Generalization of the Harary–Palmer power group theorem to all irreducible representations of object and color groups- color combinatorial group theory

The Harary–Palmer classic power group enumeration theorem applies to a group G acting on a set D of objects such as vertices, edges, faces and simultaneously with a group H acting on another set R of colors such that the power group $\hbox {H}^\mathrm{G}$ acts on the set $\hbox {R}^\mathrm{D}$ of all functions from D to R. In this paper we show for the first time that the power group enumeration can be generalized to all irreducible representations of the object group G of D and also all irreducible representations of the color group H of R. We have also provided interpretation of various power group generating functions for different irreducible representations in the context of color symmetry group theory. Special cases of the power group enumeration with all irreducible representations of G keeping the color group representation fixed to the totally symmetric representation are shown to have important enumerative combinatorial applications in a number of problems of chemistry, physics and biology that involve color symmetry (color inversions) such as magnetism, neutron imaging, NMR, ESR spectroscopy, catalytic functions of non-rigid disordered proteins, and quantum chromodynamics of strong interactions of fundamental particles.     

Combinatorial multinomial generators for colorings of 4D-hypercubes and their applications

Combinatorial enumeration of colorings of vertices, edges, faces and cells of 4D-hypercubes are considered for all irreducible representations of the 4D-hyperoctahedral group. We have developed computer codes that enumerate colorings of such hypercubes up to 25 dimensions and up to 10 different colors, only limitation being the combinatorial explosion. Tables are obtained for all irreducible representations and colorings for a four-dimensional hypercube with four-colorings. Several applications to spectroscopy, molecular dynamics, boolean algebra, DNA with unnatural pairs, parallel computing and rovibronic states and nuclear spin statistics of non-rigid molecular tetramers such as (H₂O)₄, (D₂O)₄, (Cl₂O)₄, etc., are pointed out.     

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.     

Screened potential in π-electron systems and its application to the estimation of excitation energies

A diagrammatic technique was developed for the estimation of the screened potential of -electron systems. The screened potential was expanded in terms of the polarization propagators which were constructed from either the singlet, , or triplet vertex part, . These vertex parts correspond to the singlet or triplet excitations, respectively, in the Random Phase Approximation (RPA) containing exchange diagrams. The excitation energies were calculated by using the screened potential in the framework of RPA with exchange. The excitation energies of several conjugated molecules with or without a hetero atom are in agreement with the experimental data.     

Continuous symmetry measures of irreducible representations: application to molecular orbitals

The formalism of continuous symmetry measures is extended to describe the extent to which a function, such as a molecular orbital, transforms under the symmetry operations of a given point group according to each irreducible representation of this group. For distorted molecules we are able to calculate the degree to which any molecular orbital transforms with respect to the irreducible representations of the pseudosymmetry group that is valid for a higher symmetry, idealized geometry, showing which irreducible representations participate in each molecular orbital upon symmetry loss in the geometry of the nuclear framework.     

Application of Angular Momentum Theory to Constructing Basis Functions of Irreducible Representations of Icosahedral Group

A new method based on angular momentum theory was proposed to construct the basis functions of the irreducible representations(IRs) of point groups. The transformation coefficients, i. e. , coefficients S, are the components of the eigenvectors of some Hermitian matrices, and can be made as real numbers for all pure rotation point groups. The general formula for coefficient S was deduced, and applied to constructing the basis functions of single-valued irreducible representations of icosahedral group from the spherical harmonics with angular momentum j≤7.     

Finite group theory for large systems. 2. Generating relations and irreducible representations for the icosahedral point group, I(h)

Generating relations and irreducible representations are given for the icosahedral point group, I(h), that are suited to computerized projection of symmetry-adapted bases of arbitrary spaces invariant to the group. With 120 elements I(h ) is a good prototype for symmetry-adaptation to a large finite nonabelian group. The four- and five-dimensional irreducible representations are obtained by coupling direct products of the three-dimensional irreducible representations using the symmetry-adaptation algorithm. The method is applied to the Hückel treatment of icosahedral C(20) fullerene.     

Randić index and lexicographic order

Let T be a tree and consider the Randi index (T)= ), where v _i –v _j runs over all edges of T and (v _i ) denotes the degree of the vertex v _i . Using counting arguments we show that the Randi index, is monotone increasing over the well (lexicographic order) ordered sequence of trees with unique branched vertex.     

Analytical expressions for topological properties of polycyclic benzenoid networks

Quantitative structure‐activity and structure‐property relationships of complex polycyclic benzenoid networks require expressions for the topological properties of these networks. Structure‐based topological indices of these networks enable prediction of chemical properties and the bioactivities of these compounds through quantitative structure‐activity and structure‐property relationships methods. We consider a number of infinite convex benzenoid networks that include polyacene, parallelogram, trapezium, triangular, bitrapezium, and circumcorone series benzenoid networks. For all such networks, we compute analytical expressions for both vertex‐degree and edge‐based topological indices such as edge‐Wiener, vertex‐edge Wiener, vertex‐Szeged, edge‐Szeged, edge‐vertex Szeged, total‐Szeged, Padmakar‐Ivan, Schultz, Gutman, Randić, generalized Randić, reciprocal Randić, reduced reciprocal Randić, first Zagreb, second Zagreb, reduced second Zagreb, hyper Zagreb, augmented Zagreb, atom‐bond connectivity, harmonic, sum‐connectivity, and geometric‐arithmetic indices. In addition we have obtained expressions for these topological indices for 3 types of parallelogram‐like polycyclic benzenoid networks.     

