A conjecture for some partial differential operators on L 2( R n)

OnX =L ²(R ⁿ), letQ = (Q ₁,Q ₂,…,Q _n) andP = (P ₁,P ₂, …,P _n) be the operators given by (Q _jf) (x) =x _jf(x),P _j = - i∂/∂x _j. For anyC ^∞ functionh:R ⁿ →R putH ₀ =h(P) andH =H ₀ + (1 +Q ²)^-δ, where δ > 1/2. By the method of scattering theory we prove thatH _ac, the absolutely continuous part ofH is unitarily equivalent toH ₀ when (a)n = 1 and (b) forn ≥ 2, whenh is in a large class of polynomials. It is conjectured that the results are true for any polynomialh. We use the techniques of Enss’ method and the idea of bound states for momentum.     

Low-temperature nematic phase in asymmetrical 1,3,4-oxadiazole bent-core liquid crystals possessing lateral methoxy group

Asymmetrical bent-core molecules based on 1,3,4-oxadiazole bent-core unit have been synthesised as a new design with a lateral methoxy group at outer phenyl ring of the molecule. These new asymmetrical bent-core molecules resemble hockey-stick shaped due to the presence of two different arms of different lengths. One arm of these molecules is elongated having two phenyl rings and possesses a 4-n-alkyloxy chain of a different number of carbon atoms (n = 4, 8, 12 and 18) and other arm is short and has one phenyl ring with fixed 4-n-octyloxy chain. The bent-core molecules possess a lateral polar methoxy group at the elongated arm of the molecule. These bent-core compounds exhibited fluorescence emission in the UV wavelength region (~377–386 nm) whereas in acetonitrile and dimethylformamide, solvent displays blue emission peak with a large stoke shift.The bent-core molecules with the number of carbon atoms (n = 4, 8 and 12) at the elongated arm exhibited monotropic nematic phase at low temperature, while the 4-n-octadecyloxy chain at the elongated arm displayed smectic A phase. Dielectric studies were performed in the nematic phase of the bent-core mesogens confirm the formation of the cybotactic cluster in the nematic mesophases.     

Coordination Polyhedra RhX6 (X = F,Cl, Br) in Crystal Structures

Voronoi–Dirichlet polyhedra (VDP) and the method of intersecting spheres were used to perform crystal-chemical analyses of compounds containing complexes [Rh _a X _n ] ^z– (X = F, Cl, Br). It was found that, irrespective of oxidation number (+3, +4, or +5), rhodium atoms always exhibit the coordination number 6 with respect to the halogen atoms and have octahedral coordination. The influence of site symmetry and the valence state of Rh on the distortion of RhX₆ octahedra are considered. The electronic configuration of the Rh atoms is shown to influence the symmetry of their valence-force field within the crystal structure.     

Quantum-Chemical Estimation of the Stability and Reactivity of Diphosphonium Salts

For a series of diphosphonium salts containing two positively charged covalently bonded phosphorus atoms, X_{/sub> n} Y_3-n P⁺P⁺X _n Y_3-n (X = alkyl substituent, Y = amino group, n = 0-3), the stability, reactivity, and P-P bond strength were evaluated by various physicochemical methods. The P-P bond energy is appreciably influenced by both steric factors and donor properties of the substituents. The calculations confirmed that transformations of diphosphonium salts can involve cleavage of both P-P and P-N (or P-C) bonds.     

Beiträge zur Chemie des Phosphors. 212. Tetraisopropyl-dodecaphosphan(4), P12i-Pr4 – Darstellung,Eigenschaften und Moleküldynamik

Contributions to the Chemistry of Phosphorus. 212. Tetraisopropyldodecaphosphane(4), P₁₂i-Pr₄ – Preparation, Properties, and Molecular Dynamics According to an earlier crystal structure analysis, tetraisopropyldodecaphosphane(4) ( 1 ) exhibits the symmetry C₂, and the substituents are arranged in all-trans position [3]. We have now found by NMR spectroscopic studies that in solution a second configurational isomer of the symmetry C_S ( 1b ) exists in addition to the molecule present in the crystal ( 1a ). The transformation of 1a into 1b , which can only occur through a quasi synchronous inversion at the atoms P³ and P⁴ or P⁹ and P¹⁰, takes place at a noticeable rate already below room temperature.     

