Orbital structure of vibrations of nanoparticles, clusters, and coordination polyhedra

The necessity of the development of the orbital structure of vibrations of nanoparticles, clusters, and coordination polyhedra is dictated by synthesis of clusters, supermolecules, and other structures of nanoscale dispersion for which translational symmetry is absent and the crystal system is inapplicable. The composition of complicated molecules, polynuclear complexes, and clusters is described, in addition to the chemical formula, by the composition equations derived from analysis of the symmetry properties of molecular structures. This analysis enables the derivation of analytical formulas for the types of molecular orbitals of structures with arbitrary groups of symmetry. Here, we use the representation of nanoscale structures described by the orbital system as a set of concentric nested spherical orbits of atoms, orbits of faces of different order, and orbits of edges. The orbits are grouped into shells shaped as polyhedra with vertices, edges, or faces accommodating atoms with different types of packing. In such a way, the sets of molecular orbitals of all high-, intermediate-, and low-symmetry groups have been determined depending on the number of atoms in the axial, planar, and primitive orbits.     

A note on the cyclical edge-connectivity of fullerene graphs

A fullerene graph is a 3-regular (cubic) and 3-connected spherical graph that has exactly 12 pentagonal faces and other hexagonal faces. The cyclical edge-connectivity of a graph G is the maximum integer k such that G cannot be separated into two components, each containing a cycle, by deletion of fewer than k edges. Došlić proved that the cyclical edge-connectivity of every fullerene graph is equal to 5. By using Euler’s formula, we give a simplified proof, mending a small oversight in Došlić’s proof. Further, it is proved that the cyclical connectivity of every fullerene graph is also equal to 5.     

原子构造原理与过渡元素原子的价电子组态

众所周知,元素周期系过渡元素基态原子价电子组态的多样化是已往的原子构造理论无论是从定性还是从定量方面都是迄今难以给出圆满理论解释的原子构造现象之一^[1-11]。本文介绍新近提出的原子构造的对称性原理以及依据这一原理对上述原子构造现象所作的系统理论解释。     

Fullerene Graphs with Exponentially Many Perfect Matchings

We show that for all sufficiently large even p there is a fullerene graph on p vertices that has exponentially many perfect matchings in terms of the number of vertices. Further, we show that all fullerenes with full icosahedral symmetry group have exponentially many perfect matchings and indicate how such results could be extended to the fullerenes with lower symmetry.     

The number of spanning trees in an (r,s)‐semiregular graph and its line graph

For a graph G, a “spanning tree” in G is a tree that has the same vertex set as G. The number of spanning trees in a graph (network) G, denoted by t(G), is an important invariant of the graph (network) with lots of decisive applications in many disciplines. In the article by Sato (Discrete Math. 2007, 307, 237), the number of spanning trees in an (r, s)‐semiregular graph and its line graph are obtained. In this article, we give short proofs for the formulas without using zeta functions. Furthermore, by applying the formula that enumerates the number of spanning trees in the line graph of an (r, s)‐semiregular graph, we give a new proof of Cayley's Theorem. © 2013 Wiley Periodicals, Inc.     

Symmetry properties of chemical graphs. IX. The valence tautomerism in the P73− skeleton

We investigate the intramolecular rearrangement of the P₇^3? ion in which P? P bonds are continuously broken and reestablished resulting in a dynamic structure with all seven phosphorus atoms being chemically equivalent. This leads to 7/3 = 1680 valence tautomers. The P₇^3? ion is a typical fluctuating molecule, distantly related to C₁₀H₁₀ bullvalene. In order to determine the symmetry of the process we first consider the problem of the construction of the rearrangement graph. Initial steps toward this goal are illustrated, and additional characterizations of the graph are derived after its construction by computer. A number of properties of this rather complicated graph having 1680 vertices and 2520 edges are discussed: in particular the occurrence of various cycles, the maximal distance between vertices, the count of neighbors at different distance, and finally its symmetry.     

On discerning symmetry properties of graphs

A procedure is outlined which allows the symmetry properties of graphs to be systematically and rigorously investigated. It is based on a search for all the automorphisms of a graph and this is accompanied by suitably applying the procedure for recognizing identical graphs. It consists in finding all the distinctive labeling of the vertices of graph associated with the smallest binary code derived by a particular interpretation of the associated adjacency matrix. No prior cognizance of symmetry operations is required which is in contrast to the usual discussions of the symmetry properties of molecules which are based on the knowledge of pertinent symmetry operations. This is important since in graphs it is neither apparent nor generally possible to simply enlist those permutations of labels which leave the connectivity invariant (i.e., do not alter the form of the adjacency matrix). The procedure is applied to the Petersen graphs and the Desargues-Levi graph, both associated with isomerizations of trigonal bipyramidal complex and other chemical transformations. It is shown that these graphs of high symmetry belong to symmetry groups of order 120 and 240 respectively. The approach can also provide a basis for the development of the symmetry properties of non-rigid molecules in which connectivity is preserved.     

