Chapter 3 Group Theory p. 1  Remark: This is only a brief summary of most important results of groups theory with respect


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1 Chapter 3 Group Theory p Compact Course: Groups Theory emark: Ths s only a bref summary of most mportant results of groups theory wth respect to the applcatons dscussed n the followng chapters. For a more detaled descrpton see references. 3. Defnton of a group We assume a set of elements G {A, B, C,...}. Furthermore, we assume that there s a defnton of a combnaton of two elements AB, whch we denote as the product of two elements. G s a group f the followng condtons are satsfed:. Closure: AB G The product of two elements of the group s also an element of the group.. Identty element: The s an element E G, such that AE EA A for all A G. A neutral element exsts, whch has no effect on the other elements f the group. 3. Assocate law: A ( BC ) ( AB ) C for all A, B, C G. 4. Inerse element: The s an element A G, such that A A E for all A G. In nerse element exsts for all elements of the group, whch nerts the acton of a gen element. For some, but not for all groups, the commutate law holds (commutate law: AB BA for all A G ). These groups are called Abelan groups. Wthout proof (see e.g. F. A. Cotton): C ( ABC ) A B. (3.: Example for a group) 3. rder of a group The number of elements of a group s called the order of the group.
2 Chapter 3 Group Theory p ultplcaton table For any group, we can set up a multplcaton table, whch tabulates the results of the products of two elements. Wthout proof: Eery lne and column contans eery group element exactly once. No lne or column s dentcal to another one. (3.: How many types of groups are there wth three elements? Dere the multplcaton tables.) 3.4 Cyclc groups In a cyclc group all n elements are generated by powers of the frst element n n G {E( A ), A, A,...A }. (3.3: Example of a cyclc group) An mportant property of cyclc groups s that they are Abelan (as n m n+ m A A A A m A n ). 3.5 Subgroups A subset of the elements of the group G can tself form a group U. We call U a subgroup of G. (3.4: Example of a subgroup) 3.6 Symmetry groups The complete set of symmetry elements of a molecule, surface or crystal has the mathematc structure of a group. The set s called the symmetry group. (3.5: Example of a symmetry group. H molecule: show that the symmetry elements behae lke a group). 3.7 Classes We defne a smlarty transformaton B X AX
3 Chapter 3 Group Theory p. 3  whch transform some element A by means of another element X nto some other element B. If A and X are elements of the group G, the elements are called conugated elements. A complete set of elements, whch s conugated to one another s called a class of elements of the group. The classes hae a fgurate meanng: Those symmetry operatons belong to the same class, whch can be reached by a transformaton of the coordnate system, whch s part of the symmetry group. (3.6: Example: Dde the elements of the symmetry group C 4 nto classes). (The defnton of classes wll greatly smplfy the work wth symmetry groups). 3.8 epresentaton of symmetry operatons by matrces We can represent all symmetry operatons dscussed so far n the form of a matrx. In the smplest case, these matrces act on ponts X r n threedmensonal space and assgn a new poston bass coordnated r r X X (Note: f nstead we consder a bass transformaton defnng the new B r n terms of the old one as X r are r r  X A X ): (3.7: Examples for matrx representatons of symmetry operatons). r r B AB, the coordnates of the pont n the new 3.9 epresentatons of a group A set of matrces whch upon multplcaton behaes analogous to the elements of a group s called a representaton of the group. Example: We consder the transformaton of a pont X r n three dmensonal space accordng to the symmetry operatons of group C V.
