Chemical applications of topology and group theory. XXI: Chirality in transitive skeletons and qualitative completeness [1]

This paper unifies the following ideas for the study of chirality polynomials in transitive skeletons: (1) Generalization of chirality to permutation groups not corresponding to three-dimensional symmetry point groups leading to the concepts of signed permutation groups and their signed subgroups; (2) Determination of the total dimension of the chiral ligand partitions through the Frobenius reciprocity theorem; (3) Determination of signed permutation groups, not necessarily corresponding to three-dimensional point groups, of which a given ligand partition is a maximum symmetry chiral ligand partition by the Ruch-Schönhofer partial ordering, thereby allowing the determination of corresponding chirality polynomials depending only upon differences between ligand parameters; such permutation groups having the point group as a signed subgroup relate to qualitative completeness. In the case of transitive permutation groups on four sites, the tetrahedron and polarized square each have only one chiral ligand partition, but the allene and polarized rectangle skeletons each have two chiral ligand partitions related to their being signed subgroups of the tetrahedron and polarized square, respectively. The single transitive permutation group on five sites, the polarized pentagon, has a degenerate chiral ligand partition related to its being a signed subgroup of a metacyclic group with 20 elements. The octahedron has two chiral ligand partitions, both of degree six; a qualitatively complete chirality polynomial is therefore homogeneous of degree six. The cyclopropane (or trigonal prism or trigonal antiprism) skeleton is a signed subgroup of both the octahedron and a twist group of order 36; two of its six chiral ligand partitions come from the octahedron and two more from the twist group. The polarized hexagon is a signed subgroup of the same twist group but not of the octahedron and thus has a different set of six chiral ligand partitions than the cyclopropane skeleton. Two of its six chiral ligand partitions come from the above twist group of order 36 and two more from a signed permutation group of order 48 derived from the P₃[P ₂] wreath product group with a different assignment of positive and negative operations than the octahedron.     

Far infrared spectra (500–50 cm−1) of the 2,2′-bipyridine complexes of palladium and platinum halides

Summary The far-i.r. spectra of the title complexes have been examined. Band assignments are based on the shifts induced by ligand deuteration and halide substitution. Deuteration of bipyridine causes large shifts ( >10 cm^–1) in internal ligand modes, intermediate shifts between 2 and 9 cm^–1) in metal-nitrogen stretching and bending modes and small to zero shifts in metal-halide stretching and bending vibrations. Generally, the requirements for square planarC _2v synanetry [two (M–N) and two (M–X) bands] are observed. Previous ambiguities in the assignment of the (M–N) bands have been resolved by the isotopic labelling technique employed in this study.     

Systematics of bonding in non-icosahedral carbon clusters

General formulas are presented for the vertex numbers, , of pentagon+hexagon polyhedra of icosahedral, tetrahedral or dihedral symmetries. Criteria for uniqueness of representation, isomer counts and grouping of pentagons are established. All polyhedra with 256 vertices or less and belonging to T, D ₅, D₆or their supergroups are listed. With the addition of C₃ to the dihedral and higher groups, at least one pentagon+hexagon cluster is found for all even 20 except for = 22 which is unrealisable in any symmetry, and = 46 (for which a C₃ polyhedron exists). Carbon clusters with closed electronic shells are shown to be generated by a geometrical leapfrog procedure: for all = 60+6k (where k is zero or greater than one) at least one closed shell structure is predicted. In dihedral symmetry closed shells also exist for some other values of . Separation of the 12 pentagonal faces is not sufficient to ensure a closed electronic shell but appears to be a necessary condition in dihedral or tetrahedral symmetry.     

Analysis of the Temperature and Composition Dependence of Viscosimetric Properties of 2-Butanone <Emphasis Type="Bold">+</Emphasis> 2-Butanol Solvent Mixtures

Kinematic viscosities were measured for 2-butanone + 2-butanol binary liquid mixtures with a capillary Ubbelohde routine viscometer in the temperature range from 273.15 to 353.15 K at atmospheric pressure, and covering the whole miscibility field (0x_i1). Experimental data have been correlated by means of different empirical or semiempirical relationships, such as =(T), =(x_i), and =(T, x_i). Viscosity deviations, , from ideal behavior are negative at all experimental conditions, confirming that structure breaking effects prevail in the liquids. Furthermore, the thermodynamics of viscous flow and excess Gibbs energy of activation of viscous flow, G*^E, have been calculated. As an alternative and complementary approach to such investigations, the fluidity () of this binary system has been analyzed by the modified—Batschinski theory. The results are discussed in terms of the specific molecular interactions between the mixture components.     

