DOCSLIB.ORG

Symmetry Operations, Symmetry Elements, and Point Groups

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Symmetry Operations, Symmetry Elements, and Point Groups 1.3 Summary of Symmetry Operations, Symmetry Elements, and Point Groups. Rotation axis. A rotation by 360˚/n that brings a three-dimensional body into an ^ equivalent configuration comprises a Cn symmetry operation. If this operation is performed a ^ ^ ^ 2 second time, the product CnCn equals a rotation by 2(360˚/n), which may be written as Cn . If n ^ ^ m is even, n/2 is integral and the rotation reduces to Cn/2. In general, a Cn operation is reduced ^ 6 ^ 2 by dividing m and n by their least common divisor (e.g., C9 = C3 ). Continued rotation by 360˚/n generates the set of operations: ^ ^ 2 ^ 3 ^ 4 ... ^ n Cn, Cn , Cn , Cn , Cn ^ n ^ ^ n+m ^ m where Cn = rotation by a full 360˚ = E the identity. Therefore Cn = Cn . Operations resulting from a Cn symmetry axis comprise a group that is isomorphic to the cyclic group of order n. If a molecule contains no other symmetry elements than Cn, this set constitutes the symmetry point group for that molecule and the group specified is denoted Cn. When additional symmetry elements are present, Cn forms a proper subgroup of the complete symmetry point group. Molecules that possess only a Cn symmetry element are rare, an example being + Co(NH2CH2CH2NH2)2Cl2 , which possesses a sole C2 symmetry element. N Cl N Co Cl N N ^ One special type of Cn operation exists only for linear molecules (e.g., HCl). Rotation by any angle around the internuclear axis defines a symmetry operation. This element is called ^ f C• axis and an infinite number of operations C• are associated with the element where f denotes rotation in decimal degrees. We already saw that molecules may contain more than one rotation axis. In BF3, depicted below, a three-fold axis emerges from the plane of the paper intersecting the center of the 52/bs[[1[1]dgr [01end66 61006200 2 2811I 62006200 980112023 169640 11205692 6580 80 DSt 620020572] 620] 20[1 1696] Ichemdict 5692 572 begin DSt Db SP %userdict/chemdictL/gr/grestoreL/tr/transformxl0pplSARA}{6-1st}b/OrA{py-8dppycpnpgsfilleo5awyInx w 02 dp/cYcwpy1-1gbWChemDrawCopyRight dpp cmfillpA sc8DA}{cwgr-1pp}{sqrt0m2p2oix dv1p l 2 -9.6r lt{1-2mv m p2px scsmv180 cmHA}{dL0gr}{pp}{gs a}b/PT{8-1dp exec}{aldvsms{dp dvm gmvdv dprO dv-.6 x WIpy -1 exec0sc smnegb2 lp 12stCAnp wF5 0pn mvpy0 at bdp LB 1.2scpxneg grm a}{ex1986, pmvest 2.251-9.6Laser plOA}{1emarcn 2L/gs/gsavep clip}b/Ct{bspp naLneg}if/pypexlcpgssl0L/xl/translatenpOBatmvgr 5 pmst}b/HA{lW lwy12 0op mv 1 3.375px-1 0 fillmvn/ex81ro gsnSA16.8 90lppy npl1451st}{0 accounttomark{bs 1987,Prep OB/bLwxaR bWixsg -1dpbd rpyscpx ac gr0.3 DA}{dL0.6pxro rad8 dv1 aL