Chemical applications of topology and group theory

The 60 even permutations of the ligands in the five-coordinate complexes, ML ₅, form the alternating group A ₅, which is isomorphic with the icosahedral pure rotation group I. Using this idea, it is shown how a regular icosahedron can be used as a topological representation for isomerizations of the five-coordinate complexes, ML ₅, involving only even permutations if the five ligands L correspond either to the five nested octahedra with vertices located at the midpoints of the 30 edges of the icosahedron or to the five regular tetrahedra with vertices located at the midpoints of the 20 faces of the icosahedron. However, the 120 total permutations of the ligands in five-coordinate complexes ML ₅ cannot be analogously represented by operations in the full icosahedral point group I _h, since I _his the direct product I×C₂ whereas the symmetric group S ₅ is only the semi-direct product A ₅S₂. In connection with previously used topological representations on isomerism in five-coordinate complexes, it is noted that the automorphism groups of the Petersen graph and the Desargues-Levi graph are isomorphic to the symmetric group S ₅ and to the direct product S ₅×S ₂, respectively. Applications to various fields of chemistry are briefly outlined.     

Chemical applications of group theory and topology

The following procedure is described for investigating the qualitative dynamics of simple chemical systems: 1) A so-called influence diagram is generated representing the relationships between the reference reactants (phase-determining intermediates); 2) This influence diagram is used to generate a truth table indicating possible transitions between state vectors representing the signs of the time derivatives of of the reference reactant concentrations; 3) The truth table is used to determine a state transition diagram representing the flow topology around unstable equilibrium points; 4) The characteristic equation of the adjacency matrix of the influence diagram is solved in order to determine the presence of such unstable equilibrium points. The two types of qualitative dynamics possible for chemical systems containing two reference reactants and one feedback circuit are bifurcation between two attracting regions (bistability) and limit cycle oscillation. However, in two reference reactant systems oscillation requires an additional self-activating loop to generate the unstable equilibrium point required for its realization. Bistability and limit cycle oscillation are also two of the possible types of qualitative dynamics for chemical systems containing three reference reactants. However, chemical systems with three reference reactants and two or more feedback circuits can also contain interlocking limit cycles, which can lead to toroidal oscillations or chaos. The influence diagrams are given for the systems exhibiting these various types of dynamic behavior along with a summary of the important properties of all 729 possible influences for simple chemical systems containing three reference reactants.     

Chemical applications of topology and group theory

Possible convex polyhedra for three-dimensional water networks in clathrate and semiclathrate hydrates are discussed in this paper. All such polyhedra have all vertices of order three. Therefore, the number of vertices (v), edges (e), and faces (f) must satisfy the equalities e=3v/2 and f=(4+v)/2. Possible polyhedra of this type with exclusively quadrilateral, pentagonal, and hexagonal faces and with up to 18 faces are examined. Many of these polyhedra are duals of various triangulated coordination polyhedra studied in previous papers of this series. In order to minimize angular strain, polyhedra with the maximum number of pentagonal faces are favored for water networks in clathrate and semiclathrate hydrates subject to the presence of sufficiently large cavities to accommodate the guest molecule.

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Zusammenfassung In dieser Arbeit werden mögliche konvexe Polyeder für dreidimensionale Wasser-Netzwerke in Klathrat- und Semiklathrathydraten diskutiert. Daher muß die Anzahl der Scheitelpunkte (v), Kanten (e) und Flächen (f) den Gleichungen e=3v – und f=(4+v) – genügen. Es werden mögliche Polyeder dieses Typs mit bis zu 18 Flächen, die ausschließlich quadrilateral, pentagonal und hexagonal sein sollen, untersucht. Viele dieser Polyeder sind Zwillinge von verschiedenen, aus Dreiecken zusammengesetzten Koordinationspolyedern, die in früheren Arbeiten dieser Reihe untersucht wurden. Um die Winkeldeformation auf ein Mindestmaß zurückzuführen, werden im Falle von Wassernetzwerken in Klathrat- und Semiklathrathydraten Polyeder mit der maximalen Anzahl von pentagonalen Flächen bevorzugt, weil so ausreichend große Hohlräume zur Aufnahme des Gastmoleküls entstehen.

