Subduction of Q-conjugacy representations and characteristic monomials for combinatorial enumeration

A new method of combinatorial enumeration is presented. The subduction of Q-conjugacy representations gives a characteristic subduction table and a characteristic monomial table. A cycle index is defined on the basis of such monomials and used for combinatorial enumeration of isomers. Received: 10 October 1997 / Accepted: 13 February 1998 / Published online: 17 June 1998     

Möbius function and characteristic monomials for combinatorial enumeration

After the definitions of amplified representations and number-theoretical vectors, the markaracter table of a cyclic subgroup is converted into the corresponding Q-conjugacy character table. The conversion is shown to necessitate an interconversion matrix that contains M?bius functions as elements. Since the interconversion matrix gives characteristic monomials for cyclic groups, all the powers appearing in each of the characteristic monomials are shown to be integers. Characteristic monomials for finite groups are then built up by starting from those of cyclic groups. This procedure clarifies the fact that all the powers appearing in each characteristic monomial for finite groups are integers. The relationship between characteristic monomial tables and unit-subduced-cycle-index tables is discussed with respect to their application to isomer enumeration. Received: 15 July 1998 / Accepted: 16 December 1998 / Published online: 16 March 1999     

Multinomial Combinatorial Group Representations of the Octahedral and Cubic Symmetries

We consider the full multinomial combinatorics of all irreducible representations of the octahedral (cubic) symmetry as a function of partitions for vertex, face and edge colorings. Full combinatorial tables for all irreducible representations and all multinomial partitions are constructed. These enumerations constitute multinomial expansions of character-based cycle index polynomials, and grow in combinatorial complexity as a function of edge or vertex coloring partitions.     

Relativity and the Jahn–Teller,Berry pseudorotations of TBP clusters: group theory,spin–orbit and combinatorial nuclear spin statistics of TBP Desargues–Levi isomerization graph

Jahn–Teller and Berry pseudorotations in transition metal and main group clusters such as Hf₅, Ta₅, W₅ and Bi₅ are interesting because of the competition between relativistic effects and pseudorotations. Topological representations of various isomerization pathways arising from the Berry pseudorotation of pentamers constitute the edges of the Desargues–Levi graph. We have computed the combinatorics for multinomial colorings of the vertices, edges and 10-faces of the Desargues–Levi isomerization graph for all irreducible representations and the nuclear spin statistics of spin-7/2 ¹⁸¹Ta₅ as well as the TBP composite cluster particles. Topological insights into Jahn–Teller and Berry pseudorotations and relativistic effects are provided.     

A theory for the construction of the irreducible representations of finite groups

It is shown how the irreducible representations of a finite group can be calculated from the irreducible characters (the latter can be calculated exactly by using Dixon's method). All elements of the matrix, representing a group element, lie in the rational field of polynomials of ξ = exp (2πi/e), where e is the exponent of the group.     

Factor groups,semidirect product and quantum chemistry

In this article, we prove some general theorems about representations of finite groups arising from the inner semidirect product of groups. We show how these results can be used for standard applications of group theory in quantum chemistry through the orthogonality relations for the characters of irreducible representations. In this context, conditions for transitions between energy levels, projection operators, and basis functions were determined. This approach applies to composite systems and it is illustrated by the dihedral group related to glycolate oxidase enzyme. © 2014 Wiley Periodicals, Inc.     

Bases for the irreducible representations of O(3) symmetry adapted to a crystallographic point group

We construct bases for the irreducible representations of the rotation group O(3) which are symmetry adapted to a Crystallographic point group. We obtain explicit expressions for the cubic groups, which are valid for arbitrary values of the angular momentum quantum number l. Our method yields an efficient algorithm for both analytical and numerical work. An explicit formula for the multiplicities of an irreducible representation for the cubic groups in an arbitrary angular momentum term l is also derived.     

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.     

dN离子在三角场中的基矢及有关矩阵元

通过选用合适的单电子基矢, 使按群链[(O?D_3?C_3)×SU(2)]~N构成的多电子基矢, 成为dN体系D_3?C_3旋量群不可约表示的基矢或其两个一维不可约表示直和的基矢。讨论了有关矩阵元和顺磁参数的计算。     

Symmetry groups and representations of hamiltonians for several coupled degrees of freedom: Application to non-rigid molecular vibrations

We present a systematic method of determining finite symmetry groups for multiple coordinate hamiltonians including coupling terms. Groups are first derived separately for terms involving single coordinates. When these terms are added together, direct products of the initial groups are taken, and more symmetry operations may need to be included which involve several coordinates. When coupling terms are included, some symmetry operations must be discarded and subgroups of the product group determined. The determination of the classes, the induction and subduction of irreducible representations and characters, and assignment of molecular properties to these representations are performed along with the group derivation. We discuss the application of these methods to hamiltonians for non-rigid molecular vibrations in four membered ring molecules, molecules with two methyl groups, and the pseudorotation interconversion motion of some MX₅ molecules.     

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.     

Vibration and NMR spectra of dihedral fullerenes

With the help of the topological structures of the dihedral (D_p, D_ph, D_pd; p=5, 6) fullerenes, we have derived the formulas for the decomposition of their nuclear motions into irreducible representations. So their infrared and Raman active modes have been obtained. Furthermore, by use of the concept of the representative patch we have also derived their nuclear magnetic resonance spectra. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 68: 211–217, 1998     

PuN基态分子势能函数与热力学函数的理论计算

在Pu的相对论有效原子实势近似和N原子6-311G*全电子基函数下,用密度泛函B3LYP方法计算得到PuN分子基态X6∑+的结构与势能函数、力常数与光谱数据.同时计算得到PuN(g)分子在298 K时的标准生成热力学函数△fH0、△S0和△fG0,分别为-487.239 kJ/mol、95.345 J/mol K和-515.6661 kJ/mol.     

