‐type EPR signals for 1 ⁺– 4 ⁺. Accordingly, the DFT‐calculated Mulliken spin densities of 1 ⁺ (1.152 for two Q, Ru: ?0.179, pap: 0.031) confirm Q‐based spin. Complex ions 1 ⁺– 4 ⁺ exhibit two near‐IR absorption bands at about λ=2000 and 920 nm in addition to intense multiple transitions covering the visible to UV regions; compounds [ 1 ]ClO₄–[ 4 ]ClO₄ undergo one oxidation and three separate reduction processes within ±2.0 V versus SCE. The crystal structure of the neutral (one‐electron reduced) state ( 2 ) was determined to show metal‐based reduction and an EPR signal at g=1.996. The electronic transitions of the complexes 1 ⁿ– 4 ⁿ (n=+2, +1, 0, ?1, ?2) in the UV, visible, and NIR regions, as determined by using spectroelectrochemistry, have been analyzed by TD‐DFT calculations and reveal significant low‐energy absorbance (λ_max>1000 nm) for cations, anions, and neutral forms. The experimental studies in combination with DFT calculations suggest the dominant valence configurations of 1 ⁿ– 4 ⁿ in the accessible redox states to be [Ru^III(pap⁰)(Q^.?)(Q⁰)]²⁺ ( 1 ²⁺– 4 ²⁺)→[Ru^III(pap⁰)(Q^.?)₂]⁺ ( 1 ⁺– 4 ⁺)→[Ru^II(pap⁰)(Q^.?)₂] ( 1 – 4 )→[Ru^II(pap^.?)(Q^.?)₂]^? ( 1 ^?– 4 ^?)→[Ru^III(pap^.?)(Q^2?)₂]^2? ( 1 ^2?– 4 ^2?).     

Metal Ion‐tuned Complexes based on Benzene‐1, 2, 4, 5‐tetracarboxylic Acid and the Semirigid Ligand 1, 4‐Bis(2‐methylbenzimidazol‐1‐ylmethyl) Benzene

Abstract. Two metal‐organic coordination polymers [Co(bmb)(btc)_0.5]_n( 1 ) and {[Zn(bmb)_0.5(btc)_0.5(H₂O)] · 0.5bmb · H₂O}_n ( 2 ) [H₄btc = benzene‐1, 2, 4, 5‐tetracarboxylic acid, bmb = 1, 4‐bis(2‐methylbenzimidazol‐1‐ylmethyl) benzene] were prepared under hydrothermal conditions. Single‐crystal X‐ray diffraction indicates that both complexes have a 2D framework structure with (4 · 6²) (4² · 6² · 8²) topology. Interestingly, the hydrogen bonds in 2 form a fascinating meso‐helix. The catalytic activity of 1 for oxidative coupling of 2, 6‐dimethylphenol (DMP) and the photoluminescence properties of 2 were investigated. Furthermore, the complexes were investigated by IR spectroscopy and thermogravimetric analysis.     

Structure,Optical, and Magnetic Properties of (PPN+)2(C702−)⋅2 C6H4Cl2, which Contains Dianionic Polymeric (C702−)n Chains

A new salt, (PPN⁺)₂(C₇₀^2?) ? 2 C₆H₄Cl₂ ( 1 ), which contains C₇₀^2? dianions, has been obtained as single crystals (PPN⁺=bis(triphenylphosphine)iminium cation). The C₇₀^2? dianions form polymeric zigzag (C₇₀^2?)_n chains, in which the fullerene units are bonded through single C? C bonds of length 1.581(5)–1.586(6) Å. The distance between the centers of neighboring C₇₀^2? units is 10.441 Å. The optical and magnetic properties of (C₇₀^2?)_n have also been studied. Decreasing the symmetry of C₇₀ in the polymer activate about 20 new IR bands in addition to the 10 IR‐active bands of the starting C₇₀. The polymeric structure shows absorptions in the visible and NIR regions, with three main bands at 890, 1200, and 1550 nm, instead of one band of isolated C₇₀^2? dianions at 1165–1184 nm. We concluded that the (C₇₀^2?)_n polymer was diamagnetic, with a negative molar magnetic susceptibility of ?3.82×10^?4 emu mol^?1 per C₇₀^2? dianion. The polymer is EPR silent and a weak narrow EPR signal in salt 1 is due to impurities, which only constitute 0.84 % of spin S=1/2 of the total amount of fullerene C₇₀.     