     

Chemical applications of topology and group theory

Earlier approaches to the analysis of chemical dynamic systems using kinetic logic are refined to deal more effectively with systems having the two or more feedback circuits required for chaos. The essential kinetic features of such a system can be represented by a directed graph (called aninfluence diagram) in which the vertices represent the internal species and the directed edges represent kinetic relationships between the internal species. Influence diagrams characteristic of chaotic chemical systems have the following additional features: (1) They are connected; (2) Each vertex has at least one edge directed towards it and one edge directed away from it; (3) There is at least one vertex, called a turbulent vertex, with at least two edges directed towards it. From such an influence diagram a state transition diagram representing the qualitative dynamics of the system can be obtained using the following 4-step procedure: (1) A logical relationship is assigned at each turbulent vertex; (2) A local truth table is generated for each circuit in the influence diagram; (3) The local truth tables are combined to give a global truth table using the logical relationships at the turbulent vertices; (4) The global truth table is used to determine the corresponding state transition diagram using previously described methods. This refined procedure leads to a more restricted set of influence diagrams having the interlocking cycle flow topology required for chaos than the procedure described earlier. Systems with 3 internal species are examined in detail using the refined procedure. All systems with 3 dynamic variables shown in the simulation studies of Rössler to give chaotic dynamics correspond to influence diagrams which give inter-locking cycle (chaotic) flow topologies by the refined procedure. In addition, two models for the Belousov-Zhabotinskii reaction are examined using the refined procedure. The results are potentially informative concerning possible mechanisms for the limitation of the accumulation of autocatalytically produced HBrO₂ (one of the internal species) during the course of this reaction.     

Chemical applications of topology and group theory. 17. An information theoretical approach to polyhedral symmetry [1]

Information theoretic parameters are described which measure the asymmetry of polyhedra based on partitions of their vertices, faces, and edges into orbits under action of their symmetry point groups. Such asymmetry parameters are all zero only for the five regular polyhedra and are all unity for polyhedra having no symmetry at all, i.e. belonging to the C ₁ symmetry point group. In all other cases such asymmetry parameters have values between zero and unity. Values for such asymmetry parameters are given for all topologically distinct polyhedra having five, six, and seven vertices; all topologically distinct eight-vertex polyhedra having at least six symmetry elements; and selected polyhedra having from nine to twelve vertices. Effects of polyhedral distortions on these asymmetry parameters are examined for the tetrahedron, trigonal bipyramid, square pyramid, and octahedron. Such information theoretic asymmetry parameters can be used to order site partitions which are incomparable by the chirality algebra methods of Ruch and co-workers.     

Chemical applications of topology and group theory 13. Chirality and framework groups [1]

Pople has recently introduced the concept of a framework group to specify the full symmetry properties of a molecular structure. Furthermore, Pople has developed powerful algorithms for the use of framework groups to generate all distinguishable skeletons with a given number of sites. This paper studies the systematics of chirality arising from different framework groups. In this connection framework groups can be classified into four different types: linear, planar, achiral, and chiral. Chiral framework groups lead to chiral systems for any ligand partition including that with all ligands equivalent. Linear framework groups are never chiral even for the ligand partition with all ligands different. Planar framework groups are also never chiral since all sites are in the same plane, which therefore remains a symmetry plane for any ligand partition. However, the mirror symmetry of the molecular plane of a planar framework group can be destroyed by a process called polarization; this process can be viewed as the mathematical analogue of complexing a planar aromatic hydrocarbon to a transition metal. The chirality of four-, five-, and six-site framework groups is discussed in terms of the maximum symmetry ligand partitions resulting in removal of all of the symmetry elements corresponding to improper rotations S _n (including reflections S ₁ and inversions S ₂) from achiral and polarized planar framework groups. The Ruch-Schönhofer group theoretical algorithms for the calculation of chiral ligand partitions and pseudoscalar polynomials of lowest degree (“chirality functions”) are adapted for use with these framework groups. Other properties of framework groups relevant to a study of their chirality are also discussed: these include their transitivity (i.e. whether all sites are equivalent or not), their normality (i.e. whether proper rotations correspond to even permutations and improper rotations correspond to odd permutations), and the number of sites in their symmetry planes.     

