Symmetry and group theory

Symmetry, symmetry elements, Symmetry operations, point groups.

1. Symmetry and Group Theory Dr. BASAVARAJAIAH S. M. Assistant Professor and Coordinator P. G. Department of Chemistry Vijaya College. [email protected] 9620012975

2. Understanding of symmetry is essential in discussions of molecular spectroscopy and calculations of molecular properties . consider the structures of BF 3 , and BF 2 H , both of which are planar BF bond distances are all identical (131 pm) trigonal planar the BH bond is shorter (119 pm) than the BF bonds (131 pm). pseudo-trigonal planar

3. T he molecular symmetry properties are not the same In this chapter , Symmetry Element , Symmetry Operation , Point Group Group theory is the mathematical treatment of symmetry.

4. Symmetry Elements and Symmetry Operations Identity (E) Proper Axis of Rotation ( C n ) Mirror Planes ( σ ) Center of Symmetry ( i ) Improper Axis of Rotation ( S n )

5. Identity, E All molecules have Identity. This operation leaves the entire molecule unchanged. A highly asymmetric molecule such as a tetrahedral carbon with 4 different groups attached has only identity, and no other symmetry elements.

6. Proper Axis of Rotation ( C n ) The symmetry operation of rotation about an n -fold axis ( the symmetry element ) is denoted by the symbol C n , in which the angle of rotation is: where n = 2, 180 o rotation n = 3, 120 o rotation n = 4, 90 o rotation n = 6, 60 o rotation n = , (1/ ) o rotation principal axis of rotation, C n

7. In water there is a C 2 axis so we can perform a 2-fold (180 °) rotation to get the identical arrangement of atoms. 180 ° For H 2 O

8. For NH 3

9. Applying this notation to the BF 3 molecule BF 3 molecule contains a C 3 rotation axis

10. If a molecule possesses more than one type of n -axis, the axis of highest value of n is called the principal axis ; it is the axis of highest molecular symmetry. For example, in BF 3 , the C 3 axis is the principal axis .

11. Ethane, C 2 H 6 Benzene, C 6 H 6 The principal axis is the three-fold axis containing the C-C bond. The principal axis is the six-fold axis through the center of the ring.

12. Mirror planes ( σ ) s h => mirror plane perpendicular to a principal axis of rotation s v => mirror plane containing principal axis of rotation s d => mirror plane bisects dihedral angle made by the principal axis of rotation and two adjacent C 2 axes perpendicular to principal rotation axis The symmetry operation is one of reflection and the symmetry element is the mirror plane (denoted by  ). If reflection of all parts of a molecule through a plane produces an indistinguishable configuration, the plane is a plane of symmetry.

13. The reflection of the water molecule in either of its two mirror planes results in a molecule that looks unchanged. The subscript “v” in σ v , indicates a vertical plane of symmetry. This indicates that the mirror plane includes the principal axis of rotation (C 2 ).

14. Benzene Ring

15. Reflection through a plane of symmetry (mirror plane)

16. Center of Symmetry ( i ) If reflection of all parts of a molecule through the centre of the molecule produces an indistinguishable configuration, the centre is a centre of symmetry , also called a centre of inversion ; it is designated by the symbol i . CO 2 SF 6 B enzene [x, y, z] i [-x, -y, -z]

17. Improper Axis of Rotation ( S n ) If rotation through about an axis, followed by reflection through a plane perpendicular to that axis, yields an indistinguishable configuration, the axis is an n -fold rotation – reflection axis, also called an n-fold improper rotation axis . It is denoted by the symbol S n .

18. Improper Rotation in a Tetrahedral Molecule

19. S 1 and S 2 Improper Rotations

20. For example, in planar BCl 3 , the S 3 improper axis of rotation corresponds to rotation about the C 3 axis followed by reflection through the  h plane.

21. Summary Table of Symmetry Elements and Operations

22. Group theory is the mathematical treatment of symmetry.

23. Group A group is a set,  G , together with an operation • (called the  group law  of  G ) that combines any two elements  a  and  b  to form another element, denoted  a  •  b  or  ab . To qualify as a group, the set and operation, ( G , •), must satisfy four requirements known as the  group axioms : Closure For all  a ,  b  in  G , the result of the operation,  a  •  b , is also in  G . Associativity For all  a ,  b  and  c  in  G , ( a  •  b ) •  c  =  a  • ( b  •  c ). Identity element There exists an element  e  in  G  such that, for every element  a  in  G , the equation  e  •  a  =  a  •  e  =  a  holds. Such an element is unique , and thus one speaks of  the identity element.

24. Inverse element For each  a  in  G , there exists an element  b  in  G , commonly denoted  a −1  (or − a , if the operation is denoted "+"), such that  a  •  b  =  b  •  a  =  e , where  e  is the identity element. The result of an operation may depend on the order of the operands. In other words, the result of combining element  a  with element  b  need not yield the same result as combining element  b  with element  a ; the equation a  •  b  =  b  •  a

25. Abelian and Non- abelian Group An abelian group is a set,  A , together with an operation • that combines any two elements  a  and  b to form another element denoted  a  •  b . The symbol • is a general place holder for a concretely given operation. To qualify as an abelian group, the set and operation, ( A , •), must satisfy five requirements known as the  abelian group axioms : Closure Associativity Identity element Inverse Commutativity For all  a ,  b  in  A ,  a  •  b  =  b  •  a A group in which the group operation is not commutative is called a "non- abelian group" or "non-commutative group".

