Group Theory and Spectroscopy CHEM 305: Group Theory and Spectroscopy Part 1. Group Theory   © Dr. Jaime Martell

Molecular Symmetry: Group Theory   Symmetry Elements and Symmetry Operations Point Groups and Multiplication Tables Reducible and Irreducible Representations Application to Molecular Vibrations and Infrared and Raman Spectroscopy

Symmetry Elements and Symmetry Operations A Symmetry Element is a point, line or plane with respect to which a symmetry operation is performed. A Symmetry Operation is a physical manipulation of an object which places it in a new orientation that is indistinguishable from the original. NOTE the difference- the symmetry operation is an action, the symmetry element is a geometric entity about which an action takes place.

Symmetry operations may be of two kinds Symmetry operations that move the entire molecule from one location to another such that it moves to a location previously occupied by an identical molecule are required in crystallographic applications of group theory. Symmetry operations that leave the molecule in the same physical place but reorient it in space represent the subset of symmetry operations required for analysis of bonding.

Elements Operations Point-------------------- Inversion through the point. n-fold Proper Axis------ Rotation by 2/n about the axis. Rotation by 2/n about the axis followed by reflection in a plane perpendicular to the axis. n-fold Improper Axis--- Plane---------------- Reflection in the plane. Do nothing to the object is also a symmetry operation. None (or All) of the above--

Symmetry Operations and their Properties The Identity Operation: Chemists give this the symbol E, while mathematicians call it I. This operation makes no changes in the object. It amounts to leaving the object alone. All the other symmetry operations, if done enough times, will correspond to the identity. For example, two successive reflections in a plane correspond to leaving the object alone. Its analog in the real number system is the number 1 which can be multiplied by anything leaving the thing it is multiplied by unchanged.

THE IDENTITY OPERATION – leave alone or recover to original 1 2 3 4

Inversion (i) This operation is inversion through a center of symmetry. For each point in the molecule, move to the center, and then move the same distance to the other side.

Properties of Inversion: The operation consists of an exact reversal of all the unit vectors. Doing two successive inversions is equivalent to the identity operation. x y z

Axes of rotation (Cn) Rotation about an axis of symmetry through 360/n (or 2/n. The operation C1 is equivalent to E. An H2O molecule has a two-fold axis, C2 , and NH3 has a three axis, C3 .

Properties of n-fold Proper Rotations: If the operation is carried out m times, the operation is designated Cnm. Cnn is rotation by 2 radians and is the same as the identity. Many molecules have more than one proper rotation axis. When this happens, The rotation axis with the largest value of n is called the principal axis or the highest order rotation axis. The highest order axis is usually used as the molecular axis. When there is more than one C2 axis, they are distinguished by primes (‘). The usual convention is that C2 axes passing through atoms get a single prime (‘) while those passing between atoms get a double prime (“)

Proper Rotations z The behaviour of the unit vectors depends on the angle of rotation.  = C2(y) y x The unit vector lying along the rotation axis is always left unchanged, regardless of the value of n.

PROPER ROTATIONS: Principal Axis, Molecular Axis, and Different C2s y z  = C2(y)

The Reflection Operation: The symbol for this is the Greek letter  (German Spiegel = mirror). This corresponds to reflection in a plane. The unit vectors lying in the plane remain unchanged, the one perpendicular to the plane gets reversed in direction. Two successive reflections correspond to the identity.

Reflection in a Plane z y x

Vertical mirror planes (sv) If the plane contains the principal axis, it is called vertical and denoted sv.

Horizontal mirror planes (sh) If the plane of symmetry is perpendicular to the principal axis it is called horizontal and denoted sh . An example is benzene.

Dihedral mirror planes (sd) If a vertical plane bisects the angle between two C2 axes it is called a dihedral plane and denoted sd .

Improper Rotations (Sn) : The operation is rotation by an angle of 2/n followed by reflection in a plane perpendicular to the rotation axis. The sequential performance of m of these operations is designated Snm. If n is even, Snn corresponds to Rotation by 2 radians and An even number of reflections And therefore is equivalent to the identity. If n is odd, Snn contains an odd number of reflections and is not equivalent to the identity. If n is odd, Sn2n corresponds to Rotation by 4 radians and

Improper Rotation x y z 2/4 = C4(z) x,y S4