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 three-dmensonal 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 block-dagonal matrces. Block-dagonal matrces are multpled accordng to the scheme,.e. the multplcaton can be reduced to the multplcatons of the sub-matrces 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 one-dmensonal 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.