Orientational order of tetra n-hexylesters of perylene and tetrachloroperylene tetracarboxylic acids in low-molar-mass liquid crystal investigated using absorption and fluorescence methods

The orientational order parameters <P₂> and <P₄> of perylene-3,4,9,10-tetra-(n-hexylester) (PTHE) and 1,6,7,12-tetrachloroperylene-3,4,9,10-tetra-(n-hexylester) (4Cl-PTHE) dye molecules dissolved in low-molar-mass liquid crystal, 4-(trans-4-n-hexylylcyclohexyl)-isothiocyanobenzene, are calculated from the experimental values of polarised absorption and fluorescence anisotropies. The experimental data show that the substitution of chlorine atoms instead of hydrogen in the perylene core reduces the value of <P₂> parameter. Moreover, the <P₂> and <P₄> values for molecules of the perylene derivatives, playing the role of the guest in the liquid crystal matrix, are much less than these values for molecules of the liquid crystal host. The extremely low negative values of <P₄> for the perylene dyes are explained as the result of the preferable oblique orientation of the main molecular symmetry axis of the probe with respect to the direction of the macroscopic orientation of the liquid crystal host molecules.     

Symmetry reduction of the matrix elements of a two-particle operator

Local coordinate systems are chosen for each quadruple of atoms relative to a four-center integral, in order to avoid linear combinations of orbitals when symmetry operations perform on an orbital. This choice can utilize the complete molecular symmetry to attain the optimal number of symmetry-unique integrals and to construct two-particle matrix elements by multiplying symmetry-unique integrals, called the “standard four-center integrals,” by the corresponding coefficients, called the “C coefficients.” A simple algorithm to use the complete molecular symmetry to reduce calculations of molecular matrix elements is outlined for general highly symmetric molecules. A tetrahedral molecule is analyzed. © 1996 John Wiley & Sons, Inc.     

2-(4-Alkylphenyl)-5-(alkenyloxy)pyrimidines: Synthesis,liquid crystal transition temperatures and some physical properties

Abstract

We have recently reported the introduction of a carbon-carbon double bond into a wide variety of 5-n-alkyl-2-(4-n-alkoxyphenyl)pyrimidines to produce the corresponding alkenyloxy derivatives. The position and nature (E/Z) of the double bond were varied systematically and the effect on the liquid crystal transition temperatures studied. The position and nature (E/Z) of the double bond changed the conformation of the alkenyloxy chain substantially. This resulted in higher smectic C and nematic transition temperatures for compounds with a trans-double bond (E) at an even number of carbon atoms from the molecular core. Significantly lower transition temperatures (including the melting point) were observed for materials with a cis-double bond (Z) at an odd number of carbon atoms from the molecular core. We have now performed the same operation on the related 2-(4-n-alkylphenyl)-5-n-alkoxypyrimidines to produce the corresponding alkenyloxy derivatives. An interesting feature of the new results is the high melting points of the trans-substituted materials and the low melting points of the terminally substituted compounds. The smectic C transition temperatures of both series are high. No nematic phases could be observed. However, in admixture with other smectic C components, the new compounds lead to surprisingly fast switching times, high smectic C transition temperatures and low melting points/crystallization temperatures in experimental mixtures designed for electro-optic display devices based on ferroelectric effects.     

Orientational disorder in Λ‐cobalt(III) sepulchrate trinitrate

The crystal structure of Λ‐(1,3,6,8,10,13,16,19‐octaazabicyclo[6.6.6]eicosane)cobalt(III) trinitrate, [Co(C₁₂H₃₀N₈)](NO₃)₃, consists of a sepulchrate moiety that serves as a macrobicyclic nitrogen cage for the Co³⁺ cation, which is six‐coordinated by N atoms, and three nitrate anions. The Co–sepulchrate group lies on a threefold axis (site symmetry 32), as do two symmetry‐related and ordered nitrate groups (site symmetry 3), with which it is connected via N—H...O hydrogen bonds [Co—N = 5.1452 (12) Å]. The third nitrate group is disordered as a result of symmetry requirements around the origin (site symmetry 32), and is further away from the Co–sepulchrate cage [Co—N = 6.3160 (8) Å]. The structure is described by applying orientational disorder over six equivalent orientations for the disordered nitrate group, which is considered as an ideal planar molecule of regular trigonal geometry with its molecular plane rotated out of the ab plane and the molecular centre of gravity slightly shifted away from the origin. This new model for disorder clearly improves a previous crystal structure determination.     