fillatsc dictxm10cv0pn 0SAend}b/Db{bs{dpdp eq{DD}{DS}ie DLBx py 1negey bd1.2ne{bWlpm mtCambridge80p py120L/ie/ifelsegr -9.6 xdp px rotDA}{cw scm l}forputm/w-1mvrad rO n/ey 180 bWp 1.5cm2-1 ro p eL/S{sfpy 0b10 CAg0 dv1ac cptl 021.6chemdict180 tr/dy0lt{-1 p -10st2bs sm0-.6 SAaAdpacx a}ie}b/WW{gsa}ie}b/BW{wD pyx px120lOA}{1ne{bW dv}{bd}ie p 5npgr}{gs xl cpscdx rote 8 pyDA}{dL smm}b/dA{[3st}{Asc6 p2.23m-8 py10L/ix/indexg Scientificx/dx sc50180 0DLB-1 cml-1n/dx alp scg360 sl2 px-11624.6 0px p-1negsqrttype[]type gs2 4 12rbegin/versionac2.25r neg}if/px smp arcmv gs rpy s0dv/bds xgi0 1 scLBarcwFmv o x20 OB180e31 -4.80dp rO SAst}{00.5 al pCB0.60dy eq{DB}{DS}iepy90 gs neg1SAac wDlWComputing,mS]}b/dL{dA begings w l cw1 pprev{neg}ifac L/l/linetoDA}{2.25 r pxn/dyCB 6lp 0.5sgCA x}ifcW1622.2DA}{cwsc2 25.8e1 rObegin eeq{dp -1px-2 o m xpydvs sg0 1fill xl}{xlnp 0OA}{1 acXsgbsnp enp[{pyp prdpsetgray0dvx lp161wy fill cv0 gr lmvsbW23 l ne0filllxe 0o cmbs rlinetoaccp0arc5p SA-1.61 g/wb0wxdpgrsc}b/Ov{OrAcmpp L/mt/matrix 0n lypp}{2def/b{bind-1 0gopInc.mlgr 0 e1 rOpxcm sm16 pw gsOA}{1.5cmppycY2mv 0ne{bW pp}ifelseatg acsmslclippathgs sc mvl 2 l27 eq{gsaclp 3 2 div1smcmx stne pxmt1.5WIcp sm cmm14 270cm lW st}{Asc4 es bs0bL 1.6gr}b/OB{/bS 0gi or{4p duppywfill}b/SA{aFgipst}{px0rosm 0 SAst2smm psme bdldp SAall pAA}{1 0al 27dv/bd trrOcpdef}bind g/bblcp1grfill put st1 DA}{270mvlL/mv/movetopp 2py01st}]e st4ppOA}{11 wy1 radgs 0.4 fill4.8acZLBgr cmSlW0r mg S grrO 0DT}]opx 4lt{pp o-1 pa/pysetgray xpx1dv cptend w2setdash}d/cR x}ifdv39sc sm}b/CB{np[{[{CS}{CS}{cB}{CW}{pp}{cB}{CW}{pp}{cB}]o 125.8 onpgsapac wb0eq{dL}if SA dvsc-1pm xr}ifdpdef/L{loadxl py0round CAs pxe 0.5cp12-0.4mvl gr}b/In{px0x 3 lW18040ag OA}{14atsc rad pxcvsc 1S}if/lppp wxgix pyDA}{180lWg/cX1execst}{1.0 Bd-1 1sgro}ie}b/AA{npm/aL nHZLBg 8cm 30 L/m/mul-0.4 ppwAA}{1 0lp6a/px2-2 exec}b/CS{p gr}b/wDr24.6fillix360 rO 1-1n p4o3 accurrentmatrixxgsgr}ie}b/Cr{0 o/cXx dv xls{nH papgiodef}b/d/def-1x12 gracAc}{0.5wbsc sclp pxdxx 1 larclt{e}if-1aLCA9.63 pal lw cmpxrLB gsd/wFsg0 2.25 5-1x emv pyne{pys/wx ppdv1scL/n/neg2 cmpDA}{12018 21.6g/cYn dp/cYgscX0fill sm aR mr mv2 pp120dprOeDS Ac}{1.0210rad sdictSA sm 1.5mv1cYdv grnws 0xst}{0 3 dy cvi/nSqdp360bd09.6sgx gr}{dp m0 n OA}{1 s/wyrOAA}{10xl 1L/a/addcmpx gsxppp 0d/aFd/WI{wx bb 0 spL/np/newpath npfill rlinetoradmrCA o lnp p16.8 ac 0pxpnpsglp elOv}{0.5cwsmlW 4iX wFl w pgr0 apsx DA}{90 0px 180a10-1 gfill dxarc xeqslmv2oplW bs st gopy0cmmcv cX mv bLL/al/aload ommvpy px4.0scd/aR gr s3lpdx0 gro cW st}b/DS{npe 180opx 9.6pat eq dv/m1Ov}{Ascsmdpeq{2 nSqpyiY11.5 p90g aclWst}b/Asc{OrAo radgs lne{wysetmiterlimitCA ro2 py0mv2lpbb eqp 12 0.25rOn 6st}{Ascpx AA}]e2 m 0.3pSA4 2dp mv0pyDA}{3 L/pp/pop4pp -1 dv 5and{ppIn apac8gi ma py pgixl l OA}{1 gdym 1p LB sl8ncpd/aAL/at/atan