Chimica Inorganica 3

Transcription

1 himica Iorgaica Irreducible Represetatios ad haracter Tables Rather tha usig geometrical operatios, it is ofte much more coveiet to employ a ew set of group elemets which are matrices ad to make the rule of multiplicatio that of matrix multiplicatio. The set of matrices formig the ew elemets are said to form a represetatio of the group. a a a a 4 a a a a a 4 a a a a a 4 a

2 himica Iorgaica Two operators commute whe  ˆ ˆ  Do Ĉ4 ( z ) ad σ xz commute? Ĉ 4 ( z) σˆ xz ( x, y, z ) ˆ σ xz Ĉ4 ( z) ( x, y, z ) Ĉ 4 ( z) ( x, y, z ) σˆ xz ( y, x, z ) ' σˆ d ˆ σ d

3 himica Iorgaica a a a a a a a a a a a a b b b b b b b b b b b b c a b + a b + a b + + a b c a b + a b + a b + + a b c a b + a b + a b + + a b c a b + a b + a b + + a b c ij a i b j + a i b j + a i b j + + a i b j k k k k c c c c c c c c c c c c a k b k a k b k a k b k k a k b k a ik b kj

4 himica Iorgaica D

5 himica Iorgaica ( ) ( ) D d ij b ik c kj k D E e ml a ms d sl a ms b st c tl s F f mt a ms b st s F G g ml f mt c tl a ms b st G E t s t s t c tl t s a ms b st c tl

6 himica Iorgaica a a a a a a a a a a a a b b b c c c c a k b k ; k c i a ik b k k Let us ow cosider a elemetary use of the operatio of a matrix o a vector x y z x' y' z'

7 himica Iorgaica Let us ow cosider matrices correspodig to symmetry operatio of the v group: Ê,Ĉ,Ĉ, ˆσ v, σˆ v, σˆ v The basis set is described by the triagles vertices, poits, ad. The trasformatio properties of these poits uder the symmetry operatios of the group are:

8 himica Iorgaica Ê ˆ σ v ˆ σ v Ĉ Ĉ ˆσ v

9 himica Iorgaica Ê x y x y 0 0 x y ˆσ v ' x y x y y x x y Ĉ x y x + y y x x y ˆσ v x y x y 0 0 x y If we would have cocetrated o coordiates (the z coordiate ca be disregarded) Ĉ x y x y y + x x y ˆσ v " x y x + y y + x x y

10 himica Iorgaica Which is the differece betwee the two sets of matrices? The former set is ot the simplest oe. I particular, matrices of the first set may be reduced through a similarity trasformatio. Ê ˆσ v Ê 0 0 ˆσ v 0 0 Ĉ Ĉ σˆ v σˆ v Ĉ Ĉ ˆσ v ' ˆσ " v

11 himica Iorgaica Similarity trasformatios yield irreducible represetatios (Γ i ), which lead to a useful tool i group theory the character table. The geeral strategy for determiig Γ i is as follows:, ad are matrix represetatios of symmetry operatios of a arbitrary basis set (i.e., elemets o which symmetry operatios are performed). There is some similarity trasform operator V such that: V ˆ V ˆ * V ˆ V ˆ * V ˆ V ˆ * V uiquely produces block-diagoalized matrices, which are matrices possessig square arrays alog the diagoal ad zeros outside the blocks * * *

12 himica Iorgaica Matrices,, ad are reducible. Sub-matrices i, i ad i obey the same multiplicatio properties as,, ad. If applicatio of the similarity trasform does ot further block-diagoalize *, *, ad *, the the blocks are irreducible represetatios. The character is the sum of the diagoal elemets of Γ i. E σ v σ v ' σ " v

13 himica Iorgaica The iverse of a matrix It ca be show that the iverse of a matrix ca be obtaied oly for those cases i which 0. square matrix whose determiat is zero is called a sigular matrix; otherwise it is o-sigular. If is o-sigular ( 0), we ca defie a matrix, deoted by ad called the iverse of, which has the property that if P, the P. I words, ca be obtaied by multiplyig P from the left by. alogously, if is o-sigular the, by multiplicatio from the right, P. The iverse is oly defied for square matrices!!! I I I I

14 himica Iorgaica The cofactor ad mior of the elemet of the matrix ( ) + M

15 himica Iorgaica The cofactor ad mior of the elemet of the matrix ( ) + M We ow defie a determiat as the sum of the products of the elemets of ay row or colum ad their correspodig cofactors, e.g. + + or + + Such a sum is called a Laplace expasio.

16 himica Iorgaica ( ) ik T ( ) ik ( ) ki ( ) ij ( ) ik ( ) kj k k ( ) ki ( ) ki δ ij Fid the iverse of the matrix T T

17 himica Iorgaica Let us cosider matrices V ad V -. Do ot care how we obtaied V!!! V ; V V ; T V

18 himica Iorgaica Let us cosider matrices V ad V -. Do ot care how we obtaied V!!! V ; V V V * V uiquely produces block-diagoalized matrices, which are matrices possessig square arrays alog the diagoal ad zeros outside the blocks

19 himica Iorgaica Let us cosider matrices V ad V -. Do ot care how we obtaied V!!! V E V E * V V ( ) * V V * V σ v V ( σ v ) * V σ v V ( σ v ) * V σ v V ( σ v ) * s above, the block-diagoalized matrices do ot further reduce uder reapplicatio of the similarity trasform. ll are Γ irr s.