Synthesis,mesomorphic, photophysical and computational studies of new achiral four-ring unsymmetrical bent-core mesogens and their Copper(II) complexes

New achiral four-ring unsymmetrical bent-core mesogens derived from 2,5-dihydroxybenzaldehyde and their copper(II) complexes have been synthesised as a new design with an imine and ester linkage. These new bent-core molecules resemble hockey-stick shape, which possesses 4-n-alkyloxy chain (4-n-hexyloxy and 4-n-decyloxy) at one end and methyl or methoxy group at the other end of the molecule. The synthesis, spectroscopic characterisation, phase transition temperature and characterisation of phase behaviour are reported. The bent-core molecules exhibited monotropic nematic and smectic A phase depending on the terminal chain length. Interestingly, copper(II) complexes of bent-core molecules displayed monotropic nematic phase. This is the first report on copper(II) complexes of bent-core molecules that exhibited nematic phase. The four-ring bent-core molecule exhibited fluorescence with large stoke shift. The density functional theory calculations of bent-core molecules and their copper(II) complexes are carried out using Gaussian 09 program at B3LYP level to obtain the stable molecular conformation, dipole moment, highest occupied molecular orbital (HOMO), lowest unoccupied molecular orbital (LUMO) energies and bending angle of the compounds. The natural atomic charges and electronic configurations of the atoms of free ligands as well as the complexes have been evaluated.     

Identification of molecules in inorganic compounds from X-ray or neutron diffraction data and correction of their structures on the basis of vibrational spectroscopy data

A method of identification of molecules in inorganic crystals with asymmetric M-X...M bridges has been suggested on the basis of analysis of interatomic distances in the structure unit M (−X...M) _n , which includes the atoms of the first and second coordination spheres. This analysis makes it possible to discern molecules (complex anions or cations) as a groups of atoms linked with each other by short M-X bonds, whereas the atoms of neighboring groups are linked by long M...X bonds. The symmetry of such a group is often lower than it follows from X-ray or neutron diffraction data. Studying vibrational spectra affords information on the true symmetry of a molecule. The use of the method is exemplified by the rhombohedral BaTiO₃ phase.     

Unitary symmetry and classification of the states of <Emphasis Type="Italic">n</Emphasis>-spin clusters. Magnetic and thermodynamic parameters

The unitary symmetry and classification of spin clusters by spin momenta S are considered on the basis of reduction of the full linear group to unitary groups U _{2s + 1} and orthogonal rotation group R ₃. Reduction of the permutation group P _n of n spins to the point group of the cluster is applied to the classification of the spatial states of a spin cluster with the use of permutation quantum numbers introduced in this work and the Young diagrams of the permutation group P _n . Examples of the classification of spin systems with spins s = 1/2, 1, 3/2, 2, and 5/2 with U _{2s + 1} × P _n groups (n = 5–15) are reported. This classification is common for all spin clusters and is the same for both cyclic clusters and 3D clusters with symmetry groups of a crystal. On the basis of this classification, the magnetic and thermodynamic parameters of a spin system are calculated as a function of the number of spins and temperature. For s = 1/2 clusters, the analytical formulas are derived for magnetic susceptibility, internal energy, heat capacity, and entropy as a function of quantum numbers for a cluster with any number of spins, and their dependences on temperature and the number of atoms are considered.     

Searching new structures of beryllium-doped in small-sized magnesium clusters: Be2MgnQ (Q = 0, −1; n = 1–11) clusters DFT study

