Graphical Models of Characters of Groups

A graphical method of generating one- and (some) two-dimensional characters () has been developed on the basis of a reduced homomer set, which has been derived from a new concept of negative graphs. Thus, a homomer set H[G(/G _i )]={h₁,...,h _d–1,h _d } (d=|G|/|G _i |) has been generated from a regular body of G so that it has been governed by the coset representation G(/G _i ). The homomer set has been reduced into a reduced homomer set []={h₁,...,h _d–1}, where we have placed h _d –(h₁++h _d–1) in terms of negative graphs. The action of the symmetry operations of G on the reduced homomer set [] has graphically generated a one- or (some) two-dimensional character (). The versatility of the graphical method has been tested by using C _3v , D _2h , C _2h , C _2v , D _3h , and C _3h as examples. The graphical method has been compared with an alternative algebraic generation using marks (or markaracters), i.e., =G(/G _i )–G(/G).     

Group-theoretical Foundations for the Concept of Mandalas as Diagrammatical Expressions for Characterizing Symmetries of Stereoisomers

Group-theoretical foundations for the concept of mandalas have been formulated algebraically and diagrammatically in order to reinforce the spread of the unit-subduced-cycle-index (USCI) approach (S. Fujita, Symmetry and Combinatorial Enumeration in Chemistry, Springer-Verlag, Berlin-Heidelberg, 1991). Thus, after the introducton of right coset representations (RCR) (H\)G and left coset representations (LCR) G(/H) for the group G and its subgroup H, a regular body of G-symmetry is defined as a diagrammatical expression for a right regular representation (C ₁\)G, which is an extreme case of RCRs. The |G| substitution positions of the regular body as a reference are numbered in accord with the numbering of the elements of G and segmented into |G|/|H| of H-segments, which are governed by the RCR (H\)G. By regarding each H-segment as a substitution position, the H-segmented regular body is reduced into a reduced regular body, which can be regarded as a secondary skeleton for generating a molecule. The reference regular body (or H-segmented one) is operated by every symmetry operations of G to generate regular bodies (or H-segmented ones), which are placed on the vertices of a hypothetical regular body of G-symmetry. The resulting diagram (a nested regular body) is called a mandala (or a reduced mandala), which is a diagrammatical expression for specifying the G-symmetry of a molecule. The effect of a K-subduction on the regular bodies of a mandala (or a reduced mandala) results in the K-assemblage of the mandala (or the reduced mandala), where the resulting K-assemblies governed by the LCR G(/K) construct a |G|/|K|-membered orbit, which corresponds to a molecule of K-symmetry. The sphericity of the RCR (or the LCR) is used to characterize symmetrical properties of substitution positions and those of stereoisomers. The fixed-point vector for each mandala (or reduced mandala) in terms of row view and the number of fixed points of K-assembled mandalas (or K-assembled reduced mandalas) in terms of column view are compared to accomplish combinatorial enumeration of stereoisomers. The relationship between a mandala and a reordered multiplication table is discussed.     

On graph isomorphism and graph automorphism

The problem of graph isomorphism, graph automorphism and a unique graph ID is considered. A new approach to the solution of these problems is suggested. The method is based on the spectral decompositionA = _{i i} K _iof the adjacency matrixA. This decomposition is independent of the particular labeling of graph vertices, and using this decomposition one can formulate an algorithm to derive a canonical labeling of the corresponding graphG. Since the spectral decomposition uniquely determines the adjacency matrixA and hence graphG, the obtained canonical labeling can be used in order to derive a unique graph ID. In addition, if the algorithm produces several canonical labelings, all these labelings and only these labelings are connected by the elements of the graph automorphism groupG. In this way, one obtains all elements of this group. Concerning graph isomorphism, one can use a unique graph ID obtained in the above way. However, the algorithm to decide whether graphG andG are isomorphic can be substantially improved if this algorithm is based on the direct comparison between spectral decompositions of the corresponding adjacency matricesA andA.Research supported by the Yugoslav Ministry for Development (Grant P-339).     