cmpr aall pydv cpl alcYg2 sclapgs 4.8p erOlppsp sLB gspxppRA}{3 st}b/Ar1{gs execdvlxarc ppm1mvdp cmsml}fordp wx0eq cm ppsl}b/cB{12a -1pp}b/DT{gs gs45 14.8 ex1lp4 0px 1np}b DT}{pp}ie}ie}{pp}ie}forall}b/I{counttomark scpyL/r/rollmgs psmst}{gsand{bb pxsgdx 27-1 12cp}{psmg1gr}b/ac{arctosc scmd/nH HA}{1sl pynSq{.135a cv lWcmL/cp/closepath 01sgst}b/DD{gs filllym sst}{rO dvr1 sqrt0 end st smps cp}b/ZLB{LB}b/Ar{dpdv/m1lp 3 ey2 SAs/b1 fill360 bWgrsm l}ie}repeatg dpmv gr}b/CA{aF6-1dv25L/ro/rotateac-1 sste OBdp OA}{270cmgr d/o{1 g1 0py stsc}b/LB{9pnp 2rpxarca scgr}{pp}{DB}{gs x}if bbr ne{bW}{5 s/lxbSgs0l 4rO gr}b/RA{lWx.667 smp 3ex PT 2700py0 gigs wy dL3 mv4{pp}repeat}b/pAcmdvixlx aix}b/rot{3p p st}{Ascn/excmx g}{bbalpx 0.5ppp 3 l.865m10DS mvCAdp CAs/lyrO smL/cv/curvetolWpx ppl ixL/sc/scalene{ly lp-6sm rOst}b/SP{gs/sf gr}b/DB{gssg lW OA}{180rOwx0m/aLDA}{1 x xlpyst 5 s4 x mv p.667lW LBstey m/wfill 2m}iepy-1 3ggrxl 0pmlx 12l -1 39mgr}b/Bd{[{pp}{[{DS}{DD}{gs -1bb -2n/eygs cmbW21 l0lp grnprxdv/m2 s/b1-1 px OB1lt{pp exec}{alr}b/x{epy slxlt{Ar1}{gsaL r0.5 dv5ac -10cmsmCA 2L/sg/setgray-0.40 sscnp lp0g}ie x L/cw/currentlinewidthnppx 1 dv12rcurvetocma3 32x}if 0 sgsm dx180OA}{120cwslW x/lWaRradpy27 1x 0py pwF py-0.4OB cY d/rO{4 smfill d}b/cmrly lxn/dx2 bWlWppstsmvlW -8 ew rOCA pm m}if/bds/ly mey bW5exx/bW gr st}{pys 2 2p l dp270 dppx 1oac.135 2o DA}{1cms dv dvcmmvne{lyx ZLBCAne{bW}{bd 0roxlW dvom2 scL/sl/setlinewidth0py1 pxdy mto ne{bW}{2wyocX aatsmx/cW dpsm wpcvgt{2OA}{90 m}b/Ac{0rado-.667gs LBo F F B F currentpoint 192837465 triangular projection. Three C2 axes containing each B-F bond lie in the plane of the molecule perpendicular to the three-fold axis. The rotation axis of highest order (i.e., C3) is called the principal axis of rotation. When the principal Cn axis has n even, then it contains a C2 operation associated with this axis. Perpendicular C2 axes and their associated operations must be denoted with prime and double prime superscripts. Reflection planes. Mirror planes or planes of reflection are symmetry elements whose associated operation, reflection in the plane, inverts the projection of an object normal to the mirror plane. That is, reflection in the xy plane carries out the transformation (x,y,z) ∅ (x,y,-z). Mirror planes are denoted by the symbol s and given the subscripts v, d, and h according to the following prescription. Planes of reflection that are perpendicular to a principal rotation axis of even or odd order are named sh (e.g., the plane containing the B and 3F atoms in BF3). Mirror planes that contain a principal rotation axis are called vertical planes and designated sv. For example, in BF3 there are three sv planes, each of which contains the boron atom, fluorine atom, and is perpendicular to the molecular plane.
