Symmetry Theory in Molecular Physics with Mathematica: A new kind of tutorial book 0387734694, 9780387734699

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William Martin McClain

Symmetry Theory in Molecular Physics with Mathematica

A NEW KIND OF TUTORIAL BOOK

Symmetry Theory in Molecular Physics with Mathematica

William Martin McClain

Symmetry Theory in Molecular Physics with Mathematica A new kind of tutorial book

123

William Martin McClain Department of Chemistry Wayne State University 5101 Cass Avenue Detroit MI 48202 USA [email protected]

ISBN 978-0-387-73469-9 e-ISBN 978-0-387-73470-5 DOI 10.1007/b13137 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2009933284 c Springer Science+Business Media, LLC 2009  All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

P r efa ce D i ffer en t p eop l e h a ve d i ffer en t a tti tu d p eop l e fi n d i ts s tep -b y -s tep p a ce to b y ou a r e on e of th os e ta l en ted p eop l e, p er s on wh o u n d er s ta n d s th eor y on l y l og i ca l s tep u n d er fu l l p u b l i c s cr u ti n y .

es towa r d M a t h e m a t i c a . Som e ver y g i fted e a n i m p ed i m en t to s ci en ti fi c th ou g h t. If th i s b ook i s n ot for y ou . It i s wr i tten b y a wh en i t l ea d s to ca l cu l a ti on , wi th ever y Th a t i s wh a t M a t h e m a t i c a excel s a t.

B u t M a t h e m a t i c a a l s o p er m i ts a p l a y fu l a tti tu d e towa r d th eor y , m a k i n g p os s i b l e l i ttl e exp er i m en ts a n d exp l or a ti on s th a t a r e u s u a l l y l eft i n a s ci en ti s t' s p r i va te n oteb ook . Th es e cu r i os i ty -d r i ven excu r s i on s a r e a n es s en ti a l p a r t of th e cr ea ti ve p r oces s , wh i ch p er h a p s n ow, wi th M a t h e m a t i c a a n d wi th p r i va te web s i tes , a n d i n b ook s l i k e th i s on e, ca n b ecom e a p a r t of th e p u b l i c r ecor d of s ci en ce. Th i s b ook wa s or i g i n a l l y i n ten d ed a s a con ci s e com p en d i u m of g r ou p th eor eti c d a ta a n d a l g or i th m s . Th e con ci s e s ta tem en ts a r e i n d eed th er e, i n th e two M a t h e m a t i c a p a ck a g es th a t a r e l oa d ed a t th e top of ever y ch a p ter . A tr u e th eor eti ci a n , ver s ed i n th e M a t h e m a t i c a l a n g u a g e, wou l d n eed n oth i n g el s e. B u t h u m a n s a r e n ot com p u ter s , a n d th e d evel op m en t a n d exp os i ti on of th e p a ck a g e m a ter i a l s ta k es ti m e a n d s p a ce, a n d on e th i n g l ed to a n oth er . I h a ve tr i ed to b r ea k th e m a ter i a l s i n to l ectu r es of fi fty m i n u tes l en g th (i f m u l ti p l e exa m p l es a r e om i tted ). I r em em b er wi th g r ea t l os s th os e fr om wh om I l ea r n ed g r ou p th eor y . F i r s t, P r of. An d r ea s C. Al b r ech t ta u g h t i t to h i s s p ectr os cop i c r es ea r ch s tu d en ts a t Cor n el l . After wa r d s , I wa s a p os t-d oc wi th P r of. Cr i s top h er L on g u ett-Hi g g i n s , Ca m b r i d g e Un i ver s i ty , wh o p i on eer ed th e u s e of p er m u ta ti on g r ou p s i n fl exi b l e m ol ecu l e s p ectr os cop y . An d es p eci a l l y I r em em b er P r of. L eo F a l i cov, P h y s i cs D ep a r tm en t, Un i ver s i ty of Ca l i for n i a , B er k el ey , wh o h el p ed m e wi th m y fi r s t b y -h a n d a p p l i ca ti on of p r oj ecti on op er a tor s m a n y y ea r s a g o. An d of cou r s e, th i s b ook owes i ts ver y exi s ten ce to th e m on u m en ta l a ch i evem en t of Step h en Wol fr a m i n cr ea ti n g a n d d evel op i n g M a t h e m a t i c a over th e l a s t twen ty y ea r s . He cr ea ted a g en u i n el y “ n ew k i n d of s ci en ce” , even b efor e a p p l y i n g M a t h e m a t i c a to com p l exi ty th eor y . Th e Wol fr a m cu s tom er s u p p or t g r ou p wa s h el p fu l on m a n y occa s i on s ; I wou l d es p eci a l l y l i k e to th a n k d evel op er s Ad a m Str z eb on s k i , An d r e K u z n i a r ek , a n d B u d d i e R i ch i e, wh o p u l l ed m e ou t of s ever a l d eep h ol es . I a m g r a tefu l to s ever a l col l ea g u es for h el p wi th th eor y : P r of. K a y Ma g a a r d , Ma th D ep a r tm en t, Sch l eg el a n d V l a d i m i r Ch er n y a k , Ch em i s tr y D ep R ob er t A. Ha r r i s , Ch em i s tr y D ep a r tm en t, Un i ver s i ty

m a th em a ti cs a n d ch em i ca l Wa y n e Sta te; H. B er n a r d a r tm en t, Wa y n e Sta te; a n d of Ca l i for n i a , B er k el ey . v

S y m m e tr y T h e o r y

I th a n k Tom von F oer s n ew wa y of p u b l i s h i n g s h eet von F oer s ter i s n a m a k i n g Tom ' s a g r eem m en ti on .

ter , a b m ed en t

for m er l y of Sp r i n g er V er l a g , for ook d i r ectl y fr om M a t h e m a t i c a n a fter h i m . Ma n y s p eci a l th a n k s com e tr u e, i n s p i te of vi ci s s i tu d

a g r eei n g to th i s wi l d oteb ook s . Th e s ty l e to J ea n i n e B u r k e for es too n u m er ou s to

My wi fe, Ca r ol B l u es ton e McCl a i n , wa s m y en a b l er th r ou g h y ea r s of com p u ter a d d i cti on . Sh e i s n ow h el p i n g m e to d r y ou t i n th e F eb r u a r y s u m m er of Sou th Am er i ca . I th a n k th e s tu d en ts of Ch em 8 4 90, Wi n ter 2007, Wa y n e Sta te Un i ver s i ty , wh o h el p ed wi th s u ch i n tel l eg en t a tten ti on , s u g g es ti on s , a n d cor r ecti on s : Mi ch a el Ca to, Ar m a n d o E s ti l l or e, Ha o L i , D r . B a r b a r a Mu n k , B r i a n P s ci u k , Su s h a n t Sa h u , F a d el Sh a l h ou t, J a s on Son k , D r . J a s on Son n en b er g , Hu a l i Wa n g , a n d J i a Z h ou . Th e b ook con s i s ts of a l l th e m a ter i a l s th a t cou l d b e p r es en ted to th i s cl a s s i n on e s em es ter . Wm . Ma r ti n McCl a i n P r ofes s or , Ch em i s tr y , Wa y n e Sta te Un i ver s i ty , D etr oi t, Mi ch i g a n wm m @ ch em . wa y n e. ed u Men d oz a , Ar g en ti n a , F eb . 18 , 2009

v i

Con ten ts 1. Introduction ........................................................................................1 1.1 What is symmetry theory? ................................................. 1 1.2 Outline of the book ..................................................................1 1.3 This is an interactive book ......................................................3 1.4 Learning Mathematica ............................................................4 1.5 The human's view of Mathematica ........................................5 1.6 Does Mathematica make errors ? ..........................................6 2. A tutorial on notebooks ....................................................................7 2.1 What are Mathematica notebooks? .......................................7 2.2 A basic notebook tutorial .........................................................8 3. A basic Mathematica tutorial ........................................................13 Section 3.1 This tutorial ...............................................................13 Section 3.2 On-line help ..............................................................14 Section 3.3 Basic operations .......................................................15 Section 3.4 The FrontEnd .................................... ....................... 19 Section 3.5 The K ernel ................................................................21 Section 3.6 Mathematica graphics .............................................24 4. The meaning of symmetry .............................................................29 4.1 Symmetry and its undefined terms ......................................29 4.2 Geometric symmetry .............................................................30 4.3 Algebraic symmetry ...............................................................46 4.4 Summary and preview ...........................................................54 5. Axioms of group theory ..................................................................55 5.1 Undefined terms in the axioms ............................................55 5.2 The four axioms .....................................................................56 v ii

S y m m e tr y T h e o r y

6. Several kinds of groups ..................................................................59 6.1 Numbers under Times ...........................................................59 6.2 Matrices under Dot ................................................................60 6.3 Axial rotation groups .............................................................62 6.4 Permutations under Permute .................................................63 6.5 Fruit flies under reproduction (a non-group).......................67 7. The fundamental theorem .............................................................73 7.1 Statement and commentary ...................................................73 7.2 Proof of the fundamental theorem .......................................74 7.3 How this theorem helps ..........................................................79 8. The multiplication table ..................................................................81 8.1 The generaliz ed ‘‘multiplication’’ table ........................... 81 8.2 Latin square theorem .............................................................83 8.3 The Latin square converse ....................................................84 8.4 Automated multiplication tables ..........................................87 8.5 How MultiplicationTable works ..........................................90 8.6 Test MultiplicationTable ......................................95 8.7 Redundant groups ..................................................................96 9. Molecules ...........................................................................................99 9.1 Molecule definitions in Mathematica .................................99 9.2 Molecule "objects" ............................................................... 101 9.3 Molecule object operators ................................................... 102 9.4 How to make a molecule ..................................................... 103 9.5 Molecule graphics ................................................................ 104 9.6 StereoV iew ............................................................................ 109 9.7 AxialV iews (a 3D shop drawing) ...................................... 110 10. The point groups .......................................................................... 113 10.1 Introduction ......................................................................... 113 10.2 The unit matrix ................................................................... 113 10.3 The inversion matrix .......................................................... 114 10.4 Roto-reflection matrices .................................................... 118 10.5 Only two kinds of matrices ............................................... 132 11. Euler rotation matrices .............................................................. 137 11.1 The Euler idea .................................................................... 137 11.2 The Euler rotations visualiz ed .......................................... 138 v iii

C o n te n ts

11.3 The Euler matrix theorem ................................................. 140 11.4 The function EulerMatrix[I, T,\] .................................... 141 11.5 Euler headaches .................................................................. 142 11.6 The inverse function EulerAngles[matrix] ..................... 143 12. Lie's axis-angle rotations ........................................................... 149 12.1 Introduction ......................................................................... 149 12.2 Rotation matrices in 2D form a Lie group ..................... 149 12.3 Lie group for 2D rotation .................................................. 150 12.4 Lie group for 3D rotation .................................................. 152 12.5 The operator AxialRotation[axis, angle] ........................ 157 13. Recognizing matrices .................................................................. 163 13.1 The problem, and a strategy ............................................. 163 13.2 Recogniz e a numerical rotation matrix ........................... 165 13.3 Recogniz e a rotoreflection matrix ................................... 168 13.4 Recogniz e a reflection matrix .......................................... 168 13.5 Operator Recogniz eMatrix ............................................... 169 13.6 The complex eigenvectors ................................................ 170 14. Introduction to the character table ......................................... 173 14.1 Introduction ......................................................................... 173 14.2 The standard character table ............................................. 173 14.3 Some facts about the character table ............................... 175 14.4 The rightmost column; bases for reps ............................. 180 14.5 V arious character table operators .................................... 181 15. The operator MakeGroup ......................................................... 183 15.1 An operator for constructing groups ............................... 183 15.2 MakeGroup, step by step ................................................ 184 15.3 MakeGroup with roundoff ................................................ 189 16. Product groups ............................................................................. 195 16.1 An introductory example (group C 3 h ) ..........................195 16.2 The product group theorem .............................................. 196 16.3 How to recogniz e product groups .................................... 197 16.4 Products with C h vs. products with C i ............................ 198 16.5 Some careful checking, and a scandal ..........................200 17. Naming the point groups ........................................................... 207 17.1 Molecules and point groups .............................................. 207 17.2 Schönflies names ............................................................... 207 ix

S y m m e tr y T h e o r y

17.3 International names ............................................................ 209 17.4 Schönflies-International correspondences ...................... 212 17.5 Some interesting name correspondences ........................ 213 18. Tabulated representations of groups ...................................... 217 18.1 Cartesian representation tables ......................................... 217 18.2 V ery simple groups ............................................................ 221 18.3 Groups with a principal axis ............................................. 223 18.4 Platonic groups ................................................................... 231 19. Visualizing groups ....................................................................... 251 19.1 Constellations vs. stereograms ......................................... 251 19.2 Constellations with a principal axis ................................. 253 19.3 Constellations based on Platonic solids .......................... 257 19.4 That's it, folks. What Next? .............................................. 267 20. Subgroups ...................................................................................... 269 20.1 Definition of subgroup ...................................................... 269 20.2 A simple but nontrivial example (group C3 h ) .............. 270 20.3 Finding all the subgroups of a group ........................... 272 20.4 An orientation issue ........................................................... 273 20.5 Symmetry breaking in PCl5 (group D3h) ........................ 274 20.6 Definition of ‘‘symmetry breaking’’ ............................... 279 21. Lagrange's Theorem ................................................................... 283 21.1 Purpose ................................................................................ 283 21.2 Definition of coset ............................................................. 283 21.3 An operator for cosets ....................................................... 284 21.4 The coset lemma ................................................................ 286 21.5 Lagrange's theorem proved ............................................... 288 22. Classes ............................................................................................ 291 22.1 Qualitative meaning of classes ......................................... 291 22.2 Graphics of two "similar" elements ................................. 291 22.3 Finding all the similar elements ....................................... 293 22.4 Definition of class .............................................................. 294 22.5 A very pregnant little discussion ..................................... 294 22.6 Operator MakeClass .................................................... 295 22.7 Table lookup operators for classes .................................. 296 22.8 Theorems about classes ..................................................... 297 22.9 Class character, and other class properties ..................... 298 x

C o n te n ts

22.10 The Classify operator ...................................................... 300 22.11 Isomorphisms among order 6 point groups .................. 301 23. Symmetry and quantum mechanics ........................................ 309 23.1 Transformation of eigenfunctions ................................... 309 23.2 Delocaliz ed orbitals of benz ene ....................................... 313 23.3 Linear combinations of degenerate orbitals ................... 314 23.4 Qualitative meaning of degeneracy ................................. 319 23.5 Accidental degeneracy ...................................................... 320 23.6 Dynamic symmetry ............................................................ 321 23.7 Motion reversal symmetry ................................................ 322 24. Transformation of functions ..................................................... 329 24.1 What is a function transform ? ......................................... 329 24.2 An intuitive way (but it's wrong) ..................................330 24.3 The right way ...................................................................... 330 25. Matrix representations of groups ............................................ 335 25.1 Representation theory ........................................................ 335 25.2 Definition of representation .............................................. 335 25.3 Faithful and unfaithful reps .............................................. 336 25.4 MorphTest ........................................................................... 337 25.5 Why representations are important .................................. 338 25.6 The rep construction recipe .............................................. 339 26. Similar representations .............................................................. 343 26.1 Given two reps, are they similar? .................................... 343 26.2 Universal pseudocommuter matrices .............................. 344 26.3 Universal true commuter matrices ................................... 351 26.4 A whiff of Schur's Second Lemma .................................. 352 27. The MakeRep operators ............................................................ 357 27.1 Four MakeRep operators ................................................ 357 27.2 Polynomial basis (polynomials in x, y, z ) ...................... 358 27.3 Matrix basis (Ix, Iy, Iz) ................................................ 362 27.4 Atomic orbital basis ............................................................ 362 27.5 Molecular orbital basis ...................................................... 363 27.6 Check the characters of the reps ...................................... 363 28. Reducible representations ......................................................... 371 28.1 The Cartesian representations .......................................... 371 28.2 Block diagonal multiplication .......................................... 372 x i

S y m m e tr y T h e o r y

28.3 Reducible and irreducible representations ..................... 374 28.4 Mulliken names of irreducible representations .............. 375 28.5 Example: the rep of the benz ene orbitals ........................ 376 28.6 Reduction demo ................................................................. 377 28.7 What is the point of rep reduction? ................................. 378 29. The MakeUnitary operator ....................................................... 385 29.1 Unitary and non-unitary reps ............................................ 385 29.2 The MakeUnitary theorem ................................................ 390 29.3 The operator MakeUnitary ............................................... 394 30. Schur's reduction ......................................................................... 397 30.1 Schur's idea ......................................................................... 397 30.2 Two experimental rep reductions .................................... 398 30.3 The ReduceRep operator ............................................. 404 30.4 Tests of ReduceRep ...................................................... 405 30.5 What have we learned? ..................................................... 407 31. Schur's First Lemma .................................................................. 411 31.1 Statement of the Lemma ................................................... 411 31.2 Proof .................................................................................... 411 32. Schur's Second Lemma .............................................................. 415 32.1 Introduction ......................................................................... 415 32.2 Statement and discussion .................................................. 415 32.3 Proof, using the unitary property ..................................... 416 32.4 A whiff of the great orthogonality ................................... 418 33. The Great Orthogonality ........................................................... 421 33.1 Skewers ............................................................................... 421 33.2 Two statements of the Great Orthogonality ................... 422 33.3 Two Great Orthogonality demos ..................................... 424 33.4 Proof of the Great Orthogonality ...................................... 427 34. Character orthogonalities .......................................................... 431 34.1 Introduction and demonstrations ..................................... 431 34.2 Row orthogonality proof ................................................... 434 34.3 Column orthogonality proof ............................................. 436 34.4 Character tables are square ............................................... 438 34.5 Check the twiddle matrix numerically...........................439 34.6 Unique rows and columns ................................................ 440 34.7 A theorem of your very own ............................................ 440 x ii

C o n te n ts

35. Reducible rep analysis ................................................................ 443 35.1 The point of rep analysis ................................................... 443 35.2 Symbolic development of Analyze ............................ 444 35.3 Numerical example of rep analysis ................................. 446 35.4 The Analyze operator ................................................. 447 36. The regular representation ....................................................... 449 36.1 What is the regular representation? ................................. 449 36.2 Characters of the regular representation ......................... 452 36.3 The golden property .......................................................... 452 36.4 Demo of the golden property ........................................... 453 36.5 The d 2 corollary ................................................................ 454 37. Projection operators ................................................................... 459 37.1 Projection operators in geometry ..................................... 459 37.2 Projection operators in algebra ........................................ 460 37.3 Symbolic derivation of the projectors ............................. 464 37.4 Your own examples of ProjectET ................................... 469 37.5 Your own examples of ProjectED ................................... 470 38. Tabulated bases for representations ....................................... 473 38.1 Introduction ......................................................................... 473 38.2 Basis sets from harmonic functions ................................. 473 38.3 Example: basis functions for D6d ................................... 476 39. Quantum matrix elements ......................................................... 483 39.1 Matrix elements defined ................................................... 483 39.2 The Bedrock Theorem, and its corollary ........................ 484 39.3 Example using two polynomials ...................................... 486 39.4 Major example: benz ene transition moments .................... 492 39.5 Proof of the Bedrock Theorem ......................................... 495 39.6 Integrals over functions of pure symmetry ..................... 497 40. Constructing SALCs .................................................................... 507 40.1 What are SALCs ? ............................................................. 507 40.2 Make SALCs, step by step ............................................... 508 40.3 Automation ......................................................................... 514 41. Hybrid orbitals ............................................................................. 521 41.1 Hybrid orbitals in general ................................................. 521 41.2 Pauling's hybridiz ation strategy ....................................... 522 41.3 Implement Pauling's strategy ........................................... 524 x iii

S y m m e tr y T h e o r y

41.4 The SALC shortcut ............................................................ 529 41.5 Electron density in s p 3 orbitals ....................................... 532 42. Vibration analysis ........................................................................ 535 42.1 Problem and strategy ......................................................... 535 42.2 A step-by-step example ..................................................... 536 42.3 Automate the displacement rep ........................................ 541 42.4 Analyz e the vibrations ....................................................... 543 42.5 One-click examples ........................................................... 546 43. Multiple symmetries ................................................................... 549 43.1 Inspiration ........................................................................... 549 43.2 Reps made from products of basis functions ................. 549 43.3 Lemma: Character of a direct product ............................ 551 43.4 V ibronic examples ............................................................. 551 44. One-photon selection rules ........................................................ 565 44.1 The main idea ..................................................................... 565 44.2 Polariz ation vectors and light amplitude ........................ 565 44.3 Projection of the one-photon operator ............................ 568 44.4 Transition oscillations ....................................................... 570 45. Two-photon tensor projections ................................................ 577 45.1 The operator for two-photon processes .......................... 577 45.2 Operators for tensor projection ........................................ 580 45.3 Self-purifying tensor elements ......................................... 582 45.4 Operators for tensor projection ........................................ 583 45.5 Two-photon tensor projection table ................................ 591 45.6 Axis permutation forbiddenness ...................................... 598 45.7 Alternative forms of the tensors ....................................... 599 46. Three-photon tensor projections ............................................. 601 46.1 The operator for three-photon processes ........................ 601 46.2 The projection ..................................................................... 603 46.3 Self-purifying tensor elements ......................................... 604 46.4 Operators for tensor projection ........................................ 604 46.5 Three-photon tensor projection table .............................. 611 46.6 Axis permutation forbiddenness ...................................... 640

x iv

C o n te n ts

47. Class sums and their products .................................................. 643 47.1 Motivation ........................................................................... 643 47.2 Declaration of a "group algebra" ..................................... 643 47.3 Class sums ........................................................................... 644 47.4 ClassSum theorems ............................................................ 645 47.5 Class sum multiplication for a familiar group ............... 647 47.6 The ClassSumProduct operator ........................................ 648 47.7 ClassSumMultiplicationTable .......................................... 650 48. Make a character table ............................................................... 653 48.1 Introduction ......................................................................... 653 48.2 Make the CSMT again ...................................................... 653 48.3 Class sum product coefficients ........................................ 654 48.4 A strategy for the character table ..................................... 656 48.5 Use the master equation .. ................................................... 657 48.6 Use column orthogonality .. ............................................... 661 48.7 Final formatting .................................................................. 663 48.8 Character table of a new group ........................................ 665 A1. Mathematica packages .............................................................. 673 A2. SymbolizeExpressions ................................................... 675 A3. Matrix review .............................................................................. 677 B1. Bibliography ................................................................................. 679 B1.1 Wolfram Research, Inc. .................................................... 679 B1.2 Applied group theory books ............................................ 679 B1.3 Mathematica books ........................................................... 680 B1.4 Chemistry and Physics books .......................................... 681 B1.5 Math books ......................................................................... 682 B1.6 Data collections ................................................................. 682 B1.7 Research papers ................................................................. 682 Index ...................................................................................................... 685

x v

1. In tr od u cti on 1. 1 Wh a t i s s y m m etr y th eor y ? In th i s b ook , Sy m m s y m m etr y tr a n s for m d ea l wi th a l g eb r a p r a n d p h y s i cs u s u a l l y p r ob l em s i n cr y s ta l s Al l g r ou of a m a n s in g on l y

etr y Th eor y m ea n s g r ou p th eor y , a s a p p l i ed to g r a ti on s . Th e d eep es t r oots of m a th em a ti ca l g r ou p ob l em s a n d p er m u ta ti on p r ob l em s , b u t s tu d en ts of ch en cou n ter i t fi r s t i n th e con text of g eom etr i ca l s y a n d m ol ecu l es .

s tu d en ts of p h y s i ca l ch em i s tr y a n d m ol ecu l a r p h p th eor y , i f on l y to a voi d u tter m y s ti fi ca ti on b y p h D 6 h m ol ecu l e” or “ th e d i a m on d cr y s ta l b el on g s y u n d er g r a d u a te p h y s i ca l ch em i s tr y textb ook s th a l e ch a p ter , b u t i n or d er to d o s o th ey m u s t l ea ve ou l i s ts of th i n g s to b e m em or i z ed .

ou p th em m m

s of eor y i s tr y etr y

y s i cs m u s t l ea r n a l i ttl e r a s es l i k e “ th e B 2 u s ta tes to m 3 m ” . Th er e a r e n ow t tr ea t g r ou p th eor y i n a t a l l th e l og i c, p r es en ti n g

You s h ou l d r ea d s u ch a ch a p ter , i f y ou h a ve n ever s een on e. It wi l l ta k e y ou q u i ck l y over th e m os t com m on u s es of g r ou p th eor y i n m ol ecu l a r p h y s i cs , b u t i f y ou a r e a t a l l r es i s ta n t to th e m em or i z a ti on of a u th or i ty , i t wi l l l ea ve y ou h u n g r y for m or e. An d of cou r s e, on th e b a s i s of s u ch a ch a p ter , y ou wou l d n ever b e a b l e to th i n k of a n y th i n g n ew on y ou r own . In th i s b ook , n oth i n g wi l l b e p r es en ted fr om a u th or i ty ; th e l og i c wi l l b e for em os t, a n d y ou wi l l b e p r ep a r i n g y ou r m i n d to ca r r y s y m m etr y th eor y for wa r d i n to n ew a r ea s .

1. 2 Ou tl i n e of th e b ook

P a r t I P o in t g r o u p s a n d th e ir c o n s tr u c tio n Ch a p ter 2 a n d Ch a p ter 3 i n tr od u ce y ou to M a t h e m a t i c a , a n el ectr on i c l og i c en g i n e th a t a u tom a tes r ou ti n e ca l cu l a ti on s , s y m b ol i c a s wel l a s n u m er i c. Ch a p ter 4 th r ou g h Ch a p ter 8 d evel op th e b a s i c r el a ti on s h i p b etween s y m m etr y a n d m a th em a ti ca l g r ou p th eor y . Ch a p ter 9 i n tr od u ces th e M a t h e m a t i c a m ol ecu l e, a n d i ts a u tom a ted g r a p h i cs .

W.M. McClain, Symmetry Theory in Molecular Physics with Mathematica, DOI 10.1007/b13137_1, © Springer Science+Business Media, LLC 2009

1

S y m m e tr y T h e o r y

Ch a p ter 10 th r ou g h Ch a p ter 13 d evel op s om e m a tr i x op er a tor s n eces s a r y for con s tr u cti n g s y m m etr y tr a n s for m a ti on m a tr i ces , a n d for r ecog n i z i n g s u ch m a tr i ces wh en th ey a p p ea r a s a con s tr u cti on . Ch a p ter 14 i n tr od u ces th e ch a r a cter ta b l e, a n d th e con cep ts of Cl a s s a n d Sp eci es . N oth i n g i s d er i ved or p r oved h er e, b u t th e q u es ti on s th a t wi l l occu p y th e n ext s ever a l ch a p ter s a r e exp l i ci tl y l a i d ou t. Ch a p ter 15 th r ou g h Ch a p ter 19 s h ow h ow m ol ecu l a r p oi n t g r ou p s a r e m a d e, or g a n i z ed , n a m ed a n d vi s u a l i z ed . P a r t II R e p r e s e n ta tio n T h e o r y Ch a p ter 20 th r ou g h Ch a p ter 22 d es cr i b e s u b g r ou p s a n d cl a s s es Ch a p ter 23 exp l a i n s h ow g r ou p th eor y r el a tes to q u a n tu m m ech a n i cs , a n d wh y th e n a m es th a t a p p ea r i n th e ch a r a cter ta b l e a r e u s ed a s th e n a m es of s p ectr os cop i c s ta tes . Ch a p ter 24 th r ou g h Ch a p ter 28 i n tr od u ce r ep r es en ta ti on s b a s ed on s y m m etr i c ob j ects i n th e m os t g en er a l s en s e, a n d th e i d ea of r ed u ci b l e a n d i r r ed u ci b l e r ep r es en ta ti on s . Ch a p ter 29 th r ou g h Ch a p ter 3 3 p r es en t th e b a s i c th eor em th eor y , cu l m i n a ti n g i n th e G r ea t Or th og on a l i ty Th eor em . th e G r ea t Or th og on a l i ty a r e i n tr od u ced i n a q u a s i -exp er i m i m i ta te th e wa y th a t Sch u r d i s cover ed th em . Th i s b r i n g s th p r es en ti n g q u es ti on s fi r s t a n d a n s wer s s econ d .

s of a b s tr a ct Sch u r ' s L em m en ta l wa y th a e top i c to l i fe

g r ou p a s for t m a y a g a in ,

In Ch a p ter 3 4 to Ch a p ter 3 6 we d r a w ou t s om e con s eq u en ces of th e G r ea t Or th og on a l i ty wh i ch cl ea r u p th e fi n a l m y s ter i es a b ou t th e p oi n t g r ou p s . Th e ch a r a cter or th og on a l i ti es u n d er l i e th e p r op er ti es th a t a l l ch a r a cter ta b l es h a ve i n com m on , a n d a r e of g r ea t p r a cti ca l con s eq u en ce. Th ey p er m i t th e a u tom a ti c r ed u cti on of a n y r ed u ci b l e r ep r es en ta ti on , a n d p er m i t th e a u tom a ted con s tr u cti on of a l l p os s i b l e i r r ed u ci b l e r ep r es en ta ti on s for a n y g i ven g r ou p .

