1. 1 CH6. Symmetry Symmetry elements and operations Point groups Character tables Some applications

2. 2 Symmetry elements symmetry element: an element such as a rotation axis or mirror plane indicating a set of symmetry operations symmetry operation: an action that leaves an object in an indistinguishable state. Indicated here by boldface note: C 2 x C 2 = C 2 2 = two successive C 2 rotations Types of symmetry elements: 1. E = identity element - do nothing 2. C n = proper rotation axis = rotation by (360 / n) ° 3.  = mirror plane = reflect object in plane 4.i = inversion center or center of symmetry 5.S n = improper rotation axis, a C n axis combined with reflection through a perpendicular  identity - an element that can be multiplied by any other to leave it unchanged closure - the multiplication (successive action) of any two operations is equivalent to some other operation in the group; if a and b are group operations, then for a x b = c, c must also be a group operation association - the following holds: (a x b) x c = a x (b x c) reciprocity - for every operation a there exists a reciprical operation a -1 such that a x a -1 = E all the objects you've ever seen can probably be classified into one of 15 - 20 point groups

3. 3 Elements and operations Ex: C 3 axis (symmetry element) is associated with 3 operations: C 3 = rotation about the axis by 120° C 3 2 = C 3 x C 3 = rotation by 240° C 3 3 = rotation by 360° = E C 3 4 = C 3 and etc... Ex: S 4 axis indicates the following operations: S 4 = rotation by 90° then  S 4 2 = C 2 S 4 3 = C 4 3 x  S 4 4 = E S 4 5 = S 4 and etc... S n n = E for n even, and S n n =  for n odd Also S 2 = i

4. 4 More about symmetry elements Objects (molecules) may have more than one C n. The axis with highest n is called the principal rotation axis.  h = horizontal mirror plane, perpendicular to principal C n  v (or  d ) = vertical (or dihedral) mirror planes, parallel to (containing) the principal C n

5. 5 Point groups Point groups are true mathematical groups, exhibiting the group properties of:  identity: an operation (E) that can be multiplied by any other and leave it unchanged  closure: the multiplication of any 2 operations is equivalent to some other operation in the group; i.e., for operations a and b, if a x b = c, c must be a group operation  association: (a x b) x c = a x (b x c)  reciprocity: for every operation a there exists a reciprocal operation a -1 such that a x a -1 = E All common objects can be classified into one of 15 - 20 point groups. Your goal is to assign the point group (using Schoenflies notation) to any object, molecule, or function.

6. 6 Identifying a point group

7. 7 Point Group Examples BF 3 H 2 O NH 3 HF CO 2 CH 4 CH 3 Cl CF 2 BrCl SF 6 SF 5 Cl White cube, opposite faces black See website and assigned exercises for many more practice examples

8. 8 Symmetry rules All molecules in cubic groups, D groups, or with i, are non-polar, all others can be polar. Objects with any  or S n axis are not chiral, all others are chiral. Atoms exchanged by any symmetry operation are chemically identical, otherwise, they are chemically distinct.

9. 9 Fluxionality in amines Consider a tertiary amines with three different subsituents on N, ex: ethylmethylamine NH(CH 3 )(CH 2 CH 3 ) Point group is C 1, chiral by symmetry rules (has a non-identical mirror image). Experiments, however, show no optical activity, and no resolution of stereoisomers by chiral chromatography. Fluxionality occurs more rapidly at RT than the optical measurement or column separation. NMR (a probe with a shorter time) confirms that 2 enantiomers do exist. The inversion rate depends on the activation energy required to form the pseudo-planar intermediate, for this molecule it's less than 20 kJ/mol.

10. 10 Column headings give all symmetry operations (separated into classes). For C 4v there are E, 2C 4, etc... Classes are operations that transform into one another by another group operation. In C 4v, C 4 2 = C 2 is in a class by itself. 2C 4 is short notation for the operations C 4 and C 4 3 The order, h, is the sum of the coefficients of the headings and is total number of operations. For C 4v, h = 8. C 4v E2C 4 C2C2 2v2v 2d2d basis functions A1A1 11111z, z 2 A2A2 111 RzRz B1B1 1 11 x 2 – y 2 B2B2 11 1xy E20-200(x,y), (xz, yz), (R x, R y ) Character Tables

11. 11 Conventions The z axis contains the principal rotation axis The molecule is oriented so that bond axes are along x and y when possible a  v will contain perpedicular C 2 when present a  d will bisect perpedicular C 2 or bond axes when possible.

12. 12 Irreducible reps and characters Each row corresponds to an irreducible representation,  irred, which are orthogonal vectors in h- space The numbers are called characters,  and indicate how  irred acts under a class of operations. In the simplest case,  = 1 means that  irred is unchanged, and  = -1 means that it inverts. Ex: in C 4v, for  (A 2 ),  (C 4 ) = +1, i.e. A 2 is unchanged by the operations C 4 and C 4 3 C 4v E2C 4 C2C2 2v2v 2d2d A1A1 11111z, z 2 A2A2 111 RzRz B1B1 1 11 x 2 – y 2 B2B2 11 1xy E20-200(x,y) (xz, yz) (R x, R y ) Note: The class heading E, for the identity operation, coincidentally has the same symbol as the irreducible rep label E.

13. 13 Symmetry labels The labels on the  irred indicate some of the  values: A or B means that  (E) = 1 A is for  (C 4 ) = 1 B is for  (C 4 ) = -1 E means  (E) = 2 T means  (E) = 3 The subscript g (gerade) means that  (i) is positive, u (ungerade) that  (i) is negative. C 4v E2C 4 C2C2 2v2v 2d2d A1A1 11111z, z 2 A2A2 111 RzRz B1B1 1 11 x 2 – y 2 B2B2 11 1xy E20-200(x,y) (xz,yz) (R x,R y )

14. 14 Basis functions Basis functions have the same symmetry as atomic orbitals: x for p x, y for p y, xz for d xz, etc... or are rotations about x, y, z axes (R x,R y,R z ). They also transform as a  irred. s-orbitals are spherically symmetric and have  = 1 for any operation, so they always have the symmetry of the first  irred listed (A 1 in the C 4v point group). MO’s can also be assigned and labelled with  irred. C 4v E2C 4 C2C2 2v2v 2d2d A1A1 11111z, z 2 A2A2 111 RzRz B1B1 1 11 x 2 – y 2 B2B2 11 1xy E20-200(x,y) (xz,yz) (R x,R y )

15. 15 Assign labels to MO’s in H 2 O Molecule is in yz plane C 2v EC2C2  v (xz)  v ’ (yz) A1A1 1111z A2A2 11 RzRz B1B1 1 1 x, R y B2B2 1 1y, R x

16. 16 Orthogonality of  irred all  irred within a point group are orthogonal, their cross-products are zero. MO’s that have different symmetry labels have no net overlap For metal-ligand compounds, label symmetries of metal orbitals from basis functions, and interact with same symmetry SALC’s only.

17. 17 Symmetry labels and bonding SALC’s symmetries from SA appendix 4 (or use projection method) Sulfur orbital symmetries from the O h character table 1u 1g g

18. 18 IR and Raman selection rules In IR absorption, allowed vibrational modes have the same symmetry as the transition moment operator (x, y, or z) O h molecules have only T 1u vibration modes IR active. For Raman absorption, allowed modes have the symmetry of a polarizability operator (x 2, y 2, z 2, xy, xz, yz, or any linear combination) For O h molecules, A 1g, E g, and T 2g are the allowed symmetries. An A 1g Raman stretching mode is pictured to the left.

19. 19 O h character table