Finite group theory for large systems. 1. Symmetry-adaptation

The challenge of symmetry-adapting large basis sets to finite groups, apart from extensive calculations with large matrices, is obtaining linearly independent bases for frequently repeated irreducible representations, a process that is not determined by the group theory. The usual projection method is modified here to solve this problem efficiently and systematically. The resulting basis is suitably conditioned so that repeated irreducible representations are identical as required by the symmetry-generation theorem.     

Group theory and normal coordinate calculations

A method is proposed to obtain a set of computer generated symmetry adapted basis vectors which completely factorize a matrix associated with any operator which commutes with all the symmetry operations of a molecule. The cartesian coordinates of all atoms are used to derive the irreducible representations of the appropriate molecular point group and the transformation properties of any starting set of coordinates. Symmetry coordinates are then projected out each irreducible representation.     

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。     

Group-theoretical analysis of the possible symmetries of the Jahn-Teller systems

The total classification of the possible symmetries of the extremal points of adiabatic potential surfaces of the Jahn—Teller systems for all molecular symmetry point groups was elaborated on the basis of group-theoretical analysis. The method proceeds from the consecutive split of degenerate irreducible representations of the corresponding electron terms as a consequence of the symmetry descent.     

Density of non-Bloch electron states in perfect cubic crystals

This paper gives an abbreviated method for the calculation of the density of states of a crystal on the basis of that band theory in which the crystal electron states are represented by the standinglike wave functions classified according to the point-group symmetry species. The crystal is a large but finite sphere filled regularly with atoms, and the wave functions are quantized at the boundary of the sphere. The Bloch theorem is not satisfied in this theory since the wave functions are not basis functions of the irreducible representations of the translation subgroup. On the other hand, a theorem is established that the density of states can be made up of contributions given by all irreducible representations of the crystal point group, any contribution being proportional to the square of the dimension of the irreducible representation. In distinction to a former approach, the band structure is calculated solely from the energy eigenvalues obtained with the aid of the diagonalization process of the Wannier–Slater differential operator. A simple cubic lattice with an s atomic orbital on each lattice site is taken as an example, and the results are compared with Bloch's theory.     

A Complete Solution to a Conjecture on the β-Polynomials of Graphs

In 1990, Gutman and Mizoguchi conjectured that all roots of the -polynomial (G,C,x) of a graph G are real. Since then, there has been some literature intending to solve this conjecture. However, in all existing literature, only classes of graphs were found to show that the conjecture is true; for example, monocyclic graphs, bicyclic graphs, graphs such that no two circuits share a common edge, graphs without 3-matchings, etc, supporting the conjecture in some sense. Yet, no complete solution has been given. In this paper, we show that the conjecture is true for all graphs, and therefore completely solve this conjecture.     

On the use of higher-order irreducible vertex parts in the calculation of exciton spectra

The possibility to employ higher-order irreducible vertex parts (corresponding to virtual double-excitations) in the calculation of excitonic levels in infinite systems has been discussed. Numerical ab initio Hartree-Fock and correlation calculations in the Møller-Plesset partitioning have been performed on an alternating hydrogen chain. The results show that, while proper renormalization is important in the calculation of absolute excitation energies, the exciton binding energies are stronger influenced by the higher-order terms in the irreducible vertex part.     

Symmetrie und analytische Struktur der Additionstheoreme räumlicher Funktionen und der Mehrzentren-Molekülintegrale über beliebige Atomfunktionen

Three-dimensional functions f(r) = g(r) · Y _m ^l (, ø), which transform like an irreducible tensor, are transformed simultaneously under rotations and translations. The relationships governing the transformation reveal some general properties. If the addition theorem of a function f(r) can be represented by a one-center expansion in terms of surface spherical harmonics Y _m ^l , each expansion coefficient is given by a Clebsch-Gordan coefficient and a radial function.Because of these properties, addition theorems are especially helpful for the simplification and evaluation of quantum-mechanical matrix elements and multi-center energy integrals in molecular LCAO calculations. The application of addition theorems has two major advantages: First, because addition theorems are equivalent to translation formulas, the number of centers of an integral can be reduced by translation of orbitals and operators. Second, due to the typical analytical structure of the series expansion representing the addition theorem, the dimensionality of a molecular integral can be reduced, because the integration over the angular variables can be executed. Then, a molecular multi-center integral is represented by a series of one-center integrals over functions of the radial variable only.

Herrn Professor Dr. H. Hartmann zum 65. Geburtstag gewidmet.     

Calculation of vertical excitation energies of closed-shell molecules in the CNDO and INDO approximations

The Green's function method for the calculation of vertical excitation energies is adapted to the CNDO and INDO approximations by introducing an effective interaction into the irreducible vertex part. The computational scheme is explicitly developed for closed-shell molecules and applied to H₂O, H₂CO, HCOOH, HCONH₂.