1 Physical Chemistry III (728342) Chapter 5: Molecular Symmetry Piti Treesukol Kasetsart University Kamphaeng Saen Campus

2 Molecular Symmetry  Molecular symmetry: The classification of any molecule according to its symmetry, correlating to its molecular properties  Importance of molecular symmetry Choosing LCAO Identifying vanishing integrals Classifying orbital overlap Providing selection rules for spectroscopic transition

3 Group Theory  The systematic discussion of symmetry is called Group Theory.  The symmetry elements of objects Symmetry operation: an action that leaves an object looking the same after it has been carried out Symmetry element: an element (plane, line, point) that correlates to the specific symmetry operator (unchanged) Symmetry Operation Symmetry Elements RotationLine (axis of rotation) ReflectionPlane InversionPoint

4 Point Group  Point Group: the classification of objects according to symmetry elements corresponding to operations that leave at least one common point unchanged.  The more extensive classification, including the translation through space, is called Space Group.

5 Operations and Symmetry Elements  Five kinds of symmetry operations in Point Group The identity, E An n-fold rotation, C n A reflection,  An inversion, i An n-fold improper rotation, S n

6 The Identity, E  The identity operation is doing nothing! Every molecule is indistinguishable from it self thus they have the identity element.

7  An n-fold rotation about an n-fold axis of rotation, C n, is a rotation through 360°/n C 1 = E C 2 = 180° rotation C 3 = 120° rotation (C 3 ’ and C 3 ”) C 6 = 60° rotation (C 1 1, C 1 2 … C 1 5 )  If a molecule possesses several rotational axis, the one with the greatest value of n is called the principal axis (Z). An n-fold Rotation, C n C6C6 C2C2 C2C2 C3C3 C2C2

8 A Reflection,   A reflection is a mirror plane.  V – parallel to the principle axis  d – parallel to the principle axis and bisect the angle between two C 2 axes  h – perpendicular to the principle axis hh vv  v’ dd vv vv vv

9 An Inversion, i  An inversion through a center of symmetry If the origin point is the center of symmetry center center of symmetry

10 An n-fold Improper Rotation, S n  An n-fold improper rotation is composed of two successive transformation: Rotation through 360º/n Reflection through a plane perpendicular to the axis of that rotation. C6C6 hh S4S4 S2S2

11 The Symmetry Classification of Molecules  Molecules with the same list of elements are classified to the same group. The groups C 1, C i and C s (no rotational axis) The groups C n, C nv and C nh (n-fold axis) The groups D n, D nh, D nd (n-fold axis and n perpendicular C 2 s) The groups S n (n-fold improper axis) The cubic groups  Tetrahedral groups (T, T d, T h )  Octahedral groups (O, O h )  Icosahedral groups (I)

12 Determining the Point Group  A flowchart for determining the point group of a molecule Molecule Linear  2 C n s n>2 C n ? i ?  h ? n  d ? i ? C 5 ? DhDh CvCv TdTd IhIh OhOh  ? i ? n  v ? S 2n ?  h ? CsCs CiCi C1C1 C nh C nv S 2n CnCn D nh DnDn D nd Yes No n C 2 ? มีแกน C n (n>2) มากกว่า 2 แกน มีแกน C 2 n แกนที่ตั้งฉาก กับแกนหลัก

13 Character Table  Character Table is a table that characterizes the different symmetry types possiblein the point group.  The entries in a complete character table are derived by using the formal techniques of group theory. SO 2 (p x ) C 2V (E, C 2,  v  v ’) PSPS PAPA PBPB

14 Representations and Characters  All the operators can be written in the matrix form.  The matrix is called a representation of an operator. C 2v  The Matrix representative is called  (n), where n is the dimention of the matrix  The character of the representation matrix is the sum of diagonal elements.

15 Reduce- and Inreducible Representation  Inspection of the representatives reveals that they are all of block- diagonal form.  This shows that the p s is never mixed with the rest.  The 3-D representative matrix (  (3) ) can be separated into  (1) +  (2) )

16  According to the matrix representation, p A and p B are mixed together.  Using the LC, we can write the new basis as p 1 =p A +p B and p 2 =p A -p B

17 p(S) p(O) O O S O O S v’v’ vv c2c2 A S B A S B A S B A S B A S B A S B A S B A S B

18 A S B A S B A S B A S B

19 p(S) p(O) O O S O O S A S B S (A-B) (A+B)

