Algebraic solutions for point groups: The octahedral group for the group chain O⊃T⊃C3

Using the right‐induced technique and the eigenfunction method, concise algebraic expressions of the projection operators for both single‐valued and double‐valued representations are found for the group chain O?T?C₃ in terms of the projection operators of T?C₃. Extremely simple relations are discovered between the symmetry adapted functions (SAFs) of the groups T and O; namely the SAFs of the subgroup T which have proper symmetry are the SAFs of the group O. The projection operators and SAFs are functions of only the quantum numbers of the group chain [the analogy of ( j,m) for the group chain SO₃?SO₂], without involving any irreducible matrix elements. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 83: 259–270, 2001     

Calculation of j and jm symbols for arbitrary compact groups. I. Methodology

A systematic method is established for computing the coupling coefficients associated with arbitrary compact groups using only the general properties of the coefficients and the specific character theory of the relevant groups. The basic character theory is first outlined. Then a primitive set of 6j symbols is defined and their computation sketched with careful attention being given to matters of phase fixing. A recursive method is developed for calculating arbitrary 6j symbols using the primitive set. Finally, a primitive set of 3jm factors is set up and then the general 3jm factors computed recursively.     

Calculation of j and jm symbols for arbitrary compact groups. II. An alternate procedure for angular momentum

The various orthogonality and sum rules which the 6j and 3jm symbols satisfy are sufficient to obtain the algebraic formulas for these symbols for SO₃ ? SO₂. Character theory enters in that the j's and m's occurring in the various sums are given by the triangle rule together with information on the symmetrized product and the branching, SO₃ ? SO₂, The resulting calculation is somewhat simpler, algebraically speaking, than previous calculations and has the pedagogical advantage that only the concept of an irreducible representation of a group is required, instead of the more elaborate concept of ladder operators.     

Algebraic solutions for point groups: Cubic groups G in the group chain G⊃T⊃D2⊃C2

Concise algebraic expressions of the symmetry‐adapted functions (SAFs) for both single‐valued and double‐valued representations are derived for the group chain O⊃T⊃D₂⊃C₂ and O⊃D₄⊃D₂⊃C₂, which are functions of only the quantum numbers of the respective group chain without involving any irreducible matrix elements. It is shown that the SAFs of the cubic groups G=O,T_d,T_h,O_h can be expressed in a simple way in terms of the SAFs of the group T. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 76: 585–599, 2000     

Algebraic solutions for the octahedral group: Group chain O⊃C4

Using double‐induced representation and eigenfunction method, algebraic expressions are derived for irreducible matrices, projection operators, and symmetry‐adapted functions in the group chain O⊃C₄ for both single‐valued and double‐valued representations. The simplicity of these expressions lies in the fact that they are functions of the quantum numbers of the corresponding group chain (the analogy of j, m for the group chain SO₃⊃SO₂) instead of the irreducible matrix elements. The symmetries of the symmetry‐adapted functions are disclosed. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 74: 7–22, 1999     

Clebsch–Gordan coefficients for the group chains O ⊃ T ⊃ D2 and Td ⊃ T ⊃D2 by the method of projective representations

The spinor representations for the groups forming the chains O ? T ? D ₂ and T _d ? T ? D ₂ are constructed as projective representations. The Clebsch–Gordan coefficients are then calculated using a standard method. Projective factor systems, irreducible representations, canonical bases, and tables of Clebsch–Gordan coefficients are given. The subduction from O to D ₃ is discussed.     

