DOCSLIB.ORG

Molecular Symmetry

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

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 . σ v(vertical): plane colinear with principal axis Reflection through a planes of symmetry (mirror plane) σ σ d(dihedral) Vertical, parallel to principal axis , parallel to C n and bisecting two C2' axes 4 Reflection through a planes of symmetry (mirror plane) σ h(horizontal): plane perpendicular to principal axis Inversion, Center of Inversion (i) A center of symmetry: A point at the center of the molecule. (x,y,z) (-x,-y,-z). It is not necessary to have an atom in the center (e.g. benzene). Tetrahedrons, triangles, and pentagons don't have a center of inversion symmetry. C2H6 Ru(CO) 6 C4H4Cl 2F2 5 Rotation-reflection, Improper rotation(S n) ° This is a compound operation combining a rotation 360 /n (C n) with a σ σ reflection through a plane perpendicular to the C n axis h.(C n followed by h) σ Cn=S n ° An improper rotation (or rotation–reflection), Sn, involves rotation about 360 /n followed by reflection through a plane that is perpendicular to the rotation axis. 6 Identity (E) Simplest symmetry operation. All molecules have this element. If the molecule does have no other elements, it is asymmetric. The identity operation amounts to doing nothing to a molecule and so leaves any molecule completely unchanged. CHFClBr SOFCl Successive Operations 7 Symmetry Point Groups •Symmetry of a molecule located on symmetry axes, cut by planes of symmetry, or centered at an inversion center is known as point symmetry . •Collections of symmetry operations constitute mathematical groups . •Each symmetry point group has a particular designation. Cn, C nh , C nv Dn, D nh , D nd S2n C∞v ,D∞h Ih, I Td , Th ,T Oh,O C1,C i, C s 8 HCl F2 C D ∞∞∞v ∞∞∞h Linear molecular species can be classified according to whether they possess a centre of symmetry (inversion centre) or not. 9 Tetrahedral Geometry P4 B4Cl 4 Octahedral Geometry Icosahedral Geometry 2−−− [W(CO) 6] [B 12 H12 ] 10 Identify the symmetry elements that are present in benzene. Scheme for assigning point groups of molecules and molecular ions 11 Cn Point Groups n = Cn E PBrClF C1 H2O2 C2 As(C 6H5)3 C3 M(NH 2CH 2CO 2)4 C4 Cnh Point Groups The direction of the C n axis is take as vertical, so a symmetry σ plane perpendicular to it is a horizontal plane, h. HOCl NH 2F BBrClF The point group is called C s (C 1h ) B(OH) N2F2 3 C C2h 3h 12 Cnv Point Groups If a mirror plane contains the rotational axis, the group is called a C nv group. H2O SF 4 C2v C2v NF SF Cl 3 CHCl 3 5 C3v C3v C4v Dn and Dnh Point Groups Adding a C 2 axis perpendicular to a C n axis generates one of the dihedral groups. There must be n C2 axes perpendicular to Cn Angle between rings not 0 ° or 90° D2 D3 σ Adding a h to a D n group generates a D nh group. 2- C2H4 [PtCl 4] D2h D4h 13 BF 3 D3h Dnd Point Groups Adding a vertical mirror plane to a Dn group in such a fashion to bisect adjacent C 2 axes generates a Dnd group. D2d Ferrocene Fe(C 5H5)2 D5d 14 Sn groups n σ For odd n, (S n) = h n For even n, (S n) = E The S n for odd n is the same as the C nh . 1,3,5,7-tetrafluorocyclooctatetraene Absence of mirror planes distinguish S n groups from D nd groups. Linear Groups H-Cl O=O C∞v D∞h σ Has a h High Symmetry Molecules CCl 4 SF 6 C60 Td O h Ih 15 The letter ‘A’. tetrachloroplatinate(II) GeH 3F HOCl SF 6 H O SeH 3F B(OH) 3 B H O O AsBr 5 (trigonal bipyramid) H H BH 3 B trans rotamer of Si H 2 6 H H Ni(en) 3 Crown-shaped S 8 16 Characteristic symmetry elements of some important classes of point groups. 