2

Ch a p ter 3 7 p r es en ts th e con cep t of p r oj ecti on op er a tor s , wh i ch ca n m a th em a ti ca l ob j ect of n o s y m m etr y wh a ts oever a n d d i vi d e i t s y s tem i n to p a r ts th a t h a ve s i m p l e s y m m etr i es . Th es e op er a tor s h a ve l on g b een th eor eti ca l l y , b u t th ei r a p p l i ca ti on wi th ou t a u tom a ted m a th em a ti ca l ted i ou s a n d u n cer ta i n . N ow, for th e fi r s t ti m e, th e op er a tor s a n d ta b u l a th i s b ook m a k e th em ea s y to u s e. It i s q u i te fa i r to s a y th a t th i s b ook wa s to m a k e th i s ch a p ter p os s i b l e.

ta k e a a ti ca l l y k n own h el p i s ti on s i n wr i tten

O u tlin e

P a r t III A p p lic a tio n s , a n d a u to m a te d c o n s tr u c tio n o f c h a r a c te r ta b le s Ch a p ter 3 8 to Ch a p ter 4 6 d ea l wi th a p p l i ca ti on We m a k e exten s i ve u s e of a l l th e a u tom a ted op II. Su d d en l y th e fog l i fts , a n d y ou s ee h ow s tr u ctu r e of g r ou p th eor y , th e a p p l i ca ti on s of u n i fi ed th em e, a n d a r e ea s y a n d tr a n s p a r en t. In Ch a p ter 4 7 a n d Ch a p ter 4 8 th a t d es cr i b es a g r ou p , y ou ca ter ta b l e for th a t g r ou p . Th i s th e m os t u s efu l a l g or i th m s i n ti m e i n th e s p ectr os cop y of q u i te u r g en tl y n eed ed .

s of g r ou p th eor y er a tor s d evel op ed tr u e i t i s th a t i f i t i n m ol ecu l a r th

y ou l ea r n th a t i f y ou n a p p l y m a ch i n er y th i s a l i ttl e a d va n ced , th e b ook . N ew p er m fl exi b l e m ol ecu l es , a

in in y ou eor

ch em i s tr y . P a r ts I a n d k n ow th e y fol l ow a

h a ve th e m u l ti p l i ca ti on ta b l e a t wi l l tu r n i t i n to th e ch a r a cb u t i t i s p oten ti a l l y of on e of u ta ti on g r ou p s a p p ea r a l l th e n d th ei r ch a r a cter ta b l es a r e

A p p e n d ic e s a n d B ib lio g r a p h y Th e b ook en d s wi th th r ee a p p en d i ces a n d a b i b l i og r a p h y . Th e a p p en d i ces d ea l wi th ca l cu l a ti on a l or m a th em a ti ca l i s s u es th a t a r i s e i n th e b ook , b u t wh i ch d o n ot l i e on th e m a i n p a th . Th ey m a y b e r ea d on l y b y i n ter a cti ve r ea d er s .

1. 3 Th i s i s a n i n ter a cti ve b ook If y ou a r e r ea d i n g th i s on p a p er , y ou m a y n ot k n ow ver y m u ch y et a b ou t M a t h e m a t i c a . It i s a com p u ter p r og r a m th a t ca r r i es ou t th e d eta i l s of a l g eb r a , ca l cu l u s , a n d n u m er i ca l com p u ta ti on . It i s ch a n g i n g th e wa y th a t s ci en ti s ts l ea r n a n d u s e m a th em a ti cs , b eca u s e i t fr ees th e m i n d to con cen tr a te on th e i s s u es of l og i ca l s tr a teg y , r a th er th a n on d eta i l s of s y m b ol m a n i p u l a ti on . Its s p eed a n d u n er r i n g a ccu r a cy g i ves on e th e en er g y a n d con fi d en ce to for g e a h ea d wi th n ew a n d u n exp ected r es u l ts , r a i s i n g th e cr ea ti ve p r oces s to n ew l evel s . Th er e i s a p a p er ver s i on of th i s b ook a n d a CD (com p a ct d i s k ) ver s i on . Th e p a p er ver s i on i s a r efer en ce wor k , p r ovi d i n g m os t of th e m a ter i a l on th e CD . B u t y ou ca n p u t th e CD i n to a com p u ter a n d r ea d th e u n a b r i d g ed ver s i on fr om th e s cr een , l etti n g M a t h e m a t i c a r eca l cu l a te a l l th e r es u l ts i n th e b ook a s y ou g o. If y ou d o th i s , th e M a t h e m a t i c a p r og r a m wi l l a l wa y s b e p r ep a r ed to ca r r y ou t exp er i m en ts th a t a r e s u g g es ted i n th e text, or (even b etter ) th a t y ou m a y th i n k of on y ou r own . Th i s i n ter a cti ve m od e m a y m a k e th i s b ook on e of th e fi r s t th a t p eop l e wi l l p r efer to s tu d y fr om a com p u ter r a th er th a n fr om p a p er . (Of cou r s e, for q u i ck r efer en ce p u r p os es , n oth i n g b ea ts r ea ch i n g for a b ook on a s h el f. Th a t i s wh y th er e i s a h a r d cop y . )

3

S y m m e tr y T h e o r y

You r com p u ter m u s t b i n ter a cti vel y . M a t h e m l i cen s es th a t a r e ver y p r i ce i s l es s th a n th e a u

e a b l e to r u n a t i c a i s n ow con ven i en t to th or on ce p a i d

M a th e m a wi d el y a u s e, or a for a s l i d

t i c a b efor e y ou ca n va i l a b l e th r ou g h i n s s s ta n d -a l on e cop i es e r u l e m a d e of b a m b

u s e th e b ook ti tu ti on a l s i te . Th e s tu d en t oo a n d i vor y .

If y ou wor k wi th th i s b ook i n ter a cti vel y , y ou wi l l l ea r n M a t h e m a t i c a a n d s y m m etr y th eor y tog eth er , b u t i t i s th e l og i c of s y m m etr y th eor y , n ot th e l og i c of M a t h e m a t i c a , th a t d r i ves th e b ook . We i n tr od u ce M a t h e m a t i c a op er a tor s on l y wh en we n eed th em for s y m m etr y th eor y , s o a u s efu l a p p l i ca ti on i s a l wa y s a t h a n d for a n y p r og r a m m i n g s k i l l s y ou l ea r n . Wh en y ou fi n i s h , y ou r k n owl ed g e of s y m m etr y th eor y wi l l b e q u i te s ol i d , a s fa r a s th i s b ook ta k es y ou . You wi l l feel th a t i t i s ea s y a n d n a tu r a l to ca r r y ou t cer ta i n g r ou p th eor eti c ca l cu l a ti on s wh i ch i n th e p a s t wer e a l wa y s ted i ou s a n d p r on e to h u m a n er r or , th e d om a i n on l y of s p eci a l i s ts . Th e i n ter a cti ve m od e of th i s p a r tl y i m i ta tes th e wa y i t wa s p r ovi d es a xi om s , d efi n i ti on s , exp l or ed b efor e d efi n i ti ve a n M a t h e m a t i c a y ou r s el f to d o n s ol u ti on s b efor e th ey a r e p r es ca l s k i l l s r eq u i r ed for a l l r es ea tr y th eor y .

1. 4

b ook l ets y ou l ea r n g r ou p th eor y i n a wa y th a t d i s cover ed b y r es ea r ch m a th em a ti ci a n s . Th e b ook a n d l ea d i n g q u es ti on s . Th e q u es ti on s a r e th en s wer s a r e g i ven , a n d y ou a r e en cou r a g ed to u s e u m er i ca l exp er i m en ts ; to s ee i f y ou ca n a r r i ve a t en ted . Th u s th i s b ook tea ch es th e g en er a l a n a l y ti r ch , a s wel l a s tea ch i n g M a t h e m a t i c a a n d s y m m e-

L ea r n i n g M a t h e m a t i c a

P eop l e often s a y th a t M a t h e m a t i c a h a s a ver y s teep fi ve y ea r s of i n tr od u ci n g u n d er g r a d u a tes to M a t h e fou n d th i s to b e tr u e. After a few h ou r ' s i n s tr u cti on Ch a p ter 3 of th i s b ook , a l on g wi th m od el exa m p s tu d en ts ca n b eg i n to op er a te wi th M a t h e m a t i c a .

l ea r n i n g cu r ve, b u t i n over m a t i c a , th e a u th or h a s n ot , a s g i ven i n Ch a p ter 2 a n d l es of p a r ti cu l a r p r ob l em s ,

It i s m or e a ccu r a te to s a y th a t M a t h e m a t i c a , l i k e m a th em a ti cs i ts el f, h a s a ver y l on g l ea r n i n g cu r ve. Step h en Wol fr a m , th e ch i ef a r ch i tect of M a t h e m a t i c a , fr eq u en tl y s a y s th a t h e i s a l wa y s l ea r n i n g to b e a b etter u s er of M a t h e m a t i c a . Th e p r og r a m wa s d es i g n ed to g r ow, a n d i n m or e th a n a d eca d e of d evel op m en t b y a l a r g e s ta ff of p r ofes s i on a l s , i t h a s b ecom e a va s t b u t s tr i ctl y or d er l y p r og r a m . It n ow en com p a s s es m u ch of th e k n owl ed g e con ta i n ed i n th e m a th em a ti ca l ta b l es a n d h a n d b ook s th a t l i n e th e r efer en ce s h el ves of u n i ver s i ty l i b r a r i es , a n d i n a for m m u ch ea s i er to u s e.

4

Th e on l y wa y to l ea r n M a t h e m a t i c a i s to u s e i t. After y ou r u n th r ou g h th e b a s i c tu tor i a l s of Ch a p ter 2 a n d Ch a p ter 3 a n d wor k wi th th e exa m p l es i n th e fi r s t few s ecti on s of th e b ook , y ou wi l l b e b eg i n to b e a b l e to u s e M a t h e m a t i c a on y ou r

L e a r n in g M a th e m a tic a

own . If y ou fi n i s h th e b ook , y ou wi l l h q u i te com p eten t u s er of M a t h e m a t i c a . M a t h e m a t i c a ' s own on -l i n e d ocu m en ta m i g h t n eed , a n d y ou wi l l feel th a t y ou th a t a r i s e i n y ou r own r es ea r ch .

a ve l ea r n ed en ou g h tr i ck s Al s o, y ou wi l l h a ve l ea r n ti on to tea ch y ou r s el f a n y ca n u s e M a t h e m a t i c a to ta

to m a k e y ou a ed h ow to u s e n ew a r ea y ou ck l e p r ob l em s

Th e p r i m a r y p r i n ted r efer en ce on M a t h e m a t i c a i s Step h en Wol fr a m ' s Th e Ma th em a ti ca B ook . It d es cr i b es M a t h e m a t i c a u p to th e i n tr od u cti on of V er s i on 6. Th e n ew V er s i on 6 op er a tor s a r e m i s s i n g , b u t ver y l i ttl e i n i t i s ob s ol ete. Cu r r en tl y th e p r i m a r y r efer en ce i s th e on l i n e D o c u m e n t a t i o n C e n t e r , a cces s i b l e a s th e fi r s t i tem i n th e Hel p m en u of M a t h e m a t i c a i ts el f. It p r ovi d es a d ocu m en t for ever y op er a tor , a s wel l a s m a n y con ci s e tu tor i a l s on s p eci fi c a r ea s . It s ets a n ew g ol d s ta n d a r d for com p u ter p r og r a m d ocu m en ta ti on . Cl i ck h er e for a ver y s el ecti ve l i s t of Ma th em a ti ca b ook s , a l l m or e tu tor i a l th a n Wol fr a m ' s b ook , b u t a l l l es s com p l ete.

1. 5 Th e h u m a n ' s vi ew of M a t h e m a t i c a Th er e i s a ver s i on of M a t h e m a t i c a for ever y m a j or k i n d of com p u ter . F r om a h u m a n ' s p oi n t of vi ew th ey a l l wor k th e s a m e wa y , a n d th ey a l l p r od u ce th e s a m e k i n d of Text fi l es , ca l l ed n oteb ook s . Hu m a n s ca n exch a n g e n oteb ook s el ectr on i ca l l y wi th ou t con cer n a b ou t d i ffer en ces a m on g com p u ter s . Wh en a p m a ch i n e' s va r i a ti on s i n d i ca te m b eg i n n er s n ot j u s t M

er s on fi r s t b eg i n s to u s e a com p u ter , i t i s d i ffi cu l t to g et u s ed to th e com b i n a ti on of b r i l l i a n ce a n d s tu p i d i ty . Th e s i m p l es t, m os t ob vi ou s y ou m a k e i n s y n ta x a r e j u s t n ot u n d er s tood (l i k e u s i n g th e l etter x to u l ti p l i ca ti on ). Th e i r on d i s ci p l i n e of s ti ck i n g to a p r eci s e s y n ta x i s for th e h a r d es t th i n g to l ea r n . B u t th i s i s tr u e of a l l com p u ter s p r og r a m s , a th e m a tic a .

Wh en y ou fi r s t l a u n ch M a t h e m a t i c a , on l y th e “ fr on t en d ” i s a cti va ted . A n oteb ook wi l l a p p ea r , ei th er b l a n k or wi th m a ter i a l i n i t, d ep en d i n g on h ow y ou l a u n ch . Wi th th e fr on t en d a l on e, y ou m a y r ea d th e n oteb ook a n d ed i t i ts p r os e. B u t y ou ca n n ot ca l cu l a te u n ti l y ou s ta r t th e M a t h e m a t i c a “ k er n el ” . Th e k er n el s ta r ts u p a u tom a ti ca l l y th e fi r s t ti m e y ou a s k i t to eva l u a te s om eth i n g . (Th er efor e, th e fi r s t a n s wer m a y b e s l ow to com e u p . ) M a t h e m a t i c a p r ovi d es ty p es et m a th em a ti ca l n ota ti on . Th i s oth er a l p h a b ets , s p eci a l i z ed m a th em a ti ca l ch a r a cter s , a n d s ta two-d i m en s i on a l n ota ti on for i n teg r a l s , p ower s , a n d fr a cti on ces a n d vector s , a n d oth er ob j ects . Her e, for i n s ta n ce, M a t h e m a t i c a i n p u t-ou tp u t p a i r :

in n d s, is

cl u d es G r eek a r d m a th em a a s wel l a s m a n i cel y ty p

a n d ti ca l a tr i es et

5

S y m m e tr y T h e o r y

s i n x  x p

x 

It m ic a a n y ca s

1 2

p

x

p

x 2 

p  1,  x   x  p   x  p p  1,  x 

i g h t ta k e y ou s om e ti m e to fi n d th a t i n a ta b l e of i n teg r a l s . B u t M a t h e m a t d i d n ot l ook i t u p i n a ta b l e; i t u s es a g en er a l s y m b ol i c a l g or i th m , u n l i k e th i n g n or m a l l y ta u g h t to h u m a n s , a n d i t h a n d l es va r i a n ts a s ea s i l y a s s i m p l e es :



s i n a x  b  x p

x 1 2a

x p

a

2

p

x 2

p  1,  a x  cos b    s i n b   a x  p 

 a x  p p  1,  a x  cos b    s i n b 

Th a t m i g h t n ot b e i n a n y ta b l e, even th e l a r g es t. You cou l d of cou r s e d ed u ce i t fr om th e r es u l t a b ove, b u t i t wou l d ta k e s om e ti m e, a n d y ou cou l d ea s i l y m a k e m i s ta k es i n th e p r oces s . Wel com e to th e wor l d of com p u ter -a s s i s ted m a th em a ti cs !

1. 6 D oes M a t h e m a t i c a m a k e er r or s ? Th er e i s n o wa y to m a k e a n a b s ol u te g u a r a n tee a g a i n s t er r or i n p r og r a m . Con fi d en ce th a t a p r og r a m i s er r or -fr ee com es fr om y ea u n d er a l l k i n d s of u n for es een ci r cu m s ta n ces . Th e p u b l i c u s e of a p i t th r ou g h m or e u n for es eea b l e twi s ts a n d tu r n s th a n a n y tes ti n g p cou l d . Th e p a r ts of M a t h e m a t i c a we wi l l u s e i n th i s b ook h a ve wi th u s e for over ten y ea r s , a n d we h a ve g r ea t con fi d en ce th a t a n y er r or s cen tr a l cod e h a ve l on g s i n ce b een fou n d a n d cor r ected .

a com p u ter r s of tes ti n g r og r a m p u ts r og r a m ever s tood p u b l i c i n th e b a s i c

6

N ever th el es s , th e wi s e u s er of M a t h e m a t i c a s ets u p fr eq u en t ch eck s for s el fcon s i s ten cy . Th i s i s on e of th e th i n g s we wi l l tea ch a s we g o. We h a ve n ever fou n d a M a t h e m a t i c a er r or wi th th es e ch eck s , b u t we h a ve fou n d p l en ty of h u m a n er r or s . Con s i s ten cy ch eck i n g i s th e wa y th a t r i g or ou s m i n d ed s ci en ti s ts d ea l wi th a l l th e b l a ck b oxes th ey m u s t u s e i n th i s el ectr on i c a g e.

2. A tu tor i a l on n oteb ook s  If y ou wa n t to r ea d th i s b ook “ l i ve” (a s i n ten d ed ) y ou wi l l n eed to r ea d th i s ch a p ter on s cr een a n d wi th M a t h e m a t i c a r u n n i n g , a n d d o th e th i n g s i t s a y s to d o. You won ' t g et m u ch ou t of i t b y j u s t r ea d i n g th e h a r d cop y , b u t h er e i t i s for q u i ck r efer en ce :

2. 1. Wh a t a r e M a t h e m a t i c a n oteb ook s ? E ver y d ocu m en t p r od u ced b y th e M a t h e m a t i c a fr on t en d i s a “ n oteb ook ” , a n d ever y ch a p ter of th e CD ver s i on of th i s b ook i s a “ n oteb ook ” . Th e n oteb ook a l l ows a d ocu m en t to p r es en t i ts el f i n ou tl i n e for m , wi th ea ch top i c a n d s u b top i c op en i n g u p for r ea d i n g on com m a n d , or cl os i n g u p a g a i n on com m a n d . N oteb ook s r eq u i r e a l i ttl e exp l a n a ti on , b u t th ey s oon s eem ver y n a tu r a l . N oteb ook s a r e d i vi d ed i n d i vi d u a l cel l s a r e u s u l en g th . Wh en y ou r ea b r a ck ets on th e r i g h t s i d th e on l y on e th a t d i s p l a y

i n to cel l s , wh i ch a r e r a th er l i k e p a r a g r a p h s . Th e a l l y g r ou p ed b y en cl os i n g b r a ck ets of i n cr ea s i n g d th i s b ook on -s cr een , y ou wi l l s ee n es ted b l u e e of th e s cr een . In th e p r i n ted b ook , th i s ch a p ter i s s th e n es ted b r a ck ets . L ook 

Th i s p a r a g r a p h -l i k e cel l i s g r ou p ed b y a s econ d a r y Secti on b r a ck et th a t en cl os es a l l of Secti on 2. 1. Al l Secti on b r a ck ets a r e i n tu r n en cl os ed b y a n ou ter b r a ck et th a t b eg i n s a t th e top of th e n oteb ook a n d r u n s a l l th e wa y to th e b ottom , en cl os i n g a l l of Ch a p ter 2. If y ou cl i ck on ce on th e ou ter b r a ck et, i t wi l l tu r n d a r k a n d y ou wi l l h a ve “ s el ected ” th e wh ol e n oteb ook , a s i f for Cop y i n g or (G od for b i d ! ) D el eti n g . Wel l , of cou r s e, y ou ca n d el ete th e l oa d ed cop y fr om R AM, b u t n ot fr om th e CD . If y ou cl i ck t w i c e on y ou wi l l b e l eft wi th on th e ou ter fol d ed p oi n ti n g h a l f a r r ow),

th on b a n

e ou ter m os t b r l y th e Ch a p ter r a ck et (wi th a d th e n oteb ook

a ck et, th e n oteb ook wi l l fol d u p a n d ti tl e s h owi n g . To r eop en , cl i ck twi ce s m a l l d a r k tr i a n g l e, l i k e a d own wi l l r ea p p ea r .

G o a h ea d . D o i t. Th e wh ol e r ea s on for p u tti n g th i s b ook on a com p u ter i s to l et y ou tr y th i n g s a s we g o. To op en th e Secti on h ea d i n g b el ow (“ 2. 2. A b a s i c n oteb ook tu tor i a l ” ), cl i ck twi ce on th e m i d d l e b r a ck et to i ts r i g h t, th e on e wi th th e l i ttl e d a r k W.M. McClain, Symmetry Theory in Molecular Physics with Mathematica, DOI 10.1007/b13137_2, © Springer Science+Business Media, LLC 2009

7

S y m m e tr y T h e o r y

tr i a n g l e a t th e b ottom Th ey op en s i m i l a r l y . To fol d It wi l l r ea p p ea cel l s ca

It wi l l u n fol d , s h owi n g th e s u b s ecti on h ea d er s .

i t u p a g a i n , cl i ck twi ce on th e s a m e (b u t n ow exp a n d ed ) b r a ck et. s h r i n k a n d th e l i ttl e d a r k tr i a n g l e (or d own wa r d h a l f-a r r ow) wi l l r , i n d i ca ti n g th a t th e b r a ck et i s exp a n d a b l e. Al l m u l ti p l y b r a ck eted n b e u n fol d ed a n d fol d ed u p a g a i n i n th i s wa y .

2. 2. A b a s i c n oteb ook tu tor i a l 2. 2. 1. Ma g n i fi ca ti on E ver y n oteb ook wi n d ow h a s a m a g n i fi ca ti on op ti on on i ts b ottom ed g e, j u s t l eft of th e h or i z on ta l s cr ol l b a r . B y d efa u l t, th e ch a p ter s of th i s b ook a r e s et for 200% . Cl i ck th e l i ttl e d a r k tr i a n g l e a n d oth er m a g n i fi ca ti on ch oi ces wi l l d r op d own . Tr y s om e oth er s i z es .

2. 2. 2. Ma k e a n ew cel l wi th 1 + 1 i n i t Move th e cu h or i z on ta l b a ta l “ i n s er ti on ty p i n g . F or th e q u otes ) :

r s or a r wi th l i n e” i n s ta n

r ou n d u n ti l , b etween two exi s ti n g s p l i t en d s . Cl i ck on ce, a n d i t tu r n s . Th i s i s wh er e th e n ew cel l wi l l a ce, y ou ca n d o i t j u s t b el ow, ty p i n

cel l s , i t tu r n s to a to a l on g h or i z on p p ea r , i f y ou s ta r t g “ 1+1” (wi th ou t

Th e “ 1+1” y ou j u s t ty p ed i s i n a n I n p u t cel l , d i s ti n g u i s h ed i ts B o l d C o u r i e r ty p e fa ce. Th i s i s th e k i n d of cel l y ou wa n t wh en p r ep a r i n g i n p u t for ca l cu l a ti on s . (G o a h ea d , d o i t! N oth i n g ter r i b l e ca n h a p p en ! ) N ow we d i s cu s s h ow to g et M a t h e m a t i c a to com p u te 1 + 1 for y ou .

2. 2. 3

Th e i n i ti a l com p u ta ti on

8

P u t y ou r cu r s or a n y wh er e i n th e I n p u t cel l y ou j u s t cr ea ted , a n d h i t - or -. On e of th es e s h ou l d ca u s e s om eth i n g to h a p p en ; r em em b er wh i ch on e d oes th e j ob . We wi l l ca l l i t th e P r o c e s s com m a n d .

N o te b o o k tu to r ia l

I n t h is b o o k , t h e P r o c e s s c o m m a n d w ill b e r e p r e s e n t e d b y .

On a Ma ci n tos h , y ou m a y u s e  b y i ts el f, or th e - com b i n a ti on , or th e - com b i n a ti on . On P Cs , th e  i s r eq u i r ed . If y ou a r e s ta r ti n g u p a n ew n oteb ook , y ou m a y n ow exp er i en ce a l i ttl e d i s tr a cti on . A q u es ti on wi l l a p p ea r i n a b ox :

D o y o u w a n t to e v a lu a te a ll th e in itia liz a tio n c e lls in th is n o te b o o k ? F or n oteb ook s fr om th i s b ook , y ou m u s t a l wa y s cl i ck YE S. Oth er wi s e, th e n oteb ook m a y n ot wor k p r op er l y .

In th i s b ook , th e i n i ti a l i z a ti on cel l s a r e m os tl y i n th e P r e l i m i n a r i e s s ecti on a t th e top of ever y ch a p ter . (You ca n op en th em a n d s ee th em , b u t th ey a r e h a r d to u n d er s ta n d ; d on ' t tr y u n ti l y ou k n ow m or e. ) After th e i ts I n p u t I n O u

q u es ti on d i s a p p ea r s , a cel l . Th es e cel l ty p es p u t cel l s h a ve B o l d t p u t cel l s h a ve P la

n O ca n C o in

u t p u t cel l a p p ea r s , b r a ck eted wi th b e d i s ti n g u i s h ed a t a g l a n ce: u r i e r ty p efa ce; C ou r ier ty p efa ce.

After p r oces s i n g , n oti ce th e a p p ea r a n ce of th e b l u e I n [ ] a n d O u t [ ] cel l l a b el s . Th es e wi l l p r ove ver y u s efu l . Un p r oces s ed I n p u t cel l s (a m a j or ca u s e of er r or for b eg i n n er s ) wi l l n ot h a ve th e b l u e l a b el . Tr y a c e s s a ct a s in n e r m exa m p

2. 2. 4

few oth er s i m p l e s u m s u n ti l com m a n d . On a Ma ci n tos h a P r oces s com m a n d . Th en o s t b r a c k e t of ea ch n ew cel l l e s ta n d i n g . It' s u p to y ou .

y ou a r e ver y fa m i l i a r n ote th a t b oth  a n cl ea n u p a fter y ou r s a n d h i t th e d e l e t e k ey

wi th th e P r o d - ca el f. Sel ect t h . Or , l ea ve on

 n e e

Text cel l s

N oteb ook s h a ve T e x t cel l s a s wel l a s I n p T e x t cel l s p r ovi d e a p l a ce for com m en ta r y on m os t of th e text of th i s b ook ). Th i s cel l i s a T e h a ve a s h or t h or i z on ta l l i n e n ea r th e top of th e i n

u t a n d O u t y ou r ca l cu l a x t cel l . Al l n er m os t cel l b

p u t cel l s . ti on (or for T e x t cel l s r a ck et.

9

S y m m e tr y T h e o r y

To m a k e a n ew T e x t cel l , m a n ew cel l wi l l a p p ea r , a n d i t wi l l few ch a r a cter s , s top a n d --7 , or --7 ) . Th g u i s h ed b y i ts 10-p oi n t Ti m es ty p

k e a n i n s er ti on l i n e a n d s ta r t ty p i n g . A b e, b y d efa u l t, a n I n p u t cel l . After a is s u e O p t i o n - C o m m a n d - 7 (th a t' s e cel l wi l l tu r n to a T e x t cel l , d i s ti n e fa ce.

2. 2. 5 Hea d i n g cel l s N oteb ook s i n s er ti on l i n is s u eO p t i a cel l of ty p

a ls oh ea n d o n - C eS e c

a ve h ea d i n g s ta r t ty p i n g o m m a n d - 4 t i o n , in 1 4

cel l s of s ever a l d i ffer en t ty p es . Ma k e a n a h ea d i n g . After a few ch a r a cter s s top a n d (--4 , or --4 ). It wi l l b ecom e p oi n t ty p e.

Oth er h ea d i n g s a r e m a d e b y com m a n d s of th e for m O p t i o n - C o m  m a n d - d i g i t (--d i g i t or --d i g i t ), wh er e d i g i t i s a n y d i g i t fr om 1 to 7 . In th e v o n F o e r s t e r s ty l e s h eet u s ed b y th i s b ook , d i g i ts 1 - 6 a r e h ea d i n g s of d ecr ea s i n g r a n k , a n d 7 i s T e x t , 8 i s a m a g n i fi ca ti on tog g l e, a n d 9 i s I n p u t . You m a y s wi tch a cel l fr eel y a m on g a n y of th es e ty p es b y p u tti n g th e cu r s or a n y wh er e i n s i d e i t, h ol d i n g d own O p t i o n - C o m m a n d ( -, or - ), a n d ty p i n g d i ffer en t d i g i ts . Tr y i t.

In th i s b ook ever y ch a p ter u s es th e h i er a r ch y C h a p t e r L i n e (a u ton u m b er ed , --1 ) S e c t i o n (a u ton u m b er ed , --4 ) S u b s e c t i o n (a u ton u m b er ed , --5 ) S u b s u b s e c t i o n (n ot n u m b er ed , --6 )

2. 2. 6 P r oces s i n g cel l g r ou p s Sel ect a n y g r ou p i n g b r a ck et, op en or n ot, a n d i s s u e a P r o c e s s com m a n d . Th i s wi l l p r oces s i n g a l l th e s el ected cel l s . If y ou s el ect th e ou ter m os t b r a ck et th a t en cl os es th e wh ol e n oteb ook , y ou wi l l P r o c e s s th e wh ol e n oteb ook wi th on e h i t.

1 0

N o te b o o k tu to r ia l

2. 2. 7 Th e a cti ve wa y to r ea d a n oteb ook Wi th th e cu r s or r i g h t h er e i n th i s cel l , d o a - . Th e 1 + 1 b el ow wi l l b e s el ected a n d a n oth er - s en d s i t to th e p r oces s or . A th i r d s el ects th e n ext eva l u a ta b l e cel l , a n d a fou r th eva l u a tes th a t. Tr y i t : 1 1 2 2 You ca n cl i ck d ep r es s ed a n d a p r e-wr i tten n d own . You ca

d own th e wh ol e n oteb ook b y k eep i n g th e S h i f t k ey r ep ea ted l y ty p i n g E n t e r . Th i s i s a g ood wa y to eva l u a te oteb ook a t a h u m a n p a ce, fol l owi n g th e l og i c a s y ou cl i ck n s top a t a n y p oi n t a n d ca r r y ou t exp er i m en ts of y ou r own .