Steady-state radiolysis and product analysis of aqueous diphenyloxide in the presence of air and N2O

The radiation-induced decomposition and product analysis of diphenyloxide (DPO; diphenylether), selected as a representative of the toxic polyaromatic, volatile, hydrocarbons produced by combustion of coal, was investigated in aqueous solution. In the presence of air an initial degradation yield, G_i(-DPO)0=4.6 was obtained. Phenol (G_i=1.3) appeared to be the major decomposition product in addition to a number of hydroxy-compounds as well as a mixture of aldehydes and carboxylic acids. DPO as well as the resulting products can be decomposed at an absorbed radiation dose of about 5 kGy.By performing analogous studies in solution saturated with N₂O an initial yield of G_i(-DPO)=5.0 and similar radiolytic products were found. Again phenol (G_i=1.5) is observed as the main product. In this case a much lower radiation dose (∼2.5 kGy) is found to be sufficient for the decomposition of DPO and its radiolytic products. Most likely reaction mechanisms are presented for explanation of the observed products.     

Graph-theoretical interpretation of Ugi's concept of the reaction network

The concept of a reaction network, initially suggested by Ugi and coworkers, in the framework of the graph-theoretical model of organic chemistry is elaborated. The reaction network for a pair of isomeric educt molecular (G _E) and product molecular graphs (G _P) is determined as an oriented graph. Its edge, beginning at a graph-vertexG _i –1 and ending at a graph-vertexG _i , corresponds to a feasible transformation (chemical reaction) constrained by a condition of descending chemical distance from the product graphG _P, i.e.CD(G _i –1,G _P) >CD(G _i ,G _P). In the reaction network, an oriented path which begins at G_E and ends atG _P corresponds to the decomposition of the overall transformationG _E G _P into a sequence of elementary transformationsG ₀ =G _E G ₁ G ₂ ... G _i–1 G _i =G _P that may be assigned to intermediates of the overall transformation.     

Nitrous Oxide as Carrier Gas in Capillary Gas-Liquid Chromatography

On use of nitrous oxide as carrier gas the retention factors of the chromatographed compounds decrease linearly with increasing average column pressure. Other retention characteristics (relative retention, retention index) change linearly. This effect was demonstrated by using a capillary column coated with nonpolar polydimethylsiloxane phase SE-30. As shown for capillary GLC the linear correlation is valid for the same column:k_i(G₁,P₁) = A k_i(G₂, P₂) + B, where k_i(G₁, P₁) and k_i(G₂,P₂) are the retention factors of compound i at average column pressures P₁ and P₂ when using carrier gases G₁ and G₂, respectively; A and B are coefficients.     

Kirchhoff index of linear hexagonal chains

The resistance distance r_ij between vertices i and j of a connected (molecular) graph G is computed as the effective resistance between nodes i and j in the corresponding network constructed from G by replacing each edge of G with a unit resistor. The Kirchhoff index Kf(G) is the sum of resistance distances between all pairs of vertices. In this work, according to the decomposition theorem of Laplacian polynomial, we obtain that the Laplacian spectrum of linear hexagonal chain L_n consists of the Laplacian spectrum of path P_2n+1 and eigenvalues of a symmetric tridiagonal matrix of order 2n + 1. By applying the relationship between roots and coefficients of the characteristic polynomial of the above matrix, explicit closed‐form formula for Kirchhoff index of L_n is derived in terms of Laplacian spectrum. To our surprise, the Krichhoff index of L_n is approximately to one half of its Wiener index. Finally, we show that holds for all graphs G in a class of graphs including L_n. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2008     

Catalytic decomposition of hydrogen peroxide in alkaline solutions

Catalytic activity of carbon, platinum-supported on high-area carbon, platinum, lead ruthenate, and ruthenium oxide towards hydrogen peroxide decomposition in alkaline solution is investigated using the rotating disk electrode technique. The heterogeneous rate constant for peroxide decomposition on these catalysts is determined from the slope of log(i_L) versus time, where i_L is the diffusion-limiting current corresponding to the concentration of peroxide at a given time. The order of catalytic activity is found to be platinum>lead ruthenate>ruthenium oxide>carbon. A general reaction mechanism for the peroxide decomposition on these catalysts is also proposed.     

Theoretical refinements of theendo and theexo structures of tetraborane(8) carbonyl

The molecular structures of theendo (1a) andexo (1b) isomers of B₄H₈CO have been optimized at the ab initio MP2(Full)/6-31G^* level of theory. The agreement of the computed geometrical parameters with the recently published electron-diffraction (GED) data is very good, even though a number of geometrical constraints were applied in the experimental determination. The IGLO (individual gauge for localized orbitals)¹¹B NMR chemical shifts, calculated at the II//MP2/6-31G^* level, are also in accord with experiment. The formation of1a and1b by association of B₄H₈ and CO is computed to be exothermic by 22.8 and 22.2 kcal/mol, respectively, at the MP2(Full)/6-31G^*//MP2(Full)/6-31G^* + ZPE(6-31G^*) level of theory. The Lewis acid strength of B₄H₈ toward CO is comparable to that of BH₃.     