Polyhedral structures with an odd number of vertices: nine-coordinate metal compounds

The stereochemistry of nine-coordinate transition-metal and rare-earth compounds has been studied by means of continuous shape measures (CShM) and related tools. Several reference nine-vertex polyhedra have been defined and their minimal distortion interconversion paths established. A theoretical shape map is presented in which the structures can be placed according to their distances in CShM space to the capped square antiprism and the tricapped trigonal prism, which are the most common polyhedra in nine-coordinate compounds. The structures of almost 2000 metal coordination spheres in molecular and extended solid-state compounds have been analyzed. Clear stereochemical trends can be established for subsets of these compounds grouped according to the nature of their ligands, which include families of compounds spread along the interconversion paths between the capped square antiprism and the capped cube, or between the tricapped trigonal prism and the tridiminished icosahedron.     

Polyhedral structures with an odd number of vertices: nine-atom clusters and supramolecular architectures

The geometries of metal clusters and supramolecular architectures that contain nine metal atoms are analyzed within the framework of continuous shape measures (CShM). The most common polyhedra in nine coordinate complexes, the capped square antiprism and the tricapped trigonal prism, are also found among these families of compounds, even if much more scarcely. In addition, a variety of new shapes, not found among coordination polyhedra, can be identified and their proximity to the ideal geometries quantified. These include a linear chain, two types of trigonal columns, the planar regular enneagon, two-dimensional hexagonal and square grids, fragments of a close-packed structure, the triangular cupola, the tridiminished icosahedron or different fragments of the icosahedron. Among the nine-atom boranes and related clusters of the groups 13 and 14 elements, those having between 18 and 20 framework electrons present the structure of the tricapped trigonal prism, the expected closo structure. However, clusters with 21 and 22 framework electrons present a variety of structures with geometries covering nearly all the path that takes one from the capped square antiprism (nido form) to the tricapped trigonal prism (closo form).     

Chemical applications of topology and group theory. XXII : Lowest degree chirality polynomials for regular polyhedra [1]

The lowest degree chirality polynomials for the regular octahedron, cube, and regular icosahedron are discussed. All three of these regular polyhedra are chirally degenerate since they have more than one lowest degree chiral ligand parition by the Ruch-Schönhofer scheme. The two lowest degree chirality polynomials for the octahedron have degree 6 and can be formed from three degree 3 generating polynomialsf,g, andh through the relationshipsf(g +h) andf(g –h), wheref,g, andh measure the effects of the three separating reflection planes (h), the four threefold rotation axes, and the three fourfold rotation axes, respectively. The permutation groups of the vertices of the cube and icosahedron contain only even permutations, which leads to a natural pairing of their chiral ligand partitions according to equivalence of the corresponding Young diagrams upon reflection through their diagonals. The two lowest degree chirality polynomials for the cube have degree 4 and can be formed from two degree 4 generating polynomialsf andg through the relationships –2g andf –2g, wheref andg measure the effects of theS ₆ improper rotation andC ₄, proper rotation axes. respectively. The four lowest degree chiral ligand partitions for the icosahedron have degree 4 and lead naturally to a single degree 4 chirality polynomial with 120 terms of the general type (x –y)² (z –w)². This chirality polynomial for the icosahedron cannot be broken down into simpler generating polynomials, in contrast to the lowest degree chirality polynomials for the octahedron and cube. This appears to relate to the origin of the icosahedral group from the simple alternating groupA ₅. The full icosahedral chirality polynomial can be simplified to give a chirality polynomial for the chiral boron-monosubstituted ortho and meta carboranes of the general formula B₂C₁₀H₁₁X.     

Complexes of nickel(II) and palladium(II) with thiobenzamide

Summary The electronic and vibrational spectra of Ni^II and Pd^II complexes with thiobenzamide, L, are discussed. L acts as a sulphur donor ligand. The Pd^II compounds and (NiL₄)(ClO₄)₂ are square planar. PdL₂Cl₂ has acis-structure, while PdL₂X₂ (X=Br or I) istrans; NiL₄Cl₂ istrans-octahedral. The i.r. bands due to(M.S) and(MX) have been assigned. The influence of the anions on the properties of the complexes, both in solution and in the solid state, is discussed.     

