# 1 Character Tables. 2 Each point group has a complete set of possible symmetry operations that are conveniently listed as a matrix known as a Character.

## Presentation on theme: "1 Character Tables. 2 Each point group has a complete set of possible symmetry operations that are conveniently listed as a matrix known as a Character."— Presentation transcript:

1 Character Tables

2 Each point group has a complete set of possible symmetry operations that are conveniently listed as a matrix known as a Character Table. C 2V EC2C2  v (xz)  ’ v (yz) A1A1 1111 A2A2 11 B1B1 1 1 B2B2 1 1 4 characters: İrreducible represention of B 2 Point Group LabelSymmetry Operations – The Order is the total number of operations Symmetry Representation Labels In C 2v the order is 4: 1 E, 1 C 2, 1  v and 1  ’ v Character Character Tables

3 Karakter Çizelgesi İndirgenemez gösterimler (İG) Irreducible representations (IR) Simetri işlemleri Grup derecesi h = 6 (h = 1 +2 +3) 3 sınıf mevcuttur 000112 111111 111111 ''' 3 2 1 2 333    vvvv CCEC  Mulliken Sembolleri A1A1 A2A2 E 012 111 111 32 33   vv CEC  Eşdeğer elemanlar ve eşdeğer atomlar sınıf oluşturur.

4 A veya B tek boyutlu İG E iki boyutlu İG T ( veya F) üç boyutlu İG A baş dönme eksenine göre simetrik (+) B baş dönme eksenine göre antisimetrik ( − ) Alt indis g (gerade) evirme işlemine göre simetrik (  = +1) u (ungerade) evirme işlemine göre antisimetrik (  = −1) Üst indis ' (tek üs)  h düzlemine göre simetrik (+) '' (çift üs) “ antisimetrik (−) Alt indis 1 C 2 (  C n ) eksenine, yoksa  v işlemine göre simetrik (  = +1) 2 C 2 (  C n ) eksenine, yoksa  v işlemine göre antisimetrik (  = -1) Mulliken Sembolleri

5 Mulliken labels

6 Karakter Çizelgeleri ve Mulliken Sembolleri-1 C 1 group. Consists of a single operation E; thus its order h=1 and number of classes is 1. There is a single irreducible representation. C s group. Consists of two operations, E and  h ; thus its order h is 2 and the number of classes is 2. There are two irreducible representations. C i group. Consists of two operations, E and i. Both its order h and number of classes is 2. Similarly to C s, the group includes two irreducible one-dimensional representations. C1C1 E 1 CsCs E hh 11x,y, R z 1z,R x,R y CiCi Ei 11RxRx 1x,y,z

7 C 2v EC 2  xz  yz A 1 +1+1+1+1T z A 2 +1+1-1-1R z B 1 +1-1+1-1T x or R y B 2 +1-1-1+1T y or R x C 3v E2C 3 3  v A 1 +1+1+1T z A 2 +1+1-1R z E+2-1 0(T x, T y ) or (R x, R y ) ÖRNEK: C 2v ve C 3v nokta gruplarının karakter çizelgelerindeki Mulliken sembollerini belirleyiniz. Karakter Çizelgeleri ve Mulliken Sembolleri-2

8 C 4v nokta grubunun tam karakter çizelgesi ),(),(),,( 0 0 2 02 11111 1 1111 111 11, 1 11 11 222 2 22 1 2 222 1 244 yzxzRRyxE xyB yxB RA zyxzA CCEC yx z dvv       İkili fonksiyonlar ( d orbitalleri) Tekli fonksiyonlar (p orbitalleri) Karakter Çizelgeleri ve Mulliken Sembolleri-3

9 Atom Orbitallerinin Simetrileri-1 Atom orbitaliTransformasyon sx 2 +y 2 +z 2 pxpx x pypy y pzpz z d z2 z 2, 2z 2 -x 2 -y 2 d x2-y2 x 2 -y 2 d xy xy d xz xz d yz yz Simetrileri aynı olan atom orbitalleri bağ yaparak molekül orbitallerini oluşturur. Merkez atoma ait orbitallerin simetrileri ve dejenerelikleri karakter çizelgesinden öğrenilir. Her karakter çizelgesinin ilk indirgenemez gösterimi s orbitaline karşılık gelir. Tamamen simetrik

10 Atom Orbitallerinin Simetrileri-2 C 2v A1A1 zx 2, y 2, z 2 A2A2 RzRz xy B1B1 x, R y xz B2B2 y, R x yz Atomic orbitalMulliken labels C 2v D 3h D 4h TdTd OhOh s pxpx pypy pzpz d z2 d x2-y2 d xy d xz d yz D 3h A1’A1’x 2 +y 2, z 2 A2’A2’RzRz E’(x,y)(x 2 -y 2, xy) A1”A1” A2”A2”z E”(R x,R y )(xz, yz) TdTd A1A1 x 2 +y 2 +z 2 A2A2 E(2z 2 -x 2 -y 2, x 2 -y 2 ) T1T1 (R x,R y,R z ) T2T2 (x,y,z)(xz, yz, xy) OhOh A1gA1g x 2 +y 2 +z 2 EgEg (2z 2 -x 2 -y 2, x 2 -y 2 ) T 1g (R x,R y,R z ) T 2g (xz, yz, xy) T1uT1u (x,y,z) … D 4h A 1g x 2 +y 2, z 2 B 1g x 2 -y 2 B 2g xy EgEg (R x,R y )(xz, yz) A 2u z EuEu (x, y)

