Monster g roup F ro m W ikipedia , the f ree en c y cl opedia This - PowerPoint PPT Presentation

Monster g roup F ro m W ikipedia , the f ree en c y cl opedia This arti cl e is a b out the l ar g est o f the sporadi c si m p l e g roups . F or the kind o f in fi nite g roup kno w n as a Tarski m onster g roup , see Tarski m onster g roup . I n the area o f m odern a lg e b ra kno w n as g roup theory , the Monster g roup M ( a l so kno w n as the F is c her – Griess Monster , or the F riend l y Giant ) is the l ar g est sporadi c si m p l e g roup , ha v in g order 2 ^46 · 3^ 20 · 5 ^9 · 7 ^6 · 11 ^ 2 · 1 3^3 · 17 · 1 9 · 2 3 · 2 9 · 3 1 · 4 1 · 4 7 · 5 9 · 71 = 8 0 8, 017 ,4 2 4, 7 94, 512 ,8 75 ,886,4 5 9,9 0 4,96 1 , 710 , 757 , 005 , 75 4,368, 000 , 000 , 000 ≈ 8 × 10 ^ 5 3. The fi nite si m p l e g roups ha v e b een c o m p l ete l y cl assi fi ed . Ev ery su c h g roup b e l on g s to one o f 1 8 c ounta bl y in fi nite f a m i l ies , or is one o f 2 6 sporadi c g roups that do not f o ll o w su c h a syste m ati c pattern . The Monster g roup c ontains a ll b ut si x o f the other sporadi c g roups as su bq uotients . R o b ert Griess has c a ll ed these 6 e xc eptions pariahs , and re f ers to the other 20 as the happy f a m i l y . 10

6 Geometric / physical classi fi cation Three kinds o f g roups w hi c h are easy to v isua l ise , and see m to e x haust the physi c a l syste m s en c ountered • Mo l e c u l ar g roups – shapes f ro m fi nite l ayout w ith fix ed c entre • L atti c e g roups – S pa c e fill in g unit c e ll s w ith spe c i fi ed l ayouts • Groups o f re g u l ar po l yhedra 6.1 Sch ö n fl ies classi fi cation for molecular groups • C n 2 π/ n rotation a x is usua ll y the z a x is • C n v R e fl e c tion p l ane c ontainin g a x is o f rotation y- z and / or x - z p l ane C nh R e fl e c tion p l ane ⊥ to a x is o f rotation • x - z p l ane • D n 2- f o l d a x es o f sy mm etyr l yin g in p l ane perpendi c u l ar to the rotation a x is . D is f or “ dihedra l”. • D nh Additiona l re fl e c tion p l anes c ontainin g the a x is o f sy mm etry , b ut b ise c tin g the an gl es b et w een C n v p l anes w hi c h c ontain an ato m 11

• S n I n cl udin g i m proper rotations 6.2 P ole fi gures – projecti v e mnemonic 12

Introduction to Group Theory with Applications in Molecular and Solid State Physics Karsten Horn Fritz-Haber-Institut der Max-Planck-Gesellschaft 030 8412 3100, e-mail horn@fhi-berlin.mpg.de Symmetry - old concept, already known to Greek natural philosophy Group theory: mathematical theory, developed in 19th century Application to physics in the 1920’s : Bethe 1929, Wigner 1931, Kohlrausch 1935 Why apply group theory in physics? “It is often hard or even impossible to obtain a solution to the Schrödinger equation - however, a large part of qualitative results can be obtained by group theory. Almost all the rules of spectroscopy follow from the symmetry of a problem ” E.Wigner, 1931

Outline 1. Symmetry elements and point groups 4. Vibrations in molecules 1.1. Symmetry elements and operations 4.1. Number and symmetry of normal modes in molecules 1.2. Group concepts 4.2. Vibronic wave functions 1.3. Classification of point groups, including the Platonic Solids 4.3. IR and Raman selection rules 1.4. Finding the point group that a molecule belongs to 5. Electron bands in solids 2. Group representations 5.1. Symmetry properties of solids 2.1. An intuitive approach 5.2. Wave functions of energy bands 2.2. The great orthogonality theorem (GOT) 5.3. The group of the wave vector 2.3. Theorems about irreducible representations 5.4. Band degeneracy, compatibility 2.4. Basis functions 2.5. Relation between representation theory and quantum mechanics 2.6. Character tables and how to use them 2.7. Examples: symmetry of physical properties, tensor symmetries 3. Molecular Orbitals and Group Theory 3.1. Elementary representations of the full rotation group If you come up with a symmetry- 3.2. Basics of MO theory related problem from your own 3.3. Projection and Transfer Operators 3.4. Symmetry of LCAO orbitals work, bring it in and we can discuss 3.5. Direct product groups, matrix elements, selection rules it (time permitting) 3.6. Correlation diagrams