Kirchhoff index of linear pentagonal chains

The resistance distance r_ij between two vertices v_i and v_j of a connected graph G is computed as the effective resistance between nodes i and j in the corresponding network constructed from G by replacing each edge of G with a unit resistor. The Kirchhoff index Kf(G) is the sum of resistance distances between all pairs of vertices. In this article, following the method of Yang and Zhang in the proof of the Kirchhoff index of liner hexagonal chain, we obtain the closed‐form formulae of the Kirchhoff index of liner pentagonal chain P_n in terms of its Laplacian spectrum. Finally, we show that the Kirchhoff index of P_n is approximately one half of its Wiener index. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2010     

Dehydration of solids in different modes as a proof of their primary congruent dissociative vaporization

The main purpose of this paper is to prove the applicability of the mechanism of congruent dissociative vaporization (CDV) to the solid-state decomposition kinetics through the comparison of the fundamental theoretical relationship E_i/E_e=(a+b)/a resulted from this mechanism with experiment. It has been shown that the ratios of E_i and E_e parameters of the Arrhenius equation measured in the isobaric and equimolar modes (in the presence and absence of H₂O vapour) for 22 reactants with the general formula aSalt⋅bH₂O or aOxide⋅bH₂O are in agreement with the values of (a+b)/a. The relative standard deviation is only 17% and the correlation coefficient is close to 0.99. A probability of accidental correlation for all set of the E parameters taken from the literature is lower than 4⋅10^–16 . This strongly supports the validity of the CDV mechanism. The problem of stability of polyatomic molecules of inorganic salts in the gaseous state, which are the primary decomposition products of crystalline hydrates, was also discussed on the basis of recent mass spectroscopy studies. It was concluded that any doubts in the applicability of the CDV mechanism as a general mechanism of solid-state decomposition reactions are unsound.     

Model for the fracture energy of glassy polymer–polymer interfaces

The increase in the interfacial fracture energy (G_c) with increasing interfacial width (a_i) goes through a transition at a critical value of a_i that is unique to each polymer–polymer system. This transition point does not scale with the bulk entanglement spacing (d_t) for different systems, implying that the role of chain friction in reinforcing these interfaces is more important than previously thought. A theoretical model has been developed to calculate G_c as a function of the interfacial stress transfer due to individual polymer chains. When including the effects of chain friction only, the model reproduces the nonuniversal behavior of G_c with respect to a_i/d_t but yields poor fits for a_i/d_t > 1. The effects of entanglements are then added by calculating the fraction of entangled chains as a function of a_i/d_t. This contribution, although not material specific, matches the qualitative behavior of G_c for large values of a_i/d_t. When both contributions are included in the model, excellent fits are obtained for all data sets. © 2002 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 40: 2377–2386, 2002     

2,3-dimethoxyperhydro-1,4,2,3-dioxadiazine: Synthesis,conformation, and conversion to 1-(2-hydroxyethoxy)-2-methoxydiazene-1-oxide. Crystal structure of 1-[2-(para-nitrobenzoyloxy)-ethoxy]-2-methoxydiazene-1-oxide

The oxidation of 1,2-bis(methoxyaminooxy)ethane (3) with PbO₂ afforded 2,3-dimethoxyperhydro-1,4,2,3-dioxadiazine (4) in the form of the 55:45 mixture of thetrans-4ee- andcis-4 isomers. The barrier of the ring conversion (G₂₃₈ =11.3 kcal/mole) of thecis-4 isomer was determined by the method of¹³C NMR. The regio- and stereospecific stereoelectronically controlled reaction of (4) withpara-nitrobenzoic acid gives the equimolar mixture of methylpara-nitrobenzoate and the 1,2-dialkoxydiazene oxide (5), the structure of which, and the (E)-configuration, were shown by the x-ray structural data of its acylation product — the diazene oxide (6).N. N. Semenov Institute of Chemical Physics, Russian Academy of Sciences, 117977 Moscow. Translated fromIzvestiya Akademii Nauk, Seriya Khimicheskaya, No. 11, pp. 2624–2632, November, 1992.     