Darstellung,Kristallstrukturen und Schwingungsspektren von [Pt(N3)6]2– und [Pt(N3)Cl5]2–, 195Pt‐ und 15N‐NMR‐Spektren von [Pt(N3)nCl6–n]2– und [Pt(15NN2)n(N215N)6–n]2–, n = 0–6

Synthesis, Crystal Structures, and Vibrational Spectra of [Pt(N₃)₆]^2– and [Pt(N₃)Cl₅]^2–, ¹⁹⁵Pt and ¹⁵N NMR Spectra of [Pt(N₃)_nCl_6–n]^2– and [Pt(¹⁵NN₂)_n(N₂¹⁵N)_6–n]^2–, n = 0–6 By ligand exchange of [PtCl₆]^2– with sodium azide mixed complexes of the series [Pt(N₃)_nCl_6–n]^2– and with ¹⁵N‐labelled sodium azide (Na¹⁵NN₂) mixtures of the isotopomeres [Pt(¹⁵NN₂)_n(N₂¹⁵N)_6–n]^2–, n = 0–6 and the pair [Pt(¹⁵NN₂)Cl₅]^2–/[Pt(N₂¹⁵N)Cl₅]^2– are formed. X‐ray structure determinations on single crystals of (Ph₄P)₂[Pt(N₃)₆] ( 1 ) (triclinic, space group P1, a = 10.175(1), b = 10.516(1), c = 12.380(2) Å, α = 87.822(9), β = 73.822(9), γ = 67.987(8)°, Z = 1) and (Ph₄As)₂[Pt(N₃)Cl₅] · HCON(CH₃)₂ ( 2 ) (triclinic, space group P1, a = 10.068(2), b = 11.001(2), c = 23.658(5) Å, α = 101.196(14), β = 93.977(15), γ = 101.484(13)°, Z = 2) have been performed. The bond lengths are Pt–N = 2.088 ( 1 ), 2.105 ( 2 ) and Pt–Cl = 2.318 Å ( 2 ). The approximate linear azido ligands with N^α–N^β–N^γ‐angles = 173.5–174.6° are bonded with Pt–N^α–N^β‐angles = 116.4–121.0°. In the vibrational spectra the PtCl stretching vibrations of (n‐Bu₄N)₂[Pt(N₃)Cl₅] are observed at 318–345, the PtN stretching modes of (n‐Bu₄N)₂[Pt(N₃)₆] at 401–428 and of (n‐Bu₄N)₂[Pt(N₃)Cl₅] at 408–413 cm^–1. The mixtures (n‐Bu₄N)₂[Pt(¹⁵NN₂)_n(N₂¹⁵N)_6–n], n = 0–6 and (n‐Bu₄N)₂[Pt(¹⁵NN₂)Cl₅]/(n‐Bu₄N)₂[Pt(N₂¹⁵N)Cl₅] exhibit ¹⁵N‐isotopic shifts up to 20 cm^–1. Based on the molecular parameters of the X‐ray determinations the vibrational spectra are assigned by normal coordinate analysis. The average valence force constants are f_d(PtCl) = 1.93, f_d(PtN^α) = 2.38 and f_d(N^αN^β, N^βN^γ) = 12.39 mdyn/Å. In the ¹⁹⁵Pt NMR spectrum of [Pt(N₃)_nCl_6–n]^2–, n = 0–6 downfield shifts with the increasing number of azido ligands are observed in the range 4766–5067 ppm. The ¹⁵N NMR spectrum of (n‐Bu₄N)₂[Pt(¹⁵NN₂)_n(N₂¹⁵N)_6–n], n = 0–6 exhibits by ¹⁵N–¹⁹⁵Pt coupling a pseudotriplett at –307.5 ppm. Due to the isotopomeres n = 0–5 for terminal ¹⁵N six well‐resolved signals with distances of 0.03 ppm are observed in the low field region at –201 to –199 ppm.     

The conformations of two copper(I) complexes of 1H‐benzimidazole‐2(3H)‐thione and thiosaccharinate

(Acetonitrile‐1κN)[μ‐1H‐benzimidazole‐2(3H)‐thione‐1:2κ²S:S][1H‐benzimidazole‐2(3H)‐thione‐2κS]bis(μ‐1,1‐dioxo‐1λ⁶,2‐benzothiazole‐3‐thiolato)‐1:2κ²S³:N;1:2κ²S³:S³‐dicopper(I)(Cu—Cu), [Cu₂(C₇H₄NO₂S₂)₂(C₇H₆N₂S)₂(CH₃CN)] or [Cu₂(tsac)₂(Sbim)₂(CH₃CN)] [tsac is thiosaccharinate and Sbim is 1H‐benzimidazole‐2(3H)‐thione], (I), is a new copper(I) compound that consists of a triply bridged dinuclear Cu—Cu unit. In the complex molecule, two tsac anions and one neutral Sbim ligand bind the metals. One anion bridges via the endocyclic N and exocyclic S atoms (μ‐S:N). The other anion and one of the mercaptobenzimidazole molecules bridge the metals through their exocyclic S atoms (μ‐S:S). The second Sbim ligand coordinates in a monodentate fashion (κS) to one Cu atom, while an acetonitrile molecule coordinates to the other Cu atom. The Cu^I—Cu^I distance [2.6286 (6) Å] can be considered a strong `cuprophilic' interaction. In the case of [μ‐1H‐benzimidazole‐2(3H)‐thione‐1:2κ²S:S]bis[1H‐benzimidazole‐2(3H)‐thione]‐1κS;2κS‐bis(μ‐1,1‐dioxo‐1λ⁶,2‐benzothiazole‐3‐thiolato)‐1:2κ²S³:N;1:2κ²S³:S³‐dicopper(I)(Cu—Cu), [Cu₂(C₇H₄NO₂S₂)₂(C₇H₆N₂S)₃] or [Cu₂(tsac)₂(Sbim)₃], (II), the acetonitrile molecule is substituted by an additional Sbim ligand, which binds one Cu atom via the exocylic S atom. In this case, the Cu^I—Cu^I distance is 2.6068 (11) Å.     