26. Determining the point group of a molecule or molecular ion

27. We can use a flow chart such as this one to determine the point group of any object. The steps in this process are: 1. Determine the symmetry is special (e.g. octahedral). 2. Determine if there is a principal rotation axis. 3. Determine if there are rotation axes perpendicular to the principal axis. 4. Determine if there are mirror planes. 5. Assign point group. IDENTIFYING POINT GROUPS

28. Decision Tree

29. Point group Symmetry operations Simple description of typical geometry Example 1 Example 2 C 1 E no symmetry, chiral Bromofluorochloro methane C 2 H 2 F 2 Cl 2 Dichlorodifluoroethane C s E, σ h mirror plane, no other symmetry SOCl 2 Thionyl dichloride Chloroiodomethane C i E,   i inversion center meso -Tartaric acid (S,R) 1,2-dibromo-1,2-dichloroethane COMMON POINT GROUPS

30. Point group Symmetry operations Simple description of typical geometry Example 1 Example 2 C 2 E, C 2 "open book geometry," chiral Hydrogen peroxide C 3 E, C 3 Propeller, chiral PPh 3   Triphenylphophine C ∞v E, 2C ∞,  ∞ σ v Linear HCl , HCN, HI, CO, NC, NCS, HCN, HCCH

31. Point group Symmetry operations Simple description of typical geometry Example 1 Example 2 C 2v E, C 2,  σ v (xz ), σ v '(yz) Angular H 2 O Sulfur dioxide (SO 2 ), Dichloromethane CH 2 Cl 2 C 3v E, 2C 3,  3 σ v Trigonal pyramidal or Tetrahedral Ammonia (NH 3 ) Phosphane ( PH 3 ) Chloroform (CHCl 3 ) C 4v E, 2C 4  , C 2  , 2 σ v  , 2 σ d Square pyramidal Xenon oxytetrafluoride   (XeOF 4 ) Pentaborane (B 5 H 9 )

32. Point group Symmetry operations Simple description of typical geometry Example 1 Example 2 C 2h E C 2   i   σ h P lanar with inversion center trans -1, 2-Dichloroethylene B(OH) 3 C 3h E, C 3 ,C 3 2 ,σ h , S 3, S 3 5 Propeller Boric acid Phloroglucinol D 2 E, C 2 (x ), C 2 (y ), C 2 (z ) twist, chiral Biphenyl Cyclohexane (twist) D 3 E, C 3 (z ), 3C 2, triple helix, chiral Tris ( ethylenediamine ) cobalt(III) cation

33. Point group Symmetry operations Simple description of typical geometry Example 1 Example 2 D 2h E , C 2 (z) , C 2 (y) , C 2 (x ) ,   i ,  σ(xy ) , σ(xz ) , σ(yz) Planar with inversion center Ethylene (C 2 H 4 ) Diborane (B 2 H 6 ) D 3h E, 2C 3 , 3C 2 , σ h , 2S 3 ,3σ v Trigonal planar or trigonal bipyramidal Boron trifluoride (BF 3 ) (PCl 5 ) D 4h E, 2C 4 , C 2   ,2C 2 ' 2C 2  i  2S 4  σ h 2σ v  2σ d Square planar Xenon tetrafluoride

34. Point group Symmetry operations Simple description of typical geometry Example 1 Example 2 D 6h E 2C 6  2C 3  C 2  3C 2 ' 3C 2 ‘’  i  2S 3 2S 6  σ h  3σ d  3σ v Hexagonal Benzene (C 6 H 6 ) Coronene (C 24 H 12 ) D 2d E, 2S 4 ,C 2 , 2C 2 ' , 2σ d 90° twist Allene D 3d E, 2C 3   , 3C 2   , i   ,2S 6   3, σ d 60° twist Ethane (Staggered) D ∞h E, C ∞,  ∞ σ v, ∞C 2 , i Linear N 2 , O 2 , F 2 , H 2 , Cl 2 , CO 2 , BeH 2 , N 3

35. Point group Symmetry operations Simple description of typical geometry Example 1 Example 2 S 2 E, 2S 2   , C 2 - Tetraphenylmethane T d E, 8C 3   , 3C 2   , 6S 4   , 6σ d Tetrahedral Methane Phosphorus pentoxide O h E, 8C 3   ,6C 2   ,6C 4   , 3C 2   , i   , 6S 4   ,8S 6 3σ h   ,6σ d Octahedral  or cubic Sulfur hexafluoride I h E 12C 5  12C 5 2  20C 3  15C 2   i 12S 10  12S 10 3  20S 6  15σ Icosahedral or Dodecahedral Buckminsterfullerene