Hg3Se3O10, a mercury(II) compound with mixed‐valence oxoselenium(IV/VI) anions

The title compound, trimercury(II) bis­[selenite(IV)] selen­ate(VI), contains three crystallographically inequivalent Hg^II cations with coordination numbers of eight (denoted Hg1 and Hg2) and five (denoted Hg3). The corresponding coordination polyhedra around the metal atoms might be described as intermediates between a square antiprism and a triangulated dodecahedron for both Hg1 and Hg2, and a strongly distorted truncated octahedron for Hg3. [HgO_8/2] layers of edge‐sharing [HgO₈] polyhedra propagate parallel to the bc plane, and are connected via Se^VIO₄ tetrahedra and [Hg3O₅] polyhedra along the a axis, forming an arrangement with channels propagating parallel to the b axis. The two independent Se^IVO₃ pyramids bridge the Hg atoms, and the non‐bonding orbitals of the Se^IV ions protrude into the channels from opposite sides.     

α BB Resonance in the ν8 Band of Borazine –10B

The high-resolution FTIR spectrum of the fundamental ₈ of borazine ¹⁰B₃ ¹⁴N₃ ¹H₆ was reanalyzed taking into account the ^BB resonance with the combination band (₁₀ + ₁₇). A parameter set for the states ₈ = 1 and ₁₀ = ₁₇ = 1, respectively, is given, reproducing the observed spectrum at least up to J = 30 with experimental accuracy.     

Ligand effects on metal-oxygen stretching frequencies in dioxomolybdenum(VI) complexes

Summary Several new seven-coordinated dioxomolybdenum(VI) heterochelates, [MoO₂(AA)(BBB)], where AA = bidentate ligand = ethylenediamine, 2-aminoethylpyridine,o-pheny-lenediamine,o-phenanthroline or 2,2-dipyridyl; BBB = tridentate ligand (the Schiff base derived from benzoylhydrazide and salicylaldehyde), have been synthesized and characterized by elemental analysis, molar conductance, molecular weight, i.r. spectra and magnetic susceptibility measurements. The principle of coordination number expansion of a six coordinate homochelate, [MoO₂(H₂O)(BBB)] was used to prepare the complexes, which are nonelectrolytes, monomers and diamagnetic. The i.r. spectral data reveal that they possess acis-MoO₂ structure. The _sym(OMoO) and _asym(OMoO) shifts and the difference between _sym(OMoO) and _asym(OMoO) have been related to an increase in electron density at the molybdenum atom therein.     

Polyhedral structures with three-, four-, and five fold symmetry in metal-centered ten-vertex germanium clusters

Studies using density functional theory (DFT) at the hybrid B3LYP level indicate that the relative energies of structures with three-fold, four-fold, and five-fold symmetry for centered 10-vertex bare germanium clusters of the general type M@Ge(10) (z) depend on the central metal atom M and the skeletal electron count. For M@Ge(10) clusters with 20 skeletal electrons the DFT results agree with experimental data on the isoelectronic centered 10-vertex bare metal clusters. Thus the lowest energy structure for Ni@Ge(10), isoelectronic with the known Ni@In(10) (10-), is a C(3v) polyhedron derived from the tetracapped trigonal prism. However, Zn@Ge(10) (2+) is isoelectronic with the known cluster Zn@In(10) (8-), which has the lowest energy structure, a D(4d) bicapped square antiprism. For the clusters Ni@Ge(10) (2-), Cu@Ge(10) (-), and Zn@Ge(10) that have 22 skeletal electrons the lowest energy structures are the D(4d) bicapped square antiprism predicted by the Wade-Mingos rules. For the clusters Ni@Ge(10) (4-), Cu@Ge(10) (3-), and Zn@Ge(10) (2-) that have 24 skeletal electrons the lowest energy structures are C(3v) polyhedra with 10 triangular faces and 3 quadrilateral faces derived from a tetracapped trigonal prism by extreme lengthening of the edges of the capped triangular face of the underlying trigonal prism. For the clusters Cu@Ge(10) (5-) and Zn@Ge(10) (4-) that have 26 skeletal electrons the lowest energy structures are the D(5d) pentagonal antiprisms predicted by the Wade-Mingos rules and the C(3v) tetracapped trigonal prism as a somewhat higher energy structure. However, for the isoelectronic Ni@Ge(10) (6-) the relative energies of these two structure types are reversed so that the C(3v) tetracapped trigonal prism becomes the global minimum. The effects of electron count on the geometries of the D(5d) pentagonal prism and D(4d) bicapped square antiprism centered metal cluster structures are consistent with the bonding/antibonding characteristics of the corresponding HOMO and LUMO frontier molecular orbitals.     