计算机辅助结构解析中自同构群的生成算法

在结构解析过程中,自同构群的生成是必须的。通过全通道拓扑等价性算法,得到了化合物的自同构群,并比较了几种拓扑等价性算法,结果发现,全通道算法能够正确划分化合物的结构图,从而为ＥＳＥＳＯＣ系统的立体异构体穷举生成奠定了基础。     

Maximum symmetrical split of molecular graphs. Application to organic synthesis design

Whereas the potential symmetry of a molecule may be a feature of importance in synthesis design, this one is often difficult to detect visually in the structural formula. In the present article, we describe an efficient algorithm for the perception of this molecular property. We have addressed this problem in terms of graph theory and defined it as the Maximum Symmetrical Split of a molecular graph. A solution is obtained by deleting in such a graph a minimum number of edges and vertices so that the resulting subgraph consists of exactly two isomorphic connected components that correspond to a pair of synthetically equivalent synthons. In view to reduce the search space of the problem, we have based our algorithm on CSP techniques. In this study, we have found that the maximum symmetrical split is an original kind of Constraint Satisfaction Problem. The algorithm has been implemented into the RESYN_Assistant system, and its performance has been tested on a set of varied molecules which were the targets of previously published synthetic studies. The results show that potential symmetry is perceived quickly and efficiently by the program. The graphical display of this perception information may help a chemist to design reflexive or highly convergent syntheses.     

Symmetry calculation for molecules and transition states

The symmetry of molecules and transition states of elementary reactions is an essential property with important implications for computational chemistry. The automated identification of symmetry by computers is a very useful tool for many applications, but often relies on the availability of three‐dimensional coordinates of the atoms in the molecule and hence becomes less useful when these coordinates are a priori unavailable. This article presents a new algorithm that identifies symmetry of molecules and transition states based on an augmented graph representation of the corresponding structures, in which both topology and the presence of stereocenters are accounted for. The automorphism group order of the graph associated with the molecule or transition state is used as a starting point. A novel concept of label‐stereoisomers, that is, stereoisomers that arise after labeling homomorph substituents in the original molecule so that they become distinguishable, is introduced and used to obtain the symmetry number. The algorithm is characterized by its generic nature and avoids the use of heuristic rules that would limit the applicability. The calculated symmetry numbers are in agreement with expected values for a large and diverse set of structures, ranging from asymmetric, small molecules such as fluorochlorobromomethane to highly symmetric structures found in drug discovery assays. The new algorithm opens up new possibilities for the fast screening of the degree of symmetry of large sets of molecules. © 2014 Wiley Periodicals, Inc.     

Molecular symmetry perception

An algorithm for molecular symmetry perception is presented. The method identifies the full set of molecular symmetry elements (proper and improper) and determines their coordinates. The algorithm eliminates the necessity to explore the entire graph automorphism group; as a result its computer application is extremely effective. Application to several dendrimers and fullerenes with high topological symmetry is presented.     

Delone sets with congruent clusters

In the paper, we present a short survey and new results of the local theory of regular systems, more precisely the part that establishes for a given Delone set the link between congruence of fragments (‘clusters’) of the set and its global symmetry. The theory describes ‘local rules’ of a Delone set which imply its regularity or multi-regularity, i.e., imply that a Delone set is an orbit or the union of several orbits, respectively, under its symmetry group. We will discuss a cycle of new results of the local theory for Delone sets which have centrally symmetrical fragments: clusters or patches.     

Burnside Rings: Application to Molecules of Icosahedral Symmetry

The Burnside ring, B(G), of a group G is the set of isomorphism classes of orbits of G together with the operations of addition and product. The addition is defined as the disjoint union, and the product as the Cartesian product. This paper describes basic facts about this algebraic structure and develops some applications in chemistry, as the labelling of atoms in molecules of high symmetry and the construction of symmetry-adapted functions. For illustrating such applications, the concept of Burnside ring is applied to the icosahedral symmetry. Sets of points which are isomorphic to the orbits of the I group are described and the multiplication table of B({I}) is obtained from the table of marks. This multiplication table allows us to obtain an elegant labelling of the atoms of the buckminsterfullerene which is consistent with the icosahedral symmetry. Also, we obtain complete sets of symmetry-adapted functions for the buckminsterfullerene which span the Boyle and Parker's icosahedral representations.     