17 Character Tables •Character tables contain, in a highly symbolic form, information about how something of interest (an bond, an orbital, etc.) is affected by the operations of a given point group . •Each point group has a unique character table, which is organized into a matrix. •Column headings are the symmetry operations , which are grouped into classes . •Horizontal rows are called irreducible representations of the point group. •The main body consists of characters (numbers), and a section on the right side of the table provides information about vectors and atomic orbitals . σ σ C2v EC2 v(xz) v’ (yz) 2 2 2 A1 1 1 1 1 z x , y , z A2 1 1 -1 –1 Rz xy B1 1 -1 1 –1 x, R y xz B2 1 –1 –1 1 y, R x yz σ C3v E 2C 3 3 v 2 2 2 A1 1 1 1 z x + y , z A2 1 1 -1 Rz 2 2 E 2 -1 0 (x,y), (R x, R y) (x -y , xy)(xz, yz) •Symmetry elements possessed by the point group are in the top row •Left hand column gives a list of symmetry labels •Gives information about degeneracies (A and B indicate non- degenerate, E refers to doubly degenerate, T means triply degenerate) •Main part of table contains characters (numbers) to label the symmetry properties (of MO’s or modes of molecular vibrations) 18 To obtain from this total set the representations for vibration only, it is necessary to subtract the representations for the other two forms of motion: rotation and translation . z z O y z x z y H y H y x x x Cartesian displacement vectors for a water molecule Translational Modes Rotational Modes A mode in which A mode in which all atoms are atoms move to moving in the rotate (change same direction, the orientation equivalent to of) the molecule. moving the There are 3 molecule. rotational modes for nonlinear molecules, and 2 rotational modes for linear molecules. 19 Selection Rules: Infrared and Raman Spectroscopy Infrared energy is absorbed for certain changes in vibrational energy levels of a molecule. -for a vibration to be infrared active , there must be a change in the molecular dipole moment vector associated with the vibration. For a vibration mode to be Raman active , there must be a change in the net polarizability tensor -polarizability is the ease in which the electron cloud associated with the molecule is distorted For centrosymmetric molecules, the rule of mutual exclusion states that vibrations that are IR active are Raman inactive, and vice versa The transition from the vibrational ground state to the first excited state is the fundamental transition . The two bending modes require the same amount of energy and are therefore degenerate. 20 σ σ C2v EC2 v(xz) v’ (yz) 2 2 2 A1 1 1 1 1 z x , y , z A2 1 1 -1 –1 Rz xy B1 1 -1 1 –1 x, R y xz B2 1 –1 –1 1 y, R x yz If the symmetry label (e.g. A 1, B 1, E) of a normal mode of vibration is associated with x, y, or z in the character table, then the mode is IR active . If the symmetry label (e.g. A 1, B 1, E) of a normal mode of vibration is associated with a product term ( x2, xy ) in the character table, then the mode is Raman active . σ σ C2v EC2 v(xz) v’ (yz) 2 2 2 A1 1 1 1 1 z x , y , z A2 1 1 -1 –1 Rz xy B1 1 -1 1 –1 x, R y xz B2 1 –1 –1 1 y, R x yz 21 σΞ 3σΞ D 3h E 2C 3 3C 2 hh 2S 3 3 v v ′ 2 2 2 A1 1 1 1 1 1 1 x + y , z ′ A 2 1 1 -1 1 1 -1 R z ′ E 2 -1 0 2 -1 0 (x , y ) (x 2 – y 2, 2 xy ) ′′ A 1 1 1 1 -1 -1 -1 ′′ A 2 1 1 -1 -1 -1 1 z ′′ E 2 -1 0 -2 1 0 (R x , R y )(xy , yz ) 22 The vibrational modes of CH 4 (Td), only two of which are IR active.
Recommended publications
  • 11 Practice Lesso N 4 R O U N D W H Ole Nu M Bers U N It 1