Density functional theory study of nine-atom germanium clusters: effect of electron count on cluster geometry

Density functional theory (DFT) at the hybrid B3LYP level has been applied to the germanium clusters Ge(9)(z) clusters (z = -6, -4, -3, -2, 0, +2, and +4) starting from three different initial configurations. Double-zeta quality LANL2DZ basis functions extended by adding one set of polarization (d) and one set of diffuse (p) functions were used. The global minimum for Ge(9)(2)(-) is the tricapped trigonal prism expected by Wade's rules for a 2n + 2 skeletal electron structure. An elongated tricapped trigonal prism is the global minimum for Ge(9)(4)(-) similar to the experimentally found structure for the isoelectronic Bi(9)(5+). However, the capped square antiprism predicted by Wade's rules for a 2n + 4 skeletal electron structure is only 0.21 kcal/mol above this global minimum indicating that these two nine-vertex polyhedra have very similar energies in this system. Tricapped trigonal prismatic structures are found for both singlet and triplet Ge(9)(6)(-), with the latter being lower in energy by 3.66 kcal/mol and far less distorted. The global minimum for the hypoelectronic Ge(9) is a bicapped pentagonal bipyramid. However, a second structure for Ge(9) only 4.54 kcal/mol above this global minimum is the C(2)(v)() flattened tricapped trigonal prism structure found experimentally for the isoelectronic Tl(9)(9)(-). For the even more hypoelectronic Ge(9)(2+), the lowest energy structure consists of an octahedron fused to two trigonal bipyramids. For Ge(9)(4+), the global minimum is an oblate (squashed) pentagonal bipyramid with two pendant Ge vertices.     

An index of electrotopological state for atoms in molecules

A new method for molecular structure quantitation is described, in which both electronic and topological attributes are united. The method uses the hydrogen-suppressed skeleton to represent the structure and leads to a graph invariant index for the individual atoms and hydride groups of the molecular skeleton. An intrinsic atom value is calculated for each atom asI = ( + 1)/, in which and are the counts of valence and sigma electrons of atoms in the molecular skeleton, that is, exclusive of bonds to hydrogen atoms. The electrotopological state valueS _i for an atomi is defined asS _i =I _i + I _i, where the influence of atom j on atom i, I _i, is given as (I _i-j _j)/r ²;r is the graph separation between atoms i and j, counted as number of atoms, includingi andj. The information in the electrotopological state values is revealed by examples of various types of organic structures, including chain branching and heteroatom variation. The relation of the E-state value to NMR chemical shift is demonstrated for a series of carbonyl compounds.     

Vibrational spectra and structures of the cis and trans isomers of linear benzindigo

The resonance Raman and IR spectra of the cis and trans isomers of linear benzindigo were investigated, and the frequencies of the vibrations of the multiple bonds were assigned. The large splitting of the frequencies of the symmetrical (_s) and asymmetrical (_as) vibrations of the carbonyl groups of the trans isomer of benzindigo and indigo is associated with the electronic interaction of these groups in the process of vibration.Translated from Khimiya Geterotsiklicheskikh Soedinenii, No. 4, pp. 501–504, April, 1981.     

Silver(I) complexes ofN-ethylthiourea. An infrared study

Summary The following silver(I) complexes ofN-ethylthiourea, L, have been isolated in the solid state: AgLX (X = Cl, Br, ClO₄, BF₄, and CF₃CO₂), Ag₂L₃X₂ (X=ClO₄, BF₄ and CF₃CO₂), AgL₂ClO₄, AgL₃X (X = Cl, Br, I, ClO₄, BF₄ and CF₃CO₂). The decrease of the(CS) and (SCN₂) frequencies in the i.r. spectra of the complexes indicates S-coordination of the ligands. The(AgS) frequencies lie in the 267–252 cm^–1 range for terminal and 240-210 cm^–1 range for bridged or long Ag-S bonds. The following structures may be proposed for the complexes. AgLX (X = Cl and Br), polymeric trigonal coordination with bridging halide ions; Ag₂L₃A₂ (A = ClO₄ and BF₄), polymeric trigonal coordination with three bridging sulphur atoms; AgL₂ClO₄, polymeric tetrahedral coordination with four bridging sulphur atoms; AgL₃A (A = ClO₄ and BF₄), dimeric tetrahedral coordination with two terminal and two bridging sulphur atoms; AgLO₂CCF₃ and Ag₂L₃(O₂CCF₃)₂, trigonal and/or tetrahedral coordination with terminal sulphur bonds and bridging CO₂ groups; AgL₃O₂CCF₃, tetrahedral coordination with terminal sulphur bonds and a monodentate CO₂ group. In the trifluoroacetato complexes, a(AgO) band is observed at 185–209 cm^–1. The AgL₃X complexes (X = Cl, Br and I) show two(AgX) and two(AgS) bands with rather low frequencies corresponding to long metal-ligand bond distances.     