4 Chapter 3 Group Theory p. 4  E ; C ; ; Wth respect to matrx multplcaton, these matrces follow the multplcaton table of group C V. 3. educble and rreducble representatons As specfc case of matrces are so called blockdagonal matrces. Blockdagonal matrces are multpled accordng to the scheme,.e. the multplcaton can be reduced to the multplcatons of the submatrces of lower dmenson: y b x a y x b a For the specfc example consdered, the matrces are completely dagonal,.e. all blocks are of dmenson. Accordngly we can reduce the three dmensonal representaton gen aboe nto three onedmensonal representatons, whch agan are representatons of the symmetry group C V : E E E ; C C C ; ; We consder a representaton of a group by a set of matrces of dmenson n. Addtonally, we consder a bass transformaton to a new coordnate system B B r r A, wth the coordnates of a ector n the new bass n terms of the old coordnates X X r  A. The representaton of the group n the new bass s A A A . For any representaton, we can search for the bass transformaton, whch yelds a set of representatons wth lowest possble dmenson. We
5 Chapter 3 Group Theory p. 5  denote a representaton wth the lowest possble dmenson as an rreducble representaton and a representaton wth hgher than mnmum dmenson as a reducble representaton. The example shows that there are rreducble representatons (bref: rreps) of dfferent type,.e. behang dfferently wth respect to the symmetry operatons contaned n the group. 3. Character of a matrx We defne the character χ of a matrx Γ as the sum oer the dagonal elements: Γ χ Γ. Γ (3.8: Character of matrces). The character of a matrx has an mportant property: It s narant upon a transformaton of the bass. (3.9: Character of matrces). Ths s qute handy, as n the followng t allows us to work wth characters nstead of the full representaton matrces, rrespecte of a specfc choce of the bass. 3. Propertes of rreducble representatons: GT great orthogonalty theorem (for proof see textbooks) Γ ( ) mnγ ( ) m n * δ δ δ mm nn h l l wth h : order of the group : symmetry operaton of the group Γ ( ) : matrx representaton for operaton of the rreducble representaton of type l : dmenson of the th type of rreducble representaton
6 Chapter 3 Group Theory p. 6  The ectors consstng of correspondng elements of the representaton matrces are orthogonal and normalzed. There are a number of smpler conclusons followng from the GT, whch can be easly proen (see e.g. A. F. Cotton), e.g: l χ ( E) h ; sum oer the dmenson squares of the rreps (sum oer the character squares of the dentty element) s equal to the order of the group. χ ( ) h ; sum oer character sqares oer all symmetry operaton for a gen type of representaton s equal to the order of the group. ( ) ( ) χ χ hδ : Character ectors of dfferent rreps are orthogonal. The characters of representaton matrces for a gen type of rrep for operatons belongng to a common class are dentcal. The number of classes s equal to the number of rreps. (3.: Deelop the characters and representaton matrces for the symmetry group C V from the aboe statements). 3.3 Analyss of reducble representatons The followng dea s a key pont for a large number of applcatons n the next chapters of ths course. We assume that Γ ( ) s a reducble representaton of the symmetry group G wth the correspondng characters χ ( ). We would lke to know, how many rreducble representatons of symmetry type are contaned n Γ ( ). For ths reason we assume that we hae transformed Γ ( ) to ts blockdagonal form ( ) As the characters are narant wth respect to ths transformaton, we obtan: χ ( ) χ ( ) a χ ( ) wth ( ) χ : character of th rrep of group Γ. a : number of tmes that th rrep s contaned n Γ ( ) By multplyng wth χ ( ) and summng oer all operatons of the group:
7 χ ( ) χ ( ) a χ ( ) χ ( ) a a χ ha h χ ( ) χ ( ) ( ) χ ( ) hδ a χ( ) χ ( ) Chapter 3 Group Theory p. 7  Here, all we need as an nput s the characters of the rreps of the group. These are lsted n the so called character tables. 3.4 Character tables ost mportant nformaton whch s requred to work wth a gen symmetry group s summarzed n the co called character table. Example: C 4V group name (Schoenfless) symmetry operatons ordered by classes symmetry propertes of some functons and ther classfcaton by rreps C 4V E C 4 C d A z x +y, z A   z B   x y B   (x, y) xy E  ( x, y ) (xz, yz)
8 Chapter 3 Group Theory p. 8  st of rreducble representatons: ullken notaton: () dm. rreps: A, B dm. rreps: E 3 dm. rreps: T 4 dm. rreps: G 5 dm. rreps: H () A/B: symmetrc / antsymmetrc wth respect to rotaton by π/n around prncple axs C n. (3) Index /: symmetrc / antsymmetrc wth respect to rotaton by π around C axs (perpendcular to C n ). (4) or : symmetrc / antsymmetrc wth respect to h. (5) g/u: symmetrc / antsymmetrc wth respect to.
v a 1 b 1 i, a 2 b 2 i,..., a n b n i.
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