Thermochemical properties from G3MP2B3 calculations,Set‐2: Free radicals with special consideration of CH2CH?C•CH2, cyclo−•C5H5, •CH2OOH,HO?•CO,and HC(O)O•

After a set of 32 free radicals was presented (Int J Chem Kin 34, 550–560, 2002), an additional 60 free radicals (Set‐2) were studied and characterized by energy minimum structures, harmonic vibrational wave numbers ω_e, moments of inertia I_A, I_B, and I_C, heat capacities C^o_p(T), standard entropies S^o(T), thermal energy contents H^o(T) ? H^o(0), and standard enthalpies of formation Δ_fH^o(T) at the G3MP2B3 level of theory. Thermodynamic functions at T = 298.15 K are presented and compared with recent experimental values where these are available. The mean absolute deviation between calculated and experimental Δ_fH^o(298.15) values by the previous set of 32 radicals is 3.91 kJ mol^?1. For the sake of comparison, only 49 species out of the 60 radicals of Set‐2 are characterized by experimental enthalpies of formation, and the corresponding mean absolute deviation between calculated and experimental Δ_fH^o(298.15) values is 8.96 kJ mol^?1. This situation is cause for demand of more and also more accurate experimental values. In addition to the above properties, parent molecules of a large set of the respective radicals are calculated to obtain bond dissociation energies D^o(298.15). Radical stabilization owing to resonance is discussed using the complete sets of total atomic spin densities ρ as a support. In particular, a short review about recent developments of the first‐order Jahn–Teller radical c‐C₅H₅^? is presented. In addition, radicals with negative bond energies are described, such as ^?CH₂OOH where the reaction path to CH₂O + HO^? has been calculated, as well as radicals which have two different parent molecules, for example C?N? O^?. For the reaction HO^? + CO → H^? + CO₂, two reaction paths are characterized by a total of 14 stationary points where the intermediate radicals HO? ^?CO and HC(O)O^? are involved. © 2004 Wiley Periodicals, Inc. Int J Chem Kinet 36: 661–686, 2004     

Chiral Fluorinated α‐Sulfonyl Carbanions: Enantioselective Synthesis and Electrophilic Capture,Racemization Dynamics,and Structure

Enantiomerically pure triflones R¹CH(R²)SO₂CF₃ have been synthesized starting from the corresponding chiral alcohols via thiols and trifluoromethylsulfanes. Key steps of the syntheses of the sulfanes are the photochemical trifluoromethylation of the thiols with CF₃Hal (Hal=halide) or substitution of alkoxyphosphinediamines with CF₃SSCF₃. The deprotonation of RCH(Me)SO₂CF₃ (R=CH₂Ph, iHex) with nBuLi with the formation of salts [RC(Me)? SO₂CF₃]Li and their electrophilic capture both occurred with high enantioselectivities. Displacement of the SO₂CF₃ group of (S)‐MeOCH₂C(Me)(CH₂Ph)SO₂CF₃ (95 % ee) by an ethyl group through the reaction with AlEt₃ gave alkane MeOCH₂C(Me)(CH₂Ph)Et of 96 % ee. Racemization of salts [R¹C(R²)SO₂CF₃]Li follows first‐order kinetics and is mainly an enthalpic process with small negative activation entropy as revealed by polarimetry and dynamic NMR (DNMR) spectroscopy. This is in accordance with a C_α? S bond rotation as the rate‐determining step. Lithium α‐(S)‐trifluoromethyl‐ and α‐(S)‐nonafluorobutylsulfonyl carbanion salts have a much higher racemization barrier than the corresponding α‐(S)‐tert‐butylsulfonyl carbanion salts. Whereas [PhCH₂C(Me)SO₂tBu]Li/DMPU (DMPU = dimethylpropylurea) has a half‐life of racemization at ?105 °C of 2.4 h, that of [PhCH₂C(Me)SO₂CF₃]Li at ?78 °C is 30 d. DNMR spectroscopy of amides (PhCH₂)₂NSO₂CF₃ and (PhCH₂)N(Ph)SO₂CF₃ gave N? S rotational barriers that seem to be distinctly higher than those of nonfluorinated sulfonamides. NMR spectroscopy of [PhCH₂C(Ph)SO₂R]M (M=Li, K, NBu₄; R=CF₃, tBu) shows for both salts a confinement of the negative charge mainly to the C_α atom and a significant benzylic stabilization that is weaker in the trifluoromethylsulfonyl carbanion. According to crystal structure analyses, the carbanions of salts {[PhCH₂C(Ph)SO₂CF₃]Li? L }₂ ( L =2 THF, tetramethylethylenediamine (TMEDA)) and [PhCH₂C(Ph)SO₂CF₃]NBu₄ have the typical chiral C_α? S conformation of α‐sulfonyl carbanions, planar C_α atoms, and short C_α? S bonds. Ab initio calculations of [MeC(Ph)SO₂tBu]^? and [MeC(Ph)SO₂CF₃]^? showed for the fluorinated carbanion stronger n_C→σ* and n_O→σ* interactions and a weaker benzylic stabilization. According to natural bond orbital (NBO) calculations of [R¹C(R²)SO₂R]^? (R=tBu, CF₃) the n_C→σ*_{S? R} interaction is much stronger for R=CF₃. Ab initio calculations gave for [MeC(Ph)SO₂tBu]Li ? 2 Me₂O an O,Li,C_α contact ion pair (CIP) and for [MeC(Ph)SO₂CF₃]Li ? 2 Me₂O an O,Li,O CIP. According to cryoscopy, [PhCH₂C(Ph)SO₂CF₃]Li, [iHexC(Me)SO₂CF₃]Li, and [PhCH₂C(Ph)SO₂CF₃]NBu₄ predominantly form monomers in tetrahydrofuran (THF) at ?108 °C. The NMR spectroscopic data of salts [R¹(R²)SO₂R³]Li (R³=tBu, CF₃) indicate that the dominating monomeric CIPs are devoid of C_α? Li bonds.     