If y ou wa n t to d o fu r th er wor k i n a n ewl y op en ed n oteb ook , s el ect ever y th i n g a b ove y ou r s ta r ti n g p oi n t a n d . It wi l l r u n d own to y ou r s ta r ti n g p oi n t, m a k i n g a l l n eces s a r y th e d efi n i ti on s a s i t g oes . Th en y ou ca n s ta r t wor k .

2. 2. 8

F ol d a n d u n fol d g r ou p s of cel l s

Sel ect a g r ou p i n g b r a ck et a n d d o a --} (s h i ft-com m a n d -cl os eB r a ck et). Th e s el ected b r a ck et (a n d a l l s u b -b r a ck ets wi th i n i t) wi l l cl os e. If y ou s el ect th e wh ol e n oteb ook , y ou ca n cl os e th e wh ol e n oteb ook wi th on e s tr ok e, l ea vi n g on l y th e ti tl e s h owi n g . Th en cl i ck twi ce on th e ou ter m os t b r a ck et, a n d y ou g et a n i ce com p a ct i n d ex of Secti on s for th e wh ol e n oteb ook . Si m i l a r l y , --{ (s h i ft-com m a n d -op en B r a ck et) op en s s ecti on , a n d a l l i ts s u b s ecti on s . a

s el ected

Th i s i s r ea l l y a l l y ou n eed to k n ow to g et s ta r ted u s i n g n oteb ook s .

1 1

3 . A b a s i c M a t h e m a t i c a tu tor i a l  If y ou wa n t to r ea d th i s b ook “ l i ve” (a s i n ten d ed ) y ou wi l l n eed to r ea d th i s ch a p ter on s cr een a n d wi th M a t h e m a t i c a r u n n i n g , a n d d o th e th i n g s i t s a y s to d o. You won ' t g et m u ch ou t of i t b y j u s t r ea d i n g th e h a r d cop y , b u t h er e i t i s for q u i ck r efer en ce :

P r el i m i n a r i es

Secti on 3 . 1 Th i s tu tor i a l Th i s tu tor i a l i s for V er s i on 6 of M a t h e m a t i c a . It i s a q u i ck l y over th e b a s i c ter m s a n d con cep ts . It i s p r ob a s ta r ted , b u t y ou wi l l n ot r ea l l y l ea r n th es e th i n g s excep th em u p i n y ou r con s ci ou s n es s fr om y ou r own m em or y . b ecom es ver y a u tom a ti c, b u t on l y wi th p r a cti ce.

fr a m ewor k for b l y a g ood wa t b y p r a cti ce, L ik erid in g a b

ru n n in g y to g et b rin g in g i cy cl e, i t

If y ou h a ve n ot y et r ea d Ch a p ter 2, th e N oteb ook s Tu tor i a l , i t wi l l h el p a l ot i f y ou d o s o n ow. In th i s tu tor i a l , n oth i n g on a col or ed b a ck g r ou n d wi l l eva l u a te or cop y . You s h ou l d r ety p e i t i n a cel l of y ou r own b etween th e col or ed cel l s , a n d th en . Ty p i n g for ces y ou to n oti ce i m p or ta n t d eta i l s th a t es ca p e y ou i f y ou m er el y r ea d .

Ty p e i n ever y th i n g exa ctl y a s s h own (p a r ti cu l a r l y p u n ctu a ti on , a n d

u p p er a n d l ower ca s es ).

After y ou s ee th e g i ven exa m p l e wor k cor r ectl y ,

m a k e va r i a ti on s of y ou r own . As y ou wor k th e exa m p l es , p a s s n oth i n g over wi th a s h r u g ; l et n oth i n g m y s y ou ; ever y th i n g th a t h a p p en s , h a p p en s on y ou r or d er s . If y ou s ee s om eth s tr a n g e, exa m i n e y ou r l a s t i n p u t for ty p os b efor e a n y th i n g el s e. A s m a l l a cci d ta l s p a ce i n th e m i d d l e of a va r i a b l e n a m e, or s om e oth er tr i vi a l ty p o, ca n m a l l th e d i ffer en ce. E xp l a n a ti on s , wh en n eed ed , fol l ow th e exa m p l es .

W.M. McClain, Symmetry Theory in Molecular Physics with Mathematica, DOI 10.1007/b13137_3, © Springer Science+Business Media, LLC 2009

ti fy in g en a k e

1 3

S y m m e tr y T h e o r y

Secti on 3 . 2 On -l i n e h el p 3 . 2. 1 Qu i ck , b a s i c h el p : th e q u es ti on m a r k N o on e ca n r em em b er a l l th e d eta i l s of a l l th e M a t h e m a t i c a op er a tor s , s o th er e i s a n excel l en t on -l i n e h el p fa ci l i ty . J u s t p r ep en d a q u es ti on m a r k to th e n a m e of th e op er a tor : ? S o l v e



You wi l l s ee th e s y n ta x of S o l v e , a n d a b a s i c d es cr i p ti on of i ts a cti on . Her e i s a n exa m p l e : S o l v e [ x - 2 y 0 , x ] Th e ou tp u t i s a l i ttl e s u r p r i s i n g , b u t y ou wi l l l ea r n to r ea d i t. F or h u m a n s , th e l i ttl e a r r ow “ ” b a s i ca l l y m ea n s “ = ” . F or th e com p u ter i t h a s a m ea n i n g y ou wi l l l ea r n b el ow. Som eti m es th e exa ct n a m e i s wh a t y ou h a ve for g otten . Ty p e ? a n d th en s om e p a r t of th e n a m e th a t y ou r em em b er , s u r r ou n d ed b y wi l d ca r d a s ter i s k s : ? * P l o t * Th i s b r i n g s u p ever y op er a tor th a t con ta i n s P l o t a n y wh er e i n i ts n a m e. Cl i ck on th e on e y ou wa n t, a n d h el p wi l l a p p ea r . F or a g ood r el eva n t exa m p l e, cl i ck on P l o t i ts el f a n d tr y to r ea d wh a t com es u p .

3 . 2. 2 Mor e d eta i l ed h el p N o com p u ter p r og r a m h a s m or e d eta i l ed h > > s y m b ol a t th e en d of th e th u m b n a i l s d evoted to th e P l o t op er a tor wi l l a p p ea r . m a k e i t a p a r t of th i s d ocu m en t, s o wh en cl i ck i t cl os ed a n d com e b a ck h er e.

1 4

el p th a n M a t h e m a k etch a b ove, a n d D o i t. Th e n oteb y ou h a ve fi n i s h ed

tic a . a wh ook l ook

Cl i ck on ol e n oteb i s too l on i n g a t i t,

th e ook g to ju s t

O n -lin e h e lp

3 . 2. 3 Th e D ocu m en ta ti on Cen ter Th e fi r s t i tem u n d er th e Hel p m en u i s th e D ocu m en ta ti on Cen ter (D C). Cl i ck on i t a n d y ou wi l l s ee a cl i ck a b l e ou tl i n e of ever y th i n g i n th e D C. B u t th e b es t wa y to u s e i t i s to ty p e a ter m i n to th e l on g th i n b ox a t th e top , a n d . M a t h e m a t i c a wi l l s ea r ch th e D C for a l l occu r r en ces of y ou r ter m , a n d p r es en t y ou wi th a cl i ck a b l e l i s t of D C d es ti n a ti on s . It i s ver y s i m i l a r to a G oog l e s ea r ch .

Secti on 3 . 3

B a s i c op er a ti on s

3 . 3 . 1 Si m p l e a r i th m eti c a n d a l g eb r a

(a ) A d d , s u b tr a c t, m u ltip ly , a n d d iv id e 2 + 3



2 - 3



6 / 3



2 * 2



Th e * m ea n s m u l ti p l y , a s i n m os t l a n g u a g es

2 2 Th e  a l s o m ea n s m u l ti p l y (on l y i n M a t h e m a t i c a ) Ma k e y ou r own cel l a n d tr y th es e th i n g s ou t. Cl i ck a b ove or b el ow th i s cel l to g et th e i n s er ti on l i n e, a n d th en s ta r t ty p i n g th e m a ter i a l fr om th e b oxed cel l a b ove. (b ) P o w e r s Th e ol d -fa s h i on ed (a l l -on -th e-b a s el i n e) wa y to wr i te a p ower i s

2 ^ 3

Th e ca r a t ^ m ea n s “ to th e p ower ” . It i s th e u p p er ca s e 6 k ey .

Her e i s th e wa y to g et th e p ower u p off th e b a s el i n e (wh er e i t b el on g s ! ) :

2

2 ^ 3

Ma k es a s u p er s cr i p t tem p l a te on th e 2 , l i k e 2



.

F ill it:



Com e b a ck to th e b a s el i n e wi th  .

1 5

S y m m e tr y T h e o r y

3 . 3 . 2 Th r ee u s efu l p a l ettes

T h e B a s ic I n p u t p a le tte F ol l ow th e cl i ck ch a i n P a l e t t e s M e n u B a s i c M a t h i n p u t . A p a l ette wi l l a p p ea r , con ta i n i n g tem p l a tes for a n u m b er of m a th em a ti ca l n ota ti on s .

D r a g i t to th e s i d e b y i ts top b a r . Ma k e a n i n s er ti on l i n e a n d

cl i ck on a p a l ette i tem . Th e i tem exi s ti n g cel l , th e n ew i tem

wi l l a p p ea r i n a n ew In p u t cel l . In s i d e a n

wi l l a p p ea r wh er e ever y ou l ea ve th e b l i n k i n g

cu r s or .

T h e S p e c ia lC h a r a c te r s p a le tte F ol l ow th e cl i ck ch a i n P a l e t t e s M e n u S p e c i a l C h a r a c t e r s . You m a y cl i ck to i n s er t a n y ch a r a cter th a t M a t h e m a t i c a h a s . Un d er L etter s h i s th er e a r e fi ve a l p h a b et s ty l es ; u n d er Sy m b ol s th er e a r e s even wh ol e p a l ettes of m a th s y m b ol s . Ma n y of th es e wer e i n th e p a s t a va i l a b l e on l y to p r ofes s i on a l ty p es etter s . L ook a t th em

a ll.

T h e A lg e b r a ic M a n ip u la tio n p a le tte F ol l ow th e cl i ck ch a i n P a l e t t e s M e n u A l g e b r a i c M a n i p u l a t i o n . con ta i n i n g a

In

s q u a r e of s om e fom u l a , s el ect th e for m u l a

a n

i n p u t cel l a n d

cl i ck

E xp a n d . Th en s el ectth e r es u l t a n d cl i ck on F a ctor . If y ou a r e i n a n

on

In p u t

cel l , i t wi l l h a p p en r i g h t i n -p l a ce. If n ot, a n ew In p u t cel l wi l l op en , con ta i n i n g th e exp r es s i on , a s ch a n g ed . Th en tr y s om e of th e oth er op er a ti on s .

1 6

B a s ic o p e r a tio n s

3 . 3 . 3 Th e m os t b a s i c op er a tor s

S e t (th e " = "

s ig n )

Th e m a th em a ti ca l eq u a l s i g n (= ) i s l eft-r i g h t s y m m etr i c for a ver y g ood r ea s on : Th e exp r es s i on s th a t s ta n d on ei th er s i d e of i t m u s t eva l u a te to th e s a m e n u m er i ca l va l u e. B u t th i s i s N OT h ow th e " = " s i g n i s u s ed i n com p u ter l a n g u a g es . Th e ea r l i es t l a n g u a g es u s ed i t i n com m a n d s l i k e a M e m o r y L o c a t i o n

=

a N u m b e r

Th i s s en t th e g i ven n u m b er i n to s tor a g e a t th e g i ven l oca ti on . Th i s i s n ot m a th em a ti ca l eq u a l i ty , b u t th i s u s a g e ca n n ot b e s top p ed . " = " is S e t .

So th e n a m e of

In M a t h e m a t i c a , m em or y l oca ti on s a r e r efer r ed to b y s y m b ol i c

n a m es , l i k e q . Tr y q 

If th e l oca ti on q i s fr ee, th e s y m b ol q wi l l b e r etu r n ed . If n ot, s ee C l e a r (b el ow).

q = 7

L oca ti on q i s S e t to 7 , a n d 7 i s r etu r n ed .

q

Th i s ti m e, th e n u m b er 7 i s r etu r n ed .

Un ti l th e con ten t of l oca ti on q i s ch a n g ed , M a t h e m a t i c a wi l l a l wa y s a u tom a ti ca l l y tu r n s y m b ol q i n to 7 . Th u s 7 h a s b ecom e th e " m ea n i n g " of q .

(e ) C le a r C l e a r [ q ] 

Th i s u n d oes th e S e t

op er a ti on , fr eei n g

u p

th e g i ven

m em or y l oca ti on .

1 7

S y m m e tr y T h e o r y

(f) E q u a l (th e d o u b le " = = " s ig n ) Si n ce th e " = "

i s a l r ea d y

ta k en

b y

S e t , m a th em a ti ca l eq u a l i ty

n eed s

a n oth er s y m b ol . It i s two eq u a l s i g n s ty p ed tog eth er wi th n o s p a ce : 1 = = 1 1 = = 2 a = = b

M a th e m a tic a

r etu r n s a n

a n s wer wh en

i t ca n , a n d

r ep ea ts th e

q u es ti on wh en i t ca n ' t. Un l i k e p r i m i ti ve n u m er i ca l l a n g u a g es , M a t h e m a t i c a

ca n

p u t s y m b ol i c

exp r es s i on s i n to m em or y l oca ti on s . You ca n g i ve a n a m e to a n eq u a ti on : e q N a m e

=

2

x = = 3

y



e q N a m e

(e ) U s e a d e c im a l o r n o t? I t m a tte r s . S i n [ 2 . ]



N ote th e d eci m a l p oi n t.

S i n [ 2 ]



N ote th e a b s en ce of th e d eci m a l p oi n t

Th i s r eq u i r es exp l a n a ti on . a ccep t m a ch i n e a ccu r a cy ; r es u l t. Th a t i s wh y i t r efu n a l n u m b er th a t h a s n o exa

Th e d eci m a l p oi n t tel l s M m a th a t y ou a r e wi l l i n g to th e a b s en ce of th e d eci m a l m ea n s y ou wa n t a n exa ct s es to d o a n y th i n g wi th S i n [ 2 ] , wh i ch i s a n i r r a ti oct r ep r es en ta ti on .

(f) S y m b o ls fo r ir r a tio n a l n u m b e r s c 2

c

2





Th e

 tem p l a te i s ty p ed a s 2 .

Th e c b eh a ves a s i t s h ou l d wh en s q u a r ed .

(g ) R u le a n d R e p la c e A ll C l e a r [ x ]



R u l es a r e s i m i l a r to eq u a l i ti es , excep t th a t th ey h a ve a r i g h t-p oi n ti n g a r r ow “ ” i n th em i n s tea d of a n eq u a l s i g n . We wr i te a r u l e th a t we n a m e a s x R u l e . Us e - > to m a k e th e n i ce l i ttl e a r r ow. Th a t' s e s c a p e - m i n u s - g r e a t e r t h a n e s c a p e ):

1 8

B a s ic o p e r a tio n s

x R u l e R u l e s wou l d two-s tr ok e com R u l e to b e a exp r es s i on to b

x 2



b e u s el es s wi th ou t th e R e p l a c e A l l op er a tor . It i s wr i tten a s a b i n a ti on / . b etween a n exp r es s i on a n d a R u l e . It ca u s es th e p p l i ed to th e exp r es s i on ; i n th i s ca s e i t ca u s es ever y x i n th e e r ep l a ced b y 2 .

. 2

x

=

x R u l e



You d on ' t h a ve to g i ve n a m es to R u l e s . You ca n j u s t s a y x

. 3

x 3



Th e g r ea t th i n g a b ou t R u l e a n d R e p l a c e A l l i s th a t i t d oes n oth i n g p er m a n en t to th e s y m b ol on th e l eft s i d e of th e R u l e (th e x , i n th i s ca s e). As k wh a t M a t h e m a t i c a h a s n ow for th e va l u e of x : x



N oth i n g ! It i s s ti l l a n u n s et s y m b ol , th ou g h we u s ed i t a b ove a s a tem p or a r y con ta i n er for two d i ffer en t n u m er i ca l va l u es . Th i s wi l l b e m or e u s efu l th a n y ou ca n p os s i b l y i m a g i n e r i g h t n ow.

Secti on 3 . 4

Th e F r on tE n d

M a t h e m a t i c a con s i s ts of two execu ta b l e p r og r a m s , th e F r o n t E n d a n d th e K e r n e l . Th i s s ecti on d es cr i b es th e F r o n t E n d ; th e n ext s ecti on d es cr i b es th e K e r n e l. Th e F r o n t E n d con s tr u cts a l l th e d i s p l a y s on y ou r s cr een a n d i n ter p r ets a l l i n p u t fr om th e k ey b oa r d a n d th e m ou s e. Th e K e r n e l d oes a l l s y m b ol i c a n d n u m er i c ca l cu l a ti on . Wh en y ou cl i ck twi ce on a n oteb ook i con to op en i t, on l y th e F r o n t E n d s ta r ts u p . Th e K e r n e l s ta r ts a u tom a ti ca l l y th e fi r s t ti m e y ou s en d a n y th i n g to th e p r oces s or . F or i n s ta n ce, 1  wi l l s ta r t th e K e r n e l . It ta k es a p er cep ti b l e ti m e to s ta r t. Th e n ext ti m e y ou op en a n ew n oteb ook , l ook for th i s .

1 9

S y m m e tr y T h e o r y

3 . 4 . 1 A few F r on tE n d tr i ck s G r e e k le tte r s , a n d o th e r s tr a n g e c h a r a c te r s G r eek l etter s m a y b e cl i ck ed fr om k ey b oa r d .

a p a l ette, or m a y b e i n p u t r i g h t fr om

th e

G r e e k le tte r s , a n d o th e r s tr a n g e c h a r a c te r s Th e F i l e M e n u  P a l e t t e s  C o m p l e t e C h a r a c t e r s

p a l ette con -

ta i n s m a n y s p eci a l i z ed ch a r a cter s a va i l a b l e i n th e p a s t on l y to p r ofes s i on a l ty p es etter s . It p r ovi d es a n a l ter n a ti ve wa y to g et G r eek l etter s , a m on g m a n y oth er s , b y cl i ck i n g . Ch eck i t ou t. Th er e a r e k ey b oa r d s h or tcu ts for ever y th i n g on a l l p a l ettes .

Th e G r eek

l etter s a r e p a r ti cu l a r l y ea s y ; j u s t ty p e th e L a ti n eq u i va l en t, s u r r ou n d ed b y es ca p es : 

s ta n d s for th e es ca p e k ey , e s c . Tr y

a 

G r eek a l p h a

b 

G r eek b eta

p 

If y ou wa n t y ou r p i i n G r eek , ty p e th i s .

p h 

Φ Som e n eed two s tr ok es .

p s 

Ψ

S 

Ca p i ta l s , too

S i n [ p i  /

3 ]



Π h a s a p r ea s s i g n ed m ea n i n g , th e s a m e a s

“ P i ” . S i n [ P i  /

3 ]



b u t i s n ot Π.

S u b s c r ip ts a n d s u p e r s c r ip ts We h a ve a l r ea d y s een h ow to m a k e a s u p er s cr i p t tem p l a te wi th ^ (c o n t r o l - c a r a t ) to r a i s e a n exp r es s i on to a p ower . Th e s u b s cr i p t i s q u i te s i m i l a r , th ou g h th er e i s a d i s ti n cti on to b e m a d e. B u t fi r s t, s om e p a i n fu l l y d eta i l ed i n s tr u cti on s on h ow to ty p e a s u b s cr i p t. Ty p e a s y m b ol , th en ty p e - (c o n t r o l - m i n u s ). Th i s p r od u ces a s el ected s u b s cr i p t tem p l a te b el ow a n d to th e r i g h t of th e s y m b ol . F i l l th e tem p l a te wi th

2 0

F r o n tE n d

s om s p a sy m a tr u c

eth i n g c e ). b ol . L es y m 1

c 1

s i m p l e. N ow com e u p to th e m a i n l i n e wi th - ( c o n t r o l N ow y ou h a ve a s u b s cr i p ted ob j ect th a t a cts for m os t p u r p os es l i k e a a ter y ou wi l l l ea r n to S y m b o l i z e i t wh en n eed ed s o th a t i t b ecom es b ol .

2 . 0 ; , c 2 

You ca n u s e i t a s i n p u t to fu n cti on s : L o g c 1

, c

2 1



T h e h ig h -lo w to g g le

( 5 )

Her e i s a h a n d y tr i ck . Su p p os e y ou wa n t to wr i te c 1 s q u a r ed . If y ou wr i te th e s u b s cr i p t fi r s t a n d th en th e s u p er s cr i p t, i t wi l l l ook l i k e c 1 2 . Th e i s n ot ter r i b l e, b u t s om eti m es y ou wou l d r a th er h a ve th e s u p er s cr i p t d i r ectl y over th e s u b s cr i p t. F or th i s , wr i te c 1 , a n d th en wi th th e cu r s or b l i n k i n g b eh i n d th e s u b - 1 , d o a 5 ( c o n t r o l - 5 ). Th i s wi l l cr ea te a n i n s er ti on b ox d i r ectl y a b ove th e s u b s cr i p t. Actu a l l y , y ou m a y a l s o d o i t i n th e op p os i te or d er - th e c o n t r o l - 5 op er a ti on i s a tog g l e b etween m a tch i n g h i g h a n d l ow i n s er ti on b oxes . It a l s o tog g l es b etween o v e r s c r i p t a n d u n d e r s c r i p t b oxes .

Secti on 3 . 5 Th e K er n el 3 . 5. 1 In tr od u cti on Th e K e r n e l k n ows n oth i n g a b ou t wh a t th e F r o n t E n d y ou g i ve th a t l i n e a n E n t e r or S h i f t - R e t u r n ( ). Th i s i s fr es h l y op en ed n oteb ook th a t h a s l ots of In -Ou t p a i r s i n d i s p l a y s th em , b u t th e K e r n e l k n ow n oth i n g a b ou t th em es t m i s ta k e i n M a t h e m a t i c a i s to for g et th i s . E ver y b od y d i s l ea r n to r ecog n i z e th e s y m p tom s a n d s cr ol l b a ck u p to r s k i p p ed .

h a s on a n y l i n e u n ti l p a r ti cu l a r l y tr u e of a i t. Th e F r o n t E n d . Th e ver y com m on oes i t; a l l y ou ca n d o u n th e l i n es th a t wer e

If y ou wa n t th e K er n el to b e a wa r e of a l l th e wor k th a t y ou d i d i n a n ol d n oteb ook , y ou m u s t r er u n i t F R OM THE TOP . Th i s i s ea s y . J u s t s el ect th e ou ter m os t b r a ck et for th e wh ol e n oteb ook , a n d h i t E n t e r or S h i f t - R e t u r n ( ). To d o th i s a t a m or e h u m a n p a ce, h ol d d own S h i f t wh i l e r ep ea ted l y h i tti n g

2 1

S y m m e tr y T h e o r y

E n t e r . Th i s j u m p s y ou fr om on e execu ta b l e cel l to th e n ext (a n d th en execu tes i t), s o y ou ca n fol l ow wh a t i s g oi n g on , fi x er r or s a s th ey a r e en cou n ter ed , etc.

3 . 5. 2 F i ve k i n d s of b r a ck ets 1 . S q u a r e b r a c k e t s [ ...] ( O p e r a t o r s ) Th e p r oces s or wor k s o n l y on s q u a r e b r a ck ets . In d eed , s q u a r e b r a ck ets m u s t n ever b e u s ed for a n y th i n g oth er th a n a n O p e r a t o r [ o p e r a n d ] con s tr u cti on . In p a r ti cu l a r , th ey m u s t n ever b e u s ed to g r ou p a l g eb r a i c exp r es s i on s . Ma n y exp r es s i on s con ta i n n es ted s q u a r e b r a ck ets . Th e p r oces s or l ook s for th e i n n er m os t p a i r a n d p r oces s es i t fi r s t. Th en th e exp r es s i on i s s i m p l y r ecy cl ed b a ck i n to th e p r oces s or a fr es h , a n d a g a i n th e i n n er m os t p a i r i s l oca ted . Wh en a l l th e s q u a r e b r a ck ets a r e g on e, th e fu l l y eva l u a ted exp r es s i on i s d i s p l a y ed . Or d i n a r y fu n cti on s a r e j u s t op er a tor s th a t op er a te on n u m b er s , s o we ca n m a k e a s i m p l e exa m p l e of th i s b y n es ti n g s ever a l fu n cti on s a r ou n d a s i n g l e n u m b er : F i r s t eva l u a te A r c T a n [ L o g [ S i n [ 1 . 2 3 ] ] ] . Th en d o th em

on e a t a ti m e: E va l u a te S i n [ 1 . 2 3 ] , th en p a s te th e n u m er i -

ca l va l u e i n to th e s q u a r e b r a ck ets of L o g [ ] , a n d th en th e L o g

i n to th e

A r c T a n , to ver i fy th e " i n n er m os t fi r s t" r u l e.

2 . P a r e n t h e s e s ( ... )

(A lg e b r a ic g r o u p in g )

P a r en th es es a r e u s ed excl u s i vel y for a l g eb r a i c g r ou p i n g : ( u + v ) / ( f

( b ^ 2

+

c ^ 2 ) )

Th i s m ea n s y ou wi l l h a ve to g i ve u p f ( x ) m ea n s “ f ti m es x ” a n d n ever b e wr i tten wi th s q u a r e b r a ck ets a s f [ n ot ever tr y to g r ou p s y m b ol s u s i n g a [ b + c ] . Th e p r oces s or u n d er s ta n d s

y ou r ol d fa vor i te, f ( x ) . In M a t h e m a t i c a , a n y th i n g el s e. Th e fu n cti on “ f of x ” m u s t x ] , or a s on e of i ts a l i a s es , l i k e x / / f . D o cu r l y b r a ces a { b + c } or s q u a r e b r a ck ets th es e exp r es s i on s i n a q u i te d i ffer en t wa y .

2 C o s Φ S i n Φ  C o s Φ 2 S i n Φ 2 After ty p i n g th e a b ove a n d l ook i n g a t i ts ou tp u t, cop y i t a n d p a s te i t i n to T r i g R e d u c e [ ] to s ee wh a t M a t h e m a t i c a k n ows a b ou t tr i g :

2 2

F iv e k in d s o f b r a c k e ts

T r i g R e d u c e 2 C o s Φ S i n Φ  C o s Φ 2  S i n Φ 2  3 . C u r ly b r a c e s { ... }

(L is ts )

Cu r l y b r a ces a r e u s ed excl u s i vel y to en cl os e l i s ts of th i n g s . Th e th i n g s l i s ted s ep a r a ted b y com m a s . Th e l i s t i s th e on e a n d on l y d a ta s tr u ctu r e u s ed M a t h e m a t i c a . It i s a th eor em th a t a n y d a ta s tr u ctu r e i s eq u i va l en t to a n es ted s tr u ctu r e, s o th er e i s n o n eed for a n y th i n g el s e. B el ow, n ote th a t a b c d s i n g l e s y m b ol , wh er ea s { a , b , c , d } i s a l i s t of fou r s y m b ol s . a b c d A B C D

=

a re b y lis t is a

{ a , b , c , d } ;

=

{ A , B , C , D } ;

Ma n y cl ever a l g eb r a i c op er a ti on s ca n b e ca r r i ed ou t on l i s ts . a b c d a b c d

. *

A B C D

( *

t h e

D o t

A B C D

( *

t h e

T i m e s

p r o d u c t

Ma n y fu n cti on s h a ve th e L i s t a b l e exa m p l e :

* )

p r o d u c t p r op er ty .

* ) Th e L o g

fu n cti on

is a n

L o g 1 . , 2 . , 3 . , 4 .   1 1 . , 2 . , 3 . , 4 . 



Her e i s a n es ted l i s t s tr u ctu r e (a m a tr i x), a n d th e b a s i c u s e for i t wi th a vector : m a t r i x v e c t o r

= =

{ { a , b } , { c , d } } ; { e , f } ;

m a t r i x . v e c t o r v e c t o r . m a t r i x

2 3

S y m m e tr y T h e o r y

4 . C o m m e n ts Som tr ou th e ti on

(* …

* )

(ig n o r e )

eti m es y ou wa n t to p u t a s h or t n ote b es i d e s om e i n p u t, wi th ou t g oi n g to th e b l e of m a k i n g a Text cel l for i t. Th i s i s d on e b y en cl os i n g y ou r com m en t i n two-s tr ok e com b i n a ti on (* a s th e op en er , a n d a n oth er two-s tr ok e com b i n a * ) a s th e cl os er . E ver y th i n g i n s i d e wi l l b e i g n or ed b y th e p r oces s or .

F a c t o r a 2

b 2



( * e x a m p l e

o f

5 . D o u b le b r a c k e t s  ...

f a c t o r i n g * )

(P a r t)

D ou b l e s q u a r e b r a ck ets , or d ou b l e-s tr u ck s q u a r e b r a ck ets , a r e th e h u m a n fr i en d l y g u i s e of th e P a r t op er a tor . In th e l i s t { a , b , c } , th e “ b ” i s P a r t 2. You b r i n g i t ou t on i ts own wi th { a , b , c } [ [ 2 ] ] Th e b etter -l ook i n g for m

 b el ow i s ty p ed a s [ [  a n d a s ] ] .