20 Character Table  The characters of all representations are tabulated in a character table. C 2v EC2C2 vv v’v’ h= 4 A1A1 1111Zz 2,y 2,x 2 A2A2 11 xy B1B1 11 Xxz B2B2 1 1yyx

21 The Structure of Character Tables C 3v E 2C 3 3v3v h= 6 A1A1 111Zz 2, x 2 + y 2 A2A2 11 E2 0 (x,y ) (xy, x 2 - y 2 ),(xz,yz) Group Symmetry Operations Class Order (# operations) Irreducible Representations Symmetry Properties (  ) # of degerneracy of each representative is specified by the symmetry property of E operation or  (E). LabelsA, B: 1-D E: 2-D T: 3-D A   (C n ) = 1 B   (C n ) = -1 1   (  v ) = 1 2   (  v ) = -1

22  NH 3 LCAO: this orbital is of symmetry species A1 and it contributes to a 1 MO in NH 3.  NO 2 LCAO: The Classification of LC of Orbitals sAsA sBsB sCsC N O O

23 Vanishing Integrals & Orbital Overlap  The value of integrals and orbital overlap is independent of the orientation of the molecule. I is invariant under any symmetry operation of the molecule, otherwise it must be zero. For I not to be zero, the integrand f 1 f 2 must have symmetry species A 1.  Example: f 1 = s B and f 2 = s C of NH 3  Problem: f 1 = s N and f 2 = s A +s B +s C of NH 3 sBsB sCsC not A 1

24  In many cases, the product of functions f 1 and f 2 spans a sum of irreducible representations.  In these cases, we have to decompose the reducible representation into irreducible representations C 2v EC2C2 vv v’v’ A2A2 11 B1B1 1 1 A 2 +B

25 Orbitals with nonzero overalp  Only orbitals of the same symmetry species may have nonzero overlap, so only orbitalsof the same symmetry species form bonding and antibonding combinations.

26 Symmetry-adapted Linear Combinations  Symmetry-adapted linear combination (SALC) are the building blocks of LCAO-MO  To construct the SALC from basis: 1.Construct a table showing the effect of each operation on each orbtial of the original basis. 2.To generate the combination of a specified symmetry species, take each column in turn and: a) Multiply each member of the column by the character of the corresponding operation. b) Add together all the orbitals in each column with the factors as determined in a). c) Divide the sum by the order of the group.

27 Example of building SALC  s-orbitals of NH 3 Original basis are s N, s A, s B, s C N A B C NH 3 Original basis sNsN sAsA sBsB sCsC EsNsN sAsA sBsB sCsC C3+C3+ sNsN sBsB sCsC sAsA C3-C3- sNsN sCsC sAsA sBsB vv sNsN sAsA sCsC sBsB v’v’ sNsN sBsB sAsA sCsC  v “ sNsN sCsC sBsB sAsA For A 1 combination (1,1,1,1,1,1) when

28 Vanishing Integrals and Selectrion Rules  Integrals of the form are common in quantum mechanics.  For the integral to be nonzero, the product must span A 1 or contain a component that span A 1.  The intensity of line spectra arises from a molecular transition between some initial state i and a final state f and depends on the electric transition dipole moment  fi. C 2v EC2C2 vv v’v’ B1B1 11 z1111 A1A A 1 zB if fzi does not span species A 1

29 O O 2p z H 1s C2C2 O-2p z O-2p y H A -1s  H B -1s H A -1s + H B -1s O-2s O-1s HAHA HBHB

30 C 2h EC 2 (z)iσhσh linear, rotations quadratic AgAg 1111 RzRz x 2, y 2, z 2, xy BgBg 11 R x, R y xz, yz AuAu 11 z BuBu 1 1 x, y C2C2 EC2C2 linear, rotations quadratic A11 z, R z x 2, y 2, z 2, xy B1 x, y, R x, R y yz, xz C 2V EC 2 (z)σvσv σ’vσ’v linear, rotations quadratic A1A zx 2, y 2, z 2 A2A2 11 RzRz xy B1B1 11 x, R y xz B2B2 1 1 y, R x yz

31 C 3V E2C 3 (z)3σ v linear, rotations quadratic A1A1 111 zx 2 +y 2, z 2 A2A2 11 RzRz E 2 0 (x, y) (R x, R y )(x 2 -y 2, xy) (xz, yz)

H 32 VV H O C2C2

E C 2 ( z) iσhσh line ar, rota tion s quadratic AgAg 1111RzRz x 2, y 2, z 2, xy BgBg 11 Rx,RyRx,Ry xz, yz AuAu 11 z BuBu 1 1x, y 33