Pressure dependence of dielectric properties of chlorinated polyethylene vulcanizate. I. α relaxation and shift factors

The complex dielectric constant was measured under elevated pressure for the α relaxation of vulcanized chlorinated polyethylene. Both temperature and pressure effects on the static dielectric constants, the activation enthalpy, and volume, and the pressure dependence of the glass-transition temperature were obtained. The dependence of shift factors on temperature was expressed by the Vogel–Fulcher–Tamman–Hesse (VFTH) equation: ?log a_T = A ? B/(T ? T₀). The parameters A, B, and T₀ for each pressure applied were calculated by minimizing the standard deviation between log a_T and experiments. The values of the parameters in the Williams–Landel–Ferry (WLF) equation: ?log a_T = C₁(T ? T_g)/[C₂ + (T ? T_g)], were also estimated from the resulting values of the VFTH parameters. All these parameters depended on pressure. The activation volume plotted against T ? T_g decreased with increasing pressure.     

1‐Methyl‐4‐nitraminopyridinium nitrate and 4‐nitraminopyridinium methanesulfonate

In the title compounds, C₆H₈N₃O₂⁺·NO₃^? and C₅­H₆­N₃­O₂⁺·­CH₃SO₃^?, respectively, the cations are almost planar; the twist of the nitr­amino group about the C—N and N—N bonds does not exceed 10°. The deviations from coplanarity are accounted for by intermolecular N—H?O interactions. The coplanarity of the NHNO₂ group and the phenyl ring leads to the deformation of the nitr­amino group. The C—N—N angle and one C—C—N angle at the junction of the phenyl ring and the nitr­amino group are increased from 120° by ca 6°, whereas the other junction C—C—N angle is decreased by ca 5°. Within the nitro group, the O—N—O angle is increased by ca 5° and one O—N—N angle is decreased by ca 5°, whereas the other O—N—N angle remains almost unchanged. The cations are connected to the anions by relatively strong N—H?O hydrogen bonds [shortest H?O separations 1.77 (2)–1.81 (3) Å] and much weaker C—H?O hydrogen bonds [H?O separations 2.30 (2)–2.63 (3) Å].     