{ a , b , c } 3   Ma n y exp r es s i on h a ve p a r ts th a t h a ve p a r ts . L i s ts .

Her e i s a n exa m p l e of a L i s t of

{ { a , b , c } , { d , e , f } } 2 , 3   P a r t 2 of th e m a i n exp r es s i on i s { d , e , f } . P a r t 3 of th a t i s f .

Secti on 3 . 6 M a t h e m a t i c a g r a p h i cs 3 . 6. 1. Two d i m en s i on a l g r a p h i cs Two- d i m en s i on a l g r a p h i cs a r e m a d e b y th e G r a p h i c s op er a tor wor k i n g on a g r a p h i c “ p r i m i ti ve” , l i k e C i r c l e : c i r c G r a p h i c s C i r c l e , I m a g e S i z e 7 2

2 4

M a th e m a tic a g r a p h ic s

Th e H e a d of th e ou tp u t i s a l s o ca l l ed G r a p h i c s ; th u s , th e ou tp u t i s s a i d to b e a “ G r a p h i cs ob j ect” . H e a d c i r c

G r a p h ic s

Th er e a r e on l y 13 G r a p h i c s p r i m i ti ves . Som e of th e m os t u s efu l a r e C i r  c l e , L i n e , A r r o w , T e x t , R e c t a n g l e , a n d P o l y g o n . L ook u p G r a p h  i c s i n th e D oc Cen ter a n d op en M o r e I n f o to s ee a l i s t of th em a l l . Us u a l l y , th e G r a p h i c s op er a tor i s a p p l i ed to a L i s t s p er s ed wi th va r i ou s d i r e c t i v e s . F i r s t, m a k e s u ch a l i s t: g r L i s P o P o P o

t P o i n t 0 i n t 4 i n t 8

i n t S i z e , 0  , B , 0  , W , 0  , Y

1 l u h i e l

1 e , t e l o

0 , P o i , P o w , P

R e n t i n o i

of p r i m i t i v e s i n ter -

d , 2 , 0  , G r e e n , t 6 , 0  , B l a c k , n t 1 0 , 0  ;

R ea d th e g r L i s t ca r efu l l y . Th e on l y p r i m i t i v e th a t a p p ea r s i s P o i n t [ x , y ] , wh i ch s a y s to d r a w a p oi n t wi th cen ter a t x , y . E ver y th i n g el s e i s a d i r e c t i v e . Th e l i s t s ta r ts wi th th e d i r ecti ve P o i n t S i z e [ 1 / 1 0 ] , wh i ch s a y s th a t ever y fol l owi n g p oi n t i s to h a ve a d i a m eter 1 / 1 0 th e tota l wi d th of th e fi g u r e. Th e col or s n a m es a r e a l s o d i r ecti ves , ea ch es ta b l i s h i n g a col or th a t wi l l a p p l y to a l l fol l owi n g p r i m i ti ves (u n ti l a n oth er col or d i r ecti ve i s g i ven ). N ow feed g r L i s t to th e G r a p h i c s op er a tor , a l on g wi th op ti on s to con tr ol th e P l o t  R a n g e , I m a g e S i z e a n d th e B a c k g r o u n d col or : b i g D o t s G r a p h i c s g r L i s t , P l o t R a n g e 2 , 1 2 , 1 , 1 , I m a g e S i z e 2 0 0 , B a c k g r o u n d G r a y L e v e l 0 . 6

Th e ou tp u t i s th e g r a p h i c i ts el f, a g r a p h i cs ob j ect n a m ed b i g D o t s . L ook a t th e H e a d of b i g D o t s .

Th en ch a n g e th ei r x , y coor d i n a tes a n d

r es tr u ctu r e th e l i s t to m a k e i n ter es ti n g p a tter n of R e d a n d B l a c k d ots wi th d i ffer en t s i z es .

2 5

S y m m e tr y T h e o r y

Ma k e a th i ck l i n e l oca ted on th e s a m e b a ck g r ou n d : t h i c T I m B a

k L h i a g c k

i n c k e S g r

e G n e s s i z e o u n d

T h i c k n e s s [ . 0 2 ] wi d th of th e fi g u r e.

r a p 0 .

2 0

G

h i 0 2 0 , r a

c s ` , L i n e 1 , 0 , 1 1 , 0  , P l o t R a n g e 2 , 1 2 , 1 , 1 , y L e v e l 0 . 6 `

d i r ects a l l fol l owi n g l i n es to h a ve a th i ck n es s 2%

of th e

G r a p h i cs ob j ects ca n b e s h own a g a i n , or com b i n ed , b y th e S h o w op er a tor . l i n e I n F r o n t S h o w b i g D o t s , t h i c k L i n e 

d o t s I n F r o n t S h o w t h i c k L i n e , b i g D o t s 

In th e S h o w l i s t, th e fi r s t ob j ect i s d r a wn fi r s t; th en l a ter ob j ects over l a y ea ch oth er i n l i s t or d er .

3 . 6. 2. Th r ee d i m en s i on a l g r a p h i cs Th r ee d i m en s i on a l g r a p h i cs wor k l i k e two d i m en s i on a l g r a p h i cs , excep t th a t th e op er a tor i s G r a p h i c s 3 D , a n d th e p r i m i ti ves a r e s p eci a l i z ed for th r ee d i m en s i on s . F or i n s ta n ce, P o i n t n ow r eq u i r es th r ee coor d i n a tes , n ot two. Oth er s , l i k e C u b o i d , C y l i n d e r , a n d S p h e r e a r e i n tr i n s i ca l l y 3 D on l y . Al s o, th e d i r ecti ves O p a c i t y , S p e c u l a r i t y , E d g e F o r m , a n d F a c e F o r m a d d r es s i s s u es th a t d o n ot a r i s e i n 2D . We C o m p oi n a s in

2 6

s h ow a s i n g l e exa m p l e, p l e x [ p t L i s t , g r L i s t ts u p fr on t i n p t L i s t , a n teg er s 1 , 2 , … . F or a d i a

m a ] . d th tom

k in g u s Th e i d en r efer i c, we n

e of a h a n d y op er a tor G r a p h i c s  ea b eh i n d i t i s to s p eci fy a l l th e 3 D to th es e p oi n ts i n th e g r L i s t s i m p l y eed on l y two p oi n ts :

M a th e m a tic a g r a p h ic s

d i a t o m p g B r R e B l

i c t L r L o w d , u e

M o i s I S n , S ,

l e c u l e G r a p h i c s C o m p l e x t 1 , 0 , 0 , 1 , 0 , 0 , T  C y l i n d e r 1 , 2 , . 1 , p h e r e 1 , 0 . 5 , S p h e r e 2 , 0 . 4  ;

Ma k e th e m ol ecu l e vi s i b l e b y wr a p p i n g i t i n G r a p h i c s 3 D , wi th a n op ti on th a t con tr ol s th e I m a g e S i z e : d M G r a p h i c s 3 D d i a t o m i c M o l e c u l e , I m a g e S i z e 8 0

B ook r ea d er s wi l l s ee n oth i n g ver y r em a r k a b l e a b ou t th i s . B u t i n ter a cti ve r ea d er s ca n u s e th e m ou s e to d r a g th i s i m a g e a r ou n d i n to a n y d es i r ed or i en ta ti on : d M , d M , d M 



Th i s i s on e of th s tr u ctu r e i s com p exa ctl y . B i och em cu l e s tr u ctu r es , b u

,

,

e ver y b es t th i n g s a b ou t M a t h e m a t i c a ' s l i ca ted y ou ca n m a n eu ver i t a r ou n d u n i s ts h a ve el a b or a te p r og r a m s for d oi n g t h er e i t i s j u s t p a r t of th e cor e of M a t h e m



3 D g r a p h i cs . If th e ti l y ou u n d er s ta n d i t th i s wi th m a cr om ol ea tic a .

2 7

4 . Th e m ea n i n g of s y m m etr y P r el i m i n a r i es

4 . 1. Sy m m etr y a n d i ts u n d efi n ed ter m s P r o p o r t i o n , h a r m o n y , b a l a n c e , a n d b e a u t y o f f o r m a r e ter m s i n vok ed i n d i cti on a r i es to exp l a i n th e m ea n i n g of th e wor d s y m m e t r y . B u t th e for m a l s tu d y of s y m m etr y m u s t b eg i n wi th a m u ch s i m p l er , m or e p r eci s e d efi n i ti on . S y m m e tr y We wi l l s a y th a t a s et of ob j ects h a s a s y m m etr y i f we k n ow of a tr a n s for m a ti on r u l e th a t l ea ves th e s et u n ch a n g ed . Th i s of m ter m ter m

i s a for m a l d od er n l og i c s . Th e wor d s i n th i s d efi n

efi n i ti on of th e wor d s y m m e t r y . On e of th e g r ea t d i s cover i es i s th a t ever y fu n d a m en ta l d efi n i ti on m u s t con ta i n u n d efi n ed s s e t , o b j e c t , a n d t r a n s f o r m a t i o n r u l e a r e th e m a i n u n d efi n ed i ti on .

F or i n s ta n ce, y ou cou l d d efi n e a s et a s a co l l ecti o n w i t h o u t r e g a r d Th i s i s a g ood i n for m a l exp l a n a ti on of th e wor d s e t , b u t i t i s n ot d efi n i ti on , b eca u s e wh a t i s th e for m a l d efi n i ti on of c o l l e c t i o n , or of o r d a r e wor s e off th a n b efor e, b eca u s e we h a ve tr a d ed on e u n d efi n ed ter m oth er s . It i s a n en d l es s cy cl e, a n d i t i s b es t to r ecog n i z e i t a s s u ch , a u p on th e ter m s th a t wi l l b e a ccep ted a s u n d efi n ed .

t o o r d er . a for m a l e r ? We for two n d a g r ee

Th e u n d efi n ed ter m s of a l og i ca l s y s tem a r e n ot a l i m i ta ti on or a fl a w; th ey a r e th e s ou r ce i ts p ower . B eca u s e th ey a r e u n d efi n ed , y ou a r e fr ee to i n ter p r et th em i n a n y wa y th a t y ou ca n m a k e s en s e of th em . B el ow, i n Secti on 4 . 2 (G eom etr i c Sy m m etr y ) a n d Secti on 4 . 3 (Al g eb r a i c Sy m m etr y ) we wi l l g i ve two d i ffer en t i n ter p r eta ti on s of th e wor d s o b j e c t a n d tr a n s fo r m a tio n r u le . Th e m os t cr ea ti ve th i n g th a t a p er s on ca n d o wi th fi n d a n ew i n ter p r eta ti on of i ts u n d efi n ed ter m s ; th i n th e s tu d y of s y m m etr y , a n d i t wi l l h a p p en a g a i n . ta ti on s for o b j e c t a n d t r a n s f o r m a t i o n r u l e , a l l th s y m m etr y th eor y a p p l y to th e n ew ob j ect, i m m ed i a

a for m a l is h a s h a If y ou ca e m a n y tel y i l l u m

l og i ca l p p en ed n fi n d n r i g or ou i n a ti n g

W.M. McClain, Symmetry Theory in Molecular Physics with Mathematica, DOI 10.1007/b13137_4, © Springer Science+Business Media, LLC 2009

s y s tem i s to m a n y ti m es ew i n ter p r es r es u l ts of a n d cl a r i fy -

2 9

S y m m e tr y T h e o r y

i n g i ts p r op er ti es . An ou ts ta n d i n g exa m p l e of th i s wa s th e i n ter p r eta ti on of s p i n a s a n ew s y m m etr i c ob j ect, a fter i t wa s d i s cover ed exp er i m en ta l l y . Al th ou g h i t i s u s el es s to g i ve for m a l d efi n r u l e" , on e ca n s ti l l d i s cu s s th em i n for m a l l y i n th i s ch a p ter , g eom etr i c ob j ects a n d a l g eb m u s t b e s om eth i n g th a t wou l d effect a ch a n a p p l i es , s o th a t th e few l eft u n ch a n g ed b y d es er ve th e s p eci a l a d j ecti ve s y m m e t r i c .

i ti on s of “ ob j ect” or “ tr a n s for m a ti on . We wi l l s ee two k i n d s of “ ob j ects ” r a i c ob j ects . A “ tr a n s for m a ti on r u l e” g e i n m os t of th e ob j ects to wh i ch i t i t a r e i n d eed ver y s p eci a l on es th a t

So a l r ea d y we h a ve a s u b tl e i n s i g h t: Accor d i n g to ou r for m a l d efi n i ti on , a s y m m etr i c s et of ob j ects i s n ever a th i n g en ti r el y on i ts own ; i t i s s y m m etr i c on l y wi th r es p ect to th e tr a n s for m a ti on r u l es th a t h u m a n s for m u l a te a n d l a y u p on i t. Wh en a n ob j ect i s “ i n tu i ti vel y ” s y m m etr i c, th e r u l e i s a s u b con s ci ou s on e. Wh en th i s p r oces s b ecom es con s ci ou s , wi th r u l es for m u l a ted a ccor d i n g to a n i n ten t, we exten d a n d r efi n e th e i d ea of s y m m etr y i n el eg a n t a n d u s efu l wa y s . We wi l l s ee exa m p l es of th i s . B u t fi r s t, i n th i s ch a p ter , th e b a s i cs .

4 . 2. G eom etr i c s y m m etr y 4 . 2. 1. L i n k s to b a ck g r ou n d b r u s h -u p m a ter i a l s If y ou h a ve d on e th e Tu tor i a l a n d y ou s ti l l feel a l i ttl e r u s ty on vector s , m a tr i ces , a n d th e D o t p r od u ct, cl i ck h er e to g o to a m or e d eta i l ed r evi ew a t th e en d of th e b ook . It i s l on g a n d com p r eh en s i ve, b u t th e fi r s t p a r ts of i t a r e wh a t y ou n eed .

4 . 2. 2. G eom etr i c m ea n i n g s for th e u n d efi n ed ter m s We n ow ta k e s p eci fi c m ea n i n g s for th e u n d efi n ed ter m s i n th e d efi n i ti on of s y m m e t r y . We l et s et o f o b j ects = = a s et o f p o i n ts i n 3 - d i m en s i o n a l s p a ce, a n d (to g et s ta r ted ) we l et tr a n s f o r m = = r o ta ti o n o r r ef l ecti o n o f a l l th e p o i n ts . Th e cen ter of g r a a n d th e r efl ecti on g eom etr i ca l tr a n s to g et u s s ta r ted . m ea n i n g s :

3 0

vi ty p la for m R ep

of th e ob j ect i s ta k en a s th e or i g i n , a n d th e r ota ti on a xes n es m u s t a l l con ta i n th e or i g i n . L a ter th i s l i s t of k i n d s of s wi l l b e exten d ed , b u t th es e two s i m p l e k i n d s a r e en ou g h ea ti n g th e d efi n i ti on of s y m m etr y wi th th es e exp l i ci t

G e o m e tr ic s y m m e tr y

" W e w i l l s a y th a t a s et o f g eo m etr i c p o i n ts h a s a s y m m etr y i f w e k n o w r o ta ti o n o r a r ef l ecti o n th a t l ea v es th e p o i n t s et u n ch a n g ed . " As d es cr i b ed a b ove, g eom etr i c s y m m etr y i s a m a tter of vi s u a on e' s m i n d . B u t p eop l e va r y a l ot i n th e th ei r vi s u a l i z a ti on s , h a r d to d es cr i b e con vi n ci n g l y to oth er p eop l e. To d o m a th em d efi n i ti on , we m u s t m a k e u s e of th e g r ea t d i s cover y of R en é ever y m a th em a ti ca l p oi n t i n 3 -d i m en s i on s m a y b e d es cr i b ed n u m b er s { a , b , c } , ca l l ed a v e c t o r .

o f a

l i z i n g th i n g s a n d vi s i on s a a ti cs wi th th D es ca r tes , th a s a tr i p l e

in re is a t of

As we wi l l s ee b el ow, b oth r ota ti on s a n d r efl ecti on s ca n b e d es cr i b ed b y m a t r i c e s of s i z e 3 3 th a t m u l ti p l y th e vector s , u s i n g th e r ow-b y -col u m n r u l e, or th e D o t p r od u ct. B u t vi s u a l i z a ti on ca n n ot a n d s h ou l d n ot b e s u p p l a n ted , a n d a l a r g e p a r t of M a t h e m a t i c a i s d evoted to g r a p h i cs op er a tor s th a t tu r n l i s ts of vector s i n to l i n e d r a wi n g s . So we h a ve th e b es t of b oth wor l d s : a u tom a ted m a tr i x m u l ti p l i ca ti on for s p eed a n d a b s ol u te a ccu r a cy i n ca l cu l a ti on ; a n d a u tom a ted , exa ct p er s p ecti ve l i n e d r a wi n g s of th e r es u l ts , á l a L eon a r d o, for vi s u a l u n d er s ta n d i n g .

4 . 2. 3 . R efl ecti on m a tr i ces D e fin itio n o f “ r e fle c tio n ” Th e “ r efl ecti on ” u s ed i n s y m m etr y th eor y i s n ot a n op ti ca l r efl ecti on . However , i t g ets i ts n a m e fr om th e op ti ca l d i a g r a m of a l i g h t r a y r efl ecti on , b el ow. A r a y fr om a s tr eet l i g h t b ou n ces fr om th e s u r fa ce of a p u d d l e a n d i n to a n ey e (or a ctu a l l y , i n to two s ter eos cop i c ey es ). Th e b r a i n th en u s es th e s tr a i g h t d a s h ed exten s i on of th e ey e' s l i n e of s i g h t to i n ter p r et th e p os i ti on of th e l i g h t a s a p oi n t b el ow th e s u r fa ce of th e g r ou n d , a t a d ep th eq u a l to th e h ei g h t of th e l i g h t a b ove th e g r ou n d . In Sy m m etr y Th eor y , we for g et a b ou t th e ey e a n d th e tr u e p a th of th e r a y , a n d ta k e th e r el a ti on b etween Sou r ce, Mi r r or , a n d Im a g e a s th e “ r efl ecti on ” .

3 1

S y m m e tr y T h e o r y

Sou r ce E y e Mi r r or

Im a g e F ig . 4 .1

O p tic a l r e fle c tio n d ia g r a m

T h e U . S . C a p ito l b u ild in g h a s r e fle c tio n

sy m m e tr y

Con s i d er a n ob j ect th a t s eem s i n tu i ti vel y s y m m etr i c; for i n s ta n ce, th e Un i ted Sta tes ca p i tol b u i l d i n g . Of cou r s e, i t i s n ot a b s ol u tel y s y m m etr i c, b u t we i d ea l i z e. If ou r for m a l d efi n i ti on ca n n ot b e a p p l i ed to th i s i d ea l i z ed ob j ect, th en i t i s n on s en s e; s o l et' s b eg i n h er e. Wh a t i s a tr a n s for m a ti on r u l e th a t l ea ves th e b u i l d i n g u n ch a n g ed ? Th i n k of a ver ti ca l p l a n e p a s s i n g th r ou g h th e cen ter of th e r otu n d a a n d d i vi d i n g th e wh ol e b u i l d i n g i n to eq u a l h a l ves , E a s t Wi n g a n d Wes t Wi n g . Th i s i s a m i r r o r p l a n e for th e b u i l d i n g , i n th e s en s e th a t th e Wes t Wi n g l ook s j u s t l i k e th e r efl ecti on of th e E a s t Wi n g i n a g i a n t m i r r or p l a ced i n th i s p l a n e. To s h a r p en th i s u p , we es ta b l i s h two a xes a xi s z r u n n i n g u p th r ou g h th e cen ter of r u n n i n g ou t th e fr on t d oor . Th en x i s a n m i r r or p l a n e, s a y , a n E a s t-Wes t a xi s th a t r on e wi n g , th r ou g h th e r otu n d a a n d m i r r or p i n th e oth er wi n g .

th a t l i e i n th e m i r r or p l a n e: ver ti ca l th e r otu n d a , a n d h or i z on ta l a xi s y a xi s th a t r u n s p er p en d i cu l a r to th e u n s i n a l on g th e cen tr a l h a l l wa y of l a n e, a n d ou t th r ou g h a s i m i l a r h a l l

Con s i d er a s et of vector s th a t p oi n t to a l l th e m a j or Ca p i tol . On e of i ts m em b er s , { x , y , z } , p oi n ts cen ter of a wi n d ow i n th e E a s t Wi n g . Mi r r or s y m { - x , y , z } p oi n ts to th e cen ter of a s i m i l a r wi n d tr a n s for m a ti on r u l e th a t l ea ves th e ob j ect u n ch a n g ed { x , y , z } In wor d s , th e r u l e i s : U tu r a l v e c to r n e g a te th e Th i s tr a n s for m ch a n g es vi ce-ver s a . If th e two s a m e b efor e a n d a fter th i n a s e t ). If th er e i s n o p

3 2

a r ch i tectu r a l el em en ts of th e fr om th e or i g i n to, s a y , th e m etr y m ea n s th a t th e vector ow i n th e Wes t wi n g . Th e is

{ - x , y , z }

s in g a n o r ig in th a t lie s in th e m ir r o r , in e a c h a r c h ite c c o m p o n e n t th a t is p e r p e n d ic u la r to th e m ir r o r p la n e . ever y E a s t-wi n g vector i n to a Wes t-wi n g vector , a n d wi n g s h a ve tr u e m i r r or s y m m etr y , th e vector s et i s th e e tr a n s for m (i g n or i n g th e or d er of th e vector s , a s a l wa y s i ctu r e b el ow, s el ect a n d en ter th e l i ttl e cl os ed cel l b el ow.

G e o m e tr ic s y m m e tr y

x E a s t wi n d ow E y e R otu n d a y

G i a n t m i r r or ed g e vi ew

Wes t wi n d ow

F ig . 4 .2

A r c h ite c tu r a l s y m m e tr y

A m m o n ia h a s r e fle c tio n s y m m e tr y Th er e i s a n Mol ecu l es ` p we a d op t a s el em en t n a m a tom ta g .

ob vi ou s a n a l og y wi th a ck a g e ou r s p eci fi ca ti on s ta n d a r d for a l l m ol ecu e, i ts Ca r tes i a n p os i ti on

m ol ecu l es . B el ow we ca l l u p fr om th e of th e a m m on i a m ol ecu l e, i n a for m a t th a t l es . E a ch r ow s p eci fi es on e a tom b y i ts i n th e m ol ecu l e (i n An g s tr om s ), a n d i ts

M o l e c u l e T o L i s t " a m m o n i a " ;   C o l u m n F o r m N , 0 , 0 , H ,  2

2 4 69 5 0 0

, 1 , C 3 v 

9 0 3 2 5 0 0 0

, 0 , 1 2 4 69

H ,  5

,

H ,  5

, 

2 4 69 0 0 0 2 4 69 0 0 0

1 67 3 0 0 0 0 3

, 1

5 0 0 0 2 4 69 5 0 0 0

, 1 

3

1 67 3 0 0 0 0

, 1

, 2 

1 67 3 0 0 0 0

, 3 

N ote th a t we u s e n o d eci m a l a p p r oxi m a ti on s i n th e s p eci fi ca ti on of m ol ecu l es . Th e a b ove i s b a s ed on a fou r -d i g i t exp er i m en ta l r es u l t for th e a m m on i a m ol ecu l e, b u t we h a ve r a ti on a l i z ed th e Ca r tes i a n coor d i n a tes a n d u s ed th e exa ct i r r a ti on a l 3

to m a k e th e th r eefol d r ota ti on a l s y m m etr y exa ct.

3 3

S y m m e tr y T h e o r y

Th e op er a ti on of r efl ecti on { x , Th i s m a y b e ca l l ed ei th er p l a n e. We g en er a l i z e th i s d R e fle c tio G i ven a n two a xes n eg a ti on

i n th e x , z p l a n e i s p er for m ed b y th e R u l e y , z } { x , - y , z } . a r efl ecti on p a r a l l el to y , or a r efl ecti on i n th e x z efi n i ti on :

n p la n e y p l a n e, a Ca r tes i a n coor d i n a te s y s tem ca n b e d efi n ed s u ch th a t l i e i n th e p l a n e a n d th e th i r d i s p er p en d i cu l a r . R efl ecti on i s th e of th e p er p en d i cu l a r coor d i n a te.

Is n eg a ti on of th g i ven a b ove? B com p on en t i s ch wi th ta g s 2 a n d 3

e y a xi s a s y m y i n s p ecti on , a n g ed , th e fi r s (th e l a s t two a

m etr y op er a ti on for th e a m m on i a m ol ecu l e, a s on e ca n s ee th a t i t i s . If th e s i g n of ea ch y t two a tom s a r e u n ch a n g ed , wh i l e th e H -a tom s tom s i n th e l i s t) m er el y exch a n g e p l a ces .

Th i s ch a n g es th e L i s t , of cou r s e, b eca u s e or d er i s a n i m p or ta n t p r op er ty of a L i s t . B u t we r eg a r d th e m ol ecu l e a s a s e t of a tom s , n ot a s a L i s t of a tom s , a n d or d er m ea n s n oth i n g i n a s e t . Th i s i d ea i s en for ced b y a l l ou r fu n cti on s th a t op er a te on m ol ecu l e ob j ects . We h a ve g r a p h i cs fu n cti on s , a m ol ecu l a r wei g h t fu n cti on , a cen ter -of-m a s s fu n cti on , a n i n er ti a ten s or fu n cti on , a n d oth er s . B u t i n a l l of th em th e ou tp u t i s i n d ep en d en t of th e or d er of th e a tom s i n th e m ol ecu l e ob j ect. To s ee i f two m ol ecu l e ob j ects r ep r es en t th e s a m e m ol ecu l e, a p p l y S o r t to ea ch m ol ecu l e a n d th en a p p l y th e E q u a l q u es ti on . If th ey a r e tr u l y i d en ti ca l excep t for a tom or d er , th e a n s wer wi l l b e T r u e . M a tr ix fo r m u la fo r r e fle c tio n in th e y d ir e c tio n In a com p u ter , a n ea s y wa y to ch a n g e vector { x , y , z } i n to vector { x , - y , z } i s to m u l ti p l y i t b y a s i m p l e m a tr i x. If th e vector i s wr i tten a s a col u m n , i t m u s t b e wr i tten on th e r i g h t s i d e of th e m a tr i x. Th e u s u a l r ow-b y -col u m n m u l ti p l i ca ti on r u l es a r e em b od i ed b y th e D o t op er a tor i n M a t h e m a t i c a , wr i tten q u i te l i ter a l l y a s a n or m a l p u n ctu a ti on p e r i o d b etween th e m a tr i x a n d th e col u m n vector . We a p p l y M a t r i x F o r m to th e r es u l t to h a ve i t wr i tten a s a col u m n 1 0 0 x y z

3 4

0 1 0

x 0

 M a t r i x F o r m y

. 0 1

z

G e o m e tr ic s y m m e tr y

In a m ol ecu l e wi th fou r a tom s , y ou ca n d o a l l th e a tom s a t on ce i f th e a tom p os i ti on vector s a r e wr i tten a s th e col u m n s of a 3 -b y -4 m a tr i x. 1

0 1 0

0 0 x 1 y1 z1

x 1

x 2

x 3

x 4

y 1

y 2

y 3

y 4

z 1

z 2

z 3

z 4

0 . 0 1 x 2 y2 z2

x 3 y3 z3

 M a t r i x F o r m

x 4 y4 z4

Ma tr i x m u l ti p l i ca ti on i s a n extr em el y fa s t op er a ti on , s o even p i ctu r es wi th th ou s a n d s of p oi n ts ca n b e tr a n s for m ed th i s wa y ver y q u i ck l y . Ma n y of th e com p u ter g r a p h i cs a n i m a ti on s th a t y ou s ee i n m ovi es a n d tel evi s i on a d ver ti s i n g a r e b a s ed on fr a m e-b y -fr a m e m u l ti p l i ca ti on of a 3 3 tr a n s for m m a tr i x (on th e l eft) d otted i n to a 3  N p oi n t m a tr i x (on th e r i g h t ). Th i s op er a ti on i s s o fa s t i t ca n b e d on e i n r ea l ti m e i n s om e m ovi es th a t y ou s ee i n com p u ter g a m es . L a ter , we wi l l s ee a s i m i l a r m a tr i x s ch em e, of s i z e ( 4  4 ) . ( 4  N ) , for tr a n s l a ti n g a n d r ota ti n g a t th e s a m e ti m e. It i s tr a d i ti on a l to g i ve a l l r efl ecti on m a tr i ces a n a m e th a t s ta r ts wi th Σ. In th e S y m m e t r y ` p a ck a g e we d efi n ed m a tr i ces Σx , Σy , a n d Σz , a n d i n th e p r el i m i n a r i es we m a d e th em eq u i va l en t to th e s u b s cr i p ted for m s Σx , Σy , a n d Σz : M a p M a t r i x F o r m , Σx , Σy , Σz 



1 0 0 0

0 1

0 0

1

1 0 , 0

0 1 0

0 0

1

0

0

0

1 0

0

0 1

, 1



N u m e r ic a l e x a m p le o f m o le c u le r e fle c tio n We l ook a t a r ea l exa m p l e. Th e { x , y , z } coor d i n a tes of th e a m m on i a m ol ecu l e ca n b e b r ou g h t u p b y th e com m a n d R e s t P o i n t s " a m m o n i a "  M a t r i x F o r m 0 2 4 69 2 5 0 0

2

4 69 5 0 0 0

2

4 69 5 0 0 0

9 0 3 2 5 0 0 0  1 67 3 1 0 0 0 0

0 0 2 4 69

3

5 0 0 0



2 4 69 5 0 0 0

3



1 67 3 1 0 0 0 0



1 67 3 1 0 0 0 0

B u t a b ove, th e coor d i n a tes of ea ch a tom a r e wr i tten a s a r ow. We n eed th em col u m n s , wh i ch i s ea s i l y d on e b y th e T r a n s p o s e op er a tor :

a s

3 5

S y m m e tr y T h e o r y

a m m o n i a C o o r d i n a t e s T r a n s p o s e R e s t P o i n t s " a m m o n i a "

;   M a t r i x F o r m 0

2 4 69 2 5 0 0

0

0

9 0 3 2 5 0 0 0

1

2

2

4 69 5 0 0 0

2 4 69

4 69 5 0 0 0

3



5 0 0 0

1 67 3 0 0 0 0

1

0

0

9 0 3 2 5 0 0 0

 1 1 0 670 0 3 0

Σy . a m m o n i a C o o r d i n a t e s ;

2

2

4 69 5 0 0 0



1 67 3 0 0 0 0

th e l eft b y th e r efl ecti on m a tr i x Σy :

r e f l e c t e d A m m o n i a   M a t r i x F o r m 2 4 69 2 5 0 0

3

5 0 0 0

1

1 67 3 0 0 0 0

N ow we ca n m u l ti p l y fr om

0

2 4 69

2 4 69 5 0 0 0

 1 1 0 670 0 3 0

Col u m n s 3 a n d 4 a r e b oth H a m ea n i n g l es s (a s l on g a s b oth m ol ecu l e i s u n ch a n g ed . Th e i n v a r i a n t u n d er th e g i ven tr a n s

3

4 69 5 0 0 0

2 4 69

3

5 0 0 0

 1 1 0 670 0 3 0

tom s , a n d h a ve tr a d ed p l a ces . B u t col u m n or d er i s col u m n s r efer to th e s a m e k i n d of a tom ) s o th e tech n i ca l l a n g u a g e for th i s i s th a t th e a tom s et i s for m .