At the end of this week, having followed the course, you should be able to • determine the point group of a solid object such as a molecule or a crystalline unit cell or the space group of a translational periodic pattern • determine the symmetry properties (in terms of irreducible representations) of ! tensorial properties of solids and derive those tensor elements which are “zero by symmetry” example of a wallpaper group; ! atomic wave functions in a crystal field applies to surface problems ! molecular orbital wave functions ! molecular vibrational wave functions Bloch waves in a periodic solid ! • derive symmetry selection rules for vibrational (infrared, Raman) and electronic (Vis-UV, photoemission) transition matrix elements • identify molecular orbital and electronic band degeneracies and apply the “no -crossing- rule” • and much more... What we do not cover here is the Complete Nuclear Permutation Inversion Group - see book by P. R. Bunker and Per Jensen: Fundamentals of Molecular Symmetry,IOP Publishing, Bristol, 2004 (ISBN 0-7503-0941-5). However, given the successful mastering of the material discussed in this block course you should be able to extend your knowledge to this topic

Material about symmetry on the Internet Character tables: http://symmetry.jacobs-university.de/ The platonic solids: http://www.csd.uwo.ca/~morey/archimedean.html Wallpaper groups: http://www.clarku.edu/~djoyce/wallpaper/seventeen.html Point group symmetries: http://www.staff.ncl.ac.uk/j.p.goss/symmetry/index.html Students Online Resources of the book by Atkin s & de Paula: “Physical Chemistry”, 8e at http://www.oup.com/uk/orc/bin/9780198700722/01student/tables/tables_for_group_theory.pdf Other symmetry-related links: http://www.staff.ncl.ac.uk/j.p.goss/symmetry/links.html

application: vibrational transitions in metal clusters Photoelectron spectroscopy and quantum mechanical calculations have shown that anionic Au 20 ! is a pyramid and has T d symmetry. This structure has also been suggested to be the global minimum for neutral Au 20 (14). The FIR-MPD spectrum we measured of the Au 20 Kr complex (Fig. 2A) was very simple, with a dominant absorption at 148 cm !1 , which already pointed to a highly symmetric structure. The calculated spectrum of tetrahedral Au 20 was in agreement with the experiment (Fig. 2C)... The strong absorption at 148 cm ! 1 corresponds to a triply degenerate vibration (t 2 ) in bare Au 20 with T d symmetry. Theory predicts a truncated trigonal pyramid to be the minimum energy structure for neutral Au 19 (27), for which the removal of a corner atom of the Au 20 tetrahedron reduces the symmetry from T d to C 3v . As a direct consequence, the degeneracy of the t 2 vibration of Au 20 is lifted, and this mode splits into a doubly degenerate vibration (e) and a nondegenerate vibration (a 1 ) inAu 19 . This splitting was observed in the vibrational spectrum of neutral Au 19 (Fig. 2)… The truncated pyramidal structure of Au 19 can thus be inferred directly from the IR spectrum. Structures of Neutral Au 7 , Au 19 , and Au 20 Clusters in the Gas Phase Ph. Gruene, D. M. Rayner, B. Redlich,3 A. F. G. van der Meer, J. T. Lyon, G. Meijer, A. Fielicke, Science 329 , 5889 (2008)

application: band structure in solids, including spin-orbit coupling from: Dresselhaus, Dresselhaus and Jorio, Group Theory - Application to the Physics of Condensed Matter Springer 2008 (figure given without references)

Why should we care about symmetry properties in physics and chemistry ? • Think of an surface system, e.g. a nickel atom in a (111) surface. How should we classify the d orbitals of that atom ? d z etc.? 2 How should we classify molecular vibrations? In terms of their • geometrical distortions? • How can we classify electronic states in a molecular orbital?

1.3 Classification of point groups (in Schoenflies notation) 1.3.1 The groups C 1 , C s , C i . C 1 : element E(C 1 ) C s : E and a mirror plane C i : E and an inversion centre (C i ) Chloro-bromo-fluoro-methane 1.3.2 The groups C n 2 3 n-1 Contain E and a rotation by 2  /n. C n generates C n ,C n , C n . Example: C 2 = {E,C 2 } H 2 O 2 1.3.3 The groups S n Phenol C s Contain E and only an improper rotation by 2  /n. If there are other symmetry elements, the object does not belong to S n . Example: 1,3,5,7 tetrafluorocyclooctatetraene S 4

1.3.4 The groups C nv (frequent !) Contain E, C n and n mirror planes  v which all contain the C n axis. Ammonia C 3v v stands for vertical. The rotation axis corresponding to C n with the largest n is always taken as vertical: Example: C 2v = {E, C 2 ,  v ’,  v ’’} 1.3.5 The groups C nh Contain E, C n and a horizontal mirror plane. h stands for horizontal. The rotation axis corresponding to C n with the largest n is always taken as vertical. For n even an inversion center exists. peroxide C 2h planar hydrogen C’ C’  e D , n C’  C’ ’axes. ane D 3h