Introduction to group theory
- 1. GROUP THEORY TONY FRANCIS DEPARTMENT OF CHEMISTRY St. MARY'S COLLEGE, MANARKADU
- 2. Mathematical study of symmetry is called Group Theory • Symmetry Element– A symmetry element is a geometrical entity such as a point, a line or a plane about which a symmetry operation is performed. • Symmetry operation– A symmetry operation is a movement such as inversion about a point, rotation about a line or a reflection about a plane in order to get an equivalent orientation.
- 3. An equivalent orientation is an orientation similar to the original orientation but not the identity. Equivalent orientation Identity
- 4. Symmetry Elements Element Symmetry Operation Symbol Identity E Proper axis Rotation by 2π/n Cn Plane of symmetry Reflection σ Center of symmetry Inversion i Improper axis of Rotation by 2π/n Sn symmetry followed by reflection perpendicular to the axis of rotation
- 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. Centre of symmetry (i) • It is a point within the molecule from which lines drawn to opposite direction meet similar points at exactly the same distance and direction.
- 7. Proper axis of symmetry • It is an axis passing through the molecule about which the molecule is rotated through 360◦, if we get n times equivalent orientations the molecule has an n-fold axis of symmetry.
- 9. Principal axis • If there are more than one axis of symmetry in many cases one of the axis is identified as principal axis. The selection will be on the following basis:- 1. Highest order axis 2. Unique axis 3. The axis passing through maximum no of molecule. 4. The axis perpendicular to the plane of the molecule The other axis are known as subsidiary axis
- 10. Plane of symmetry • Plane of symmetry is a plane which divide the molecule into two equal halves such that one half is the mirror image of the other half. • On the basis of the principal axis they are of two types vertical and horizontal plane. • HP:-plane perpendicular to the principal axis(σh ) • VP:-plane which is along the principal axis or involving the principal axis (σv )
- 12. Rotational axes and mirror planes of the water molecule: C2 principal axis C2 C2σv mirror plane σv mirror plane The water molecule has only one rotational axis, its C2 axis, which is also its principal axis. It has two mirror planes that contain the principal axis, which are therefore σv planes. It has no σh mirror plane, and no center of symmetry.
- 13. A rotation-reflection operation (Sn) required rotation of 360° /n, followed by reflection through a plane perpendicular to the axis of rotation.
- 14. Equivalent and non-equivalent operations 14 O H H C2 sv sv’ H N H H C3 sv sv sv • sv : No atom moves sv ‘: H atoms interchange • sv : Two atoms move Other two don’t • sv and sv ‘ do not interchange by C2 • The three sv planes interchange by C3 Non-equivalent planes Equivalent planes (Same class)
- 16. Boron trifluoride C3 principal axis C3 principal axis σh σh σv σv C2 C2 C2 boron trifluoride has a C3 principal axis and three C2 axes, a σh mirror plane three σv mirror planes, but no center of inversion E,2C3,3C2,3σv,σh,2S3—D3h
- 26. C6 principal axis C2 C2 C2 C6 C2 σv σv Rotational axes and mirror planes of benzene σh C6 principal axis C6 principal axis
- 37. Distinct operations • D.O are operations that cannot be represented by any other axis of lower symmetry.
- 38. • Order- Total number of symmetry operation • Classes- It is the number of distinct symmetry operations