Load more

Recommended publications
  • In Order for the Figure to Map Onto Itself, the Line of Reflection Must Go Through the Center Point
    3-5 Symmetry State whether the figure appears to have line symmetry. Write yes or no. If so, copy the figure, draw all lines of symmetry, and state their number. 1. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. The figure has reflectional symmetry. In order for the figure to map onto itself, the line of reflection must go through the center point. Two lines of reflection go through the sides of the figure. Two lines of reflection go through the vertices of the figure. Thus, there are four possible lines that go through the center and are lines of reflections. Therefore, the figure has four lines of symmetry. eSolutionsANSWER: Manual - Powered by Cognero Page 1 yes; 4 2. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. The given figure does not have reflectional symmetry. There is no way to fold or reflect it onto itself. ANSWER: no 3. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. The given figure has reflectional symmetry. The figure has a vertical line of symmetry. It does not have a horizontal line of symmetry. The figure does not have a line of symmetry through the vertices. Thus, the figure has only one line of symmetry. ANSWER: yes; 1 State whether the figure has rotational symmetry. Write yes or no. If so, copy the figure, locate the center of symmetry, and state the order and magnitude of symmetry.
    [Show full text]
  • Proofs with Perpendicular Lines
    3.4 Proofs with Perpendicular Lines EEssentialssential QQuestionuestion What conjectures can you make about perpendicular lines? Writing Conjectures Work with a partner. Fold a piece of paper D in half twice. Label points on the two creases, as shown. a. Write a conjecture about AB— and CD — . Justify your conjecture. b. Write a conjecture about AO— and OB — . AOB Justify your conjecture. C Exploring a Segment Bisector Work with a partner. Fold and crease a piece A of paper, as shown. Label the ends of the crease as A and B. a. Fold the paper again so that point A coincides with point B. Crease the paper on that fold. b. Unfold the paper and examine the four angles formed by the two creases. What can you conclude about the four angles? B Writing a Conjecture CONSTRUCTING Work with a partner. VIABLE a. Draw AB — , as shown. A ARGUMENTS b. Draw an arc with center A on each To be prof cient in math, side of AB — . Using the same compass you need to make setting, draw an arc with center B conjectures and build a on each side of AB— . Label the C O D logical progression of intersections of the arcs C and D. statements to explore the c. Draw CD — . Label its intersection truth of your conjectures. — with AB as O. Write a conjecture B about the resulting diagram. Justify your conjecture. CCommunicateommunicate YourYour AnswerAnswer 4. What conjectures can you make about perpendicular lines? 5. In Exploration 3, f nd AO and OB when AB = 4 units.
    [Show full text]
  • Symmetry and Tensors Rotations and Tensors
    Symmetry and tensors Rotations and tensors A rotation of a 3-vector is accomplished by an orthogonal transformation. Represented as a matrix, A, we replace each vector, v, by a rotated vector, v0, given by multiplying by A, 0 v = Av In index notation, 0 X vm = Amnvn n Since a rotation must preserve lengths of vectors, we require 02 X 0 0 X 2 v = vmvm = vmvm = v m m Therefore, X X 0 0 vmvm = vmvm m m ! ! X X X = Amnvn Amkvk m n k ! X X = AmnAmk vnvk k;n m Since xn is arbitrary, this is true if and only if X AmnAmk = δnk m t which we can rewrite using the transpose, Amn = Anm, as X t AnmAmk = δnk m In matrix notation, this is t A A = I where I is the identity matrix. This is equivalent to At = A−1. Multi-index objects such as matrices, Mmn, or the Levi-Civita tensor, "ijk, have definite transformation properties under rotations. We call an object a (rotational) tensor if each index transforms in the same way as a vector. An object with no indices, that is, a function, does not transform at all and is called a scalar. 0 A matrix Mmn is a (second rank) tensor if and only if, when we rotate vectors v to v , its new components are given by 0 X Mmn = AmjAnkMjk jk This is what we expect if we imagine Mmn to be built out of vectors as Mmn = umvn, for example. In the same way, we see that the Levi-Civita tensor transforms as 0 X "ijk = AilAjmAkn"lmn lmn 1 Recall that "ijk, because it is totally antisymmetric, is completely determined by only one of its components, say, "123.