4‐Carboxypiperidinium 1‐carboxycyclobutane‐1‐carboxylate

The title salt, C₆H₁₂NO₂⁺·C₆H₇O₄⁻ or ISO⁺·CBDC⁻, is an ionic ensemble assisted by hydrogen bonds. The amino acid moiety (ISO or piperidine‐4‐carboxylic acid) has a protonated ring N atom (ISO⁺ or 4‐carboxypiperidinium), while the semi‐protonated acid (CBDC⁻ or 1‐carboxycyclobutane‐1‐carboxylate) has the negative charge residing on one carboxylate group, leaving the other as a neutral –COOH group. The –⁺NH₂– state of protonation allows the formation of a two‐dimensional crystal packing consisting of zigzag layers stacked along a separated by van der Waals distances. The layers extend in the bc plane connected by a complex network of N—H...O and O—H...O hydrogen bonds. Wave‐like ribbons, constructed from ISO⁺ and CBDC⁻ units and described by the graph‐set symbols C₃³(10) and R₃³(14), run alternately in opposite directions along c. Intercalated between the ribbons are ISO⁺ cations linked by hydrogen bonds, forming rings described by the graph‐set symbols R₆⁶(30) and R₄²(18). A detailed analysis of the structures of the individual components and the intricate hydrogen‐bond network of the crystal structure is given.     

Bis(μ‐succinato‐O,O′:O′′,O′′′)bis[bis(2‐amino‐1,3‐benzothiazole‐N3)copper(II)]

In the title dimeric complex, [Cu₂(C₄H₄O₄)₂(C₇H₆N₂S)₄], which possesses a centre of symmetry, the Cu atoms are enclosed in a 14‐membered ring. They adopt a distorted square‐bipyramidal (4+2) coordination. The four closest donor atoms are two N atoms of 2‐amino­benzo­thiazole ligands and two O atoms of the succinate carboxylate groups. They form a square‐planar cis arrangement, with an average Cu—N distance of 2.003 (3) Å and Cu—O distances of 1.949 (3) and 1.965 (3) Å. Two longer Cu—O bonds of 2.709 (3) and 2.613 (3) Å involving the remaining O atoms of the carboxylate groups complete the sixfold coordination of the Cu atoms. The H atoms of each amino group of the 2‐amino­benzo­thiazole molecules form intra‐ and inter­molecular N—H?O hydrogen bonds. A nearly perpendicular inter­molecular C—H?Cg interaction (Cg is the centroid of the imidazole ring) is observed. The intramolecular Cu?Cu distance is 6.384 (2) Å.     

13C NMR spectra of metal σ-cyclopentadienyls

Proton decoupled ¹³C NMR spectra have been measured for the cyclopentadienyl compounds C₅H₅Si(CH₃)_nCl_3?n(n = 1, 2, 3), C₅H₅Ge(CH₃)₃, CH₃C₅H₄Ge(CH₃)₃, C₅H₅Sn(CH₃)₃, σ-C₅H₅Fe(CO)₂-π-C₅H₅ and C₅H₅HgCH₃. A fast metallotropic rearrangement occurring in the compounds causes the spectra to be temperature dependent for the Si, Ge, Sn and Fe derivatives. For the derivatives of silicon or germanium, the olefinic signals are unsymmetrically broadened by the 1,2-shift at lower migration rates. Line widths of the ring carbon signals have been measured to give an estimate for the activation parameters of the rearrangement in C₅H₅Ge(CH₃)₃ (E_a = 10·7 ± 0·9 kcal/mole, ΔG^? = 13·4 ± 0·9 kcal/mole) and C₅H₅Sn(CH₃)₃ (E_a = 6·8 ± 0·7 kcal/mole, ΔG^? = 7·1 ± 0·7 kcal/mole). At room temperature, the spectrum of C₅H₅HgCH₃ displays just one narrow signal responsible for the cyclopentadienyl ligand. The spectrum of CH₃C₅H₄Ge(CH₃)₃ at –30° demonstrates that two isomers containing methyl in the vinylic position are present, the ratio being ca. 2:1. The ¹³C spectra of the vinylic isomers have been analysed in the case of C₅H₅Si(CH₃)_nCl_3?n.     