    11 Practice Lesso N 4 R O U N D W H Ole Nu M Bers U N It 1

    CC04MM RPPSTG TEXT.indb 11 © Practice and Problem Solving Curriculum Associates, LLC Copying isnotpermitted. Practice Lesson 4 Round Whole Numbers Whole 4Round Lesson Practice Lesson 4 Round Whole Numbers Name: Solve. M 3 Round each number. Prerequisite: Round Three-Digit Numbers a. 689 rounded to the nearest ten is 690 . Study the example showing how to round a b. 68 rounded to the nearest hundred is 100 . three-digit number. Then solve problems 1–6. c. 945 rounded to the nearest ten is 950 . Example d. 945 rounded to the nearest hundred is 900 . Round 154 to the nearest ten. M 4 Rachel earned $164 babysitting last month. She earned $95 this month. Rachel rounded each 150 151 152 153 154 155 156 157 158 159 160 amount to the nearest $10 to estimate how much 154 is between 150 and 160. It is closer to 150. she earned. What is each amount rounded to the 154 rounded to the nearest ten is 150. nearest $10? Show your work. Round 154 to the nearest hundred. Student work will vary. Students might draw number lines, a hundreds chart, or explain in words. 100 110 120 130 140 150 160 170 180 190 200 Solution: ___________________________________$160 and $100 154 is between 100 and 200. It is closer to 200. C 5 Use the digits in the tiles to create a number that 154 rounded to the nearest hundred is 200. makes each statement true. Use each digit only once. B 1 Round 236 to the nearest ten. 1 2 3 4 5 6 7 8 9 Which tens is 236 between? Possible answer shown.
  • NOTES on FINITE GROUP REPRESENTATIONS in Fall 2020, I

    NOTES on FINITE GROUP REPRESENTATIONS in Fall 2020, I

    NOTES ON FINITE GROUP REPRESENTATIONS CHARLES REZK In Fall 2020, I taught an undergraduate course on abstract algebra. I chose to spend two weeks on the theory of complex representations of finite groups. I covered the basic concepts, leading to the classification of representations by characters. I also briefly addressed a few more advanced topics, notably induced representations and Frobenius divisibility. I'm making the lectures and these associated notes for this material publicly available. The material here is standard, and is mainly based on Steinberg, Representation theory of finite groups, Ch 2-4, whose notation I will mostly follow. I also used Serre, Linear representations of finite groups, Ch 1-3.1 1. Group representations Given a vector space V over a field F , we write GL(V ) for the group of bijective linear maps T : V ! V . n n When V = F we can write GLn(F ) = GL(F ), and identify the group with the group of invertible n × n matrices. A representation of a group G is a homomorphism of groups φ: G ! GL(V ) for some representation choice of vector space V . I'll usually write φg 2 GL(V ) for the value of φ on g 2 G. n When V = F , so we have a homomorphism φ: G ! GLn(F ), we call it a matrix representation. matrix representation The choice of field F matters. For now, we will look exclusively at the case of F = C, i.e., representations in complex vector spaces. Remark. Since R ⊆ C is a subfield, GLn(R) is a subgroup of GLn(C).
  • Symmetry and Tensors Rotations and Tensors

    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.
  • P. 1 Math 490 Notes 7 Zero Dimensional Spaces for (SΩ,Τo)