    [Show full text]
  • 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.
    [Show full text]
  • Properties of the Single and Double D4d Groups and Their Isomorphisms with D8 and C8v Groups A
    Properties of the single and double D4d groups and their isomorphisms with D8 and C8v groups A. Le Paillier-Malécot, L. Couture To cite this version: A. Le Paillier-Malécot, L. Couture. Properties of the single and double D4d groups and their isomorphisms with D8 and C8v groups. Journal de Physique, 1981, 42 (11), pp.1545-1552. 10.1051/jphys:0198100420110154500. jpa-00209347 HAL Id: jpa-00209347 https://hal.archives-ouvertes.fr/jpa-00209347 Submitted on 1 Jan 1981 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. J. Physique 42 (1981) 1545-1552 NOVEMBRE 1981, 1545 Classification Physics Abstracts 75.10D Properties of the single and double D4d groups and their isomorphisms with D8 and C8v groups A. Le Paillier-Malécot Laboratoire de Magnétisme et d’Optique des Solides, 1, place A.-Briand, 92190 Meudon Bellevue, France and Université de Paris-Sud XI, Centre d’Orsay, 91405 Orsay Cedex, France and L. Couture Laboratoire Aimé-Cotton (*), C.N.R.S., Campus d’Orsay, 91405 Orsay Cedex, France and Laboratoire d’Optique et de Spectroscopie Cristalline, Université Pierre-et-Marie-Curie, Paris VI, 75230 Paris Cedex 05, France (Reçu le 15 mai 1981, accepté le 9 juillet 1981) Résumé.
    [Show full text]
  • 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 .
    [Show full text]
  • 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].
    [Show full text]
  • 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.
    [Show full text]
  • COXETER GROUPS (Unfinished and Comments Are Welcome)
    COXETER GROUPS (Unfinished and comments are welcome) Gert Heckman Radboud University Nijmegen [email protected] October 10, 2018 1 2 Contents Preface 4 1 Regular Polytopes 7 1.1 ConvexSets............................ 7 1.2 Examples of Regular Polytopes . 12 1.3 Classification of Regular Polytopes . 16 2 Finite Reflection Groups 21 2.1 NormalizedRootSystems . 21 2.2 The Dihedral Normalized Root System . 24 2.3 TheBasisofSimpleRoots. 25 2.4 The Classification of Elliptic Coxeter Diagrams . 27 2.5 TheCoxeterElement. 35 2.6 A Dihedral Subgroup of W ................... 39 2.7 IntegralRootSystems . 42 2.8 The Poincar´eDodecahedral Space . 46 3 Invariant Theory for Reflection Groups 53 3.1 Polynomial Invariant Theory . 53 3.2 TheChevalleyTheorem . 56 3.3 Exponential Invariant Theory . 60 4 Coxeter Groups 65 4.1 Generators and Relations . 65 4.2 TheTitsTheorem ........................ 69 4.3 The Dual Geometric Representation . 74 4.4 The Classification of Some Coxeter Diagrams . 77 4.5 AffineReflectionGroups. 86 4.6 Crystallography. .. .. .. .. .. .. .. 92 5 Hyperbolic Reflection Groups 97 5.1 HyperbolicSpace......................... 97 5.2 Hyperbolic Coxeter Groups . 100 5.3 Examples of Hyperbolic Coxeter Diagrams . 108 5.4 Hyperbolic reflection groups . 114 5.5 Lorentzian Lattices . 116 3 6 The Leech Lattice 125 6.1 ModularForms ..........................125 6.2 ATheoremofVenkov . 129 6.3 The Classification of Niemeier Lattices . 132 6.4 The Existence of the Leech Lattice . 133 6.5 ATheoremofConway . 135 6.6 TheCoveringRadiusofΛ . 137 6.7 Uniqueness of the Leech Lattice . 140 4 Preface Finite reflection groups are a central subject in mathematics with a long and rich history. The group of symmetries of a regular m-gon in the plane, that is the convex hull in the complex plane of the mth roots of unity, is the dihedral group of order 2m, which is the simplest example of a reflection Dm group.