Structures and stabilities of [C3H2]+ ˙ and [C3H2]2+ ions

Ab initio molecular orbital theory using basis sets up to 6-311G^{* *}, with electron correlation incorporated via configuration interaction calculations with single and double substitutions, has been used to study the structures and energies of the C₃H₂ monocation and dication. In agreement with recent experimental observations, we find evidence for stable cyclic and linear isomers of [C₃H₂]^{+ ˙}. The cyclic structure (, a) represents the global minimum on the [C₃H₂]^{+ ˙} potential energy surface. The linear isomer (, b) lies somewhat higher in energy, 53 kJ mol^?1 above a. The calculated heat of formation for [HCCCH]^{+ ˙} (1369 kJ mol^?1) is in good agreement with a recent experimental value (1377 kJ mol^?1). For the [C₃H₂]²⁺ dication, the lowest energy isomer corresponds to the linear [HCCCH]²⁺ singlet (h). Other singlet and triplet isomers are found not to be competitive in energy. The [HCCCH]²⁺ dication (h) is calculated to be thermodynamically stable with respect to deprotonation and with respect to C? C cleavage into CCH⁺ + CH⁺. The predicted stability is consistent with the frequent observation of [C₃H₂]²⁺ in mass spectrometric experiments. Comparison of our calculated ionization energies for the process [C₃H₂]^{+ ˙} → [C₃H₂]²⁺ with the Q_min values derived from charge-stripping experiments suggests that the ionization is accompanied by a significant change in structure.     

Theoretical Studies of the Reactions CFxH3−xCOOR+Cl and CF3COOCH3+OH

The mechanism and kinetics of the reactions of CF₃COOCH₂CH₃, CF₂HCOOCH₃, and CF₃COOCH₃ with Cl and OH radicals are studied using the B3LYP, MP2, BHandHLYP, and M06‐2X methods with the 6‐311G(d,p) basis set. The study is further refined by using the CCSD(T) and QCISD(T)/6‐311++G(d,p) methods. Seven hydrogen‐abstraction channels are found. All the rate constants, computed by a dual‐level direct method with a small‐curvature tunneling correction, are in good agreement with the experimental data. The tunneling effect is found to be important for the calculated rate constants in the low‐temperature range. For the reaction of CF₃COOCH₂CH₃+Cl, H‐abstraction from the CH₂ group is found to be the dominant reaction channel. The standard enthalpies of formation for the species are also calculated. The Arrhenius expressions are fitted within 200–1000 K as k_T(1)=8.4×10^?20T ^2.63exp(381.28/T), k_T(2)=2.95×10^?21T ^3.13exp(?103.21/T), k_T(3)=1.25×10^?23T ^3.37exp(791.98/T), and k_T(4)=4.53×10^?22T ^3.07exp(465.00/T).     

Darstellung und Kristallstruktur eines Calciumcarbidchlorides mit einer C34−-Einheit,Ca3Cl2C3

Synthesis and Crystal Structure of a Calciumcarbide Chloride Containing a C₃^4? Unit, Ca₃Cl₂C₃ Ca₃Cl₂C₃ was prepared from calcium, CaCl₂ and graphite in sealed tantalum capsules. Red, transparent crystals were obtained from heating the mixture to 900°C (for one day) and annealing afterwards at 780°C for three days. The compound forms a layered structure (Cmcm, Z = 4, a = 384.24(9) pm, b = 1340.7(3) pm, c = 1152.6(3) pm, R = R_W = 0.036 for 481 independent intensities) with alternating stacks of double layers of Ca²⁺ and monolayers of Cl^?. The double layers of calcium contain allylenide ions, C₃^4?. The latter exhibit C_2v symmetry, a bond angle (C? C? C) of 169.0(6)° and a C? C separation of 134.6(4) pm.     

3‐Hydro­xy­benz­aldehyde

The title compound, C₇H₆O₂, forms infinite chains where the mol­ecules are hydrogen bonded via the hydroxyl and aldehyde groups, with an O?O distance of 2.719 (3) Å. Interchain interactions are weak. The geometry of the ring differs from the ideal form due to the effect of the substituents. Abinitio (Hartree–Fock self‐consistent field–molecular orbital and density functional theory) calculations for the free mol­ecule reproduce well the observed small distortions of the ring. In the crystal, the geometry deviates from the ideal C_s symmetry of the free mol­ecule, as given by the ab initio calculations. The aldehyde and hydroxyl groups are twisted around the single bonds which join them to the ring as a result of the intermolecular hydrogen‐bond interactions. These are also responsible for an elongation of the hydroxy C—OH bond compared with that calculated for the free mol­ecule.     

