An Introduction to Molecular Symmetry (A) E Is the Identity Operator. It
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Symmetry of Fulleroids Stanislav Jendrol’, František Kardoš
Symmetry of Fulleroids Stanislav Jendrol’, František Kardoš To cite this version: Stanislav Jendrol’, František Kardoš. Symmetry of Fulleroids. Klaus D. Sattler. Handbook of Nanophysics: Clusters and Fullerenes, CRC Press, pp.28-1 - 28-13, 2010, 9781420075557. hal- 00966781 HAL Id: hal-00966781 https://hal.archives-ouvertes.fr/hal-00966781 Submitted on 27 Mar 2014 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Symmetry of Fulleroids Stanislav Jendrol’ and Frantiˇsek Kardoˇs Institute of Mathematics, Faculty of Science, P. J. Saf´arikˇ University, Koˇsice Jesenn´a5, 041 54 Koˇsice, Slovakia +421 908 175 621; +421 904 321 185 [email protected]; [email protected] Contents 1 Symmetry of Fulleroids 2 1.1 Convexpolyhedraandplanargraphs . .... 3 1.2 Polyhedral symmetries and graph automorphisms . ........ 4 1.3 Pointsymmetrygroups. 5 1.4 Localrestrictions ............................... .. 10 1.5 Symmetryoffullerenes . .. 11 1.6 Icosahedralfulleroids . .... 12 1.7 Subgroups of Ih ................................. 15 1.8 Fulleroids with multi-pentagonal faces . ......... 16 1.9 Fulleroids with octahedral, prismatic, or pyramidal symmetry ........ 19 1.10 (5, 7)-fulleroids .................................. 20 1 Chapter 1 Symmetry of Fulleroids One of the important features of the structure of molecules is the presence of symmetry elements. -
Origin of Icosahedral Symmetry in Viruses
Origin of icosahedral symmetry in viruses Roya Zandi†‡, David Reguera†, Robijn F. Bruinsma§, William M. Gelbart†, and Joseph Rudnick§ Departments of †Chemistry and Biochemistry and §Physics and Astronomy, University of California, Los Angeles, CA 90095-1569 Communicated by Howard Reiss, University of California, Los Angeles, CA, August 11, 2004 (received for review February 2, 2004) With few exceptions, the shells (capsids) of sphere-like viruses (11). This problem is in turn related to one posed much earlier have the symmetry of an icosahedron and are composed of coat by Thomson (12), involving the optimal distribution of N proteins (subunits) assembled in special motifs, the T-number mutually repelling points on the surface of a sphere, and to the structures. Although the synthesis of artificial protein cages is a subsequently posed ‘‘covering’’ problem, where one deter- rapidly developing area of materials science, the design criteria for mines the arrangement of N overlapping disks of minimum self-assembled shells that can reproduce the remarkable properties diameter that allow complete coverage of the sphere surface of viral capsids are only beginning to be understood. We present (13). In a similar spirit, a self-assembly phase diagram has been here a minimal model for equilibrium capsid structure, introducing formulated for adhering hard disks (14). These models, which an explicit interaction between protein multimers (capsomers). all deal with identical capsomers, regularly produced capsid Using Monte Carlo simulation we show that the model reproduces symmetries lower than icosahedral. On the other hand, theo- the main structures of viruses in vivo (T-number icosahedra) and retical studies of viral assembly that assume icosahedral sym- important nonicosahedral structures (with octahedral and cubic metry are able to reproduce the CK motifs (15), and, most symmetry) observed in vitro. -
Molecular Symmetry, Super-Rotation, and Semi-Classical Motion –
Molecular symmetry, super-rotation, and semi-classical motion – New ideas for old problems Inaugural - Dissertation zur Erlangung des Doktorgrades der Mathematisch-Naturwissenschaftlichen Fakultät der Universität zu Köln vorgelegt von Hanno Schmiedt aus Köln Berichterstatter: Prof. Dr. Stephan Schlemmer Prof. Per Jensen, Ph.D. Tag der mündlichen Prüfung: 24. Januar 2017 Awesome molecules need awesome theories DR.SANDRA BRUENKEN, 2016 − Abstract Traditional molecular spectroscopy is used to characterize molecules by their structural and dynamical properties. Furthermore, modern experimental methods are capable of determining state-dependent chemical reaction rates. The understanding of both inter- and intra-molecular dynamics contributes exceptionally, for example, to the search for molecules in interstellar space, where observed spectra and reaction rates help in under- standing the various phases of stellar or planetary evolution. Customary theoretical models for molecular dynamics are based on a few fundamental assumptions like the commonly known ball-and-stick picture of molecular structure. In this work we discuss two examples of the limits of conventional molecular theory: Extremely floppy molecules where no equilibrium geometric structure is definable and molecules exhibiting highly excited rotational states, where the large angular momentum poses considerable challenges to quantum chemical calculations. In both cases, we develop new concepts based on fundamental symmetry considerations to overcome the respective limits of quantum chemistry. For extremely floppy molecules where numerous large-amplitude motions render a defini- tion of a fixed geometric structure impossible, we establish a fundamentally new zero-order description. The ‘super-rotor model’ is based on a five-dimensional rigid rotor treatment depending only on a single adjustable parameter. -