(Z)‐2‐Methylbuten‐1‐yl(aryl)iodonium trifluoromethanesulfonates containing electron‐withdrawing groups on the aryl moiety

The crystal structures of [(Z)‐2‐methyl­but‐1‐en‐1‐yl]­[4‐(tri­fluoro­methyl)­phenyl]­iodo­nium tri­fluoro­methane­sulfonate, C₁₂H₁₃F₃I⁺·CF₃O₃S^?, (I), (3,5‐di­chloro­phenyl)­[(Z)‐2‐methyl­but‐1‐en‐1‐yl]­iodo­nium tri­fluoro­methane­sulfonate, C₁₁H₁₂­Cl₂I⁺·CF₃O₃S^?, (II), and bis{[3,5‐bis­(tri­fluoro­methyl)­phenyl][(Z)‐2‐methyl­but‐1‐en‐1‐yl]­iodo­nium} bis­(tri­fluoro­methane­sulfonate) di­chloro­methane solvate, 2C₁₃H₁₂F₆I⁺·­2CF₃­O₃S^?·CH₂Cl₂, (III), are described. Neither simple acyclic β,β‐di­alkyl‐substituted alkenyl­(aryl)­idonium salts nor a series containing electron‐deficient aryl rings have been described prior to this work. Compounds (I)–(III) were found to have distorted square‐planar geometries, with each I atom interacting with two tri­fluoro­methane­sulfonate counter‐ions.     

Theoretical studies of organometallic compounds. II. All electron and pseudopotential calculations of M(CH3)nCl4 − n (M = C,Si, Ge,Sn, Pb; n = 0–4)

The performance of effective core potentials (ECP) for the main group elements of group IV has been studied by calculating the geometries and reaction energies of isodesmic reactions for the molecules M(CH₃)_nCl_{4 ? n} (M = C, Si, Ge, Sn, Pb; n = 0–4) at the Hartree–Fock level of theory. The results are compared with data from all electron calculations and experimental results as far as available. The all electron calculations were performed with a 3-21G(d) and a 6-31G(d) basis set for Si, a (43321/4321/41) basis set for Ge, and a (433321/43321/431) basis set for Sn. For the ECP calculations the potentials developed by Hay and Wadt with a configuration (n)s^a(n)p^b and the valence basis set (21/21), extended by a set of d functions, are employed. © 1992 by John Wiley & Sons, Inc.     

Complexes of N‐Thiophosphorylthiourea α‐NaphthylNHC(S)NHP(S)(OiPr)2 (HL) with Copper(I). Crystal Structures of HL and Cu(PPh3)2L

Reaction of the potassium salt of N‐thiophosphorylated thiourea α‐naphthylNHC(S)NHP(S)(OiPr)₂ ( HL ) with Cu(PPh₃)₃I in aqueous EtOH/CH₂Cl₂ leads to the mononuclear complex [Cu(PPh₃)₂L–S,S′]. By using copper(I) iodide instead ofCu(PPh₃)₃I, the polynuclear complex [Cu_n(L–S,S′)_n] was obtained. The structures of these compounds were investigated by elemental analysis, ¹H and ³¹P{¹H} NMR and IR spectroscopy. The crystal structures of HL and Cu(PPh₃)₂L were determined by single‐crystal X‐ray diffraction.     

Integrity and Structural Characteristics of the M6E8 (E=S, Se, Te) Cluster Core: Syntheses and Structures of [Co6S8(PR3)6] n+ (R=Me2Ph, n=1; R=OMe, n=0)

Two new members of the hexanuclear series [Co₆S₈(PR₃)₆] ⁿ⁺, complexes [Co₆S₈(PMe₂Ph)₆](ClO₄) (1) and [Co₆S₈P(OMe)_{3 6}] (2), have been synthesizes and characterized by X-ray diffraction analyses. Their formation process was postulated to go through trinuclear ₃--S bridged moieties. The structural characteristics of the M₆E₈P₆ skeleton of a whole series of [M₆E₈(PR₃)₆] ⁿ⁺ (M=Co, Cr, Fe, Mo; E=S, Se, Te) complexes are presented in terms of atomic distances and core volumes.     

Molecular point group adaptation of spin–free configurations

Symmetry adaptation of spin–free multishell electron configurations in molecules to general non-Abelian point groups has been carried out. Using the basis spanning the irreducible representation [2^N/2?S, 1^2S, 0^n?N/2?S] of the unitary group U(n) as primitives, the Wigner operators for point groups were applied to generate the required basis. In the process it was found that the segments of the Weyl tableaux could be handled individually. Using this and the matric algebra of permutation group a viable procedure has been developed for point groups adaptation. A program based on the procedure has been generated and implemented.