P ic tu r e s o f s y m m e tr y a n d n o n -s y m m e tr y r e fle c tio n s We h a ve d efi n ed a n p i ctu r e of a n y g i ven m a tr i x, tr a n s for m s th p i ctu r e of th e m ol ecu

op er a tor ca l l ed m ol ecu l e, extr a em b y th e g i ven l e a fter tr a n s for

on th e m { 4 , 2 , 3 l ook s d i r m a ti ca l l y

m m o n i a " . We a l s o i n cl u d e th e op ti on V i e w P o i n t

ts th e vi ewi n g ey e a t th i s p os i ti on i n 3 D s p a ce. Th e ey e th e or i g i n , a n d th e vi ew th a t i t s ees i s r en d er ed i n m a th ep ecti ve.

ol ecu l e " a } , wh i ch p u ectl y towa r d p er fect p er s

S h o w O p e r a t i o n wh i ch d r a cts th e r es t p oi n ts of i ts a tom s y m m etr y m a tr i x, a n d th en d r m a ti on . B el ow, we a s k i t to u

ws s a a ws s em

a b s a a n a tr

e fo r e 3 N a fte r i x Σy

We fi r s t m a k e a G r a p h i c s 3 D ob j ect th a t s h ows wh a t we wa n t, b u t wi th q u i te a b i t of u n wa n ted wh i te s p a ce a r ou n d i t. Th e 3 D ob j ect ca n n ot b e cr op p ed . So we R a s t e r i z e i t to m a k e i t i n to a two-d i m en s i on a l G r a p h i c s ob j ect, a n d cr op i t u s i n g P l o t R a n g e i n s i d e th e S h o w op er a tor .

3 6

G e o m e tr ic s y m m e tr y

f i g 3 S h o w O p e r a t i o n Σy , " a m m o n i a " , V i e w P o i n t 4 , 2 , 3 , S f i g 3 S h o w I m a

t y l e R R f i g g e S i

" T B " , I m a s t e r i z e f 3 R , P l o t R a z e 2 0 0 ,

a g i g n g 1 0

e S i z e 3 0 0 , 3 0 0 , T i c k s N o n e ; 3 , I m a g e R e s o l u t i o n 3 0 0 ; e 4 0 , 2 7 0 , 1 0 0 , 2 3 0 , 0 

F i g . 4 . 3 Am m on i a wi th n u m b er ed a tom s , b efor e a n d a fter r efl ecti on i n th e z , x p l a n e. H a tom s 2 a n d 3 s wi tch p l a ces . B u t th ey a r e i d en ti ca l , s o th e m ol ecu l e i s u n ch a n g ed . Th i s i s s y m m etr y .

Th e m ol ecu l e l ook s th e s a m e b e f o r e a n d a f t e r (excep t for a tom l a b el s , wh i ch d on ' t cou n t), s o a ccor d i n g to ou r d efi n i ti on Σy i s a s y m m etr y op er a ti on for th e a m m on i a m ol ecu l e- i t i s a tr a n s for m th a t l ea ves th e ob j ect u n ch a n g ed . B u t to r ea l l y u n d er s ta n d th i s , y ou m u s t a l s o s ee a n op er a ti on on a m ol ecu l e th a t i s N OT a s y m m etr y op er a ti on : f i g 4 S f i g 4 f i g 4 I m

S h t y l e R R S S a g e S

o w O

" a s t h o w i z e

p e r a T B " , e r i z f i g

2

t i o I m e f 4 R , 0 0 ,

n Σx a g e S i g 4 , P l o 1 0 0

, " i z e I m t R a 

a m m o n i a "

3 0 0 , a g e R e s o l n g e 4

, V 3 0 u t 0 ,

i e w 0 , i o n 2 7 0

P o i T i c

3 ,

n t k s 0 0 1

4 , 2 , 3 ,

N o n e ;

; 0 0 , 2 3 0 ,

F i g . 4 . 4 Am m on i a , b efor e a n d a fter r efl ecti on i n th e y , z p l a n e. Th e a tom s m oved , s o y , z r efl ecti on i s n ot a s y m m etr y of a m m on i a .

E a ch a tom “ r efl ects th r ou g h ” th e z , y p l a n e (or s ta y s p u t, i f i t l i es i n th e p l a n e). Th e m ol ecu l e a f t e r l ook s q u i te d i ffer en t th a n b e f o r e , s o Σx i s N OT a s y m m etr y op er a ti on for th i s m ol ecu l e.

3 7

S y m m e tr y T h e o r y

4 . 2. 4 . R ota ti on m a tr i ces R o ta tio n a l s y m m e tr y R ota ti on a l s y m m etr y i s a n oth er com i t i s em b od i ed b y th e r otu n d a a n d d r es t of th e b u i l d i n g . It h a s 24 i d en oth er fea tu r es , s o we s a y th a t i t h a xi s th a t r i s es ver ti ca l l y th r ou g h i ts P h y s i ca l l y , Ca p i tol wou th i s wou l d en ti r e n a ti on Ou u m top tu r

m on i n ter p r eta ti on of th e wor d “ s y m m etr y “ ; om e of th e ca p i tol , con s i d er ed a p a r t fr om th e ti ca l s eg m en ts , ea ch wi th a l i ttl e wi n d ow a n d a s “ 24 -fol d ” r ota ti on a l s y m m etr y a b ou t a z exa ct cen ter .

we m ea n th a t i f th e d om e wer e j a ck ed u p a n d tu r n ed b y 2 Π/ 2 4 , th e l d b e exa ctl y th e s a m e b efor e a s a fter . As a p u b l i c wor k s p r oj ect, cer ta i n l y b r i n g h om e th e m ea n i n g of r ota ti on a l s y m m etr y to th e .

r m od el a m m on i a m ol ecu l e h a s a s i m i l a r r ota ti on s y m m etr y . It i s a l i ttl e b r el l a , wh er e th e u m b r el l a h a n d l e i s th e z a xi s , a n d th e b l u e N a tom i s a t th e of th e u m b r el l a . B u t i n th i s ca s e th e s y m m etr y r ota ti on i s on l y a th i r d of a n . S h o w A t o m G r a p h i c s " a m m o n i a " , A t o m L a b e l s  F a l s e , B o n d G r a p h i c s " a m m o n i a " , 1 . 2 , . 0 1 5 , B o x e d  F a l s e ; A x i a l V i e w s a m m , I m a g e S i z e 2 0 0 , 2 0 0 

a m m

F i g . 4 . 5 Com p l ete i n for m a ti on on th e s tr u ctu r e of a m m on i a .

3 8

G e o m e tr ic s y m m e tr y

A m a tr ix fo r m u la fo r r o ta tio n a b o u t th e z a x is B efor e we r ota te wh ol e ob j ects , l et u s th i n k a b m a ti ca l p oi n t a r ou n d th e z -a xi s . We b eg i n b y { x , y , z } i n cy l i n d r i ca l coor d i n a tes { r , Α, z z a xi s . Th e g en er a l p oi n t b efor e tr a n s for m a ti on p t B e f o r e x , y , z  .

ou t h ow to r ota te a s i n g l e m a th ewr i ti n g a g en er a l Ca r tes i a n p oi n t } , ta k i n g th e cy l i n d er a xi s a s th e is

x r C o s Α ,

y r S i n Α 

r C os Α, r S in Α, z

B efor e tr a n s for of r , Α, a n d z . s i m p l e r ota ti on z m u s t s ta y con

m a ti on , Th e fi n th r ou g h s ta n t a n

th a l s d

e p oi n t cou l d b e a n y wh er e, a s s p eci fi ed b y p oi n t, h owever , m u s t b e r el a ted to th e i n i ti a l om e a n g l e Β a b ou t th e z a xi s . Th i s m ea n s th e a n g l e m u s t ch a n g e fr om Α to Α + Β. Th u s

p t A f t e r p t B e f o r e

.

th e va l u es p oi n t b y a th a t r a n d we wr i te

Α ΑΒ

r C os Α  Β, r S in Α  Β, z

T r i g E x p a n d r ewr i tes tr i g on om etr i c fu n cti on s of s u m s , d i ffer en ce, or m u l ti p l es of a n g l es i n ter m s of th e i n d i vi d u a l a n g l es . p t A f t e r 2 T r i g E x p a n d p t A f t e r

r C os Α C os Β  r S in Α S in Β, r C os Β S in Α  r C os Α S in Β, z

In th i s for m we ca n r ever s e th e or i g i n a l s u b s ti tu ti on , tr a d i n g off Α a n d r fa vor of th e or i g i n a l x a n d y :

in

p t A f t e r 3 p t A f t e r 2 . r C o s Α x , r S i n Α y  x C os Β  y S in Β, y C os Β  x S in Β, z

Th { x lin m a

is , ea tr

i s th e p os i ti on of th e p oi n t a fter r y , z } a n d th e r ota ti on a n g l e Β. r i n x , y , a n d z . Th er efor e, we i x ti m es th e or i g i n a l vector { x , y ,

ota E a ca n z }

ti on , i n ter m s of th e or i g i n a l p os i ti on ch of th e th r ee vector com p on en ts i s r ewr i te th i s exp r es s i on a s a r o t a t i o n .

In e f lis p u

fa ct, th e m a tr i x m a y b e extr a cted b y a h a n d y op er a tor f i c i e n t s . It ta k es two op er a n d s ; fi r s t, a l i s t of exp t of va r i a b l es , i n wh i ch th e exp r es s i on s m u s t b e l i n ea r el y m a th em a ti ca l exa m p l e. Con s i d er two exp r es s i on s

ca l l ed M a t r r es s i on s , a n d r . F i r s t, we l i n ea r i n x , y

i x O f C o  s econ d , a tr y i t on a , a n d z .

t w o E x p r e s s i o n s  c 1 1 x  c 1 2 y  c 1 3 z , c 2 1 x  c 2 2 y ;

3 9

S y m m e tr y T h e o r y

Th e m a tr i x of coeffi ci en ts extr a cted b y th e op er a ti on M a t r i x O f C o e f f i c i e n t s t w o E x p r e s s i o n s , x , y , z   M a t r i x F o r m 

c c

1 1

c

1 2

2 1

c

2 2

c

1 3

0



N ote th a t a n y m i s s i n g va r i a b l e i s a u tom a ti ca l l y a s s i g n ed a z er o coeffi ci en t, s o th e m a tr i x i s a l wa y s r ecta n g u l a r . N ow we a r e r ea d y to extr a ct th e s q u a r e m a tr i x of coeffi ci en ts for r ota ti on a b ou t th e z a xi s b y a n g l e Β : r o t M a t M a t r i x O f C o e f f i c i e n t s p t A f t e r 3 , x , y , z  ; M a t r i x F o r m r o t M a t

C os Β S in Β 0 S in Β C os Β 0 0 0 1

Th i s i s a b ea u ti fu l r es u l t. It s a y s th a t i n or d er to r ota te a n y p oi n t { x , y , z } a r ou n d th e z a xi s b y a n g l e Β, we s i m p l y l eft-m u l ti p l y th e p oi n t b y th i s m a tr i x. L et' s ch eck i t, u s i n g th e s ta r ti n g exp r es s i on p t A f t e r 3 a b ove for th e r i g h t s i d e : C o s Β S i n Β 0 S i n Β C o s Β 0 0 0 1

x y . z

x C o s Β  y S i n Β

y C o s Β  x S i n Β

z

Tr u e

E ven b etter , th e for m a t m a t r i x . o l d V e c t o r = = n e w V e c t o r i s exa ctl y th e s a m e a s for r efl ecti on . If we k eep a t i t, we ca n d evel op m a tr i ces r ep r es en ti n g a l l p os s i b l e r efl ecti on s a n d r ota ti on s . P er h a p s i t' s ob vi ou s , b u t th i s for m a t a l s o m ea n s th a t we ca n r ota te m a n y vector s a t on ce, j u s t a s we r efl ected m a n y vector s a t on ce, u s i n g for m u l a s wi th col u m n vector p oi n ts , l i k e C o s Β S i n Β 0 S i n Β C o s Β 0 0 0 1

4 0

x 1

x 2

x 3

x 4

y 1

y 2

y 3

y 4

z 1

z 2

z 3

z 4

.

 G r i d F o r m

C os Β x 1  S in Β y1

C os Β x 2  S in Β y2

C os Β x 3  S in Β y3

C os Β x 4  S in Β y4

S in Β x 1  C os Β y1

S in Β x 2  C os Β y2

S in Β x 4  C os Β y4

z1

z2

S in Β x 3  C os Β y3 z3

z4

G e o m e tr ic s y m m e tr y

Two-d i m en s i on a l r ota ti on s (a b ou t a n a xi s i m p l i ci tl y p er p en d i cu l a r to th e p a g e) m u s tb e x C o s Β S i n Β

.  y S i n Β C o s Β

C os Β x S in Β x 1

1

1

x 2

1

y 2

  G r i d F o r m

 S in Β y1  C os Β y1

C os Β x S in Β x 2

2

 S in Β y2  C os Β y2

S ta n d a r d n o ta tio n fo r r o ta tio n s

C n

d en otes a r ota ti on b y 2 Π/ n r a d i a n s a b ou t a n u n s p eci fi ed a xi s .

C

n , x

d en otes a r ota ti on b y 2 Π/ n r a d i a n s a b ou t th e x a xi s .

x Φ i s a m a tr i x for r ota ti on b y Φ r a d i a n s a b ou t th e x a xi s , a n d s i m i l a r l y for y a n d z .

C

D oes th e S y m m e t r y ` p a ck a g e k n ow th i s n ota ti on ? Su b s cr i p ts a r e for b i d d en i n p a ck a g es , b u t i n th e P r e l i m i n a r i e s we s et s om e n ota ti on a l eq u a l i ti es th a t p er m i t u s to wr i te C

Φ  G r i d F o r m x

1 0 0

0 C os Φ S in Φ

0 S in Φ C os Φ

r ig h t h a n d e d r o ta tio n P os i ti on y ou r r i g h t h a n d wi th th e th u m b exten d ed a n d p oi n ti n g i n th e p os i ti ve d i r ecti on a l on g th e a xi s of r ota ti on , a n d cu r l y ou r fi n g er s . Th e fi n g er s th en p oi n t i n th e s en s e of a r i g h t h a n d ed r ota ti on . A n o th e r w a y o f s a y in g it: R i g h t h a n d ed r ota ti on s a r e cou n ter cl ock wi s e, a s vi ewed fr om a p oi n t on th e p os i ti ve s i d e of th e a xi s of r ota ti on , l ook i n g towa r d th e or i g i n . R i g h t h a n d e d i s m ea n i n g l es s i f th e a xi s of r ota ti on d oes n ot h a ve p l u s a n d m i n u s en d s . Th u s , i f a n a xi s of r ota ti on i n a d r a wi n g i s j u s t a s i m p l e l i n e wi th n o

4 1

S y m m e tr y T h e o r y

i n d i ca ti on of th e p os i ti ve en d (s u ch a s a p l u s s i g n , or a n a r r owh ea d , or a n a xi s n a m e), th en th er e i s n o wa y to k n ow wh i ch s en s e of r ota ti on i s r i g h t h a n d ed . Th e u n d efi n ed ter m s of th e h a n d ed n es s d efi n i ti on a r e r i g h t h a n d a n d c o u n t e r c l o c k w i s e . Th i s h a s g i ven r i s e to s om e d eep th ou g h ts . Wh a t wou l d y ou u n d er s ta n d h er e i f y ou k n ew n oth i n g a b ou t h a n d s or cl ock s ? Th e d efi n i ti on r es ts on l y on h u m a n con ven ti on s , y et th e wor l d i s fu l l of h a n d ed n es s a s y m m etr i es ; m os t n ota b l y i n th e r ea l m of b i om ol ecu l es . Th e el ectr om a g n eti c for ce, th a t g over n s th e s tr u ctu r e of m ol ecu l es , h a s n o i n tr i n s i c h a n d ed n es s , s o r i g h t- a n d l efth a n d ed ver s i on s of h a n d ed m ol ecu l es h a ve exa ctl y th e s a m e en er g y i n a l l s ta n d a r d q u a n tu m ch em i ca l ca l cu l a ti on s . Th e en er g i es s p l i t i n a m a g n eti c fi el d , b u t th e effect i s ver y ti n y for fi el d s th e s i z e of th e ea r th ' s . At a m or e fu n d a m en ta l l evel , th e “ wea k ” i n ter a cti on , wh i ch d oes a p p l y to el ectr on s , i s h a n d ed . B u t i t i s s o s m a l l th a t i t ca n n ot i n fl u en ce ch em i s tr y . Yet th e a m i n o a ci d s extr a cted fr om l i vi n g m a ter i a l s a r e a l l 100% h a n d ed , a n d th e h a n d ed n es s i s a l wa y s th e s a m e th r ou g h ou t a l l of b i ol og y . Th i s a s y m m etr y r em a i n s a m y s ter y , on e of th e few th a t 20th cen tu r y s ci en ce fa i l ed to i l l u m i n a te. Mos t s ci en ti s ts b y n ow b el i eve th a t th e s ou r ce of th i s a s y m m etr y i s n ot to b e s ou g h t i n p h y s i cs , b u t i n th e extr em e ea r l y s ta g es of D a r wi n i a n evol u ti on , wh en b a r el y l i vi n g p ol y m er s y s tem s wer e fi r s t em er g i n g fr om a p r i m or d i a l ch em i ca l s ou p . If m i xed h a n d ed n es s ever exi s ted , th en s om e ea r l y evol u ti on a r y a d va n ta g e d evel op ed i n on l y on e h a n d ed n es s , i n a wa y th a t cou l d n ot b e a d op ted b y th e oth er . P er h a p s i t wa s a s m a l l p ol y m er th a t cou l d b i n d l eft-X (b u t n ot r i g h tX ) a n d con ver t i t to s om eth i n g u s efu l . So th e or g a n i s m s th a t h a d on l y r i g h t-X fel l i n to a d i s a d va n ta g e, a n d d wi n d l ed to exti n cti on . R ota ti on s a b ou t th e x -a xi s a n d th e y -a xi s m a y b e d er i ved j u s t l i k e th e z r ota ti on s , a b ove. (Th i s i s a n excel l en t exer ci s e; tr y i t! ) Th e r es u l ts a r e a l r ea d y i n th e S y m m e t r y ` p a ck a g e : Β  M a t r i x F o r m

C y

C os Β 0 0 1 S in Β 0

C z

S in Β 0 C os Β

Β  M a t r i x F o r m C os Β S in Β 0 S in Β C os Β 0 0 0 1

4 2

G e o m e tr ic s y m m e tr y

N u m e r ic a l e x a m p le o f m o le c u le r o ta tio n You n eed to s ee a n u m er i ca l exa m p l e to g et th e fu l l m ea n i n g h er e. Ab ove, we d efi n ed a m m o n i a C o o r d i n a t e s T r a n s p o s e R e s t P o i n t s " a m m o n i a "

; a m m o n i a C o o r d i n a t e s  M a t r i x F o r m 0

2 4 69 2 5 0 0

0

0

9 0 3 2 5 0 0 0

1

2

2

4 69 5 0 0 0

2 4 69

4 69 5 0 0 0

3



5 0 0 0

1

1 67 3 0 0 0 0

2 4 69

1

1 67 3 0 0 0 0

3

5 0 0 0 1 67 3 0 0 0 0

Th e fi r s t col u m n r ep r es en ts a n a tom i s on th e z a xi s , wh i ch m u s t b e th e n i tr og en a tom . Th e oth er th r ee col u m n s a r e th e H a tom s . N ow we r ota te a l l th e a tom p os i ti on s a t on ce a r ou n d th e z a xi s b y th e m a tr i x for m u l a r o t a t e d A m m o n i a C z 2 Π  3 . a m m o n i a C o o r d i n a t e s 2

2

4 69 5 0 0 0

0 0 9 0 3 2 5 0 0 0

Th e n i tr og en r ota ti on . Th col u m n or d er exp ected , a th

2 4 69

4 69 5 0 0 0

3

5 0 0 0



1 67 3 1 0 0 0 0



2 4 69



2 4 69 2 5 0 0 3

0

5 0 0 0 1 67 3 1 0 0 0 0

 M a t r i x F o r m



1 67 3 1 0 0 0 0

a tom (fi r s t col u m n ) wa s u n m e r ota ti on h a s s wi tch ed th e (a tom or d er ) i s m ea n i n g l es s , r ee-fol d r ota ti on a b ou t z i s a s

oved b eca u s e i t l i es on th r ee H -a tom col u m n s a s o th e m ol ecu l e i s u n ch y m m etr y op er a ti on for a m

th e a xi s of r ou n d , b u t a n g ed . As m on i a .

P ic tu r e s o f s y m m e tr y a n d n o n -s y m m e tr y r o ta tio n s S h o w O p e r a t i o n C z 2 Π  3 , " a m m o n i a " , V i e w P o i n t  0 , 0 , 4 , I m a g e S i z e 3 0 0

4 3

S y m m e tr y T h e o r y

F i g . 4 . 6 R ota ti on of a m m on i a b y a th i r d of a tu r n IS a s y m m etr y op er a ti on .

Wi th ou t th e l a b el s , on e ca n n ot tel l a b ove th a t a n y th i n g h a p p en ed . s y m m etr y tr a n s for m . L ook a t r ota ti on b y h a l f a tu r n :

Th i s i s a

S h o w O p e r a t i o n C z Π , " a m m o n i a " , V i e w P o i n t  0 , 0 , 4 , I m a g e S i z e 3 0 0

F i g . 4 . 7 R ota ti on of a m m on i a b y a h a l f tu r n i s N OT a s y m m etr y tr a n s for m .

Her e y ou d o n ot n eed l a b el s to s ee th a t s om eth i n g h a p p en ed . s y m m etr y tr a n s for m .

4 4

Th i s i s N OT a

G e o m e tr ic s y m m e tr y

4 . 2. 5. N ew s y m m etr i es fr om

ol d

If y ou k n ow of two tr a n s for m s th a t l ea ve a n ob j ect u n ch a n g ed , th en i t m u s t b e tr u e th a t a p p l y i n g fi r s t on e a n d th en th e oth er a l s o l ea ves th e ob j ect u n ch a n g ed . Th i s s i m p l e i d ea h a s a ver y i m p or ta n t con s eq u en ce. L ook a t a n exa m p l e. a m m o n i a C o o r d i n a t e s 0

2 4 69 2 5 0 0

0

0 

9 0 3 2 5 0 0 0

 M a t r i x F o r m

5

5

2 4 69 0 0 0

2 4 69

2 4 69 0 0 0

3

5 0 0 0



1 67 3 1 0 0 0 0

1 67 3 1 0 0 0 0



2 4 69

3

5 0 0 0



1 67 3 1 0 0 0 0

If we a p p l y fi r s t th e r efl ecti on a n d th en th e r ota ti on a b ou t z b y a th i r d of a tu r n , th e ca l cu l a ti on i s 2 Π  3 . Σy . a m m o n i a C o o r d i n a t e s  M a t r i x F o r m

C z

2

4 69 5 0 0 0

0

2 4 69

0

3



1 67 3 1 0 0 0 0



4 69 5 0 0 0

 0

5 0 0 0

9 0 3 2 5 0 0 0

2

2 4 69 2 5 0 0

1 67 3 1 0 0 0 0

2 4 69

3

5 0 0 0



1 67 3 1 0 0 0 0

Th e r es u l t i s th e s a m e m ol ecu l e, a s i t m u s t b e, th ou g h we h a ve n ot s een th i s p a r ti cu l a r or d er of th e H a tom s b efor e. Ma tr i x m u l ti p l i ca ti on i s a s s oci a ti ve, s o th i s i s th e s a m e a s C

2 Π  3 . Σy . a m m o n i a C o o r d i n a t e s z

2

4 69 5 0 0 0

0

2 4 69

0

1

9 0 3 2 5 0 0 0

2

2 4 69 2 5 0 0 3

0

5 0 0 0 1 67 3 0 0 0 0

1

1 67 3 0 0 0 0

 M a t r i x F o r m

4 69 5 0 0 0



2 4 69

1

3

5 0 0 0 1 67 3 0 0 0 0

Th i s s h ows th a t th e s i n g l e n ew m a tr i x, g i ven b y C z

2 Π  3 . Σy 3

1 3 1

2

0

0

2 2

0 2

0

 M a t r i x F o r m

1

4 5

S y m m e tr y T h e o r y

i s a l s o a s y m m etr y tr a n s for m of th e a m m on i a m ol ecu l e. L a ter we wi l l s ee th a t th i s m a tr i x r ep r es en ts r efl ecti on i n a l i n e r ota ted b y 1/3 tu r n fr om th e y -a xi s . It i s d i ffer en t fr om ei th er of th e oth er two th a t we h a ve b een con s i d er i n g , wh i ch we r ep ea t j u s t b el ow for com p a r i s on : C z

2 Π  3  M a t r i x F o r m , 2 1

 3 2

0

3



0

2

2 0

0

1

, 1

1 0 0 0

Σx

 M a t r i x F o r m 

0 0 0

1



1

Th i s i l l u s tr a tes a n i m p or ta n t p r i n ci p l e: If y ou m u l ti p l y a n y s y m m etr y tr a n s for m m a tr i ces of th e s a m e ob j ect, y ou g tr a n s for m m a tr i x of th a t ob j ect. An oth er wa y of s a y i n g i m p l y th e exi s ten ce of a th i r d s y m m etr y , wh i ch m a y b e d two, a n d wh i ch y ou m a y or m a y n ot h a ve n oti ced d i r ectl y .

two m a tr i ces et a n oth er s y i t: Two s y m i ffer en t fr om

th a t a r e m m etr y m etr i es th e fi r s t

4 . 3 . Al g eb r a i c s y m m etr y 4 . 3 . 1. Al g eb r a i c m ea n i n g s for th e u n d efi n ed ter m s We n ow ta k e s p eci fi c m ea n i n g s for th e u n s y m m e t r y . We l et o b j ect = = a l g eb r a i c ex r ep l a cem en t o f v a r i a b l es th a t b el on g to th ti on of s y m m etr y wi th th es e exp l i ci t m ea n

d efi n ed ter m s i n th e d efi n i ti on of p r es s i o n , a n d we l et tr a n s f o r m = = e exp r es s i on . R ep ea ti n g th e d efi n i in g s :

" W e w i l l s a y th a t a s et o f a l g eb r a i c ex p r es s i o n s h a s a s y m m etr y i f w e k n o w o f a r ep l a cem en t o f v a r i a b l es th a t l ea v es th e s et u n ch a n g ed . " R ep l a cem en t of va r i a b l es i s a n op er a ti on th a t l i es a t th e h ea r t of M a t h e m a t i c a , b u t u s u a l l y , i t i s N OT a s y m m etr y tr a n s for m . F or i n s ta n ce, i n th e exp r es s i on x 3 we ca n r ep l a ce x b y - x i f we wr i te . 3

x x

x x

3

Th e ou tp u t i s a l g eb r a i ca l l y d i ffer en t fr om th e i n p u t a n d th er e i s n o s y m m etr y . Wh en we s a y u n c h a n g e d we m ea n i t q u i te l i ter a l l y . L ook a t th e fol l owi n g , p er h a p s th e m os t fa m ou s of exa m p l es :

4 6

A lg e b r a ic s y m m e tr y

. 2

x

x x

2

x

N ow th er e, th a t i s wh a t we m ea n b y s y m m etr y i n a n a l g eb r a i c exp r es s i on . Th e ou tp u t exp r es s i on i s E X ACTL Y th e s a m e a s th e i n p u t exp r es s i on Or , a s we wi l l s ee, i s a l g eb r a i ca l l y eq u i va l en t to i n p u t exp r es s i on .

