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 xz commute? Ĉ 4 z) ˆ xz x, y, z ) ˆ xz Ĉ4 z) x, y, z ) Ĉ 4 z) x, y, z ) ˆ xz y, x, z ) ˆ d ˆ 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 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 c tl G E t s t s t t s

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

7 himica Iorgaica x y z Let us ow cosider matrices correspodig to symmetry operatio of the v group: Ê,Ĉ,Ĉ, ˆ v, ˆ v, ˆ 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 v ˆ v Ĉ Ĉ ˆ v

9 himica Iorgaica Ê x y x y 0 0 x y ˆ v x y x y y x x y Ĉ x y x + y y x x y ˆ 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 ˆ 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. Similarity trasformatios yield irreducible represetatios, Γ i, which lead to the 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 * * *

11 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 S v S v S v

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

13 himica Iorgaica Let us cosider matrices V ad V -. Do ot care how we obtaied V V ; V V V * matrix possessig square arrays alog the diagoal ad zeros outside the blocks

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

15 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.

16 himica Iorgaica The triagular basis set does ot ucover all Γ irr of the group defied by {E,,, v, v, 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):

17 himica Iorgaica E: R z R z : R z R z σ v xy): 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 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.

18 himica Iorgaica Five importat rules gover irreducible represetatios ad their characters: Rule The sum of the squares of the dimesios,, of irreducible represetatio Γ i is equal to the order, h, of the group. i h i 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: E )) h i i haracter uder E With referece to the previous example: E )) i i ) + ) + )

19 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 i R) ) h R haracter of Γ i uder operatio R With referece to the previous example: R) ) R ) E R) ) R ) E E R) ) R ) E ) + + ) + + ) + + ) v + v + v + ) v + v + v + ) v + v + v

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

21 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

22 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.

23 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 : + + 6, Rule : ll character tables have a totally symmetric represetatio. Thus oe of the irreducible represetatios, Γ i, possesses the character set E),, ), σ v, σ v, σ v ). The applicatio of rule implies for the secod oe-dimesioal irreducible represetatio E) E) + ) ) + σ v ) σ v ) 0 E) + ) + σ v ) 0 + ) + σ v ) 0; ) + σ v ) - ), σ v ) -

24 himica Iorgaica For the case of Γ ) 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

25 himica Iorgaica haracter Table

26 himica Iorgaica

27 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 R) 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:

28 himica Iorgaica E s v ŝ v s v v E σ v Γ 0 a k h h R R) k R) a ) a E ) + 0)