Chemical applications of topology and group theory. 34. Structure and bonding in titanocarbohedrene cages

Chemical bonding models are developed for the titanocarbohedrenes Ti14C13 and Ti8C12 by assuming that the Ti atoms use a six-orbital sd5 manifold and there is no direct Ti...Ti bonding. In the 3 x 3 x 3 cubic structure of Ti14C13, the 8 Ti atoms at the vertices of the cube are divided into two tetrahedral sets, one Ti(III) set and one Ti(IV) set, and the 6 Ti atoms at the midpoints of the cube faces exhibit square planar TiC4 coordination with two perpendicular three-center four-electron bonds. The energetically unfavorable Th dodecahedral structure for Ti8C12 has 8 equivalent Ti(III) atoms and C2(4-) units derived from the complete deprotonation of ethylene. In the more energetically favorable Td tetracapped tetrahedral structure for Ti8C12, the C2 units are formally dianions and the 8 Ti atoms are partitioned into inner tetrahedra (Ti(i)) bonded to the C2 units through three-center Ti-C2 bonds and outer tetrahedra (Ti degrees) bonded to the C2 units through two-center Ti-C bonds. The Ti atoms in one of the Ti4 tetrahedra are Ti(0) and those in the other Ti4 tetrahedron are Ti(III). Among the two such possibilities, the lower energy form has the (Ti0)o4(Ti(III))i4 configuration, corresponding to dicarbene C2 ligands with two unpaired electrons in the carbon-carbon pi-bonding similar to the multiple bond in triplet O2. This contrasts with the opposite (Ti(III)o4(Ti0)i4 configuration in the higher energy form of Th-Ti8C12, corresponding to ethynediyl ligands with full C...C triple bonds and unpaired electrons in the C sp hybrid orbitals for sigma-bonding to Ti.     

Chemical applications of topology and group theory. XXI: Chirality in transitive skeletons and qualitative completeness [1]

This paper unifies the following ideas for the study of chirality polynomials in transitive skeletons: (1) Generalization of chirality to permutation groups not corresponding to three-dimensional symmetry point groups leading to the concepts of signed permutation groups and their signed subgroups; (2) Determination of the total dimension of the chiral ligand partitions through the Frobenius reciprocity theorem; (3) Determination of signed permutation groups, not necessarily corresponding to three-dimensional point groups, of which a given ligand partition is a maximum symmetry chiral ligand partition by the Ruch-Schönhofer partial ordering, thereby allowing the determination of corresponding chirality polynomials depending only upon differences between ligand parameters; such permutation groups having the point group as a signed subgroup relate to qualitative completeness. In the case of transitive permutation groups on four sites, the tetrahedron and polarized square each have only one chiral ligand partition, but the allene and polarized rectangle skeletons each have two chiral ligand partitions related to their being signed subgroups of the tetrahedron and polarized square, respectively. The single transitive permutation group on five sites, the polarized pentagon, has a degenerate chiral ligand partition related to its being a signed subgroup of a metacyclic group with 20 elements. The octahedron has two chiral ligand partitions, both of degree six; a qualitatively complete chirality polynomial is therefore homogeneous of degree six. The cyclopropane (or trigonal prism or trigonal antiprism) skeleton is a signed subgroup of both the octahedron and a twist group of order 36; two of its six chiral ligand partitions come from the octahedron and two more from the twist group. The polarized hexagon is a signed subgroup of the same twist group but not of the octahedron and thus has a different set of six chiral ligand partitions than the cyclopropane skeleton. Two of its six chiral ligand partitions come from the above twist group of order 36 and two more from a signed permutation group of order 48 derived from the P₃[P ₂] wreath product group with a different assignment of positive and negative operations than the octahedron.     