Molecular Symmetry
Molecular Symmetry Symmetry helps us understand molecular structure, some chemical properties, and characteristics of physical properties (spectroscopy) – used with group theory to predict vibrational spectra for the identification of molecular shape, and as a tool for understanding electronic structure and bonding. Symmetrical : implies the species possesses a number of indistinguishable configurations. 1 Group Theory : mathematical treatment of symmetry. symmetry operation – an operation performed on an object which leaves it in a configuration that is indistinguishable from, and superimposable on, the original configuration. symmetry elements – the points, lines, or planes to which a symmetry operation is carried out. Element Operation Symbol Identity Identity E Symmetry plane Reflection in the plane σ Inversion center Inversion of a point x,y,z to -x,-y,-z i Proper axis Rotation by (360/n)° Cn 1. Rotation by (360/n)° Improper axis S 2. Reflection in plane perpendicular to rotation axis n Proper axes of rotation (C n) Rotation with respect to a line (axis of rotation). •Cn is a rotation of (360/n)°. •C2 = 180° rotation, C 3 = 120° rotation, C 4 = 90° rotation, C 5 = 72° rotation, C 6 = 60° rotation… •Each rotation brings you to an indistinguishable state from the original. However, rotation by 90° about the same axis does not give back the identical molecule. XeF 4 is square planar. Therefore H 2O does NOT possess It has four different C 2 axes. a C 4 symmetry axis. A C 4 axis out of the page is called the principle axis because it has the largest n . By convention, the principle axis is in the z-direction 2 3 Reflection through a planes of symmetry (mirror plane) If reflection of all parts of a molecule through a plane produced an indistinguishable configuration, the symmetry element is called a mirror plane or plane of symmetry . -
The Cubic Groups
The Cubic Groups Baccalaureate Thesis in Electrical Engineering Author: Supervisor: Sana Zunic Dr. Wolfgang Herfort 0627758 Vienna University of Technology May 13, 2010 Contents 1 Concepts from Algebra 4 1.1 Groups . 4 1.2 Subgroups . 4 1.3 Actions . 5 2 Concepts from Crystallography 6 2.1 Space Groups and their Classification . 6 2.2 Motions in R3 ............................. 8 2.3 Cubic Lattices . 9 2.4 Space Groups with a Cubic Lattice . 10 3 The Octahedral Symmetry Groups 11 3.1 The Elements of O and Oh ..................... 11 3.2 A Presentation of Oh ......................... 14 3.3 The Subgroups of Oh ......................... 14 2 Abstract After introducing basics from (mathematical) crystallography we turn to the description of the octahedral symmetry groups { the symmetry group(s) of a cube. Preface The intention of this account is to provide a description of the octahedral sym- metry groups { symmetry group(s) of the cube. We first give the basic idea (without proofs) of mathematical crystallography, namely that the 219 space groups correspond to the 7 crystal systems. After this we come to describing cubic lattices { such ones that are built from \cubic cells". Finally, among the cubic lattices, we discuss briefly the ones on which O and Oh act. After this we provide lists of the elements and the subgroups of Oh. A presentation of Oh in terms of generators and relations { using the Dynkin diagram B3 is also given. It is our hope that this account is accessible to both { the mathematician and the engineer. The picture on the title page reflects Ha¨uy'sidea of crystal structure [4]. -
Systematics of Atomic Orbital Hybridization of Coordination Polyhedra: Role of F Orbitals
molecules Article Systematics of Atomic Orbital Hybridization of Coordination Polyhedra: Role of f Orbitals R. Bruce King Department of Chemistry, University of Georgia, Athens, GA 30602, USA; [email protected] Academic Editor: Vito Lippolis Received: 4 June 2020; Accepted: 29 June 2020; Published: 8 July 2020 Abstract: The combination of atomic orbitals to form hybrid orbitals of special symmetries can be related to the individual orbital polynomials. Using this approach, 8-orbital cubic hybridization can be shown to be sp3d3f requiring an f orbital, and 12-orbital hexagonal prismatic hybridization can be shown to be sp3d5f2g requiring a g orbital. The twists to convert a cube to a square antiprism and a hexagonal prism to a hexagonal antiprism eliminate the need for the highest nodality orbitals in the resulting hybrids. A trigonal twist of an Oh octahedron into a D3h trigonal prism can involve a gradual change of the pair of d orbitals in the corresponding sp3d2 hybrids. A similar trigonal twist of an Oh cuboctahedron into a D3h anticuboctahedron can likewise involve a gradual change in the three f orbitals in the corresponding sp3d5f3 hybrids. Keywords: coordination polyhedra; hybridization; atomic orbitals; f-block elements 1. Introduction In a series of papers in the 1990s, the author focused on the most favorable coordination polyhedra for sp3dn hybrids, such as those found in transition metal complexes. Such studies included an investigation of distortions from ideal symmetries in relatively symmetrical systems with molecular orbital degeneracies [1] In the ensuing quarter century, interest in actinide chemistry has generated an increasing interest in the involvement of f orbitals in coordination chemistry [2–7]. -