Using CALYPSO method to search new structures of neutral and anionic beryllium-doped magnesium clusters followed by density functional theory (DFT) calculations, an extensive study of the structures, electronic and spectral properties of Be₂Mg_n^Q (Q = 0, −1; n = 2–11) clusters is performed. Based on the structural optimization, it is found that the Be₂Mg_n^Q (Q = 0, −1) clusters are shown by tetrahedral-based geometries at n = 2–6 and tower-like-based geometries at n = 7–11. The calculations of stability indicate that Be₂Mg₅^Q=0, Be₂Mg₅^Q=−1, and Be₂Mg₈^Q=−1 clusters are “magic” clusters with high stability. The NCP shows that the charges are transferred from Mg atoms to Be atoms. The s- and p-orbitals interactions of Mg and Be atoms are main responsible for their NEC. In particular, chemical bond analysis including molecular orbitals (MOs) and chemical bonding composition for magic clusters to further study their stability. The results confirmed that the high stability of these clusters is due to the interactions between the Be atom and the Mg₅ or Mg₈ host. Finally, theoretical calculations of infrared and Raman spectra of the ground state of Be₂Mg_n^Q (Q = 0, −1; n = 1–11) clusters were performed, which will be absolutely useful for future experiments to identify these clusters.     

Kristall- und Molekülstruktur eines neuartigen Dimetallamacrocyclus: cyclo-Di[quecksilber-μ-[1,1,1′,1′-tetraethyl-3,3′-terephthaloyl-bis-thioureato(2-)-S:S′]]

Crystal and Molecular Structure of a Novel Dimetallamacrocycle: cyclo-Di[mercury-μ-[1,1,1′,1′-tetraethyl-3,3′-terephthaloyl-bis-thioureato(2-)-S:S′]] The structure of the title compound has been determined by an X-ray structure analysis. The compound crystallizes in the monoclinic space group P2₁/n with a = 13.437, b = 9.270, c = 18.791 Å, β = 100.65° and Z = 2. The solution of the structure was performed by Patterson methods. The final R value was R = 0.090 for 3089 observed reflections. The molecule has a cyclic structure consisting of two molecules of the ligand which coordinate only by the sulphur atoms and two mercury atoms. The formed ring is a 26-membered dimetallamacrocycle. The intramolecular Hg-Hg distance is 10.39 Å. The molecule has the symmetry C_i.     

Vibrational softening of diatomic molecules

An analysis of a model molecular oscillator is presented: a vibrating diatomic molecule carrying N ₀ electrons. The energy derivatives over the number of electron (N) and the deformation (Q), ∂ ⁿ /∂N ⁿ and ∂ ⁿ /∂Q ⁿ have been analyzed up to second order (n=2), including the appropriate mixed derivatives. The effect of coupling between distortion of the electron density induced by ΔN and the vibrational deformation of the molecule has been studied. Anharmonicity of the oscillator has been shown to be a possible result of that coupling; new relations between the parameters characterizing the anharmonicity of the oscillator and the energy derivatives at density functional theory level have been obtained. Ab initio calculations for a set of diatomic molecules have been performed, yielding values for all the derivatives discussed and demonstrating the effect of coupling with vibrations. Received: 1 June 2000 / Accepted: 20 October 2000 / Published online: 21 March 2001     

The kinetics of the decarboxylation of n-butylmalonic acid in alkanols and amines

Rate constants and activation parameters are reported for the decarboxylation of n-butylmalonic acid in four normal alkanols (hexanol? 1, octanol? 1, decanol? 1, and dodecanol? 1) and in five amines (aniline, N-methylaniline, N-ethylaniline, N-n-propylaniline, and N-n-butylaniline). Both ΔH? and ΔS? of the reaction in both homologous series decrease regularly with increasing length of the hydrocarbon chain of solvent. If we compare data for the reaction in alkanol–amine pairs containing the same total number of carbon atoms in the molecule, we find that the ΔH? values are identical, but that the value of ΔS? is 0.8 eu/mole higher for the reaction in the amines as compared with the alcohol. The rate constant, at all temperatures, is 1.5 times as large in the amine as it is in the corresponding alcohol. Empirical equations are deduced relating the parameters ΔH? ΔS? ΔG? and k of the reaction to the parameters n and T, where n is the total number of carbon atoms in the solvent molecule and T is the absolute temperature. The results reported herein are compared with previously reported data for malonic acid.     