    P. 1 Math 490 Notes 7 Zero Dimensional Spaces for (SΩ,Τo)

    p. 1 Math 490 Notes 7 Zero Dimensional Spaces For (SΩ, τo), discussed in our last set of notes, we can describe a basis B for τo as follows: B = {[λ, λ] ¯ λ is a non-limit ordinal } ∪ {[µ + 1, λ] ¯ λ is a limit ordinal and µ < λ}. ¯ ¯ The sets in B are τo-open, since they form a basis for the order topology, but they are also closed by the previous Prop N7.1 from our last set of notes. Sets which are simultaneously open and closed relative to the same topology are called clopen sets. A topology with a basis of clopen sets is defined to be zero-dimensional. As we have just discussed, (SΩ, τ0) is zero-dimensional, as are the discrete and indiscrete topologies on any set. It can also be shown that the Sorgenfrey line (R, τs) is zero-dimensional. Recall that a basis for τs is B = {[a, b) ¯ a, b ∈R and a < b}. It is easy to show that each set [a, b) is clopen relative to τs: ¯ each [a, b) itself is τs-open by definition of τs, and the complement of [a, b)is(−∞,a)∪[b, ∞), which can be written as [ ¡[a − n, a) ∪ [b, b + n)¢, and is therefore open. n∈N Closures and Interiors of Sets As you may know from studying analysis, subsets are frequently neither open nor closed. However, for any subset A in a topological space, there is a certain closed set A and a certain open set Ao associated with A in a natural way: Clτ A = A = \{B ¯ B is closed and A ⊆ B} (Closure of A) ¯ o Iτ A = A = [{U ¯ U is open and U ⊆ A}.
  • Molecular Symmetry, Super-Rotation, and Semi-Classical Motion –

    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.
  • 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

    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.
  • (II) I Main Topics a Rotations Using a Stereonet B

    (II) I Main Topics a Rotations Using a Stereonet B

    GG303 Lecture 11 9/4/01 1 ROTATIONS (II) I Main Topics A Rotations using a stereonet B Comments on rotations using stereonets and matrices C Uses of rotation in geology (and engineering) II II Rotations using a stereonet A Best uses 1 Rotation axis is vertical 2 Rotation axis is horizontal (e.g., to restore tilted beds) B Construction technique 1 Find orientation of rotation axis 2 Find angle of rotation and rotate pole to plane (or a linear feature) along a small circle perpendicular to the rotation axis. a For a horizontal rotation axis the small circle is vertical b For a vertical rotation axis the small circle is horizontal 3 WARNING: DON'T rotate a plane by rotating its dip direction vector; this doesn't work (see handout) IIIComments on rotations using stereonets and matrices Advantages Disadvantages Stereonets Good for visualization Relatively slow Can bring into field Need good stereonets, paper Matrices Speed and flexibility Computer really required to Good for multiple rotations cut down on errors A General comments about stereonets vs. rotation matrices 1 With stereonets, an object is usually considered to be rotated and the coordinate axes are held fixed. 2 With rotation matrices, an object is usually considered to be held fixed and the coordinate axes are rotated. B To return tilted bedding to horizontal choose a rotation axis that coincides with the direction of strike. The angle of rotation is the negative of the dip of the bedding (right-hand rule! ) if the bedding is to be rotated back to horizontal and positive if the axes are to be rotated to the plane of the tilted bedding.
  • Superheroes Trivia Quiz Iv

