Chapter 01 Symmetry elements

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Chapter 1 – Symmetry Elements		� PAGE �13�
Why is Group Theory Important??
Consider the following reaction and the anticipated product:
Chapter 1 - Symmetry Elements or Operations
Symmetry Operations (or Elements): The various ways of moving an object such that after the movement has been carried out the orientation of the object is equivalent (or identical) to the original orientation.
The identity operation is labeled E and corresponds to a 360º rotation. All objects, no matter how low their overall symmetry, always possess the identity operation as a symmetry element. 
Inversion
Rotations
Reflections
Improper Rotations
Inversion ( i ): The symmetry operation that transforms a general atom position x, y, z to -x, -y, -z ( x, y, z ). The inversion center is a point and must lie at the center (or origin) of the molecule (object). 
in = E	when n is even
in = i	when n is odd
Examples:
	
�Problem: Identify which of the following molecules or objects have or do not have an inversion center:
Reflection (  ): The symmetry operation (plane) that transforms a general atom position x, y, z to x, y, -z 
( x, y, z ) when the plane of symmetry lies in the xy plane. The reflection plane must always pass through the origin of the molecule (object). 
n = E 	when n is even
n = 	 	when n is odd
Example: There are six reflection planes that slice through tetrahedral CCl4
�Problem: Identify all the reflection planes for the following molecules. 
�Proper Rotations ( Cn ): The symmetry operation that rotates an object about an axis passing through the center point (origin) to an equivalent orientation.
 = a rotation by m × 2/n
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
�Example: There are four C3, four C32, and three C2 rotation axes that cut through a tetrahedral CCl4
�Problem: Identify all the rotation axes for the following molecules. 
 
�Improper Rotations ( Sn ): The symmetry operation that combines a proper rotation and a reflection through a reflection plane perpendicular ( ( ) to the rotation axes. The axis about which the rotation occurs is called an improper axis. 
 = a rotation by m × 2/n
	 followed by a perpendicular reflection
Important note: for even order improper rotations neither the proper rotation axis NOR the mirror plane need exist independently!!
 when n and m are even.
	
	
	
	
	
	
	
�Example: There are S6n symmetry elements in staggered ethane. 
Odd order improper axes have two major differences (properties) from even order versions:
An odd order Sn requires that Cn and a  reflection plane perpendicular to it must exist independently.
An even order Sn generates n symmetry operations, an odd order Sn, however, generates 2n operations!
	
	
	
	
	
	
	
	
	
	
�Problem: Identify which of the following molecules has improper rotation axes and what kind (different views of some molecules shown for clarity). 
�Relations Between Symmetry Elements
The product of two proper rotations (Cn) is another proper rotation
The product of two reflections () in planes A and B intersecting at an angle (AB is a rotation by 2(AB about the axis defined by the line of the planes intersection. 
A proper rotation axis of even order (Cn) and a perpendicular mirror plane (() generate an inversion center ( i ).
				
				
The following pairs of symmetry operations always commute:
two rotations about the same axis
reflections through planes ( to each other
inversion and any reflection or rotation operation
two C2 rotations about a ( axis
rotation and reflection in a plane ( to the rotation axis (Sn case). 
�Optical Activity: A common statement about the existence of optical activity (chirality) in a molecule is that the molecule lacks a plane or center of symmetry. Also that it is non-superimposable upon a mirror image of itself. 
A more precise definition is: A molecule that does NOT possess an improper rotation axis (Sn) will be dissymmetric (chiral)
Note that one can relate reflection and inversion to the hypothetical improper rotation axes S1 and S2 in the following way: 
S1 =  and S2 = i
Consider the chiral compound binapthol:
�Qualitative Symmetry Element Classifications
1) Inversion Center: There is only one inversion operation (center) per point group (if it has one) and it resides in its own class. 
2)	The Identity operation (E) is always in a class by itself.
3)	Reflections: a horizontal reflection plane (h) is always in a class by itself.
Vertical reflection planes in the same class are designated by nv . Vertical mirror planes in separate classes either are denoted by nd or by nv’.
4)	Proper Rotations: for Cn and Cnh point groups all Cn rotations are listed in their own separate classes, e.g., 
			C5	=	C5		C52		C53		C54
			C5h	=	C5		C52		C53		C54
	For Cnv, Dn, Dnh, Dnd, O, T, and I point groups rotations are classed with their inverse, e.g.,
			C5v	=	2C5	(contains C5 & C54) 
					2C52 (contains C52 & C53)
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