4 . 3 . 2. E xa m p l es of a l g eb r a i c s y m m etr y S in e a n d c o s in e u n d e r x , y e x c h a n g e Ta k e a s et of two exp r es s i on s , th e Ca r tes i a n exp r es s i on s for s i n e a n d cos i n e, a n d th en s wa p x a n d y : y



x

 .  x y ,

, 2

x

y 2

x

x



2

y

y ,

2

x

 y2 x

2

y x 

2



 y2

Th e s econ d b ecom es th e fi r s t a n d fi r s t b ecom e th e s econ d . Th e L i s t i s d i ffer en t, b eca u s e th e or d er h a s ch a n g ed . B u t a s e t i s a col l ecti on wi th ou t r eg a r d to or d er , a n d th e s e t i s th e s a m e. F r o m

m a tr ix to r u le

E ver y g eom etr i c s y m m etr y tr a n s for m ca n b e d es cr i b ed a s ei th er a m a tr i x or a s a r ep l a cem en t r u l e. Ma n y tr a n s for m m a tr i ces a r e ta b u l a ted i n ou r S y m m e t r y ` p a ck a g e, a n d th er e i s a s i m p l e wa y to tu r n th em i n to r u l es . Her e i t i s : T h r e a d x , y , z  m a t . x , y , z 

x  m a t . x , y, z, y  m a t . x , y, z, z  m a t . x , y, z

To s ee wh a t T h r e a d d oes , ta k e i t a wa y a n d r er u n . (It tu r n s a 3 D vector r ep l a cem en t i n to th r ee 1 D s ca l a r r ep l a cem en ts . ) Ma tr i x Σz i s d efi n ed i n th e S y m m e t r y ` p a ck a g e : Σz  M a t r i x F o r m 1

0 0

1

0

0 0

0 1

4 7

S y m m e tr y T h e o r y

Th e cor r es p on d i n g r u l e i s T h r e a d x , y , z  Σz . x , y , z 

x  x , y  y, z  z

L a ter , wh en y ou a r e m or e fa m i l i a r wi th th e n a m es of g r ou p s a n d el em en ts , y ou ca n u s e a p a ck a g e op er a tor to ca l l th em d own b y th e n a m e S y m m e t r y R u l e . Ca l l u p i ts s y n ta x s ta tem en t (u s i n g th e ? ) a n d r ea d i t. D is ta n c e fr o m

th e o r ig in , u n d e r r o ta tio n

Th i s on e i l l u s tr a tes a l i ttl e d i ffer en t twi s t. We wi l l s h ow th a t th e d i s ta n ce of a p oi n t fr om th e or i g i n i s i n va r i a n t u n d er r ota ti on a b ou t th e or i g i n . Th e or i g i n a l p os i ti on i s { x , y , z } a n d i ts d i s ta n ce fr om

th e or i g i n i s x

2

y 2

z 2

. We

r ota te a r ou n d th e z a xi s b y a n g l e Α : C

Α . x , y , z  z

x C os Α  y S in Α, y C os Α  x S in Α, z

Th e cor r es p on d i n g r ota ti on a l tr a n s for m r o t R u l e

r u l es a r e

T h r e a d x , y , z  C z

Α . x , y , z 

x  x C os Α  y S in Α, y  y C os Α  x S in Α, z  z

N ow a p p l y th es e r u l es to th e r a d i u s for m u l a :

t r a n s f o r m e d R a d i u s

x

2

y 2

z 2

.

r o t R u l e

z2  y C os Α  x S in Α2  x C os Α  y S in Α2

Th i s cer ta i n l y d oes n ot l ook of th i s exa m p l e i s , i n i ti a l wh eth er , a fter r ep l a cem en t, r etu r n i t to th e or i g i n a l for m

th e s a m e b efor e a n d a fter r ep l a cem en t. Th e p oi n t a p p ea r a n ce i s n ot th e cr i ter i on . Th e cr i ter i on i s on e ca n u s e va l i d a l g eb r a i c r ep l a cem en t r u l es to :

M a p T r i g E x p a n d , t r a n s f o r m e d R a d i u s

x

2

 y2  z2

Th er e i t i s . As we k n ew, th e r a d i u s exp r es s i on i s i n d eed i n va r i a n t u n d er r ota ti on .

4 8

A lg e b r a ic s y m m e tr y

C o u lo m b fie ld o f o n e H

n u c le u s

Th e a l g eb r a i c s y m m etr y of Ha m i l ton i a n op er a tor s i s of cen tr a l i m p or ta n ce i n m ol ecu l a r p h y s i cs . If th e Ha m i l ton i a n x , y , z i s s y m m etr i c u n d er a g i ven a l g eb r a i c tr a n s for m , th en th e g r ou n d s ta te of th e m ol ecu l e Ψ0 x , y , z (b u t n ot n eces s a r i l y h i g h er s ta tes ! ) wi l l b e s y m m etr i c u n d er th e s a m e tr a n s for m . We a g a i n ta k e th e a m m on i a m ol ecu l e a s ou r exa m p l e. We l ook fi r s t a t th e p oten ti a l en er g y p a r t of th e Ha m i l ton i a n . Th e p oten ti a l en er g y of a n el ectr on a t { x , y , z } d u e to a p r oton a t th e or i g i n i s g i ven (i n a tom i c u n i ts ) b y

- 1 /

y 2

x

z 2

2

. If th e p r oton i s n ot a t th e or i g i n , b u t a t

{ a , b , c } , th i s en er g y b ecom es 1 / x  a 2  y  b 2  z  c 2 . We ca n m a k e th i s i n to a fu n cti on of th e vector { a , b , c } : C l e a r C o u l o m b E n e r g y ; C o u l o m b E n e r g y a , b , c  : 1 x  a 2  y  b 2  z  c 2 We tes t th e fu n cti on ou t: C o u l o m b E n e r g y x 1

, y 1

, z 1



1



x  x 1

2  y  y1 2  z  z1 2

C o u lo m b fie ld o f a ll th e H

n u c le i in a m m o n ia

In ou r a m m on i a m ol ecu l e th e th r ee H a tom s a r e a t p t p t p t p

H 1 H 2 H 3 t H 1

2

4 69 2 5 0 0

,

R e R e R e p t

s t s t s t H 2

, 0 ,  ,

 2

, 

4 69 5 0 0 0 4 69 5 0 0 0

n t n t n t H 3

1 67 3 1 0 0 0 0



3

, 

2 4 69

 2

P o i P o i P o i , p t

5 0 0 0 2 4 69 5 0 0 0

3

s " a m s " a m s " a m   C

1 67 3 1 0 0 0 0

, 

m o m o m o o l

n i n i n i u m

a " a " a " n F

2 ;

3 ;

4 ; o r m



1 67 3 1 0 0 0 0



4 9

S y m m e tr y T h e o r y

Th e th r ee Cou l om b en er g y exp r es s i on s m a y b e wr i tten i n a l i s t b y th e com m a n d t h r e e E n e r g i e s C o u l o m b E n e r g y p t H 1 , C o u l o m b E n e r g y p t H 2 , C o u l o m b E n e r g y p t H 3   E x p a n d A l l 1



, 

2 0 0 66 8 61 2 0 0 0 0 0 0 0

2 4 69 x 1 2 5 0

x

y  2

1 67 3 z 5 0 0 0

2

z 2

1



, 

2 0 0 66 8 61 2 0 0 0 0 0 0 0

2 4 69 x 2 5 0 0

x

2 4 69

 2

3

y

2 5 0 0

 y2 

1 67 3 z 5 0 0 0

 z2

1

 

2 0 0 66 8 61 2 0 0 0 0 0 0 0

2 4 69 x 2 5 0 0

x

2 4 69

 2

 3

2 5 0 0

y

 y2 

1 67 3 z 5 0 0 0

 z2

R o ta tio n a l s y m m e tr y o f th e C o u lo m b fie ld N ow we wa n t to s ee wh a t h a p p en s to th es e exp r es s i on s wh en we r ota te th e coor d i n a te { x , y , z } b y on e-th i r d of a tu r n . We a l r ea d y h a ve th i s tr a n s for m a b ove i n ter m s of th e g en er a l r ota ti on a n g l e Α, s o we j u s t r ep l a ce Α a p p r op r i a tel y . r o t R u l e

o n e T h i r d T u r n x  

P er for m

2

x



3 2

y

, y

3 2

x



.

2

y

Α 2 Π  3 

, z  z

th e r ep l a cem en ts :

r o t a t e d E n e r g i e s t h r e e E n e r g i e s . o n e T h i r d T u r n

5 0

 E x p a n d A l l

A lg e b r a ic s y m m e tr y

1



, 2 0 0 66 8 61 2 0 0 0 0 0 0 0



2 4 69 x 2 5 0 0

x 2



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3

y

2 5 0 0

 y2 

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 z2

1



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x

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1

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x 2



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3

2 5 0 0

y

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A ver y i n ter es ti n g th i n g h a s h a p p en ed : ea ch tr a n s for m ed i n to on e of th e oth er two.

1 67 3 z 5 0 0 0

 z2

Cou l om b

exp r es s i on

h a s b een

t h r e e E n e r g i e s 1  r o t a t e d E n e r g i e s 2 , t h r e e E n e r g i e s 2  r o t a t e d E n e r g i e s 3 , t h r e e E n e r g i e s 3  r o t a t e d E n e r g i e s 1  Tr u e, Tr u e, Tr u e

Cl ea r l y , th e s u m of th e th r ee i s th e s a m e b efor e a n d a fter th i s tr a n s for m , s o th e s u m i s a n a l g eb r a i c exp r es s i on th a t i s s y m m etr i c u n d er r ota ti on b y on e th i r d of a tu r n . R e fle c tio n s y m m e tr y o f th e C o u lo m b fie ld Th e Cou l om b en er g y s h ou l d a l s o b e s y m m etr i c u n d er th e r efl ecti on tr a n s for m y - y . You ca n s ee i t b y i n s p ecti on , b u t we m a k e i t exp l i ci t: t h r e e E n e r g i e s R e f l e c t e d E x p a n d A l l

t h r e e E n e r g i e s

.

y y



5 1

S y m m e tr y T h e o r y

1



, 

2 0 0 66 8 61 2 0 0 0 0 0 0 0

x

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y  2

1 67 3 z 5 0 0 0

2

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1



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2 0 0 66 8 61 2 0 0 0 0 0 0 0

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2 4 69

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 z2

Com p a r i n g th e r efl ected exp r es s i on to th e or i g i n a l , we s ee th a t th e l a s t two ter m s h a ve b een tr a n s for m ed i n to ea ch oth er , wh i l e th e fi r s t r em a i n s i n va r i a n t. Ag a i n , th e s u m wi l l b e i n va r i a n t, s o y -r efl ecti on i s a s y m m etr y tr a n s for m for i t.

4 . 3 . 3 . V i s u a l i z a ti on of a l g eb r a i c s y m m etr y If a l l th i s i s r ota ti on a l s y m a d d u p th e th d u e to th e th r to th e p l a n e z

tr u e, th m etr y r ee exp ee H n u = = 0

e con tou a n d a y -r r es s i on s , cl ei , a n d :

r p l ot of th i s fu n cti on s h ou l d s efl ecti on s y m m etr y . Th i s i s ea s to g i ve th e tota l p oten ti a l en er g (p r ep a r i n g to m a k e a 2-D g r a p h

A p p l y P l u s , t h r e e E n e r g i e s .

e n e r g y H 3 z 0

1

 2 0 0 66 8 61 2 0 0 0 0 0 0 0



2 4 69 x 1 2 5 0



2 4 69 x 2 5 0 0

z 0

 x 2

y 2

1 2 0 0 66 8 61 2 0 0 0 0 0 0 0

h ow a th r eefol d y to ver i fy . We y of th e el ectr on i c) we s p eci a l i z e



x 2



2 4 69



2 4 69

3

y

 y2

3

y

 y2

2 5 0 0

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x 2

2 5 0 0

Th e A p p l y op er a tor , a s u s ed a b ove, i s wor th n oti n g i n d eta i l . We ta k e a s i m p l e exa m p l e a n d wr i te i t ou t i n F u l l F o r m , to s h ow th a t A p p l y i s a n op er a tor th a t ch a n g es th e h ea d of a n exp r es s i on :

5 2

A lg e b r a ic s y m m e tr y

A p p l y P l u s ,

L i s t A , B , C

A B C

a n d ca n th er efor e b e u s ed to a d d u p th e i tem s i n a L i s t . R etu r n i n g to e n e r g y  H 3 z 0 fu n cti on , we m a k e a con tou r p l ot for i t: C o n t o u r P l o t e n e r g y H 3 z 0 , x , 2 , 2 , y , 2 , 2 , P l o t P o i n t s  4 0 , B a s e S t y l e " T R " , A x e s  T r u e , A x e s L a b e l  " x " , " y " , E p i l o g  T h i c k n e s s . 0 0 5 , R e d , L i n e 2 , 0 , 2 , 0  , B l u e , L i n e 1 ,  L i n e 1 , 3

, 1 ,  3

3

, 1 , 3

,

, I m a g e S i z e 2 0 0 

y 2

1

0

x

1

2

2

1

0

1

2

F i g . 4 . 8 Cr os s s ecti on of th e p oten ti a l of a n el ectr on i n th e th r ee p r oton s of th e a m m on i a m ol ecu l e, ta k en th r ou g h th e cen ter of g r a vi ty , p er p en d i cu l a r to th e th r eefol d a xi s .

Th e p oten ti a l d oes i n d eed h a ve th e exp ected s y m m etr i es : a th r eefol d r ota ti on a l s y m m etr y , a s wel l a s a r efl ecti on s y m m etr y i n th e r ed h or i z on ta l l i n e. Th i s p l ot a l s o s h ows two oth er s y m m etr i es , r efl ecti on s i n th e two b l u e l i n es r ota ted ± 1 / 3 tu r n fr om th e r ed h or i z on ta l l i n e. You ca n s ee r a th er i n tu i ti vel y th a t i f a n y ob j ect h a s a r efl ecti on p l a n e th a t con ta i n s a th r eefol d r ota ti on a xi s , i t h a s to h a ve th e two oth er s i m i l a r r efl ecti on p l a n es . F i n d i n g a l l th e s y m m etr i es i m p l i ed b y a s m a l l n u m b er of s y m m etr i es i s a n i m p or ta n t p a r t of g r ou p th eor y th a t we wi l l s tu d y i n d eta i l i n Ch a p ter 15 (Ma k e Ma tr i x G r ou p ).

5 3

S y m m e tr y T h e o r y

4 . 4 . Su m m a r y a n d p r evi ew We h a ve con s i d er ed two b a s i c k i n d s of s y m m etr y , r efl ecti on focu s s i n g on th e s y m m etr y tr a n s for m s th em s el ves , i n two for m s .

a n d

r ota ti on ,

1 . G eom etr i c s y m m etr y tr a n s for m s wer e ca r r i ed ou t b y m a tr i x m u l ti p l i ca ti on s . 2 . Al g eb r a i c s y m m etr y tr a n s for m s wer e ca r r i ed ou t b y s u b s ti tu ti on s of va r i a b l es . In th e n ext ch a p ter (Axi om s ) we wi l l i n tr od u ce th e con cep t of a m a th em a ti ca l g r ou p ; i n th e ch a p ter a fter th a t we wi l l s ee m a n y exa m p l es of g r ou p s ; th en i n Ch a p ter 7 we wi l l p r ove th e fu n d a m en ta l th eor em of s y m m etr y th eor y ; n a m el y , th a t a l l th e s y m m etr y tr a n s for m s of a n y s y m m etr i c ob j ect m u s t for m a g r ou p .

4 . 5. E n d N otes Atom

ta g s

Th e fu l l a tom ta g i s a l i s t. Its fi r s t (a n d often on l y ) el em en t i s a n i n teg er th a t d i s ti n g u i s h es ea ch a tom fr om oth er s of i ts own k i n d . Th i s i n teg er i s a u tom a ti ca l l y p r i n ted i n th e cen ter of ea ch a tom b y th e m ol ecu l a r g r a p h i cs fu n cti on s . B u t i n com p l i ca ted m ol ecu l es , s u ch a s p r otei n s , th e a tom ta g l i s t ca n i n cl u d e oth er i n for m a ti on , s u ch a s to wh i ch a m i n o a ci d th e a tom b el on g s , or to wh i ch ch a i n of a p r otei n com p l ex. Sp eci a l i z ed m ol ecu l a r g r a p h i cs fu n cti on s ca n extr a ct th i s i n for m a ti on to col or wh ol e a m i n o a ci d s , or wh ol e ch a i n s .

5 4

5. Axi om s of g r ou p th eor y P r el i m i n a r i es

5. 1. Un d efi n ed ter m s i n th e a xi om s In th i s b ook , we wi l l b e con cer n ed a l m os t excl u s i vel y wi th g r ou p s of s y m m etr y tr a n s for m s . B u t th e “ g r ou p ” i s on e of th e m os t g en er a l con cep ts i n m a th em a ti cs , a n d we m u s t b eg i n wi th th e a xi om s of g r ou p th eor y , s ta ted i n a for m th a t a p p l i es to a l l k i n d s of g r ou p s . We con s i d er a s et of e l e m e n t s { A , B , C , … } a n d a n o p e r a t o r wr i tten a s  wh i ch ta k es two m em b er s of th e s et a s i ts a r g u m en ts , i n exp r es s i on s l i k e A  B . Th e el em en ts of th e s et for m a “ g r ou p u n d er ” i f th e con d i ti on s of Axi om s 1 th r ou g h 4 a r e m et. C o m m e n ts: ( 1 ) Th e ter m s e l e m e n t s a n d o p e r a t o r a r e u n d efi n ed . m ea n i n g s for wh i ch th ey m a k e s en s e.

Th ey ca n ta k e on a n y

( 2 ) Th e s et of el em en ts m a y b e fi n i te or i n fi n i te i n n u m b er . ( 3 ) Wh en l a ted i n to b ecom e a n ca s es , A a n

A  B g oes to th e M a t h e m a t i c a s q u a r e b r a ck et for m l i k e f exp l i ci t f , l i k e T i m e s , D o t d B wi l l b e s i m p l e, d efi n i te th i n

a p r oces s or , [ A , B ] , wh , P l u s , or g s th a t th e op

it is a er e th P e r m er a tor

u tom a ti ca l l y tr a n s e s y m b ol i c  h a s u t e . In a l l th es e ca n wor k on .

( 4 ) Th e n u m b er of el em en ts i n a g r ou p i s ca l l ed i ts g r o u p o r d e r . Th i s h a s a b s ol u tel y n oth i n g to d o wi th s or ti n g or d er . Th e g r ou p el em en ts m a y b e n a m ed or l i s ted i n a n y s or ti n g or d er . ( 5 ) Th er e i s s om e con fu s i on wh eth er th e el em en ts { A , B , C , … } m d i ffer en t, or wh eth er r ep ea ts a r e a l l owed . Th i s con fu s i on i s tol er a ted a ffects n oth i n g b u t th e l a n g u a g e wi th wh i ch s om e of th e th eor em s Us u a l l y we wi l l m ea n th e el em en ts to b e d i s ti n ct, b u t wh en th ey a r e n ca l l th e g r ou p a r e d u n d a n t g r o u p .

u s t a ll b e b eca u s e i t a r e s ta ted . ot we wi l l

Sy m m etr y g r ou p s a r e a s s oci a ted wi th g r ou p s of m a tr i ces ca l l ed r ep r es en ta ti on s , a n d r ep r es en ta ti on s a r e often r ed u n d a n t g r ou p s .

W.M. McClain, Symmetry Theory in Molecular Physics with Mathematica, DOI 10.1007/b13137_5, © Springer Science+Business Media, LLC 2009

5 5

S y m m e tr y T h e o r y

5. 2. Th e fou r a xi om s 5. 2. 1. Cl os u r e Axi om

A x io m

1 .

If M a n d N a r e m em b er s of a g r ou p u n d er , th e op er a ti on M  N y i el d s a u n i q u e va l u e, wh i ch i s a l s o a m em b er of th e g r ou p . C o m m e n ts: ( 1 ) N o m a tter wh a t th e op er a tor  a ctu a l l y i s , i t i s con ven ti on a l l y ca l l ed m u l t i p l i c a t i o n . F or i n s ta n ce, we wi l l often s p eci fy th a t th e el em en ts of th e g r ou p a r e m a t r i c e s a n d th a t th e op er a ti on i s m a t r i x m u l t i p l i c a t i o n , In th i s ca s e M  N wi l l b e ca r r i ed ou t a s M . N , wh i ch i n M a t h e m a t i c a i s D o t [ M , N ] . Or M a n d N m i g h t b e or d er ed l i s ts , a n d M  N cou l d m ea n a r eor d er i n g of l i s t M a ccor d i n g to i n s tr u cti on s i n l i s t N . Th i s i s p er m u ta ti on , ca r r i ed ou t i n M a t h e m a t ic a a s P a r t [ M , N ] . ( 2 ) “ Mu l ti p l i ca ti on ” m a y b e com m u ta ti ve or n on -com or m a y n ot b e tr u e th a t M  N i s a l wa y s i d en ti ca l to N n u m er i ca l m u l ti p l i ca ti on , i t i s tr u e. B u t wh en  i s p er m u ta ti on of a l i s t, th e two p r od u cts a r e g en er a l l y g r ou p s a r e often ca l l ed A b e l i a n g r ou p s , a fter th e N N i el s Hen d r i k Ab el (18 02- 18 29), for ever y ou n g .

m u ta ti ve.  M . F or m a tr i x m d i ffer en t. or weg i a n

Th a t i s , i t m a y n u m b er s u n d er u l ti p l i ca ti on or Com m u ta ti ve m a th em a ti ci a n

5. 2. 2. Un i t E l em en t Axi om

A x io m

2 .

E ver y g r ou p con ta i n s a u n i t el em en t E , s u ch th a t for ever y m em b er X of th e g r ou p , X E = = E X = = X . C o m m e n ts: ( 1 ) Th e l etter E com es n ota ti on i s s o fi r m l y em ch a n g ed . However , i n M . R a th er th a n d i s r u p t th th eor y . In In p u t cel l s i t m

5 6

fr om b ed d a th e is s y a k es

th e G er m a n wor d ed i n g r ou p th eor y m a t i c a th e Ca p i ta l E s tem , we wi l l u s e a a b i g d i ffer en ce :

E in th is s tr

h e it, m a t it a b d efi n ed in g “ E ”

ea n i n g u n i t y . s ol u tel y ca n n a s th e exp on for th e E of

Th i s ot b e en ti a l g r ou p

T h e fo u r a x io m s

E , " E "   N 2 . 7 1 8 2 8 , E 

( 2 ) F or g r ou p s of n u m b er s u n d er n u m er i ca l m u l ti p l i ca ti on ,  i s i n d eed th e n u m b er 1 . F or g r ou p s of m a tr i ces ,  i s th e u n i t m a tr i x (wi th 1 ' s d own th e d i a g on a l a n d z er oes ever y wh er e el s e). ( 3 ) F or a m or e p er ver s e exa m p l e, con s i d er th e s et of a l l n u m b er s wi th  a s n u m er i ca l a d d i ti on . Her e th e n u m b er z er o p l a y s th e r ol e of th e u n i t el em en t , b eca u s e 0 + n = n + 0 = n . ( 4 ) Wh en l i s ti n g th e m em b er s of a g r ou p , th e u n i q u e el em en t  i s con ven ti on a l l y g i ven fi r s t.

5. 2. 3 . In ver s e Axi om

A x io m

3 .

F or ever y el em en t X of th e g r ou p , th er e i s a n el em en t Y s u ch th a t X Y = = Y X = = E . C o m m e n ts: ( 1 ) E l em en ts Y 1 , or Y a s

 i n A x i o i ts own i n ver

i n th e g r ou p

X a n d Y a r e s a i d to b e m u t u a l l y i n v e r s e , a n d y ou m a y wr i te X a s 1 . It i s a l l owed th a t X a n d Y b e th e s a m e. F or i n s ta n ce, s etti n g X m 3 , a b ove, we s ee a n i n s ta n t p r oof th a t Y h a s to b e , s o th a t  i s s e. X

( 2 ) Th e a l g or i th m for F or n u m b er s u n d er T i d i vi s i on . F or m a tr i ces ou t b y th e M a t h e m a t i c a

fi n d i n g th m e s , th e u n d er D o t op er a tor I

e i n ver s e d ep en d s on wh a t th e el em en ts a r e. i n ver s e of x i s 1 / x , a s ca r r i ed ou t b y l on g , i t i s th e m a tr i x i n ver s e op er a ti on , a s ca r r i ed n v e r s e [ m a t ] .

( 3 ) Th i s a xi om excl u d es 0 fr om ever y g r ou p of n u m b er s u n d er m u l ti p l i ca ti on , j u s t a s i t excl u d es s i n g u l a r m a tr i ces (m a tr i ces th a t h a ve n o m a tr i x i n ver s e) fr om a l l g r ou p s of m a tr i ces u n d er D o t .

5. 2. 4 . As s oci a ti ve Axi om

A x io m

4 .

F or a n y th r ee el em en ts A , B , a n d C of th e g r ou p , i t m u s t b e T r u e th a t A ( B  C ) ) = = ( A  B )  C .

5 7

S y m m e tr y T h e o r y

C o m m e n ts: ( 1 ) Th i s p r op er ty of th e op er a tor  i s ca l l ed a s s o c i a t i v i t y . ( 2 ) Th i s s ta tem ea ch s tep i n a n A ( B C ) , B u tin ( A B eva l u a ted .

en t m eva l u a th e i n ) C

u st ti on n er , th

b e r ea d i n th e l i g h t of th e u n i ver s a l con ven ti on th a t a t , on l y th e i n n er m os t g r ou p i n g s a r e eva l u a ted . Th u s , i n B  C eva l u a tes a s s om e D , th en A  D i s eva l u a ted . e i n n er A  B eva l u a tes fi r s t a s s om e F , th en F  C i s

( 3 ) E ver y op er a tor m u s t b e tes ted for a s s oci a ti vi ty a ccor d i n g to i ts own p r op er ti es . As s oci a ti vi ty i s a wel l k n own p r op er ty of n u m er i ca l m u l ti p l i ca ti on , a n d of m a tr i x m u l ti p l i ca ti on . ( 4 ) B eca u s e of a s s oci a ti vi ty , th e tr i p l e p r od u ct A  B  C i s u n i q u el y d efi n ed , a n d d oes n ot r ea l l y n eed a n y g r ou p i n g . ( 5 ) Th i s a xi om excl u d es n u m er i ca l d i vi s i on a s a p os s i b l e g r ou p b eca u s e, for exa m p l e, 8 / ( 4 / 2 ) i s 4 , wh er ea s ( 8 / 4 ) / 2 i s 1 .

op er a ti on ,

5. 2. 5. Cod a Th es e a xi om of a b s tr a ct g Th e ter m s “ n ot occu r i n la n g u a g ein a xi om s .

s a r e s u r p r i s i n g l y r i ch i n con ten t. r ou p th eor y tr a ce b a ck to th es e fou cl a s s ” , “ s u b g r ou p ” , “ i r r ed u ci b l e r ep th e a xi om s . Yet a l l th es e con cep th e a xi om s , a n d a l l th ei r p r op er ti es

Al l of th e m r s ta tem en ts r es en ta ti on ” ts a r e d efi n a r e l og i ca l

a n y s u b tl e th eor em s a n d to n oth i n g el s e. , a n d “ ch a r a cter ” d o ed on l y i n ter m s of con s eq u en ces of th e

Th er e a r e a l s o s u r p r i s i n g l y m a n y d i ffer en t k i n d s of m a th em a ti ca l ob j ects th a t for m g r ou p s . In th e n ext Ch a p ter , we wi l l l ook i n d eta i l a t s ever a l of th em .

5 8

6. Sever a l k i n d s of g r ou p s P r el i m i n a r i es

6. 1 N u m b er s u n d er Ti m es 6. 1. 1 G r ou p s a n d s u b g r ou p s of n u m b er s Al l com p l ex n u m b er s (excl u d i n g z er o, wh i ch h a s n o i n ver s e) for m a g r ou p u n d er T i m e s . We h a ve a l r ea d y m en ti on ed th e r el eva n t p r op er ti es i n th e Com m en ts fol l owi n g ea ch a xi om . F u r th er , cer ta i n fi n i te s u b s ets of th e com p l ex n u m b er s a l s o for m

g r ou p s .