11 C 2V EC2C2  v (xz)  v ’ (yz) Mikrodalga IRRaman A1A1 1111 zx 2, y 2, z 2 A2A2 11 RzRz xy B1B1 11 RyRy xxz B2B2 1 1RxRx yyz Spektroskopik Seçim Kuraları Dönme : R x, R y, R z Titreşim (ve öteleme) : x, y, z Raman: x 2, y 2, z 2, xy, xz, yz Mikrodalga aktif Infra-red aktif Raman aktif A 2, B 1, B 2 A 1, B 1, B 2 Hepsi C 2V

12 İ ndirgeme İşlemleri-1 n i = indirgenemez gösterim sayısı h = nokta grubu derecesi g c = simetri işlemi sayısı veya katsayısı  r = indirgenebilir temsilin karakteri  i = indirgenemez temsilin karakteri

13 C 2V EC2C2  v (xz)  ’ v (yz)  4022 C 2V EC2C2  v (xz)  ’ v (yz) A1A1 1111zx 2,y 2,z 2 A2A2 11 RzRz xy B1B1 11 x, R y xz B2B2 1 1y, R x yz  = 2A 1 + B 1 + B 2 İndirgeme İşlemi-2

14 a A 1 = (1/4)[ ( 1x9x1) + (1x-1x1) + (1x1x1) + (1x3x1)] = (12/4) =3 C 2v 3N3N EC2C2  (xz)  (yz) +9+13 a A 2 = (1/4)[ ( 1x9x1) + (1x-1x1) + (1x1x-1) + (1x3x-1)] = (4/4) =1 a B 1 = (1/4)[ ( 1x9x1) + (1x-1x-1) + (1x1x1) + (1x3x-1)] = (8/4) =2 a B 2 = (1/4)[ ( 1x9x1) + (1x-1x-1) + (1x1x-1) + (1x3x1)] = (12/4) =3  3N = 3A 1 + A 2 + 2B 1 + 3B 2 İndirgeme İşlemi-3

15 2C 3 C 3v E  v 3N3N 1503 n(A 1 ) = 1/6[(1x 15x1) + (2 x 0 x 1) + (3 x 3x 1)] = 1/6 [15 + 0+ 9] = 4 n(A 2 ) = 1/6[(1 x 15 x 1) + ( 2 x 0 x 1) + (3 x 3x –1)] = 1/6 [15 + 0 -9] = 1 n(E) = 1/6[ (1 x 15 x 2) + (2 x 0 x –1) + (3 x 3 x 0)] = 1/6[30 + 0 + 0 ] = 5  = 4A 1 + A 2 + 5E İndirgeme İşlemi-4

16 Titreşimler Bir molekül için 3N tane serbestlik derecesi vardır. ( N = atom sayısı) Doğrusal moleküllerde (nokta grubu C  v veya D  h) 3N-5, diğer moleküllerde 3N-6 tane temel titreşim bulunur. 3N-5 : 3x3-5 = 4 temel titreşim CO 2 ( D  h ) SO 2 (C 2v ) N-6 : 3x3-6 = 3 temel titreşim

17 NH 3 molekülüne ait  3N indirgenebilir temsilini indirgeyiniz. a) Dönme hareketinin simetrilerini belirleyiniz. b) Öteleme hareketlerinin simetrilerini belirleyiniz. c) Titreşim hareketlerinin simetrilerini, IR ve R aktifliklerini belirleyiniz. NH 3 E2C 3 (z)3σ v A1A1 111zx 2 +y 2, z 2 A2A2 11RzRz E2 0(x, y) (R x, R y )(x 2 -y 2, xy) (xz, yz)  3N 1202  dönme = A 2 (Rz) + E (R x,R y )  öteleme = A 1 (z) + E (x,y)  titreşim = 2A 1 (IR,R) + 2E (IR, R)  titreşim =  3N –  dönme –  öteleme

18  titreşim = 2A 1 (IR,R) + 2E (IR, R) IR spektrumu Raman spektrumu NH 3 as (asimetrik gerilme) = 3414 cm -1 (E) s (simetrik gerilme) = 3316 cm -1 (A 1 )  as (asimetrik eğilme) = 1627 cm -1 (E)  s (simetrik eğilme) = 950 cm -1 (A 1 )

19 s = 3336 cm -1 simetrik gerilme as = 3414 cm -1 asimetrik gerilme  s = 950 cm -1 simetrik eğilme  as = 1627 cm -1 asimetrik eğilme

20  3N = 3A 1 + A 2 + 2B 1 + 3B 2 H2OH2O  dönme = A 2 (R z ) + B 1 (R y ) + B 2 (R x )  öteleme = A 1 (z) + B 1 (x) + B 2 (y)  titreşim = 2A 1 (IR, R) + B 2 (IR, R)  3N-6 C 2V EC2C2  v (xz)  ’ v (yz) A1A1 1111zx 2,y 2,z 2 A2A2 11 RzRz xy B1B1 11 x, R y xz B2B2 1 1y, R x yz

21 Shown opposite are the main vibrations occurring in water. The movements are animated using the cursor. The dipole moments change in the direction of the movement of the oxygen atoms as shown by the arrows.Shown opposite are the main vibrations occurring in water. The movements are animated using the cursor. The dipole moments change in the direction of the movement of the oxygen atoms as shown by the arrows. Shown opposite are the main vibrations occurring in water. The movements are animated using the cursor. The dipole moments change in the direction of the movement of the oxygen atoms as shown by the arrows.Shown opposite are the main vibrations occurring in water. The movements are animated using the cursor. The dipole moments change in the direction of the movement of the oxygen atoms as shown by the arrows. vas, 3490 cm -1 vs, 3280 cm -1 , 1644 cm -1 H2OH2O 3N-6 = 3 titreşim Raman spektrum

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