Algebraic expressions of Clebsch–Gordon coefficients for the group chain O† ⊃ C

Using the algebraic expressions of the projection operators for the group chain O ? C, concise algebraic expressions of the Clebsch–Gordon (CG) coefficients are derived in the group chain O ? C for both single‐valued and double‐valued representations. The simplicity of the expressions is that they are merely functions of the quantum numbers of the group chain O ? C. The symmetry of the CG coefficients is also derived. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003     

The Chemical Bond in C2

Quantum chemical calculations using the complete active space of the valence orbitals have been carried out for H_nCCH_n (n=0–3) and N₂. The quadratic force constants and the stretching potentials of H_nCCH_n have been calculated at the CASSCF/cc‐pVTZ level. The bond dissociation energies of the C?C bonds of C₂ and HC≡CH were computed using explicitly correlated CASPT2‐F12/cc‐pVTZ‐F12 wave functions. The bond dissociation energies and the force constants suggest that C₂ has a weaker C?C bond than acetylene. The analysis of the CASSCF wavefunctions in conjunction with the effective bond orders of the multiple bonds shows that there are four bonding components in C₂, while there are only three in acetylene and in N₂. The bonding components in C₂ consist of two weakly bonding σ bonds and two electron‐sharing π bonds. The bonding situation in C₂ can be described with the σ bonds in Be₂ that are enforced by two π bonds. There is no single Lewis structure that adequately depicts the bonding situation in C₂. The assignment of quadruple bonding in C₂ is misleading, because the bond is weaker than the triple bond in HC≡CH.     

Two C3‐Symmetric Dy3III Complexes with Triple Di‐μ‐methoxo‐μ‐phenoxo Bridges,Magnetic Ground State,and Single‐Molecule Magnetic Behavior

Two series of isostructural C₃‐symmetric Ln₃ complexes Ln₃ ? [BPh₄] and Ln₃ ? 0.33[Ln(NO₃)₆] (in which Ln^III=Gd and Dy) have been prepared from an amino‐bis(phenol) ligand. X‐ray studies reveal that Ln^III ions are connected by one μ₂‐phenoxo and two μ₃‐methoxo bridges, thus leading to a hexagonal bipyramidal Ln₃O₅ bridging core in which Ln^III ions exhibit a biaugmented trigonal‐prismatic geometry. Magnetic susceptibility studies and ab initio complete active space self‐consistent field (CASSCF) calculations indicate that the magnetic coupling between the Dy^III ions, which possess a high axial anisotropy in the ground state, is very weakly antiferromagnetic and mainly dipolar in nature. To reduce the electronic repulsion from the coordinating oxygen atom with the shortest Dy?O distance, the local magnetic moments are oriented almost perpendicular to the Dy₃ plane, thus leading to a paramagnetic ground state. CASSCF plus restricted active space state interaction (RASSI) calculations also show that the ground and first excited state of the Dy^III ions are separated by approximately 150 and 177 cm^?1, for Dy₃ ? [BPh₄] and Dy₃ ? 0.33[Dy(NO₃)₆], respectively. As expected for these large energy gaps, Dy₃ ? [BPh₄] and Dy₃ ? 0.33[Dy(NO₃)₆] exhibit, under zero direct‐current (dc) field, thermally activated slow relaxation of the magnetization, which overlap with a quantum tunneling relaxation process. Under an applied H_dc field of 1000 Oe, Dy₃ ? [BPh₄] exhibits two thermally activated processes with U_eff values of 34.7 and 19.5 cm^?1, whereas Dy₃ ? 0.33[Dy(NO₃)₆] shows only one activated process with U_eff=19.5 cm^?1.