20 himica Iorgaica reducible represetatio, Γ red, has bee decomposed uder a similarity trasformatio ito a ( ) ad ( ) block-diagoalized irreducible represetatios, Γ i. The traces (i.e. sum of diagoal matrix elemets) of the Γ i s uder each operatio yield the characters (idicated by χ) of the represetatio. Takig the traces of each of the blocks: Note: characters of operators i the same class are idetical This collectio of characters for a give irreducible represetatio, uder the operatios of a group is called a character table. s this example shows, from a completely arbitrary basis ad a similarity trasform, a character table is bor.

21 himica Iorgaica The triagular basis set does ot ucover all Γ irr of the group defied by {E,,, σ v, σ v, σ v }. triagle represets artesia coordiate space (x, y, z) for which the Γ i s were determied. May choose other basis fuctios i a attempt to ucover other Γ i s. For istace, cosider a rotatio about the z-axis The trasformatio properties of this basis fuctio, R z, uder the operatios of the group (will choose oly operatio from each class, sice characters of operators i a class are idetical):

22 himica Iorgaica E: R z R z : R z R z σ v (xz): R z -R z These trasformatio properties geerate a Γ i that is ot cotaied i a triagular basis. ew ( x ) basis is obtaied, Γ, which describes the trasform properties for R z. summary of the Γ i for the group defied by E,,, σ v, σ v, σ v is: from triagular basis, i.e. (x, y, z) from R z Is this character table complete? Irreducible represetatios ad their characters obey certai algebraic relatioships. From these 5 rules, we ca ascertai whether this is a complete character table for these 6 symmetry operatios.

23 himica Iorgaica Five importat rules gover irreducible represetatios ad their characters: Rule The sum of the squares of the dimesios, l, of irreducible represetatio Γ i is equal to the order, h, of the group. i i h Sice the character uder the idetity operatio is equal to the dimesio of Γ i (sice E is always the uit matrix), the rule ca be reformulated as: i ( χ ( E )) i h haracter uder E With referece to the previous example: ( χ ( E )) i i ( ) + ( ) + ( )

24 himica Iorgaica Five importat rules gover irreducible represetatios ad their characters: Rule The sum of squares of the characters of irreducible represetatio Γ i equals h R ( χ i ( R) ) h haracter of Γ i uder operatio R With referece to the previous example: R ( χ ( R) ) ( χ ( R) ) ( ) E R ( χ E ( R) ) R ( ) E ( ) E ( ) + + ( ) + + ( ) + + ( ) σ v +σ v ' +σ v " + ( ) ' " σ v +σ v +σ v + ( ) σ v +σ v ' +σ v "

25 himica Iorgaica Five importat rules gover irreducible represetatios ad their characters: Rule Vectors whose compoets are characters of two differet irreducible represetatios are orthogoal ( ) ( ) χ i R χ R j 0 for i j R With referece to the previous example: χ ( R) ( R) R χ χ ( R) ( R) R χ E χ ( R) ( R) R χ E ( ) ( ) + ( ) ( ) + E ( ) E ( ) ( ) E ( ) + + ( )( ) σ v +σ v ' + ( ) ( ) + +σ v " ( )( 0) σ v +σ v ' +σ v " + ( ) ( ) + + ( )( 0) σ v +σ v ' +σ v "

26 himica Iorgaica Five importat rules gover irreducible represetatios ad their characters: Rule 4 For a give represetatio, characters of all matrices belogig to operatios i the same class are idetical

27 himica Iorgaica Five importat rules gover irreducible represetatios ad their characters: Rule 5 The umber of Γ i s of a group is equal to the umber of classes i a group.

28 himica Iorgaica With these rules oe ca algebraically costruct a character table. Returig to our example, let s costruct the character table i the absece of a arbitrary basis: Rule 5: (E); (, ) (σ v, σ v, σ v ) classes Γ i s Rule : l + l + l 6 l l, l Rule : ll character tables have a totally symmetric represetatio. Thus oe of the irreducible represetatios, Γ i, possesses the character set χ (E), χ (, ), χ (σ v, σ v, σ v ). The applicatio of rule implies for the secod oe-dimesioal irreducible represetatio χ (E) χ (E) + χ ( ) χ ( ) + χ (σ v ) χ (σ v ) 0 χ (E) + χ ( ) + χ (σ v ) 0 + χ ( ) + χ (σ v ) 0; χ ( ) + χ (σ v ) - χ ( ), χ (σ v ) -

29 himica Iorgaica For the case of Γ (l ) there is ot a uique solutio to Rule + χ ( ) + χ (σ v ) 0 We the eed a secod idepedet equatio. + [χ ( )] + [χ (σ v )] 6 (Rule ) We the obtai [χ ( )] - ad [χ (σ v )] 0, i.e., the same result previously obtaied

30 himica Iorgaica haracter Table 7/04/85-7/05/98 07/06/896 - /0/986

31 himica Iorgaica

32 himica Iorgaica Give a reducible represetatio Γ, it is straightforward to get the umber of times a specific irreducible represetaio Γ k cotribute to Γ. a k h h R χ ( R) χ ( k R) Let us assume as basis for a reducible represetatio Γ the s orbitals of three hydroge atoms positioed at the vertices of a regular triagle. These orbitals may be labeled,, ad. Matrices correspodig to the differet symmetry operatios are:

33 himica Iorgaica E s v ŝ v ' s " v v E σ v Γ 0 a k h h R χ ( R) χ k ( R) a 6 ( ) a E 6 ( + 0 ( ) + 0)