Chapter 1 – Symmetry of Molecules – P. 1
Chapter 1 – Symmetry of Molecules – p. 1 - 1. Symmetry of Molecules 1.1 Symmetry Elements · Symmetry operation: Operation that transforms a molecule to an equivalent position and orientation, i.e. after the operation every point of the molecule is coincident with an equivalent point. · Symmetry element: Geometrical entity (line, plane or point) which respect to which one or more symmetry operations can be carried out. In molecules there are only four types of symmetry elements or operations: · Mirror planes: reflection with respect to plane; notation: s · Center of inversion: inversion of all atom positions with respect to inversion center, notation i · Proper axis: Rotation by 2p/n with respect to the axis, notation Cn · Improper axis: Rotation by 2p/n with respect to the axis, followed by reflection with respect to plane, perpendicular to axis, notation Sn Formally, this classification can be further simplified by expressing the inversion i as an improper rotation S2 and the reflection s as an improper rotation S1. Thus, the only symmetry elements in molecules are Cn and Sn. Important: Successive execution of two symmetry operation corresponds to another symmetry operation of the molecule. In order to make this statement a general rule, we require one more symmetry operation, the identity E. (1.1: Symmetry elements in CH4, successive execution of symmetry operations) 1.2. Systematic classification by symmetry groups According to their inherent symmetry elements, molecules can be classified systematically in so called symmetry groups. We use the so-called Schönfliess notation to name the groups, Chapter 1 – Symmetry of Molecules – p. 2 - which is the usual notation for molecules. -
Can Every Face of a Polyhedron Have Many Sides ?
Can Every Face of a Polyhedron Have Many Sides ? Branko Grünbaum Dedicated to Joe Malkevitch, an old friend and colleague, who was always partial to polyhedra Abstract. The simple question of the title has many different answers, depending on the kinds of faces we are willing to consider, on the types of polyhedra we admit, and on the symmetries we require. Known results and open problems about this topic are presented. The main classes of objects considered here are the following, listed in increasing generality: Faces: convex n-gons, starshaped n-gons, simple n-gons –– for n ≥ 3. Polyhedra (in Euclidean 3-dimensional space): convex polyhedra, starshaped polyhedra, acoptic polyhedra, polyhedra with selfintersections. Symmetry properties of polyhedra P: Isohedron –– all faces of P in one orbit under the group of symmetries of P; monohedron –– all faces of P are mutually congru- ent; ekahedron –– all faces have of P the same number of sides (eka –– Sanskrit for "one"). If the number of sides is k, we shall use (k)-isohedron, (k)-monohedron, and (k)- ekahedron, as appropriate. We shall first describe the results that either can be found in the literature, or ob- tained by slight modifications of these. Then we shall show how two systematic ap- proaches can be used to obtain results that are better –– although in some cases less visu- ally attractive than the old ones. There are many possible combinations of these classes of faces, polyhedra and symmetries, but considerable reductions in their number are possible; we start with one of these, which is well known even if it is hard to give specific references for precisely the assertion of Theorem 1. -
On Polyhedra with Transitivity Properties
Discrete Comput Geom 1:195-199 (1986) iSCrete & Computathm~ ~) 1986 Springer-V©rlagometrvNew York Inc. IF On Polyhedra with Transitivity Properties J. M. Wills Mathematics Institute, University of Siegen, Hoelderlinstrasse 3, D-5900 Siegen, FRG Abstract. In a remarkable paper [4] Griinbaum and Shephard stated that there is no polyhedron of genus g > 0, such that its symmetry group acts transitively on its faces. If this condition is slightly weakened, one obtains" some interesting polyhedra with face-transitivity (resp. vertex-transitivity). In recent years several attempts have been/made to find polyhedral analogues for the Platonic solids in the Euclidean 3-space E 3 (cf. [1, 4, 5, 7-14]). By a polyhedron P we mean the union of a finite number of plane polygons in E 3 which form a 2-manifold without boundary and without self-intersections, where, as usual, a plane polygon is a plane set bounded by finitely many line segments. If the plane polygons are disjoint except for their edges, they are called the faces of P. We require adjacent faces to be non-coplanar. The vertices and edges of P are the vertices and edges of the faces of P. The genus of the underlying 2-manifold is the genus of P. A flag of P is a triplet consisting of one vertex, one edge incident with this vertex, and one face containing this edge. If, for given p- 3, q >- 3 all faces of P are p-gons and all vertices q-valent, we say that P is equivelar, i.e., has equal flags (cf. -
