Symmetry and Molecular Spectroscopy
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1 PG510 Symmetry and Molecular Spectroscopy Lecture no. 3 Group Theory: Point Groups Giuseppe Pileio 1 Learning Outcomes By the end of this lecture you will be able to:!! Understand the concepts of Point Group!! Classify all the possible point group from symmetry elements!! Understand the concept of classes of symmetry operations and the tricks to arrange them in classes!! Find molecular point groups by a simple but systematic procedure 2
2 Symmetry Point Group What is a Symmetry Point Group? It is a complete set of symmetry operations where complete means that every product between operations is still a member of the group Does a complete set of symmetry operations form a group?! Rule 1 is satisfied by the meaning of complete! Rule 2 is satisfied with the operation E as identity! Rule 3 is satisfied since product is associative! Rule 4 is satisfied since it is always possible to find an operation with does the opposite (inverse):!!=e; ii=e; C nm C n =E; S nm S n =E (n even, m even or odd) S n m = C n m (n odd, m even); S n m = C nm! * C n! (n,m odd) 3 Point Groups Asymmetry one high-order axes A.! o symmetry operations but E h=1 C 1 One symmetry element B.! Symmetry element: plane Symmetry operations: E,!" h=2 C s (cyclic) C.! Symmetry element: inversion point Symmetry operations: E, i h=2 C i (cyclic) 4
3 D.! Symmetry element: proper axis, C n Symmetry operations: E, C n, C n2,, C n " h=n C n (cyclic) E.! Symmetry element: improper axis, S n (n even) Symmetry operations: E, S n, C n/2, S n3,, S n h=n S n F.! Symmetry element: improper axis, S n (n odd) Symmetry operations: E, S n,, S n,! h, C n,, C n h=2n C nh 5 Two or more symmetry elements G.! Symmetry elements: C n + perpendicular C 2 Symmetry operations: E, C n,, C n, n C 2 h=2n D n H.! Symmetry elements: C n +! h Symmetry operations: E, C n,, C n,! h, S n,, S n h=2n C nh I.! Symmetry elements: C n +! v Symmetry operations: E, C n,, C n, n! v (n even) E, C n,, C n, n/2! v, n/2! d (n odd) h=2n C nv 6
4 J.! Symmetry elements: C n + perpendicular C 2 +! h Symmetry operations: E, C n,,c n, nc 2,! h, n! v, S n,, S n h=4n D nh K.! Symmetry elements: C n + perpendicular C 2 +! d Symmetry operations: E, C n,,c n, nc 2, n! d, S 2n,, S 2n 2 h=4n D nd 7 Infinite order axes Linear Molecules Two equivalent halves A-A or A-X-A Two different halves A-B or A-X-B L. elements: C! +! h + C 2(perp) operations: E, C!,,C!!-1,!C 2,! h,!! v, S n,, S n!-1 M. elements: C! +! v operations: E, C!,,C!!-1,!! v h=! D!h h=! C!v 8
5 More than one high-order axes Platonic Solids or Regular Polyhedra T d T h T O h O I h I 9. elements: 3S 4, 4 C 3, 6! d " operations: E, 8C 3, 3C 2, 6S 4, 6! d h=24 T d O. elements: 3S 4, 4 C 3 " operations: E, 8C 3, 3C 2 h=12 T P. elements: 3S 4, 4 C 3, 3! h " operations: E, 8C 3, 3C 2, 9S 4, 3! h h=24 T h 10
6 Q. elements: 3S 4, 3 C 4, 4 S 6 " operations: E, 8C 3, 6C 4, 9C 2, 6S 4, 8S 6, i, 6! d, 3! h h=48 O h R. elements: 3S 4, 3 C 4, 4 S 6 " operations: E, 8C 3, 6C 4, 9C 2 " h=24 O S. elements: 6S 10, 10 S 6, 6 C 5 " operations: E, 24C 5, 20C 3, 15C 2, 24S 10, 20S 6, i, 15! h=120 I h T. elements: 6S 10, 10 S 6, 6 C 5 " operations: E, 24C 5, 20C 3, 15C 2 " h=60 I 11 Classes of symmetry operations Group theory: a set of group element (symmetry operations in the point group case) that are conjugate to one another form a class Operational: A set of equivalent operations can be arranged in a class A and B are called equivalent operations if they can be inter-converted by a third operation, C. 12
7 Symmetry operations and classes: i)! Inversions: only one is possible in a molecule and it always occurs in a class by itself ii)! Reflections:! h is always in a class by itself. A set of n dihedral planes in the same class is indicated by n! d. The same for n! v if they are all in the same class otherwise the notation n! v, m! v etc is used iii)! Proper Rotations: if the group is cyclic they are all in separate classes otherwise C n m will fall in the same class with C n iv)! Improper Rotations: S m n will fall in the same class with S n 13 Systematic Classification start Special Groups a)!linear molecule: C!v, D!h b)!multiple high order axes: T,T h,t d,o,o h,i,i h c)!o rotation at all: C 1,C s,c i d)!only S n (n even): S 4,S 6,S 8 C n + nc 2 perp. C nh! h! h D nh C nv n! v C n D n n! d D nd 14
8 What did we learn in this lecture?! The concept of point group and its link with group theory! How various combinations of symmetry elements (and operations) generate all the possible molecular point groups! How to arrange operation in classes! A systematic procedure to assign the point group to any molecular system Pictures from: 15
Chem Identifying point groups
Special cases: Perfect tetrahedral (T d ) e.g. P 4, C 4 Perfect octahedral ( h ) e.g. S 6, [B 6 6 ] -2 Perfect icosahedral (I h ) e.g. [B 12 12 ] -2, C 60 Low symmetry groups: nly* an improper axis (S
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