From symmetry-labeled quotient graphs of crystal nets to coordination sequences

The combinatorial topology of crystal structures may be described by finite graphs, called symmetry-labeled quotient graphs or voltage graphs, with edges labeled by symmetry operations from their space group. These symmetry operations themselves generate a space group which is generally a non-trivial subgroup of the crystal space group. The method is an extension of the so-called vector method, where translation symmetries are used as vector labels (voltages) for the edges of the graph. Non-translational symmetry operations may be used as voltages if they act freely on the net underlying the crystal structure. This extension provides a significant reduction of the size of the quotient graph. A few uninodal and binodal nets are examined as illustrations. In particular, various uninodal nets appear to be isomorphic to Cayley color graphs of space group. As an application, the full coordination sequence of the diamond net is determined.     

Oriented 2-cell embeddings of a graph and their symmetry classification: generating algorithms and case study of the möbius-kantor graph

We discuss a method to derive all symmetry-distinct oriented 2-cell embeddings of a given graph and classify them based on their symmetry. As an example, we apply the algorithm to the highly symmetrical trivalent M?bius-Kantor graph. Considering the derived 2-cell embeddings as carbon networks leads to some interesting negative curvature carbon allotropes.     

Broder and Karlin's formula for hitting times and the Kirchhoff Index

We give an elementary proof of an extension of Broder and Karlin's formula for the hitting times of an arbitrary ergodic Markov chain. Using this formula in the particular case of random walks on graphs, we give upper and tight lower bounds for the Kirchhoff index of any N‐ vertex graph in terms of N and its maximal and minimal degrees. We also apply the formula to a closely related index that takes into account the degrees of the vertices between which the effective resistances are computed. We give an upper bound for this alternative index and show that the bound is attained—up to a constant—for the barbell graph. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

New N-representability results with symmetry in molecular quantum chemistry

We prove a new type of N-representability result: given a totally symmetric density function ρ, we construct a wavefunction Ψ such that the totally symmetric part of $\rho \Psi $ (its projection over the totally symmetric functions) be equal to ρ, and, furthermore, such that Ψ belongs to a given class of symmetry associated to the symmetry group of a molecule. Our proof uses deformations of density functions and which are solutions of a “Jacobian problem”. This allows us to formalize rigorously an idea of A. Görling (Phys. Rev. A 47 (1993) 2783), for Density-Functional Theory in molecular quantum chemistry, by defining a density functional that takes into account the symmetry of the molecule under study.     

Globally attractive oscillations in open monosubstrate allosteric enzyme reactions

Here we study the dynamical properties of glycolytic and other similar biochemical oscillation-generating processes by means of the analysis of a model proposed by Golbdeter and Lefever (Bioph J 13:1302–1315, 1972) in a reduced form proposed by Keener and Sneyd (Mathematical physiology, chap 1, Springer Verlag, Berlin, 2009). After showing that the orbits of the system are bounded, we give some conditions for the existence of oscillations and for the global arrest of them. Then, after deriving an equivalent Lienard-Newton’s equation we assess uniqueness and the global stability of the arising limit cycle. Finally, we shortly investigate the possibility of breaking of the spatial symmetry. Some biological remarks end the work.     

Application of DNMR,group theory and molecular mechanics in the elucidation of the inversion process in the flexible molecule perhydrotetra-azapyrene

The inversion process in cis-10b,10c-perhydro-1H,6H-3a,5a,8a,10a-tetraazapyrene (1) has been investigated by ¹H and ¹³C NMR. The free energy of activation is found to be 14.95 ± 0.2 kcal mole^?1 at 45°. Application of group theoretical techniques led to a graph representing the essential symmetry properties of the potential energy surface for conformational change. The energies of intermediates on this graph were then estimated using molecular mechanics calculations. This combined approach suggests that the total inversion proceeds via a conformation of C_2v symmetry with two non-chair piperazine rings, calculated to be 6.8 kcal mole^?1 less stable than the ground state conformation (C₂ symmetry).     

The onset of chaos in the vibrational dynamics of LiNC/LiCN

Recent advances in vibrational spectroscopy have greatly enhanced the possibilities of research of highly excited states in molecular systems of moderate size. At sufficiently high level of excitation the correspondence principle holds, and classical mechanical arguments constitute a useful interpretative tool. The corresponding dynamics often become very complex specially in systems with floppy degrees of freedom, and periodic motion plays an important role for its understanding. In this paper, we present a computational procedure to systematically calculate periodic orbits of LiNCLiCN with a given symmetry, that has the additional advantage of providing a useful insight into the onset of chaos in this system.