On the evaluation of the geometrical factors utilized in ligand polarization calculations

The utility of the Ligand polarization model in solving many physical problems in quantum mechanics has been appreciated among scientists during the last years. Problems such as electric dipole strength, vibronic electric dipole strength, optical activity calculations have been carried out within the framework of a dynamic coupling mechanism. Taking advantage of the irreducible tensor method put forward by Griffith in the case of molecular symmetry groups, both the molecular states and relevant operators can be classified in terms of irreducible representations of the molecular group in question, and therefore it is most convenient to express the relevant operators involved in any specific calculation in a symmetry adapted form. As a starting point, we may classify our molecular states and operators in the 0-rotation group and lower symmetry groups may also be studied by using simple correlation properties. Here we aim to deal with d-d and f-f type of transitions, and hence the 2² (electric quadrupole), 2⁴ (electric hexadecapole) and the 2⁶-multipoles are considered in some detail. We have adopted, the octahedral set of functions as given by Griffith to define the 2^itl (l = 2, 4, 6) multipoles and obtain the corresponding geometrical factors for the various irreducible representations.     

群的对称性偶合系数计算的新方法

本文根据群G和子群g的不可约表示基向量标准化的性质,提出了直接计算子群g的V系数和W系数的新方法。对于SO(3)群-点群的情况,可由标准化基向量与尽可能低j值的SO(3)群-点群V系数和变换系数,以及3j、 6j符号等关系,分别导出计算点群V系数和W系数的公式。并以正二十四面体Ih群为例,计算了三角组分系、五角组分系的所有V系数和W系数,计算结果列于附表。     

Unitary group tensor operator algebras for many-electron systems: I. Clebsch-Gordan and Racah coefficients

A basis for the Racah-Wigner algebra of irreducible representations of the unitary group U(n) that are pertinent to quantum chemical models of many-electron systems is developed. Standard Clebsch-Gordan coefficients and isoscalar factors (also called coupling factors or reduced Wigner coefficients) for both symmetric (S _N ) and unitary [U(n)] groups are extended to transformation coefficients and corresponding isoscalar factors relating canonical Young-Yamanouchi or Gel'fand-Tsetlin bases to simple partitioned bases. All these different types of isoscalar factors are interrelated using the well-known reciprocity between the S _N and U(n) tensor representations, and general expressions relating these different factors are given. For the two-column representations characterizing the many-electron theory, detailed explicit expressions are presented for both the above-mentioned relationships and for all relevant U(n) isoscalar factors. Finally, U(n) Racah coefficients are introduced and explicit expressions derived for certain special classes of these coefficients.Killam Research Fellow 1987–89.     

Polyurethane-TiO2 nanocomposite coatings from sunflower- oil-based amide diol as soft segment

Abstract

Vegetable oil based environmentally friendly polyurethane-TiO₂ nanocomposite coatings have been synthesized by using sunflower oil derived diol, toluene diisocyanate and TiO₂ nanoparticles. The chemical structure was confirmed by FTIR and NMR techniques while physico-chemical testing was carried out by standard laboratory methods. Physico-mechanical and anticorrosive tests of the coatings (in different corrosive media) have been investigated by standard methods. In addition to this the morphology and thermal stability behavior of the coatings have been carefully investigated by different techniques like XRD, TEM, TGA/DTG and DSC. The comparison of the performance of nanocomposites with the respective virgin polyurethane coatings reveals that the dispersion of nanoTiO₂ enhanced the mechanical, corrosion and thermal stability behavior of the polymer. The synthesized nanocomposites can be used safely upto 250–275?°C. These sunflower oil derived polyurethane nanocomposites can be used in the world of protective coatings, as an alternative of petroleum derived corrosion protective coating materials.     

Pseudo-point groups and chronality in enumeration of reaction pairs

Summary The pseudo-point group is constructed to characterize the symmetry of a basic pair of hexagonal reaction graphs having no par-bonds on its edges. Any pairs of reaction graphs (reaction pairs) are considered to be obtained by adding par-bonds to the edges of the basic pair; they are counted by the USCI (unit-subduced-cycle-index) approach. Thus, the six edges of the basic pair are assigned to the coset representation (/C _2v ). After the subduction of the is calculated, the partial-cycle-index method of the USCI approach is applied to the combinatorial enumeration of reactions pairs.Reaction pairs are classified to two categories, i.e. isoenergetic and anisoenergetic. An isoenergetic pair is concluded to be a self-reaction pair, while an anisoenergetic pair corresponds to a non-self-reaction pair. The concept of chronality is also discussed to clarify the symmetrical nature of the resulting orbits.     

High symmetries in quantum chemistry

A method of simplifying the solution of secular equations occurring in electronic structure, normal mode and nuclear spin state calculations for molecules possessing symmetry is illustrated by applying it to the π-electrons of the recently discovered C₆₀ cluster. In contrast to the usual procedure, the method of reduction to characters does not require actual matrix realizations of the irreducible representations of the symmetry group but only the information contained in the character and multiplication tables. In the case of C₆₀ it leads to explicit algebraic solutions for all the π-orbital energies.