A comment on SCF energy partitioning schemes

Starting from the symmetry groupG × SU(2) for the special case of negligible spin-orbit coupling, a general character formula is derived for them-tuplet representation, which is realized in the space of state functions ofn-electron systems in fields with symmetryG. Apart from the characters of the initial representation for the single electron space function, the formula only contains the number of particlesn and the multiplicitym.

Herrn Professor Dr. Hermann Hartmann zu seinem 60. Geburtstag gewidmet.     

The synthesis and liquid crystalline behaviour of 2-(4-n-alkoxyphenyl)-5-methylpyridines

We have synthesized a new series of polyphenylene compounds with a pyridine ring, 2-(4-n-alkoxyphenyl)-5-methylpyridines (CH₃C₅H₃NC₆H₄OC _i H_2i1) (1a-i ; the carbon number, i, of the alkoxy group is 1-9), and studied their phase transitions and mesogenicity using DSC and polarizing microscopy. In the homologous series of 1a-i , no mesomorphic phase appeared for i=1-5 and only a monotropic nematic phase was observed for i=6-9. The mesomorphic potential is relatively weaker for the 1a-i than for the 2a-i and 3a-i homologues, which are pyridine-containing three- and four-ring similar systems. This lowering of the mesogenicity may be due to the lack of a phenyl ring between the pyridine ring and the methyl group in the 1a-i homologues.     

A density functional representation of quantum chemistry. I. Motivation and general formalism

The molecular N-electron problem is formulated in terms of the particle density ??( r ) = ∑σ( r ? q _n) and its canonically conjugated field. Weyl's representation of the canonical commutation relations is used and a characteristic functional of the molecular system is introduced as the expectation value of Weyl's functional exp {i(α, Q) + i(β, P)}. The Hamiltonian equation of motion for the characteristic functional is derived and its cumulant approximation is discussed. The first paper is restricted to a general discussion, independent of a special choice of the Hamiltonian. The density functional representation may be considered as a step towards a formulation of a renormalized quantum chemistry. The lowest order cumulants of the characteristic functional are accessible to actual precision measurements; on the other hand, most experiments are correctly represented by a characteristic functional of the Gaussian type. The significance of the higher order correlations is neither theoretically nor experimentally disclosed yet, their possible importance for biological systems is mentioned, and the question is raised of how far quantum mechanics is empirically confirmed for extremely complex systems.     

Anab initio quantum-chemical study of proton addition to F-, Ch3-, and CF3-substituted ethylene derivatives

Protonated forms of the molecules of ethylene derivatives with the general formula C₂X₂Y₂ (X=Y=H) (1), F (2), CH₃ (3) CH₃ (4); X=F, Y=H:cis-(5)trans- (6)) were calculated by theab initio MP2/6-31 G^* method with full geometry optimization. The minima and saddle points located on the potential energy surface (PES) of the protonated ethylene molecule correspond to the stationary states and transition states of proton migration, respectively. The stationary states are characterized by a nonclassical geometry of carbocations similar to that of π-complexes, whereas the transition states have a classical structure. Unlike1, the carbocations of molecules2–6 have the classical structure. The saddle points on the PES of the ethylene derivatives correspond to the structures of the π-complex type, which are the transition states of proton migration between the C atoms of the ethylene bond. The barrier to rotation about, the C−C bond depends essentially on the substituent nature. Published inIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 8, pp. 1333–1337, August, 2000.     

Extending the applicability of TG measurements to industry and to quality ensuring by dimensionless analysis

TheI _i=E _i/RT _i dimensionless evaluation is very suitable for describing the TG measurements according to theE _i/RT _i=lnA +n[ln(1–)_i]–ln(d/dr)_i equation. TheI _i andE _i functions make the comparison of the different TG measurements possible quantitatively in the case of more DTG peaks as well.TheI _i andE _i values as function of (1-)_i and 1/T _i open new way for further theoretical and practical studies by TG measurements. Such types of results are the quantitative determination of the effect of the measuring conditions, the measuring of the mechanochemical effect of grinding and among others the explanation of the self-hardening process of fly ashes of power stations.Strict connections exist between theI _i functions and the constants of the compensation effect (CE). These constants (tan, axis intersect) can be calculated directly from the average of the measured data of theI _i function making the introduction and theoretical and practical application of the idea of general activation energy (¯E) possible. The quantitative characterisation of the examined materials of the fine structure ofCE and of the thermal processes together proves the extending importance of TG measurements from industrial and material qualification aspects as well.The author thanks gratefully to Professor Márta Fehér mathematicien, Head of the Department of Philosophy at the Technical University of Budapest for the consultations and for the encouragements.