Vibrational spectra and structures of CCl2 and SiCl2 molecules stabilized in an Ar matrix

Conclusions A study has been made of the isotopic structures of the ₁ and ₃ IR absorption bands of dichlorocarbene and dichlorosilylene, each stabilized in an Ar matrix at 15–20°K. A determination has been made of the valence angles, force constants, and vibration frequencies, ₁, ₂, and ₃ for various isotopic CCl₂ and SiCl₂ molecules.Translated from Izvestiya Akademii Nauk SSSR, Seriya Khimicheskaya, No. 10, pp. 2236–2242, October, 1977.The authors would like to thank S. P. Kolesnikov and B. L. Perl'mutter who supplied the C₄H₈O₂·GeCl₂ complex, and E. A. Chernyshev, N. G. Komalenkova and S. A. Bashkirova who supplied the Si₂Cl₆.     

Establishment of The Power—Time Curve Equation of Bacterial Growth in the Log Phase

The power-time curves of two species of bacteria, Vibro metschnikovii, Vibro bollisae were determined calorimetrically by using a 2277 bioactivity monitor. The power-time curve equation of bacterial growth in the log phase can be expressed as

Tetrastrontium dimanganese copper nonaoxide,Sr4Mn2CuO9

Single crystals of Sr₄Mn_2.09Cu_0.91O₉ have been grown by flux synthesis and the structure, closely related to the hexagonal perovskite 2H, was solved from single‐crystal X‐ray data in space group P321. The structure of Sr₄Mn₂CuO₉ is composed of chains of face‐sharing polyhedra with a sequence of two octahedra and one trigonal prism. The octahedra are filled by Mn atoms and the Cu atoms are randomly distributed at the centres of the square faces of the trigonal prism. A stacking fault is observed within one of the two chains, which can be attributed to a shifting of the chain along the c axis.     

Synthesis and Structure of New Cobalt–Carbonyl Complexes Containing Tellurium Atoms

Co₂(CO)₈ and Te₂O react to form the well known Co₄(CO)₁₀Te₂, Co₄(CO)₁₁Te₂ complexes and the two new cluster complexes CCo₆(CO)₁₂Te₂(1), and CCo₆(CO)₁₀Te₂(Te₃) (2). The structures of 1 and 2 were determined by X-ray analysis, together with the triphenylphosphine derivative of 1, CCo₆(CO)₁₁(PPh₃)Te₂(3), which was analyzed to clarify the disordered structure of the parent compound. Complex 1 is formed by a prismatic cluster of cobalt atoms with a carbon embedded in the cage; two tellurium atoms cap the triangular faces of the prism and each cobalt atom links two terminal carbonyl groups. The complex 2 has a similar prismatic cage CCo₆; two ₄-Te atoms cap two rectangular faces of the prism, while other two Te atoms bridge two edges of the triangular faces and are linked each other through a third Te atom. Electron counting gives for complex 2 92 electrons: the presence of two long Co–Co distances suggests that the two excess electrons are located on Co–Co antibonding orbitals. Crystal data for 1, space group C2/c, a = 12.845(2) Å, b = 13.449(2) Å, c = 13.246(2) Å, = 91.95(2)°, Z = 4, R = 0.097 for 2555 reflections; for 2, space group Pnna, a = 17.219(5) Å, b= 14.969(6) Å, c = 9.178(4) Å, Z = 4,R = 0.037 for 3103 reflections; for 3, space group P2₁/c, a = 9.288(2) Å, b = 14.920(6) Å, c = 26.300(9) Å, = 99.99(2)°, Z = 4, R = 0.037 for 4300 reflections. The vibrational analysis of the complex 1 was performed and most of the (CO), (₆C–Co), (Co–Co) and (Co–Co) modes were assigned. The (Co–Te) modes were interpreted on the basis of the intermolecular coupling, due to the close contact between neighboring clusters in one distinct direction in the crystal.