Chemical applications of topology and group theory. XXII : Lowest degree chirality polynomials for regular polyhedra [1]

The lowest degree chirality polynomials for the regular octahedron, cube, and regular icosahedron are discussed. All three of these regular polyhedra are chirally degenerate since they have more than one lowest degree chiral ligand parition by the Ruch-Schönhofer scheme. The two lowest degree chirality polynomials for the octahedron have degree 6 and can be formed from three degree 3 generating polynomialsf,g, andh through the relationshipsf(g +h) andf(g –h), wheref,g, andh measure the effects of the three separating reflection planes (h), the four threefold rotation axes, and the three fourfold rotation axes, respectively. The permutation groups of the vertices of the cube and icosahedron contain only even permutations, which leads to a natural pairing of their chiral ligand partitions according to equivalence of the corresponding Young diagrams upon reflection through their diagonals. The two lowest degree chirality polynomials for the cube have degree 4 and can be formed from two degree 4 generating polynomialsf andg through the relationships –2g andf –2g, wheref andg measure the effects of theS ₆ improper rotation andC ₄, proper rotation axes. respectively. The four lowest degree chiral ligand partitions for the icosahedron have degree 4 and lead naturally to a single degree 4 chirality polynomial with 120 terms of the general type (x –y)² (z –w)². This chirality polynomial for the icosahedron cannot be broken down into simpler generating polynomials, in contrast to the lowest degree chirality polynomials for the octahedron and cube. This appears to relate to the origin of the icosahedral group from the simple alternating groupA ₅. The full icosahedral chirality polynomial can be simplified to give a chirality polynomial for the chiral boron-monosubstituted ortho and meta carboranes of the general formula B₂C₁₀H₁₁X.     

Chemical applications of topology and group theory. 26. The sextuple diamond-square-diamond rearrangement of the icosahedron through a cuboctahedron intermediate

The details of the symmetry factoring of the graphs corresponding to the icosahedron and the cuboctahedron are presented. Such symmetry factoring procedures use the sequence of two-foldC ₂ and threefoldC ₃ elementsC ₂ xC ₂ x CZ x C3 to give disconnected graphs having eigenvalue spectra similar to those of the original polyhedra but with components having only one and two vertices. In addition, the same symmetry factoring sequence is used to determine the eigenvalue spectrum of an intermediate in the sextuple diamond-square process for conversion of the icosahedron to the cuboctahedron.This paper is dedicated to Professor Frank Harary in recognition of his pioneering work in areas of graph theory closely related to chemical problems. For part 25 of this series, see ref. [1].     

Chemical applications of topology and group Theory. XXIV: Chiralization of chemically significant polyhedra [1]

Permutation group-theoretical methods are used to study the chiralization of achiral polyhedral skeletons with v vertices by successive ligand replacement. Starting from the fully symmetrical ligand partition (), such chiralization processes may be characterized either by the minimum number of ligand replacement steps m, or the minimum number of different kinds of ligandsi, required to destroy all improper rotations. These parameters are trivially related to the lowest degree chiral ligand partition(s) as determined by the subduction of the skeleton point group G into the corresponding symmetric groupS by the procedure of Ruch and Schönhofer. Two different chiralization pathways with different values ofm andi are found for the octahedron, cube, hexagonal bipyramid, and icosahedron. Many less symmetrical chemically significant polyhedra have the degree 2 ligand partition (v - 2, 2) as the lowest degree chiral ligand partition and thus have only one chiralization pathway. Such polyhedra include the bicapped tetrahedron, trigonal prism, capped octahedron, bisdisphenoid, square antiprism, 4, 4, 4-tricapped trigonal prism, 4-capped square antiprism, 4,4-bicapped square antiprism, and the cuboctahedron.     