    [Show full text]
  • 20. Geometry of the Circle (SC)
    20. GEOMETRY OF THE CIRCLE PARTS OF THE CIRCLE Segments When we speak of a circle we may be referring to the plane figure itself or the boundary of the shape, called the circumference. In solving problems involving the circle, we must be familiar with several theorems. In order to understand these theorems, we review the names given to parts of a circle. Diameter and chord The region that is encompassed between an arc and a chord is called a segment. The region between the chord and the minor arc is called the minor segment. The region between the chord and the major arc is called the major segment. If the chord is a diameter, then both segments are equal and are called semi-circles. The straight line joining any two points on the circle is called a chord. Sectors A diameter is a chord that passes through the center of the circle. It is, therefore, the longest possible chord of a circle. In the diagram, O is the center of the circle, AB is a diameter and PQ is also a chord. Arcs The region that is enclosed by any two radii and an arc is called a sector. If the region is bounded by the two radii and a minor arc, then it is called the minor sector. www.faspassmaths.comIf the region is bounded by two radii and the major arc, it is called the major sector. An arc of a circle is the part of the circumference of the circle that is cut off by a chord.
    [Show full text]
  • 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).
    [Show full text]
  • Geometry Course Outline
    GEOMETRY COURSE OUTLINE Content Area Formative Assessment # of Lessons Days G0 INTRO AND CONSTRUCTION 12 G-CO Congruence 12, 13 G1 BASIC DEFINITIONS AND RIGID MOTION Representing and 20 G-CO Congruence 1, 2, 3, 4, 5, 6, 7, 8 Combining Transformations Analyzing Congruency Proofs G2 GEOMETRIC RELATIONSHIPS AND PROPERTIES Evaluating Statements 15 G-CO Congruence 9, 10, 11 About Length and Area G-C Circles 3 Inscribing and Circumscribing Right Triangles G3 SIMILARITY Geometry Problems: 20 G-SRT Similarity, Right Triangles, and Trigonometry 1, 2, 3, Circles and Triangles 4, 5 Proofs of the Pythagorean Theorem M1 GEOMETRIC MODELING 1 Solving Geometry 7 G-MG Modeling with Geometry 1, 2, 3 Problems: Floodlights G4 COORDINATE GEOMETRY Finding Equations of 15 G-GPE Expressing Geometric Properties with Equations 4, 5, Parallel and 6, 7 Perpendicular Lines G5 CIRCLES AND CONICS Equations of Circles 1 15 G-C Circles 1, 2, 5 Equations of Circles 2 G-GPE Expressing Geometric Properties with Equations 1, 2 Sectors of Circles G6 GEOMETRIC MEASUREMENTS AND DIMENSIONS Evaluating Statements 15 G-GMD 1, 3, 4 About Enlargements (2D & 3D) 2D Representations of 3D Objects G7 TRIONOMETRIC RATIOS Calculating Volumes of 15 G-SRT Similarity, Right Triangles, and Trigonometry 6, 7, 8 Compound Objects M2 GEOMETRIC MODELING 2 Modeling: Rolling Cups 10 G-MG Modeling with Geometry 1, 2, 3 TOTAL: 144 HIGH SCHOOL OVERVIEW Algebra 1 Geometry Algebra 2 A0 Introduction G0 Introduction and A0 Introduction Construction A1 Modeling With Functions G1 Basic Definitions and Rigid
    [Show full text]