Chapter 4 Symmetry and Chemical Bonding
Chapter 4 Symmetry and Chemical Bonding 4.1 Orbital Symmetries and Overlap 4.2 Valence Bond Theory and Hybrid Orbitals 4.3 Localized and Delocalized Molecular Orbitals 4.4 MXn Molecules with Pi-Bonding 4.5 Pi-Bonding in Aromatic Ring Systems 4.1 Orbital Symmetries and Overlap Bonded state can be represented by a Schrödinger wave equation of the general form H; hamiltonian operator Ψ; eigenfunction E; eigenvalue It is customary to construct approximate wave functions for the molecule from the atomic orbitals of the interacting atoms. By this approach, when two atomic orbitals overlap in such a way that their individual wave functions add constructively, the result is a buildup of electron density in the region around the two nuclei. 4.1 Orbital Symmetries and Overlap The association between the probability, P, of finding the electron at a point in space and the product of its wave function and its complex conjugate. It the probability of finding it over all points throughout space is unity. N; normalization constant 4.1 Orbital Symmetries and Overlap Slater overlap integral; the nature and effectiveness of their interactions S>0, bonding interaction, a reinforcement of the total wave function and a buildup of electron density around the two nuclei S<0, antibonding interaction, decrease of electron density in the region around the two nuclei S=0, nonbonding interaction, electron density is essentially the same as before 4.1 Orbital Symmetries and Overlap Ballon representations: these are rough representations of 90-99% of the probability distribution, which as the product of the wave function and its complex conjugate (or simply the square, if the function is real). -
Molecular Symmetry
Molecular Symmetry 1 I. WHAT IS SYMMETRY AND WHY IT IS IMPORTANT? Some object are ”more symmetrical” than others. A sphere is more symmetrical than a cube because it looks the same after rotation through any angle about the diameter. A cube looks the same only if it is rotated through certain angels about specific axes, such as 90o, 180o, or 270o about an axis passing through the centers of any of its opposite faces, or by 120o or 240o about an axis passing through any of the opposite corners. Here are also examples of different molecules which remain the same after certain symme- try operations: NH3, H2O, C6H6, CBrClF . In general, an action which leaves the object looking the same after a transformation is called a symmetry operation. Typical symme- try operations include rotations, reflections, and inversions. There is a corresponding symmetry element for each symmetry operation, which is the point, line, or plane with respect to which the symmetry operation is performed. For instance, a rotation is carried out around an axis, a reflection is carried out in a plane, while an inversion is carried our in a point. We shall see that we can classify molecules that possess the same set of symmetry ele- ments, and grouping together molecules that possess the same set of symmetry elements. This classification is very important, because it allows to make some general conclusions about molecular properties without calculation. Particularly, we will be able to decide if a molecule has a dipole moment, or not and to know in advance the degeneracy of molecular states. -
Symmetry in Chemistry
SYMMETRY IN CHEMISTRY Professor MANOJ K. MISHRA CHEMISTRY DEPARTMENT IIT BOMBAY ACKNOWLEGDEMENT: Professor David A. Micha Professor F. A. Cotton 1 An introduction to symmetry analysis WHY SYMMETRY ? Hψ = Eψ For H – atom: Each member of the CSCO labels, For molecules: Symmetry operation R CSCO H is INVARIANT under R ( by definition too) 2 An introduction to symmetry analysis Hψ = Eψ gives NH3 normal modes = NH3 rotation or translation MUST be A1, A2 or E ! NO ESCAPING SYMMETRY! 3 Molecular Symmetry An introduction to symmetry analysis (Ref.: Inorganic chemistry by Shirver, Atkins & Longford, ELBS) One aspect of the shape of a molecule is its symmetry (we define technical meaning of this term in a moment) and the systematic treatment and symmetry uses group theory. This is a rich and powerful subject, by will confine our use of it at this stage to classifying molecules and draw some general conclusions about their properties. An introduction to symmetry analysis Our initial aim is to define the symmetry of molecules much more precisely than we have done so far, and to provide a notational scheme that confirms their symmetry. In subsequent chapters we extend the material present here to applications in bonding and spectroscopy, and it will become that symmetry analysis is one of the most pervasive techniques in inorganic chemistry. Symmetry operations and elements A fundamental concept of group theory is the symmetry operation. It is an action, such as a rotation through a certain angle, that leave molecules apparently unchanged. An example is the rotation of H2O molecule by 180 ° (but not any smaller angle) around the bisector of HOH angle.