S u b g r o u p If  i s a g r ou p a n d  i s a s u b s et of  th a t i s a l s o a g r ou p , we s a y th a t  i s a s u b g r o u p of  . N ow y ou s h ou l d u s e th es e con cep ts on y ou r own . Th i s wi l l ca r ve th em s ton e of y ou r m i n d .

i n to th e

6. 1. 2 Som e p r a cti ce q u es ti on s ( 1 ) Wh i ch (a ) (b ) (c ) (d ) If y ou ca n h er e.

of th e fol l owi n g i s fa l s e? ( Th e op er a tor i s T i m e s i n a l l ca s es . ) Th e r ea l n u m b er s a r e a s u b g r ou p of th e com p l ex n u m b er s . Th e r a ti on a l s a r e a s u b g r ou p of th e r ea l s . Th e i n teg er s a r e a s u b g r ou p of th e r a ti on a l s . Th er e i s on e fi n i te s u b g r ou p wi th i n th e i n teg er s . ' t s ee wh i ch s ta tem en t i s fa l s e, th en a fter r ea l l y , r ea l l y tr y i n g , cl i ck

( 2 ) Th e n u m b er s 1 a n d - 1 for m a g r ou p u n d er T i m e s . P r ove th a t i f a n oth er r ea l n u m b er i s j oi n ed i n , i t i s n o l on g er a g r ou p . Wh en y ou h a ve i t, cl i ck h er e to s ee a s ta n d a r d p r oof. ( 3 ) Wh a t i s th e u n i t el em en t for th e g r ou p of a l l com p l ex n u m b er s u n d er T i m e s ? Wh a t i s a s i m p l e exp r es s i on for th e i n ver s e of a com p l ex n u m b er , i n ter m s of th e n u m b er i ts el f? D on ' t cl i ck h er e; th i s i s too ea s y . ( 4 ) Ma n y fi n i te s ets of com p l ex n u m b er s for m g r ou p s . B u t th ey a r e a l l s om ewh a t s i m i l a r . Th e i d ea i s a s tr a i g h tfor wa r d exten s i on of th e g r ou p { 1 , - 1 } for W.M. McClain, Symmetry Theory in Molecular Physics with Mathematica, DOI 10.1007/b13137_6, © Springer Science+Business Media, LLC 2009

5 9

S y m m e tr y T h e o r y

th e r ea l s . It wi l l com e to y ou i f y ou s ta r t b y th i n k i n g a b ou t wh eth er s u cces s i ve p ower s of a g i ven com p l ex n u m b er m i g r a te towa r d th e or i g i n , or a wa y fr om i t. . If, a fter s er i ou s effor t, y ou s ti l l n eed h el p , cl i ck h er e.

6. 2 Ma tr i ces u n d er D ot 6. 2. 1 Al l n on s i n g u l a r 2 2 m a tr i ces Al l s q u a r e m a tr i ces of th e s a m e s i z e (excl u d i n g s i n g u l a r m a tr i ces , wh i ch h a ve n o i n ver s e) for m a g r ou p u n d er D o t , th e m a tr i x m u l ti p l i ca ti on op er a tor . We m a k e a d eta i l ed exa m i n a ti on of a l l th e a xi om s , for 2 2 m a tr i ces : A x io m 

1 . C lo s u r e a

b c

d

. 

e

f g

h

  G r i d F o r m

a eb g

a f b h

c ed g

c f d h

Th e p r od u ct i s a n oth er 2-b y -2 m a tr i x. Th i s b y i ts el f i s n ot q u i te en ou g h to s h ow cl os u r e for n on s i n g u l a r m a tr i ces . We m u s t a l s o s h ow th a t th e p r od u ct of two n on s i n g u l a r m a tr i ces i s n on s i n g u l a r . Th i s h a s to b e tr u e, b eca u s e of two fa cts : ( 1 ) D e t [ A ] = = 0 i f a n d on l y i f m a tr i x A i s s i n g u l a r (h a s n o i n ver s e), a n d (2 ) D e t [ A . B ] ca s e : m a t 1 

D e t [ A ] * D e t [ B ] .

= = a

b c

d

;

m a t 2 

e

f g

h

Th i s i s ea s i l y ver i fi ed

for th e 2 2

;

D e t m a t 1 . m a t 2 E x p a n d D e t m a t 1 D e t m a t 2

Tr u e

Si n ce we u s ed n o n u m er i ca 2 2 ca s e : If n ei th er D e t va n i s h ei th er . Th er efor e n m a tr i ces , a n d Axi om 1 i s s a A x io m

2 . U n it e le m e n t

We tes t th e m a tr i x  el em en t : 6 0

l va l u es i n th e m a tr i ces , th i s i s a g en er a l p r oof for th e [ A ] n or D e t [ B ] va n i s h es , th en D e t [ A . B ] ca n n ot on -s i n g u l a r i ty com es d own i n th e b l ood l i n e of 2 2 ti s fi ed .

1

0 0

1

 to s ee i f i t h a s th e p r op er ti es r eq u i r ed of a u n i t

M a tr ic e s u n d e r D o t



1

0 0

1

. 

a

b c

d

 

a

b c

d

. 

1

0 0

1

 

a

b c

d



Tr u e

Th a t Tr u e p r oves i t. A x io m 3 . I n v e r s e Ta k e th e i n ver s e of a g en er a l 2-b y -2 : I n v e r s e  

d b c  a d

a

b c

d

, 

 b b c  a d

, 

c

a

b c  a d

,

b c  a d



Th i s fa i l s on l y i f th e d en om i n a tor a d - b c = 0 . B u t th i s i s th e d eter m i n a n t, D e t 

a

b c

d



b c  a d

a n d m a tr i ces wi th z er o d eter m i n a n t a r e excl u d ed b y d efi n i ti on . A x io m

4 . A s s o c ia tiv ity o f th e D o t o p e r a to r

Tr y i t, u s i n g th r ee p er fectl y g en er a l 2 2 m a tr i ces : E x p a n d  

a

b c

d

E x p a n d 

a

b c

d

.  . 

e

f g

h e

f g

h

. 

i

j k

. 



m i

j k

m



Tr u e

Al l th e a xi om s a r e s a ti s fi ed ; th er efor e th e s et of a l l n on s i n g u l a r 2 2 m a tr i ces for m a g r ou p , q . e. d .

6. 2. 2 Al l u n i ta r y 2 2 m a tr i ces It wi l l tu r n ou t th a t a l l s y m m etr y tr a n s for m Th es e a r e ca l l ed u n i t a r y m a tr i ces . If D e t [ A ]  1 . Th i s s u b s et of u l a r 2 2 m g r ou p s . Th

= ± 1 a n d D e t p r oves th a t u n i t a r y 2 2 m a tr i ces . Th i s e u n i ta r y m a tr

m a tr i ces h a ve d eter m i n a n t + 1 or - 1 .

[ B ] = ± 1 , th en D e t A . B D e t A D e t B n i ta r i ty com es d own i n th e b l ood l i n e. Th er efor e th e a tr i ces for m a g r ou p wi th i n th e g r ou p of a l l n o n s i n g u i s y ou r fi r s t exa m p l e of th e g en er a l con cep t of s u b i ces a r e a s u b g r ou p of th e n on s i n g u l a r m a tr i ces .

6 1

S y m m e tr y T h e o r y

6. 2. 3 Al l n on s i n g u l a r n n m a tr i ces Th e r es u l ts a ti ve p r oof : n . Us e th i s s p eci a l ca s e,

b ove ca n b e exten As s u m e D e t A . to p r ove i t for m s u ch a s n = 2 . It i s

R em em b er exp a n s i on b y m i n or for m a tr i ces of s i z e ( n + 1 ) ( ea ch m a tr i x, g i vi n g ever y th i n g on e ter m fr om ea ch exp a n s i on h a ve s h own th a t IF i t i s tr u e for ( n + 1 ) m a tr i ces .

d ed to s q u a r e m a tr i ces of a n B D e t A D e t B for a tr i ces of s i z e ( n + 1 ) ( n + i m p or ta n t to p r ove i t for th e g s ? Us e i t on th e exp n + 1 ) . E xp a n d b y m i n ter m s of n n m a . Th e r el a ti on h ol d s n n m a tr i ces THE N

y s i z e b y a n i n d u cm a tr i ces of s i z e n  1 ) . D on ' t u s e a n y en er a l n .

r es s i on D e t A D e t B

i n or s a l on g th e top r ow of tr i ces . Th en wor k wi th j u s t ter m -b y -ter m , s o y ou wi l l i t m u s t b e tr u e for ( n + 1 )

Wh en y ou h a ve i t for th e g en er a l n , r ea s on a s fol l ows : We k n ow b y d i r ect ca l cu l a ti on th a t n on -s i n g u l a r i ty com es d own i n th e b l ood l i n e for n = 2 . Th er efor e i t i s tr u e for n = 3 . Th er efor e, i t i s tr u e for n = 4 . An d s o on . . . . q .e .d Th i s fi n i s h es th e p r oof th a t a l l n on s i n g u l a r m a tr i ces of s i z e n n wh a tever n m a y b e.

for m

a g r ou p ,

6. 2. 4 Al l u n i ta r y n n m a tr i ces Th e i n d u cti ve p r oof i n 6. 2. 3 (Al l n on s i n g u l a r n n m a tr i ces ) ca n b e a p p l i ed wi th ou t ch a n g e to 6. 2. 2, s h owi n g th a t a l l u n i ta r y n n m a tr i ces for m a g r ou p .

6. 3 Axi a l r ota ti on g r ou p s In Ch a p ter 4 we s h owed th a t i n th r ee d i m en s i on s , r ota ti on a b ou t th e z a xi s i s p er for m ed b y th e m a tr i x C z

Φ  M a t r i x F o r m C os Φ S in Φ 0 S in Φ C os Φ 0 0 0 1

Cu tti n g b a ck to two d i m en s i on s , we d efi n e R Φ : 

6 2

C o s Φ S i n Φ

 S i n Φ C o s Φ

A x ia l r o ta tio n g r o u p s

It i s ea s y to s h ow th a t a l l th e 2 2 R m a tr i ces p r od u ced a s Φ r u n s a r ou n d th e wh ol e ci r cl e for m a g r ou p u n d er D o t , ca l l ed th e “ a xi a l r ota ti on g r ou p " .  1 , C l o s u r e 

R a . R b R a  b  T r i g E x p a n d 

Tr u e

 2 , I d e n t i t y 

R 0 1 , 0 , 0 , 1 

Tr u e

 3 , I n v e r s e 

R a . R a  T r i g E x p a n d  R 0

Tr u e

(* 4 , A s s o c ia tiv e * )

Al wa y s Tr u e for D o t

Th e n a m e “ a xi a l r ota ti on g r ou p ” i s s y m b ol i z ed b y p h y s i ci s ts a n d ch em i s ts a s C  , or b y m a th em a ti ci a n s a s S O ( 2 ) , m ea n i n g S p e c i a l O r t h o g o n a l ( 2 D ) . Th es e m a tr i ces a r e O r t h o g o n a l b eca u s e th ei r tr a n s p os es a r e th ei r i n ver s es R Φ . T r a n s p o s e R Φ

 T r i g E x p a n d 1 , 0 , 0 , 1 

a n d th ey a r e S p e c i a l b eca u s e a n y cl os ed cu r ve i n th e p l a n e h a s th e s a m e a r ea b efor e a n d a fter a n y r ota ti on .

6. 4

P er m u ta ti on s u n d er P er m u te 6. 4 . 1 Ab ou t p er m u ta ti on s

E xch a n g e of i d en ti ca l p a r ti cl es , b oth el ectr on s or n u cl ei , i s a n i m p or ta n t s y m m etr y op er a ti on of th e m ol ecu l a r Ha m i l ton i a n . Th e l a b el s th a t d i s ti n g u i s h th e p a r ti cl es a r e j u s t th e fi r s t n i n teg er s , a n d th e or d er of th i s l i s t ch a n g es a s p a r ti cl es a r e exch a n g ed . B u t th i s i s exa ctl y wh a t m a th em a ti ci a n s ca l l a p e r m u t a t i o n . P e r m u ta tio n A p er m u ta ti on of l en g th n i s a L i s t con ta i n i n g th e fi r s t n i n teg er s , i n a n y or d er . M a t h e m a t i c a h a s a fa s t a n d s i m p l e wa y of u s i n g a p er m u ta ti on a s a n i n s tr u cti on for r ea r r a n g i n g a l i s t of n ob j ects . It i s th e P a r t op er a tor . Its s i m p l es t u s e i s j u s t to p u l l on e i tem ou t of a L i s t :

6 3

S y m m e tr y T h e o r y

P a r t a , b , c , 2

b

Mor e often th i s i s wr i tten a s a , b , c 3  c

B u t P a r t a l s o r es p on d s to a l i s t of a d d r es s es : a , b , c 2 , 3 , 1  b , c , a 

In fa ct i t h a s p er m u ted th e l i s t { a , b , c } a ccor d i n g to th e i n s tr u cti on s i n { 2 , 3 , 1 } . G oi n g s tr a i g h t d own th e l i s t of i n teg er s , i t fi r s t took i tem # 2 ; th en i t took i tem # 3 , th en i t took i tem # 1 . Th i s g ets m or e i n ter es ti n g wh en th e fi r s t l i s t i s a l s o a p er m u ta ti on of th e s a m e i n teg er s : 3 , 1 , 2 2 , 3 , 1  1 , 2 , 3 

N ow we h a ve two ob j ects com b i n i n g to p r od u ce a th i r d of th e s a m e k i n d . Th i s s ou n d s l i k e i t cou l d b e a g r ou p op er a tor . We ca l l th e g r ou p op er a tor P e r m u t e , a n d d efi n e i t b y th e s ta tem en t P e r m u t e p e r m u t e e , p e r m u t e r : p e r m u t e e p e r m u t e r  Tr y i t ou t : P e r m u t e 3 , 1 , 2 , 3 , 2 , 1 

2 , 1 , 3 

Sta r e a t th i s u n ti l y ou s ee exa ctl y wh a t h a p p en ed .

6. 4 . 2 P er m u t e s a ti s fi es th e a xi om s We wi l l n ow s h ow th a t P e r m u t e ca n b e th e op er a tor for a g r ou p . C lo s u r e Th er e a r p e r m u t n o wa y a r ou n d a

e on e r th e m on

U n it e le m e n t

6 4

l y n ! p er m u ta ti on s of th e fi r s t n i n teg er s , s o th e p e r m u t e e , th e a n d th ei r p r od u ct a r e a l l m em b er s of th e s a m e fi n i te s et. Th er e i s p er m u ta ti on op er a ti on ca n wa n d er off to i n fi n i ty . It on l y h op s g th e n ! p os s i b i l i ti es , or s om e s u b s et th er eof. Cl os u r e i s a s s u r ed .

P e r m u ta tio n s u n d e r P e r m u te

Th e u n i t el em en t i s th e l i s t of th e fi r s t n i n teg er s i n n a tu r a l or d er . It m u s t b e tes ted i n b oth p os i ti on s . F i r s t we tr y i t a s th e p e r m u t e r , th en a s p e r m u t e e : u n i t 1 , 2 , 3 , 4 , 5 ; p e r m 3 , 2 , 5 , 1 , 4 ; P e r m u t e u n i t , p e r m P e r m u t e p e r m , u n i t p e r m  Tr u e

It wor k s a s a u n i t el em en t s h ou l d : E  P =

P E

P . =

In v e r se Th er e i s a p a ck a g e op er a tor th a t fi n d s th e i n ver s e of a n y p er m u ta ti on . F i r s t we d em on s tr a te i t, th en we d i s cu s s i t. Ag a i n l et p e r m 3 , 5 , 1 , 2 , 4 ; a n d a s k for i ts i n ver s e: p e r m I n v I n v e r s e P e r m u t a t i o n p e r m

3 , 4 , 1 , 5 , 2 

D oes p e r m I n v h a ve th e fu n d a m en ta l p r op er ti es of a n i n ver s e? P e r m u t e p e r m , p e r m I n v , P e r m u t e p e r m I n v , p e r m  1 , 2 , 3 , 4 , 5 , 1 , 2 , 3 , 4 , 5 

Yes ! It wor k s a s i t s h ou l d : P  P 1 = P 1  P i n ver s e a n d ta l k s l i k e a n i n ver s e, i t i s a n i n ver s e. =

E

.

If i t wa l k s l i k e a n

How d oes th e I n v e r s e P e r m u t a t i o n op er a tor wor k ? It d oes exa ctl y wh a t a h u m a n wou l d d o. (To a voi d com p l i ca ted g en er a l n ota ti on , we wor k on a n exa m p l e. ) If y ou n eed th e i n ver s e of { 3 , 5 , 1 , 2 , 4 } y ou th i n k to y ou r s el f: I wa n t th e p er m u ta ti on th a t p u ts th i s i n n a tu r a l or d er . So I ta k e th e th i r d on e (th e 1 ) fi r s t, th e fou r th on e (th e 2 ) s econ d , etc. Wor k i n g i t ou t i n y ou r h ea d , y ou s h ou l d com e u p wi th { 3 , 4 , 1 , 5 , 2 } , exa ctl y a s ca l cu l a ted a b ove. B u t th er e i s a P o s i t i o n op er a tor th a t ca n a u tom a te th i s th ou g h t p r oces s . Tr y i t on p e r m = { 3 , 5 , 1 , 2 , 4 } : P o s i t i o n p e r m , 3

1 

Yes , th e 3 i s i n p os i ti on 1 . Tr y s om e oth er s on y ou r own . We ca n d efi n e a fu n cti on n a m ed p e r m P o s i t i o n th a t d oes th i s for el em en t n of th e exa m p l e p e r m : p e r m P o s i t i o n n  : P o s i t i o n p e r m , n ; Tr y i t ou t:

6 5

S y m m e tr y T h e o r y

p e r m P o s i t i o n 4

5 

It s a y s th a t 4 i s i n th e 5 th p os i ti on of p e r m . Th i s i s tr u e. So we j u s t M a p ou r n ew op er a tor on to th e i n teg er s i n n a tu r a l or d er : p e r m I n v e r s e R a w M a p p e r m P o s i t i o n , 1 , 2 , 3 , 4 , 5 

3 , 4 , 1 , 5 , 2 

R em em b er , p e 3 r d p l a ce, th e cu r l y b r a ck ets . m u l ti p l y b r a ck s im p leL i s t :

r m wa 2 is in F or tu n eted ob

s { 3 , 5 , 1 , 2 , 4 } . So th e r es u l t a b ove s a y 4 th p l a ce, etc. Th a t i s wh a t we wa n t, excep a tel y , th er e i s a n op er a tor ca l l ed F l a t t e n j ect a n d r em oves a l l i n ter i or b r a ck ets , tu r

s : th e 1 i s i n t for a l l th os e th a t ta k es a n y n i n g i t i n to a

a u t o m a t e d I n v p e r m I n v e r s e R a w  F l a t t e n 3 , 4 , 1 , 5 , 2 

Is th i s th e i n ver s e th a t we fou n d a t th e top b y th i n k i n g i t ou t, or b y u s i n g I n v e r s e P e r m u t a t i o n ? a u t o m a t e d I n v p e r m I n v Tr u e

So i t wor k ed . B u t r em em b er , th i s i s n ot a for m a l p r oof, on l y a n exa m p l e. O n y o u r o w n Th i n k a b ou t h ow y ou cou l d tu r n th i s exa m p l e i n to a g en er a l p r oof. A s s o c ia tiv ity N u m er i ca l exa m p l es a r e n ot p r oofs , b u t th ey ca n b e s u g g es ti ve. L et a P e r m R a n d o m P e r m u t a t i o n 1 6

1 4 , 9 , 1 3 , 8 , 6, 1 2 , 1 , 2 , 1 5 , 7 , 4 , 1 0 , 1 1 , 5 , 3 , 1 6

b P e r m R a n d o m P e r m u t a t i o n 1 6

7 , 1 3 , 3 , 6, 5 , 1 2 , 1 0 , 1 1 , 1 5 , 1 6, 9 , 4 , 1 , 8 , 2 , 1 4 

c P e r m R a n d o m P e r m u t a t i o n 1 6

1 0 , 1 4 , 8 , 1 3 , 6, 4 , 1 , 7 , 9 , 1 6, 1 2 , 5 , 3 , 1 5 , 2 , 1 1 

B el ow, th e r ed p a i r s a r e ca l cu l a ted fi r s t. If P e r m u t e i s a s s oci a ti ve, th e tr i p l e p r od u cts wi l l b e th e s a m e:

6 6

P e r m u ta tio n s u n d e r P e r m u te

P e r m u t e a P e r m , P e r m u t e b P e r m , c P e r m

P e r m u t e P e r m u t e a P e r m , b P e r m , c P e r m

Tr u e

If y ou a b ove, Soci ety (4 01) 3

ever fi n d p l ea s e s , 201 Ch 3 1-3 8 4 2.

a n a P e r m , en d th em , a r l es St. , P Th ey g et a

b P e r m wi th fu r ovi d en l ot of cr

, a n d c P e r m th a t p r od u ce d i ffer en l l d eta i l s , to th e Am er i ca n Ma th ce, R I, USA P h on e (4 01) 4 55-4 a ck p ot s tu ff a n d th ey wi l l k n ow wh a

Th e a s s oci a ti ve r el a ti on P e r m u t e [ a , P e r m u t e [ b , c ] ] = = P e r m u t e [ P e r i s a fa m ou s r el a ti on , a n d a tr u e on e, b u t n o q u i ck a n B og u s p r oofs of i t a b ou n d , s om e p u b l i s h ed u n d er ver y offer ed a q u i ck a n d ea s y p r oof, tr y th i s : Su b s ti tu te D a n d s ee i f th e p r oof s ti l l s eem s to h ol d .

m u t d ea b ig i v i

t r es u l ts em a ti ca l 00, F a x t to d o.

e [ a , b ] , c ] s y p r oof i s k n own . n a m es . If y ou a r e d e for P e r m u t e ,

D i v i d e a , D i v i d e b , c

, D i v i d e D i v i d e a , b , c  

a c

a ,

b

b c



N ow th er e i s a r i g or ou s a n d com p l ete p r oof th a t D i v i d e i s n ot a s s oci a ti ve. If y ou h a ve a “ p r oof ” th a t i t i s a s s oci a ti ve, th i n k a g a i n . P r ob a b l y th e cl ea r es t va l i d p r oof th a t P e r m u t e i s a s s oci a ti ve fol l ows s k etch : F i r s t p r ove th a t th er e i s a on e-to-on e cor r es p on d en ce b etween p er m ti on s a n d p er m u ta ti on m a tr i ces (a P e r m  a M a t , b P e r m  b M a t ) (cl i ck h a n d th en th a t P e r m u t e [ a P e r m , b P e r m ] cor r es p on d s to D o t [ a M a t M a t ] . Si n ce we k n ow th a t m a tr i ces a r e a s s oci a ti ve u n d er D o t, i t fol l ows p er m u ta ti on s a r e a s s oci a ti ve u n d er P e r m u t e .

A

6. 5 F r u i t fl i es u n d er r ep r od u cti on b r ep r f l y d eta

ottl e of od u cti on A f l y i l , a xi om

fr u i t fl i es . B i ol og B = f l y b y a xi om

d oes n i ca l r ep C , b u t , is a n

ot for m a g r ou p u r od u cti on g i ves m m a n y th i n g s a r e m excel l en t exer ci s e.

n d er ea n i n is s in Tr y

th i s u ta er e), , b  th a t

(a n on -g r ou p )

th e op er a ti on of b i ol og i ca l g to th e cl os u r e exp r es s i on g . Th i n k i n g of th em a l l i n i t, th en cl i ck h er e.

6 7

S y m m e tr y T h e o r y

6. 6 E n d N otes 6. 6. 1 Th e fa l s e cl a i m Th e fa l s e cl a i m i s ( c ) . Th e i n teg er s d o n ot for m a g r ou p u n d er T i m e s b eca u s e th e i n ver s e of a n i n teg er i s n ot a n i n teg er (excep t i n two s p eci a l ca s es ; n a m el y , 1 a n d -1).

6. 6. 2 R ea l n u m b er s Con s i d er a xi om s a d oes n ' t, p ower s g

th e s et { 1 y s th e s et m s o i t i s n ' t. r ow s tea d i l y

1 , a } , wh er e a i s a n y t con ta i n a 2 i f i t i s to b ep en d i n g on th e va l u e r g er or s tea d i l y s m a l l er ,

, u s D la

oth er r ea l n u m b er . e a g r ou p u n d er T i of a , a b s ol u te va l u s o cl os u r e i s n ever p

Th e cl os u r e m e s . B u tit es of fu r th er os s i b l e.

6. 6. 3 Un i t a n d i n ver s e of com p l ex n u m b er s Th e u n i t el em en t of th e com p l ex n u m b er s u n d er T i m e s i s 1 + * 0 . In oth er wor d s , i t i s j u s t th e u s u a l 1 i ts el f. Th e i n ver s e of a + * b is a    b   a 2  b 2 . If y ou d on ' t b el i eve i t, m u l ti p l y i t ou t. You wi l l s ee.

6. 6. 4

F i n i te g r ou p s of com p l ex n u m b er s

Th e " n th r oots of u n i ty " a r e a s et of n com p l ex n u m u l ti p l i ca ti on Sta r ti n g a t { 1 , 0 } i n th e com p s p a ced a r ou n d th e u n i t ci r cl e b y 2 Π/ n r a d i a n s . th e u n i t ci r cl e m ove s tea d i l y cl os er to th e or i g i n a wa y fr om th e or i g i n . B u t p ower s of n u m b er s on ci r cl e. We g en er a te th e fi ve fi fth r oots of u n i ty : s o l n S o l v e u 5

m b er s th a t for m l ex p l a n e, th ey P ower s of a n y ; ou ts i d e, th ey th e u n i t ci r cl e s

1 , u   N ;

r o o t s 5 s o l n . u r  r

 F l a t t e n

1 . , 0 . 8 0 9 0 1 7  0 . 5 8 7 7 8 5 , 0 . 3 0 9 0 1 7  0 . 9 5 1 0 5 7 , 0 . 3 0 9 0 1 7  0 . 9 5 1 0 5 7 , 0 . 8 0 9 0 1 7  0 . 5 8 7 7 8 5 

6 8

a g r ou p u n d er a r e r eg u l a r l y n u m b er i n s i d e m ove s tea d i l y ta y on th e u n i t

E n d N o te s

We u s ed a of n u m b er / . ( u r _ p l a n e) for

l i ttl e tr i ck s a b ove. Th e ou tp u t of S o l v e i s a l i s t of R u l e s , n ot a l i s t s , b u t we con ver ted i t to a l i s t of n u m b er s b y a p a tter n -m a tch i n g tr i ck ) r . Ta k e a l ook a t th e Ar g a n d d i a g r a m (a p l ot i n th e com p l ex th es e n u m b er s :

I m a g in a r y 1

1 1

R e a l

1 F ig . 6 .1

Th e fi ve r ed d ots a r e th e fi fth r oots of u n i ty .

Ta k e th e fi fth p ower of ea ch on e, j u s t to m a k e s u r e: r o o t s 5

5

 C h o p I n t e g e r

1 , 1 , 1 , 1 , 1 

L ook a t th i s s tep b y s tep to s ee th e fu n cti on of a l l i ts p a r ts . r o o t s 5 , th en a t r o o t s 5 5 , th en wi th C h o p I n t e g e r .

F i r s t l ook a t

6. 6. 5 F r u i t fl i es 1 . Th e Cl os u r e Axi om i s n ot r ea l l y fu l fi l l ed , b eca u s e fl i es a r e ei th er m a l e or fem a l e, a n d th e exp r es s i on i s T r u e on l y for ( m a l e f l y ) ( f e m a l e f l y ) = f l y . 2 . Th e Un i t Axi om m i g f l y E i n wh i ch ever y g en cer n ed , f l y A  f l y E = s a m e a s f l y A , i t wou l d n s i ve g en es . 3 . G i ven f l y A , i s a l wa y s y i el d s f l y E z i l l i on tr i es , (or i t Men d el i a n i n h er i ta n a n y g i ven f l y A .

h t in eis r f l y ot r ea

a n i n com p l ete s en eces s i ve. Th en a s A . B u t a l th ou g h l l y b e th e s a m e; i t

s e b e fu l fi l l ed b y a p ecu l i a r fa r a s th e p h en oty p e i s con th e offs p r i n g m i g h t l ook th e wou l d h a ve a l ot m or e r eces -

th er e a f l y A i n v e r s e s u ch th a t f l y A  f l y A i n v e r s e , wi th a l l -r eces s i ve g en es ? Th i s m i g h t h a p p en on ce i n a cou l d h a p p en m or e q u i ck l y u n d er a r ti fi ci a l s el ecti on ) b u t ce i s p r ob a b i l i s ti c, a n d i t cou l d n ot h a p p en r ep ea ta b l y for

6 9

S y m m e tr y T h e o r y

4 . Th e a s s oci a ti ve a xi om d efi n i tel y d oes n ot h ol d , b eca u s e ( f l y A  f l y B )  f l y C h a s h a l f i ts g en es fr om f l y C , wh er ea s f l y A ( f l y B  f l y C ) h a s on l y a q u a r ter of i ts g en es fr om f l y C . Too b a d . G r ou p th eor y j u s t ca n n ot h el p wi th Men d el i a n g en eti cs .

6. 6. 6 P er m u ta ti on m a tr i ces We m u s t s h ow th a t th er e i s a on e-to-on e cor r es p on d en ce b etween p er m u ta ti on s a n d p er m u ta ti on m a tr i ces . B eg i n b y con s tr u cti n g a p er m u ta ti on m a tr i x P m a t , a n n n m a tr i x th a t m u l ti p l i es l eftwa r d i n to a n or d er ed L i s t of n ob j ects to p r od u ce th e r eq u i r ed p er m u ta ti on of th os e ob j ects . Tota l l y g en er a l n ota ti on for th i s g ets con fu s i n g , s o we s i m p l y ta k e a n exa m p l e. Wh a t i s th e p er m u ta ti on m a tr i x th a t ch a n g es { a , b , c , d } i n to { b , c , d , a } ? We d efi n e i t a s th e m a tr i x P m a t th a t ob ey s a , b , c , d . P m a t b , c , d , a 

( 6. 1 )

You m i g h t p r efer th a t th e l eft s i d e b e P m a t . { a , b , c , d } , b u t we h a ve a r ea s on to p u t P m a t on th e r i g h t, a n d we s ti ck fi r m l y wi th i t. We ca n s ol ve th i s for P m a t u s i n g th e M a t r i x O f C o e f f i c i e n t s op er a tor :  M a t r i x O f C o e f f i c i e n t s [ e x p r s , v a r s ] r etu r n s m a tr i x M C , wh er e M C . v a r s = e x p r s . It m a y fa i l i f th e v a r s a r e n ot s i m p l e s y m b ol s , a n d i t wi l l fa i l i f th e e x p r s a r e n ot l i n ea r i n th e v a r s .

