Why is Group Theory Important??

Consider the following reaction and the anticipated product:

Chapter 1 – Symmetry Elements 2

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:

[CO3]2

Chapter 1 – Symmetry Elements 3

Problem: Identify which of the following molecules or objects have or do not have an inversion center:

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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

Chapter 1 – Symmetry Elements 5

Problem: Identify all the reflection planes for the following molecules.

Chapter 1 – Symmetry Elements 6

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

Chapter 1 – Symmetry Elements 7

Example: There are four C3, four C32, and three C2 rotation

axes that cut through a tetrahedral CCl4

Chapter 1 – Symmetry Elements 8

Problem: Identify all the rotation axes for the following molecules.

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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.

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Example: There are S6n symmetry elements in staggered

ethane.

Odd order improper axes have two major differences (properties) from even order versions:

1) An odd order Sn requires that Cn and a reflection

plane perpendicular to it must exist independently.

2) An even order Sn generates n symmetry operations,

an odd order Sn, however, generates 2n operations!

Chapter 1 – Symmetry Elements 11

Chapter 1 – Symmetry Elements 12

Problem: Identify which of the following molecules has improper rotation axes and what kind (different views of some molecules shown for clarity).

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Relations Between Symmetry Elements

1) The product of two proper rotations (Cn) is another

proper rotation

2) The product of two reflections () in planes A and B intersecting at an angle AB is a rotation by 2AB about the axis defined by the line of the planes intersection.

3) 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:

1) two rotations about the same axis

2) reflections through planes to each other

3) inversion and any reflection or rotation operation

4) two C2 rotations about a axis

5) rotation and reflection in a plane to the rotation axis (Sn case).

Chapter 1 – Symmetry Elements 14

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:

Chapter 1 – Symmetry Elements 15

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 nv . Vertical mirror planes in separate

classes either are denoted by nd or by nv’.

4) Proper Rotations: for Cn and Cnh point groups all Cn

rotations are listed in their own separate classes, e.g.,

C5 = C5 C52 C5

3 C54

C5h = C5 C52 C5

3 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 C5

2 & C53)