    Superheroes Trivia Quiz Iv

    SUPERHEROES TRIVIA QUIZ IV ( www.TriviaChamp.com ) 1> Which superhero carries an indestructible shield? a. The Green Lantern b. Captain America c. Captain Flag d. The Red Tornado 2> Which character is often romantically paired with Batman? a. The Black Canary b. Miss America c. Catwoman d. Hawkgirl 3> Which superhero started out as a petty criminal? a. The Atom b. Spiderman c. The Blue Knight d. Plastic Man 4> Which superhero's alter ego is Raymond Palmer? a. The Atom b. Hawkman c. The Green Arrow d. The Tornado 5> Which superhero is associated with the phrase, "With great power there must also come great responsibility"? a. Spiderman b. Hell Boy c. Batman d. The Hulk 6> Which superhero is nicknamed the "Scarlett Speedster"? a. The Flash b. Speedball c. Stardust d. The Thing 7> Which superhero is dubbed the "Man without Fear"? a. Daredevil b. The Flash c. Wolverine d. Green Lantern 8> Which superhero is the medical doctor for the X-men? a. Storm b. Shadowcat c. Ice Man d. The Beast 9> Who is Batgirl's father (Barbara Gordon)? a. The Mayor b. The Governor c. Batman's Butler d. The Chief of Police 10> Which superhero gains his power from a ring? a. The Green Lantern b. Storm c. Dazzler d. The Hulk 11> Which superhero can manipulate the weather? a. The Tornado b. The Atom c. Strom d. The Thing 12> Which Island does Wonder Woman call home? a. Emerald Island b. Paradise Island c. Amazonia d. Eden Isle 13> Where does the Green Arrow operate? a.
  • The Cubic Groups

    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].
  • Euler Quaternions

    Euler Quaternions

    Four different ways to represent rotation my head is spinning... The space of rotations SO( 3) = {R ∈ R3×3 | RRT = I,det(R) = +1} Special orthogonal group(3): Why det( R ) = ± 1 ? − = − Rotations preserve distance: Rp1 Rp2 p1 p2 Rotations preserve orientation: ( ) × ( ) = ( × ) Rp1 Rp2 R p1 p2 The space of rotations SO( 3) = {R ∈ R3×3 | RRT = I,det(R) = +1} Special orthogonal group(3): Why it’s a group: • Closed under multiplication: if ∈ ( ) then ∈ ( ) R1, R2 SO 3 R1R2 SO 3 • Has an identity: ∃ ∈ ( ) = I SO 3 s.t. IR1 R1 • Has a unique inverse… • Is associative… Why orthogonal: • vectors in matrix are orthogonal Why it’s special: det( R ) = + 1 , NOT det(R) = ±1 Right hand coordinate system Possible rotation representations You need at least three numbers to represent an arbitrary rotation in SO(3) (Euler theorem). Some three-number representations: • ZYZ Euler angles • ZYX Euler angles (roll, pitch, yaw) • Axis angle One four-number representation: • quaternions ZYZ Euler Angles φ = θ rzyz ψ φ − φ cos sin 0 To get from A to B: φ = φ φ Rz ( ) sin cos 0 1. Rotate φ about z axis 0 0 1 θ θ 2. Then rotate θ about y axis cos 0 sin θ = ψ Ry ( ) 0 1 0 3. Then rotate about z axis − sinθ 0 cosθ ψ − ψ cos sin 0 ψ = ψ ψ Rz ( ) sin cos 0 0 0 1 ZYZ Euler Angles φ θ ψ Remember that R z ( ) R y ( ) R z ( ) encode the desired rotation in the pre- rotation reference frame: φ = pre−rotation Rz ( ) Rpost−rotation Therefore, the sequence of rotations is concatentated as follows: (φ θ ψ ) = φ θ ψ Rzyz , , Rz ( )Ry ( )Rz ( ) φ − φ θ θ ψ − ψ cos sin 0 cos 0 sin cos sin 0 (φ θ ψ ) = φ φ ψ ψ Rzyz , , sin cos 0 0 1 0 sin cos 0 0 0 1− sinθ 0 cosθ 0 0 1 − − − cφ cθ cψ sφ sψ cφ cθ sψ sφ cψ cφ sθ (φ θ ψ ) = + − + Rzyz , , sφ cθ cψ cφ sψ sφ cθ sψ cφ cψ sφ sθ − sθ cψ sθ sψ cθ ZYX Euler Angles (roll, pitch, yaw) φ − φ cos sin 0 To get from A to B: φ = φ φ Rz ( ) sin cos 0 1.
  • Chapter 1 – Symmetry of Molecules – P. 1

    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.
  • COXETER GROUPS (Unfinished and Comments Are Welcome)

    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.