Natural resonance theory: III. Chemical applications

We describe quantitative numerical applications of the natural resonance theory (NRT) to a variety of chemical bonding types, in order to demonstrate the generality and practicality of the method for a wide range of chemical systems. Illustrative applications are presented for (1) benzene and polycyclic aromatics; (2) CO₂, formate, and related acyclic species; (3) ionic and polar compounds; (4) coordinate covalent compounds and complexes; (5) hypervalent and electron-deficient species; (6) noncovalent H-bonded complex; and (7) a model Diels-Alder chemical reaction surface. The examples exhibit the general harmony of NRT weightings with qualitative resonance-theoretic concepts and illustrate how these concepts can be extended to many new types of chemical phenomena at a quanitative ab initio level. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 628–646, 1998     

Symmetry coordinates and group orbitals from basis functions

The gradients of the basis functions of group theory are vector-valued basis functions. When one of the components of such a gradient is evaluated at atomic positions, and the values are summed over a set of equivalent atoms, the result represents a symmetry coordinate and/or a group orbital.     

Exploiting non-abelian point group symmetry in direct two-electron integral transformations

Summary A simple and general scheme to exploit any discrete point group symmetry in two-electron integral transformations is introduced. It has been implemented together with integral pre-screening techniques in direct two-electron integral transformations. Its application has also been extended to subsequent MO integral processing steps like MP2 or solution of the coupled-perturbed Hartree-Fock equations (CPHF). Timings for representative applications are presented.     

Unitary group adapted state specific multireference perturbation theory: Formulation and pilot applications

We present here a comprehensive account of the formulation and pilot applications of the second‐order perturbative analogue of the recently proposed unitary group adapted state‐specific multireference coupled cluster theory (UGA‐SSMRCC), which we call as the UGA‐SSMRPT2. We also discuss the essential similarities and differences between the UGA‐SSMRPT2 and the allied SA‐SSMRPT2. Our theory, like its parent UGA‐SSMRCC formalism, is size‐extensive. However, because of the noninvariance of the theory with respect to the transformation among the active orbitals, it requires the use of localized orbitals to ensure size‐consistency. We have demonstrated the performance of the formalism with a set of pilot applications, exploring (a) the accuracy of the potential energy surface (PES) of a set of small prototypical difficult molecules in their various low‐lying states, using natural, pseudocanonical and localized orbitals and compared the respective nonparallelity errors (NPE) and the mean average deviations (MAD) vis‐a‐vis the full CI results with the same basis; (b) the efficacy of localized active orbitals to ensure and demonstrate manifest size‐consistency with respect to fragmentation. We found that natural orbitals lead to the best overall PES, as evidenced by the NPE and MAD values. The MRMP2 results for individual states and of the MCQDPT2 for multiple states displaying avoided curve crossings are uniformly poorer as compared with the UGA‐SSMRPT2 results. The striking aspect of the size‐consistency check is the complete insensitivity of the sum of fragment energies with given fragment spin‐multiplicities, which are obtained as the asymptotic limit of super‐molecules with different coupled spins. © 2015 Wiley Periodicals, Inc.     

Construction and applications of symmetrized valence bond wave functions

A method constructing symmetry-adapted bonded Young tableau bases is proposed, based on the symmetry properties of bonded tableaus and the projection operator associated with a point group. Several examples including the ground states and π excited states of O₃⁻, O₃, O₃⁺, and C₃⁻ are shown for instruction to construct the symmetrized valence bond (VB) wave function. Excitation energies of transitions from the ground states to π excited states of O₃⁻, C₃H₅, and C₃⁻ are calculated with an optimized symmetrized valence bond wave function in the σ–π separation approximation. Good agreement between the VB and experimental excitation energies is observed. The bonding features of the ground state and the first π excited singlet and triplet states for S₃ are discussed according to bonding populations from VB calculations. Both the singlet-biradical and the dipole structures have significant contributions to the ground state X ¹A₁ of S₃, while the excited state 1 ¹B₂ is essentially composed of the dipole structures, and the 1 ³B₂ excited state is comprised from a triplet-biradical structure. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 66 : 1–7, 1998     

Stereochemistry of optically active compounds. Problems and prospects

The review is concerned with the fundamental ideas and concepts of chiral stereochemistry, i.e., of the stereochemistry dealing with optically active compounds, from the asymmetric synthesis to the basics of mathematics, including characterization of the principal results and the current state of this branch of science.