Syntheses of (C_5H_9C_5H_4)_2LnCl(THF)_n (Ln=Nd, Sm, Er, Yb) and crystal structure of [(C_5H_9C_5H_4)_2-LnCl(THF)_n]_2(Ln=Sm, n=1; Ln=Er, n=0)

Reactions of lanthanoid trichlorides with sodium cyclopentylcyclopentadienyl in THFafford bis(cyclopentylcyclopentadienyl) lanthanoid chloride complexes (C_5H_9C_5H_4)_2LnCl(THF)_n (Ln=Nd, Sm, n=1; Ln= Er, Yb, n= 0). The compound [CP'_2SmCl(THF)]_2 (2) (Cp' =cyclopentylcy-clopentadienyl) crystallizes from mixed solvent of hexane and THF in monoclinic space group P_2_1/cwith a = 11.583 (3), b = 23.019(6), c = 8.227 (2), β= 90.26 (2)°, V= 2194 (1)~3, D_c= 1.59 g/cm~3.μ= 28.6 cm~(-1), F(000) = 1060, Z= 2 (dimers). Its crystal molecule is a dimer with a crystallographicsymmetry center. The central metal atom Sm is coordinated to two Cp' rings, two bridging chlorineatoms and one THF forming a distorted trigonal bipyramid. The crystal of [Cp'_2ErCl]_2 (3) belongs tothe triclinic space group P with a = 11.264 (2), b= 13.296(5), c = 14.296(6), a = 96.99 (3), β=112.47(2), γ= 102.78(2)°, V= 1865(1)~3, D_c= 1.67 g /cm~3, μ= 48.0 cm~(-1), F(000) = 924, Z = 2 (dimers).The molecule is a dimer consisting of two Cp'_2 ErCl species bridged by two Cl atoms. The centralmetal atom Er is coordinated to two Cp' rings and two bridging chlorine atoms forming a pseudo-tetrahedron. All these complexes are soluble in THF, DME, Et_2O, toluene and hexane.     

Approximate symmetry characteristics using fuzzy-subset theory study for chiral transitions of allene-1,3,-dihalides

Based on our study of the application of fuzzy-subset theory to the characterization of imperfect symmetry in some stable molecular systems and simple dynamic molecular systems, we analyze the internal rotation process of allene-1,3- dihalides. Allene-1,3-dihalides (CHX=C=CHY, where X and Y may be the same or different halogen atoms) are optically chiral nonplanar molecules. The two end-groups may internally rotate about the near straight linear C=C=C axis, and the molecule may change its chirality. The internal rotation process may pass through two different planar transition state (TS): cis-TS and trans-TS, which belong to C_2v and C_2h point groups (as X and Y to be same), respectively. The intrinsic reaction coordinate (IRC) corresponding to the two TS processes is denoted as cis-IRC and trans-IRC. However, for the whole IRC reaction process, only their subgroup C₂ well-defined symmetry remains. Other symmetry transformations in C_2v and C_2h point groups can only be examined in terms of imperfect symmetry, although there appear certain reaction reversal joint point group G(R_cC_2v) and G(R_tC_2h) well-defined symmetry in the dynamics through the IRC processes. If X and Y are different, the stable molecule has no conventional nontrivial point group symmetry. The internal rotation processes may pass through two different planar TS’s (cis-and trans-TS). The TS will still be a planar molecule belonging to C_S point group with the molecule plane as its symmetry plane. Other states in the IRC may belong to certain reaction reversal joint point groups, G(RM)_C and G(RM)_T. We have thus examined the approximate symmetry of MO’s related to C₂ point group. Moreover, we have also analyzed the membership functions, representation components, and their relationships shown in the MO fuzzy main representation correlation diagrams.     

The correlation equations describing retention index data on OV-101, OV-275 and PEG 20M.n-alkanes andn-alkynes

Summary The correlation equations between Kovats retention indices ofn-alkenes andn-alkynes, column temperature and number of carbon atoms in molecules of these compounds on OV-101, OV-225 and PEG 20M capillary columns have been calculated. The two variable equationI=A + B · n + C / T give a good fit and are the simplest for practical use.     

Theorie des Orbitals Moleculaires: Recherche d'une Base STO Homogene pour les Atomes Alcalins et Leur Serie Isoelectronique

A method of direct treatment of valence electrons is proposed for the various ²S, ²P,²D, ²F states of the lithium, sodium, and potassium atoms and the corresponding isoelectronic ions. The function describing the outer electron, which is orthogonal to the wave functions of the electrons in the core, is given as a linear combination of generalized Laguerre functions, with effective charges Q = 1, 2 …? equal to the charges of the core. A simpler analytical STO function, with non-integer principal quantum number n is then proposed.