Lecture 8 SYMMETRY, GROUP THEORY AND THEIR APPLICATIONS

1. Lecture 8SYMMETRY, GROUP THEORY AND THEIR APPLICATIONS 1) Symmetry elements and operations A symmetry element is a geometrical object such as a point, line or plane with respect to which some particular symmetry operation(s) can be carried out. A symmetry operation is a movement of a species such that after it is complete, the original and the product of this movement are undistinguishable. Generated by a symmetry element. Relationships between Snm and other symmetry operations: S1 = s ; S2 =i ; Sn2m = Cn2m (n ≥4)

2. 2) Molecules with symmetry elements E and i, s, C2 and S4 E 1) Symmetry element: identity, E. Symmetry operationE leaves each point on its place: (x,y,z)  (x,y,z) 2) Symmetry elements: E and center of inversion, i. Symmetry operationi moves all atoms through the center: (x,y,z)  (-x,-y,-z) 3) Symmetry elements: E and mirror plane, s. Symmetry operationsxy moves all atoms through the plane xy: (x,y,z)  (x,y,-z) 4) Symmetry elements: E and proper axis, C2. Symmetry operationsC2m rotate atoms about the axis by m·(360/2) degrees (1 ≤ m ≤ 2) 5) Symmetry elements: E, C2, C3, s, improper axis, S4. Symmetry operationsS4m rotate atoms about the axis by 360/4 degrees and then reflect in a perpendicular mirror plane. This sequence is repeated m times (1 ≤ m ≤ 4) i s C2 S4

3. 3) Groups Using four rules defined below, certain sets of symmetry operations (designated below as A, A-1, B, C, E) can be combined into symmetry point groups. 1) The combination (multiplication) of any twooperations of the group, A & B, is anotheroperationC which belongs to the group: AB = C 2) One of the operations in the group must commute with all others and leave them unchanged: EA = AE = A. This operation is identity, E. 3) The associative law must hold: A(BC) = (AB)C 4) Each operation A must have a reciprocalA-1 which is also an operation of the group; AA-1 = E Conclusion: identity operation,E, is present in each symmetry point group

4. 4) Group multiplication tables. Point groups of low symmetry • Multiplication table for each symmetry point group contains columns and rows labeled with symbols of all symmetry operations present in the group. The products of the symmetry operations are indicated in the intersections • Each row and each column of the table lists each of the group operations only once. Thus, no two rows or two columns must be identical. Each row and each column is a rearranged list of the group operations. 3 Point Groups of low symmetry: C1 E EE Cs Ci E s EEs ssE E i E E i i iE

5. 5) Point groups with n-fold rotational axis • Cn : rotational axes only; no mirror planes C2 – gauche-H2O2 (operations: E, C2) • Cnh : a horizontal plane perpendicular to Cn is present C2h – trans-HN=NH (operations: E, C2, i, s) • Cnv : vertical plane(s) containing Cn is(are) present C3v – NH3 (operations: E, C31, C32, 3sv) • Cv : infinite number of vertical planes containing C linear molecules without center of symmetry: CO, HF, N2O (operations: E, 2C2∞, ∞sv) C2 C3

6. 6) Dihedral groups Contain nC2 axes perpendicular to the principal axis Cn • Dn : no mirror planes D3 – Co(en)33+ (operations: E, 2C3, 3C2) • Dnh: mirror plane perpendicular to the principal axis Cn D2h – CH2=CH2 (operations: E, 3C2, i, 3s) D3h – PCl5 (operations: E, 2C3, 3C2, sh, 2S3, 3sv) D4h – PtCl42‑ • Dh: infinite number of nC2 axes linear centrosymmetrical molecules like H2, CO2 etc. • Dnd: mirror planes contain Cn and bisect the angle formed with adjacent C2 axes D3d – ethane/staggered D4d – S8 D6d – Cr(C6H6)2