Actu a l l y th i s i s n ot exa ctl y wh a t we wa n t. We wou l d p r efer a n M C s u ch th a t v a r s . M C = = e x p r s , b u t we ca n g et th e s a m e th i n g i f we s i m p l y a p p l y a T r a n s p o s e to th e r es u l t of th i s op er a tor . P m a t M a t r i x O f C o e f f i c i e n t s b , c , d , a , a , b , c , d   T r a n s p o s e ;   M a t r i x F o r m 0

0

1 0 0

0

0 1 0

1

0 1

0

0 0 0

P er m u ta ti on m a tr i ces a r e ea s y to r ecog n i z e a t a g l a n ce: th ey m u s t b e a l l z er oes excep t for a s i n g l e 1 i n ea ch r ow a n d col u m n . Ma k e s u r e i t r ea l l y wor k s : 0

0

1

1

0

0

0

0

1

0

0

0

0

1

0

a , b , c , d .

0

7 0

b , c , d , a 

E n d N o te s

Tr u e

We h a ve a n op er a tor th a t con ver ts a th e cor r es p on d en ce p e r m u t a t i o n a b ou t th e r ever s e cor r es p on d en ce, m r i g h t u p a b ove. G i ven a n y p er m u ta p er m u ta ti on , a n d th e a n s wer i s th e p So th e cor r es p on d en ce p e r m u t a t i o

s i n g l e p er m u ta ti on i n to a s i n g l e m a tr i x, s o

m a t r i x i s a l wa y s on e-to-on e, Wh a t a t r i x p e r m u t a t i o n ? It i s s i tti n g ti on m a tr i x, D o t i t l eftwa r d i n to th e u n i t er m u ta ti on th a t cor r es p on d s to th e m a tr i x. n  m a t r i x i s on e-to-on e b oth wa y s .

F or th e cl os u r e a xi om to h ol d we n eed to s h ow th a t th e p r od u ct of a n y two p er m u ta ti on m a tr i ces i s a p er m u ta ti on m a tr i x. In oth er wor d s , th e p r op er ty of b ei n g a p er m u ta ti on m a tr i x m u s t com e d own i n th e b l ood l i n e of m a tr i ces . Th e q u es ti on i s : If p M a t A a n d p M a t B a r e p er m u ta ti on m a tr i ces , a n d i f p M a t A . p  M a t B i s p M a t A B , ca n we b e s u r e th a t p M a t A B i s a p er m u ta ti on m a tr i x? We ca n s ettl e th i s ea s i l y u s i n g th e cor r es p on d en ce j u s t es ta b l i s h ed . Sta r t wi th two p er m u ta ti on s A th r ee p er m u ta ti on s to p er m u ta tees th a t p M a t A B i s a p er m u ta to s ee i f i t i s tr u e. We d i a g r a m p e r m A p M a t A

a n d ti on ti on th i s

 p e r m B



p M a t B

= ? =

.

B , a n d fi n d th ei r p r od u ct. Th en con ver t a l l m a tr i ces u s i n g p e r m T o M a t . Th i s g u a r a n m a tr i x. F i n a l l y , tes t th e d ot p r od u ct r el a ti on p r oces s b el ow : p e r m A B p M a t A B

D o a con cr ete exa m p l e : L et p e r m A 2 , 5 , 3 , 1 , 4 ; p e r m B 3 , 2 , 4 , 5 , 1 ;  p e r m B i s i m p l em en ted a s

Th e op er a ti on p e r m A

p e r m A B P e r m u t e p e r m A , p e r m B

3 , 5 , 1 , 4 , 2 

N ow con ver t a l l th r ee p er m u ta ti on s to m a tr i ces : p M a t A , p M a t B , p M a t A B  M a p p e r m T o M a t , p e r m A , p e r m B , p e r m A B  ;   M a t r i x L i s t  S i z e 6

0

0 1

0 0

 0 0 0

0

0

0 1

0

0

0

1 0

1

0

0

0 0

0

0

0

0 0

0

1

1 ,

1 0

0

0 0

0 0

0 0

0 0

1

0 0

1 0

0

1 0

1

0

0 0

0

0

0

0 1

0

0

0

0 0

1

0

1 ,

0 0

0

0

0

1 0



0 0

N ow p M a t A B i s k n own to b e a p er m u ta ti on m a tr i x b eca u s e i t wa s cr ea ted fr om a p er m u ta ti on b y p e r m T o M a t . So ca r r y ou t th e tes t p M a t A . p M a t B p M a t A B

7 1

S y m m e tr y T h e o r y

Tr u e

Th i s i s on l y a n exa m p l e, n ot a p r oof. B u t th er e i s n oth i n g s p eci a l a b ou t th e p e r m A a n d p e r m B we ch os e, a n d i n fa ct th e r el a ti on i s a l wa y s tr u e.

7 2

7. Th e fu n d a m en ta l th eor em P r el i m i n a r i es

7. 1. Sta tem en t a n d com m en ta r y

T h e fu n d a m e n ta l th e o r e m o f sy m m e tr y th e o r y Th e s y m m etr y tr a n s for m s of a n ob j ect for m a g r ou p u n d er th e op er a ti on of s eq u en ti a l a p p l i ca ti on . Th i s Ch a p ter p u ts tog eth er th e d efi n i ti on of s y m m e t r y g i ven i n Ch a p ter 4 , a n d th e d efi n i ti on of g r o u p g i ven i n Ch a p ter 5. You m i g h t th i n k th e s i m p l es t wa y to p r ove th i s th eor em wou l d b e j u s t to g o d own th e a xi om s a n d s h ow th a t th e s y m m etr y tr a n s for m s of a n ob j ect fu l fi l l ea ch a xi om . Th i s a l m os t wor k s , b u t i t r u n s i n to tr ou b l e a t th e In ver s e Axi om . F or i n s ta n ce, con s i d er a l l th e s y m m etr y r ota ti on s of th e cu b e; th er e a r e on l y twel ve of th em . As wi l l b e s h own i n d eta i l i n th e n ext ch a p ter , th ey ca n b e con s tr u cted b y cl os u r e fr om on l y two r ota ti on s , s o th e Cl os u r e Axi om i s s a ti s fi ed . Th er e i s a n o-r ota ti on el em en t, s o th e Un i t Axi om i s s a ti s fi ed . Al l m a tr i x m u l ti p l i ca ti on s a r e a s s oci a ti ve, s o th e As s oci a ti ve Axi om i s s a ti s fi ed . B u t h ow ca n we b e s u r e (wi th ou t d eta i l ed exa m i n a ti on , wh i ch we wa n t to a voi d i n th i s g en er a l th eor em ) th a t wi th i n th i s l i ttl e cl os ed s et, ever y el em en t h a s a n i n ver s e? Str a n g el y , n o on e h a s b een a b l e to cr ea te a s i m p l e, d i r ect l i n e of r ea s on i n g th a t a s s u r es i t. F or tu n a tel y , a cl ever tr i ck h a s b een fou n d th a t d oes th e j ob wi th ou t too m u ch d i s tr a cti on . Th e tr i ck b eg i n s b y r em em b er i n g th a t th e twel ve m a tr i ces i n q u es ti on a r e j u s t twel ve ch os en fr om th e u n i ver s e of a l l u n i ta r y 3 -b y -3 m a tr i ces , a s et wh i ch i s k n own to for m a g r ou p . Th en th er e a r e two th i n g s to s h ow: (1) E ver y m em b er of th e cl os ed s et i s a s y m m etr y tr a n s for m of th e ob j ect, a n d (2) Th e cl os ed s et i n h er i ts , fr om i ts i n fi n i te u n i ver s e g r ou p , a l l th e p r op er ti es th a t m a k e i t a fi n i te g r ou p . Th e tr i ck m a k es th i s q u i te ea s y .

W.M. McClain, Symmetry Theory in Molecular Physics with Mathematica, DOI 10.1007/b13137_7, © Springer Science+Business Media, LLC 2009

7 3

S y m m e tr y T h e o r y

7. 2. P r oof of th e fu n d a m en ta l th eor em 7. 2. 1. Th e u n i ver s e of tr a n s for m s We wi l l wr i te th i s p r oof s p eci fi ca l l y for 3 -b y -3 m a tr i x tr a n s for m s . It cou l d b e m a d e m or e g en er a l , b u t th a t wou l d i n vol ve a l ot of b l a th er th a t d oes n ot r ea l l y cl a r i fy th e es s en ti a l s . Th a t i s a va i l a b l e a l r ea d y i n p l en ty of m a th b ook s . In Ch a p ter 6 we s h owed th a t a l l u n i ta r y n -b y -n m a tr i ces for m a g r ou p u n d er D o t . In p a r ti cu l a r , th i s a p p l i es to 3 -b y -3 m a tr i ces . Th i s i n fi n i te g r ou p i s th e u n i ver s e-g r ou p fr om wh i ch ou r s y m m etr y tr a n s for m s a r e ch os en . In Ch on for

th i s a p ter th ei r b oth

p r oof, we 6, a n d s om r i g h t, a s i n k i n d s of op

wi l l s om eti m es D o t th e m a tr i ces i n to th em s el ves , a s i n eti m es we wi l l D o t th em i n to a n a r r a y of n p oi n t col u m n s Ch a p ter 4 . Th u s th e a p p l i ca ti on op er a tor  i s wel l d efi n ed er a n d s .

7. 2. 2. Th e s y m m etr i c ob j ect Her e we r evi ew, i n m or e g en er a l l a n g u a g e, s om e th i n g s th a t wer e s h own b y exa m p l e i n Ch a p ter 4 . A com p l ex g eom etr i c ob s y m b ol i z e a s  p 1 , p 2 , th a t th e con ten ts a r e l i k e a m ea n i n g l es s . Wh en a tr a n T  p

, p 1

2

j ect !. l i s t, s for m

i s r ep r es en ted b y a s Th e p oi n ted b r a ck ets excep t th a t th e or d er i n T i s a p p l i ed to s u ch a n

, ! m ea n s T  p 1

, T p 2

e t of p oi n ts , wh i ch we en cl os i n g th e s e t m ea n wh i ch th ey a r e wr i tten i s ob j ect,

, !.

G en er a l l y , T wi l l ch a n g e ea ch p oi n t i n to a n oth er p oi n t, a n d th e tr a n s for m ed s et wi l l b e d i ffer en t fr om th e or i g i n a l s et. B u t i t ca n h a p p en th a t T ei th er l ea ves a p oi n t u n ch a n g ed , or el s e j u s t ch a n g es i t to a n oth er p oi n t a l r ea d y i n th e s et. If n o ol d p oi n ts a r e l os t, a n d n o n ew on es cr ea ted , th e s et i s s y m m etr i c u n d er T , a ccor d i n g to ou r for m a l d efi n i ti on i n 4 . 1 . p

L et s y m

, p 1

s y m

T 1

2

, ! b e s y m m etr i c u n d er two tr a n s for m s ,

s y m  a n d

Th en th e a s s oci a ti ve a xi om T 1

 T 2

T

s y m 2

s y m

r eq u i r es

 s y m  T 1

 T 2

s y m s y m

a n d we s ee th a t th e p r od u ct T 1  T 2 i s a l s o a s y m m etr y op er a tor for s y m . We g i ve i t a n a m e l i k e th os e of th e oth er s y m m etr y op er a tor s T 3 T 1  T 2 . F or a

7 4

P ro o f

con cr ete exa m p l e, cl i ck b a ck to Ch a p ter 4 , “ N ew s y m m etr i es fr om

ol d ” .

7. 2. 3 . Con s tr u ct a cl os ed s et of tr a n s for m s Th e p r oces s d es cr i b ed b el ow i s s o u s efu l th a t we wi l l m a k e a M a t h e m a t i c a op er a tor M a k e G r o u p th a t ca r r i es i t ou t a u tom a ti ca l l y . B u t fi r s t, we d es cr i b e th e p r oces s for m a l l y . Sa y we s ta r t wi th a s et of two s y m m etr y tr a n s for m s , T 1 , T 2 . We m a k e n o effor t to i n cl u d e th e u n i t el em en t, or to m a k e th em m u tu a l l y i n ver s e. We th en com p u te a l l p a i r wi s e p r od u cts T 1  T 1 , T 1  T 2 , T 2  T 1 , T 2  T 2 . Th i s m i g h t cr ea te n oth i n g n ew, or i t m i g h t cr ea te a s m a n y a s fou r n ew tr a n s for m s . We a p p en d ever y th i n g n ew to th e or i g i n a l s et of two, cr ea ti n g T 1 , T 2 , T 3 , . Th en we m a k e a n ew ta b l e of p a i r wi s e p r od u cts , i n cl u d i n g p r od u cts l i k e T 1  T 3 , wh i ch we d i d n ot h a ve b efor e. We a g a i n a p p en d a n y n ew p r od u cts to th e or i g i n a l s et, a n d r ecy cl e. An d s o on , a n d on . We wi l l s ee s om e g r ou p s wh er e y ou h a ve to s ta r t wi th m or e th a n j u s t two tr a n s for m s . B u t th e cl os u r e p r oces s i s en ti r el y s i m i l a r . We a s s u m cea s es to m (Th i s a l wa “ s i m p l e” h p r oof; i t i s

e n ow th a t a fter a cer ta i n fi n i te n u m b er of i ter a ti on s , th i s p r oces s a k e n ew p r od u cts ; or i n oth er wor d s , we a s s u m e th a t th e s et cl os es . y s h a p p en s i f th e ob j ect h a s a s i m p l e g eom etr i c s y m m etr y . B u t s i n ce a s b een g i ven n o for m a l m ea n i n g , th i s s ta tem en t i s n ot p a r t of th e j u s t exp l a n a tor y . B u t y ou k n ow wh a t we m ea n . )

7. 2. 4 . F i n i te s u b g r ou p th eor em

T h e o r e m I f  i s a g r ou p , a n d  i s a fi n i te s u b s et of  th a t i s cl os ed u n d er m u l ti p l i ca ti on , t h e n  i s a g r ou p . C o m m e n t :  m a y b e fi n i te or i n fi n i te, b u t a s we wor k , th i n k of i t a s th e i n fi n i te g r ou p of a l l u n i ta r y 3 -b y -3 m a tr i ces u n d er D o t . L e m m a 1 E ver y el em en t of  i s th e b a s i s of a cy cl i c p ower s eq u en ce wi th i n . L et A b e a n y Al l m em b er s ca ti on . F u r th r ecu r r en t el em

m em b er of , a n of th e s eq u en ce m er , b eca u s e  i s en t B fr om wh i ch

d con s i d er th e i n fi n i te s eq u en ce of p ower u s t b e i n , b eca u s e  i s cl os ed u n d er m fi n i te, th er e m u s t b e a m on g th e p ower s wh ol e s eq u en ce r ep ea ts ; oth er wi s e th e n

s of A . u l ti p l i of A a u m b er

7 5

S y m m e tr y T h e o r y

of el em en ts i n  wou l d b e i n fi n i te. Su p p os e th a t B i s A m 1 , a n d th a t i t r ep ea ts wi th p er i od n . (L em m a 2 wi l l s h ow th a t A n 1 i s E . ) Th en th e s eq u en ce m u s t r ea d

B u tB is A

, A m , B , B A n , B ,

2

A ,

A

,

m 1

, ,

A

n

,

B ,



, s o we ca n r ewr i te wi th ou t u s i n g B a t a l l :

2

A ,

B A ,  , B A ,  , B A

A

m

,

A

m 1

A

, A

m 2

m n

, A

, , m 1 , , A

m n

, A

m 1

, 

N ote th a t a fter th e p ower m + n , th e p ower d r op s b a ck to m + 1 . N ow A wa s a n y el em en t, n ot or i g i n a l l y cl a i m ed a s a r ecu r r i n g el em en t. B u t l ook i n g a t th e fi r s t p a r t of th e s eq u en ce, we s ee th a t wh er e A m 1 occu r s , we ca n cou n t b a ck m p l a ces a n d we wi l l b e a t A a g a i n . So i n th i s s eq u en ce, ever y el em en t i s a r ecu r r i n g el em en t wi th p er i od m . (We or i g i n a l l y s a i d th e p er i od of B wa s n , b u t n ow we s ee th a t ever y el em en t h a s th e s a m e p er i od , s o n = m . ) So th e s eq u en ce i s r ea l l y A ,

A

2

,

A ,

, A m , A 2 , , A m , A , A 2 , , A

m

,



So th e s et of th e a l l p ower s of A th a t occu r i n i ts own p ower s eq u en ce i s j u s t a fi n i te l i ttl e s et,  [ A ] = A , A 2 ,  , A m . In oth er wor d s , ever y el em en t of  i s th e b a s i s of a cy cl i c p ower s eq u en ce, wh i ch wa s to b e p r oved . L e m m a 2 Th e p ower s eq u en ce for ever y el em en t A i n  i n cl u d es th e i d en ti ty . Si n ce A m i s fol l owed b y A , i t m u s t b e tr u e th a t A m  A A a n d r eg r ou p i n g , a l s o th a t A  A m A . B u t b y d efi n i ti on , on l y th e u n i t el em en t E b eh a ves l i k e A m i n th es e eq u a ti on s , s o A m E . L e m m a 3 Th e p ower s eq u en ce for ever y el em en t A A 1 . Si n ce E i s p r eced ed b y A m 1 , A  A m 1 E . B u t th e el em en t th i n ver s e of A . Si n ce we s ta r ted b y i n ver s e of ever y el em en t i n th e s eq u en ce.

7 6

i n  i n cl u d es th e i n ver s e

we h a ve A m 1  A E a n d r eg r ou p i n g , a t b eh a ves l i k e A m 1 i s b y d efi n i ti on A 1 , th e s a y i n g " L et A b e a n y m em b er of  " , th e s eq u en ce i s a l s o p r es en t s om ewh er e i n th e

P ro o f

P r o o f o f th e th e o r e m N oth i n g i n L em m a 1 s a y s th a t th e p ower s eq u en ce b a s ed on A ,  A [ ], con ta i n s a l l th e el em en ts of . E xcep t i n p u r e cy cl i c g r ou p s , i t d oes n ot. So i f C i s a n el em en t of  n ot con ta i n ed i n  A [ ], we s ta r t a g a i n wi th C a n d con s tr u ct i ts p ower s eq u en ce  C [ ]. We con ti n u e to con s tr u ct p ower s eq u en ces u n ti l a l l th e el em en ts of  a p p ea r i n s om e p ower s eq u en ce. Th en we com b i n e a l l th e p ower s eq u en ces , d i s ca r d i n g d u p l i ca te el em en ts . (In M a t h e m a t i c a , th i s i s d on e b y th e U n i o n op er a tor . ) Th e U n i o n of a l l p ower s eq u en ces of  i s th e s a m e a s  i ts el f, b eca u s e: ( a ) Th e p ower s eq u en ces d o n ot om i t a n y th i n g i n , b eca u s e ever y el em en t of  ei th er occu r r ed i n a p ower s eq u en ce, or wa s u s ed to s ta r t a n ew p ower s eq u en ce. ( b ) Th e p ower s eq u en ces ca n n ot con ta i n a n y th i n g n ew, b eca u s e we b eg a n b y s a y i n g " l et  b e a fi n i te s u b s et  cl os ed u n d er m u l ti p l i ca ti on " . N ow we ca n s u cces s fu l l y exa m i n e th e g r ou p a xi om s , on e b y on e. 1 . 2 . 3 . 4 .

C lo s u r e U n it In v e r se A s s o c ia tiv e

Th e s et  i s cl os ed u n d er m B y L em m a 2,  i n cl u d es th B y L em m a 3 ,  i n cl u d es a n Th e m u l ti p l i ca ti on op er a tor Si n ce  i s a g r ou p , th e op

u l ti p l i ca ti on b y e u n i t el em en t. i n ver s e p a r tn er for  i s th e s a m er a tor i s a s s oci a

h y p oth es i s . for ea ch el em en t. e a s for  . ti ve.

Al l fou r a xi om a ti c r eq u i r em en ts a r e m et. Th er efor e, i f  i s cl os ed m u l ti p l i ca ti on , th en  i s a g r ou p , a s wa s to b e s h own .

u n d er

7. 2. 5. E xa m p l e of p ower s eq u en ces i n a g r ou p Th e cen tr a l cl a i m of th e p r oof i s th a t a l l fi n i te g r ou p s ca n b e s p l i t u p i n to n on over l a p p i n g p ower s eq u en ces . J u s t to m a k e th i s a b s ol u tel y cl ea r , we ca r r y i t ou t exp l i ci tl y for a g r ou p n a m ed " D 3 h " , wh i ch i s ta b u l a ted i n th e S y m m e t r y ` p a ck a g e. Th e s ta n d a r d n a m es of i ts el em en ts a r e a l l G r o u p N a m e s

E l e m e n t N a m e s " D 3 h "

E , C 3 a , C 3 b , C 2 a , C 2 b , C 2 c , Σh , S 3 a , S 3 b , Σv a , Σv b , Σv c 

We con ver t on e of th e n a m es to th e m a tr i x i ts el f : m a t S 3 a  2

1

, 

" S 3 a " . N a m e s T o R e p M a t s " D 3 h "

2

3

, 0 ,  2

3

,  2

1

, 0 , 0 , 0 , 1 

We u s e N e s t L i s t (l ook i t u p ) to con s tr u ct th e p ower s eq u en ces b a s ed on a n

7 7

S y m m e tr y T h e o r y

el em en t n a m ed " S 3 a " . Si n ce th e g r ou p h a s on l y 12 el em en ts , we m a k e a s eq u en ce of 12 el em en ts . Th i s h a s to b e l on g en ou g h .  S 3 a N e s t L i s t ". m a t S 3 a & , m a t S 3 a , 1 2 . R e p M a t s T o N a m e s " D 3 h "

S 3 a , C 3 b , Σh , C 3 a , S 3 b , E , S 3 a , C 3 b , Σh , C 3 a , S 3 b , E , S 3 a 

Th i s p ower s eq u en ce h a s a r ep ea t l en g th of 6. We el i m i n a te th e r ep ea ts :  S 3 a  S 3 a  U n i o n C 3 a , C 3 b , E , S 3 a , S 3 b , Σh 

Wh i ch el em en ts of g r ou p D 3 h d o n ot a p p ea r i n th i s s eq u en ce? C o m p l e m e n t a l l G r o u p N a m e s ,  S 3 a

C 2 a , C 2 b , C 2 c , Σv a , Σv b , Σv c 

Si x el em en ts d i d n ot a p p ea r . Th e n ext s tep i s to m a k e th e p ower s eq u en ce b a s ed on th e fi r s t s u ch el em en t : m a t C 2 a

" C 2 a " . N a m e s T o R e p M a t s " D 3 h "

1 , 0 , 0 , 0 , 1 , 0 , 0 , 0 , 1 

 C 2 a N e s t L i s t ". m a t C 2 a & , m a t C 2 a , 1 2 . R e p M a t s T o N a m e s " D 3 h "

C 2 a , E , C 2 a , E , C 2 a , E , C 2 a , E , C 2 a , E , C 2 a , E , C 2 a 

 C 2 a  C 2 a  U n i o n C 2 a , E 

Th e r ep ea t l en g th i s on l y 2. Th i s s a y s th a t el em en t C 2 a i s i ts own i n ver s e. Th i s i s tr u e; C 2 a i s a twofol d r ota ti on . Th e s a m e i s tr u e for a n y el em en t wi th a C 2 _ n a m e. Si m i l a r l y , a l l Σ_ n a m es a r e r efl ecti on s , a n d a l s o g i ve p ower s eq u en ces wi th on l y two m em b er s . Th er efor e, th e wh ol e g r ou p i s th e U n i o n of th e p ower s eq u en ces b a s ed on s even of i ts el em en ts : S 3 a , C 2 a , C 2 b , C 2 c , Σv a , Σv b , a n d Σv c . Th i s i s a ver y ty p i ca l s i tu a ti on .

7 8

H o w th is th e o r e m h e lp s

7. 3 . How th i s th eor em

h el p s

Step h en J a y G ou l d , l a te l a m en ted n a tu r a l h i s tor y es s a y i s t a n d Ag a s s i z P r ofes s or of B i ol og y a t Ha r va r d , tol d th i s s tor y a b ou t h i s ep on y m ou s p r ed eces s or , L ou i s Ag a s s i z . P r of. Ag a s s i z a s s i g n ed a n ew s tu d en t to d r a w a fi s h p i ck l ed i n for m a l d eh y d e. At th e en d of th e d a y , Ag a s s i z g l a n ced a t th e d r a wi n g s a n d s a i d , " N o! You h a ve n ot s een on e of th e m os t es s en ti a l fea tu r es of th i s a n i m a l ! " . Th i s wen t on for s ever a l d a y s . Th e s tu d en t' s d r a wi n g s b eca m e m or e a n d m or e d eta i l ed i n fi n a n d s ca l e a n d m a r k i n g , b u t Ag a s s i z a l wa y s r ej ected th em , n ever exp l a i n i n g wh y . F i n a l l y , on th e s i xth d a y , fa i n ti n g fr om for m a l d eh y d e fu m es , th e s tu d en t s a w i t: Hi s er r or wa s n ot i n th e d eta i l s , b u t i n th e p i ctu r e a s a wh ol e. Th e fi s h h a d b i l a ter a l r efl ecti on s y m m etr y . We r ep ea t th i s g ood a t s eei n g h owever , g i ves a s k , a b s ol u tel y ti on . We d i s cu

s tor y to r em i n d ou r s el ves th a t h u m a n s a r e n ot n eces s a r i l y ver y s y m m etr y tr a n s for m s " i n s ti n cti vel y " . Th e fu n d a m en ta l th eor em , u s a wa y of fi n d i n g th e on es we fa i l to s ee. B u t wa i t, y ou m a y a l l of th em , wi th ou t fa i l ? Wel l , y es , b u t wi th on e l i ttl e r es er va s s th e r es er va ti on l a ter ; fi r s t, we d es cr i b e th e b a s i c p r oced u r e:

To con s tr u ct th e g r ou p of a n ob j ect, b eg ch os en Ca r tes i a n coor d i n a te s y s tem on tr a n s for m s th a t y ou ca n s ee, i n th e for m (You wi l l s ee m a n y exa m p l es of th es e m m i s s s om e of th e ob j ect' s s y m m etr y tr a m a tter .

i n b y exa m i n i n g th e ob j ect. P i t, a n d wr i te d own a l l of i ts s of 3 -b y -3 Ca r tes i a n tr a n s for m a tr i ces j u s t a h ea d . ) You wi l l n s for m s , b u t d on ' t wor r y ; th i s

u t a wel l y m m etr y m a tr i ces . p r ob a b l y wi l l n ot

Wh en y ou h a ve a l i s t of a s m a n y s y m m etr y m a tr i ces a s y ou ca n s ee, m a k e a m u l ti p l i ca ti on ta b l e fr om th em . An y n ew m a tr i x th a t a p p ea r s i n th e ta b l e i s a s y m m etr y m a tr i x th a t y ou m i s s ed b y d i r ect ob s er va ti on . L ook a t th e ob j ect a g a i n , a n d y ou wi l l s ee i t th i s ti m e. N ow j oi n th e n ew m a tr i ces on to th e ol d l i s t, a n d ca l cu l a te th e n ew, l a r g er ta b l e. Con ti n u e th i s u n ti l y ou g et a ta b l e wi th n oth i n g n ew i n i t. At th i s p oi n t y ou wi l l h a ve a g r o u p of s y m m etr y tr a n s for m s th a t ch a r a cter i z es th e ob j ect. N ow com es th e u n a n s wer a b l e q u es ti on : Is th i s th e s et of AL L s y m m etr y tr a n s for m s of th e g i ven ob j ect? You ca n n ever b e m a th em a ti ca l l y cer ta i n th a t i t i s , b eca u s e i t i s p os s i b l e th a t i n y ou r exa m i n a ti on , y ou fa i l ed to n oti ce a s y m m etr y el em en t th a t i s th er e, b u t com p l etel y i n d ep en d en t fr om th e on es th a t y ou d i d s ee. If, a t a l a ter ti m e, th i s n ew s y m m etr y el em en t com es to y ou r a tten ti on , y ou m a y a d d i t to y ou r ta b l e a n d exp a n d th e ta b l e u n ti l i t cl os es a g a i n . You r d es cr i p ti on of th e s y m m etr i es of th e ob j ect wi l l b e i m p r oved ; b u t th en of cou r s e, th e q u es ti on com es a g a i n : D o we h a ve AL L th e s y m m etr i es th i s ti m e? Th e a n s wer , a l a s , m u s t b e th e s a m e.

7 9

8. The multiplication table Preliminaries

8.1. The generalized “multiplication” table Consider a group with unspecified operator  and a list of n elements, named {E, A, B,..., Z}. The Closure Axiom says that a “ product” is defined for every ordered pair of these elements, giving n2 products in all. It seems natural to display them in a square “multiplication” table. We have written a display operator for such tables, called BoxUp. Here is what it produces if we take the elements to be {E, A, B,C}, and the entries to be blanks : blankMat = Table@"", 8i, 4