Introduction to Group Theory and Molecular Groups

Molecular Spectroscopy January 24, 2008

Introduction to Group Theory and Molecular Groups

Properties that define a group • A group is a collection of elements that are interrelated based on certain rules and conditions • In order for a set of elements to form a group the following four conditions must be met….

•

(1) The product of any two elements in a group and the square of each element must also be an element in the group.

•

We must define what me mean by “multiply” and “product” • We take the convention that if we choose to combine an element A with element B we denote this by writing AB. • The question as to whether the elements in a group commute is an important one. In group theory…the commutative law is not always valid. • There are some groups in which the combination of elements is commutative and such groups are said to be Albelian. • Another point that comes to mind when looking at the combination of two elements as written above…AB…is whether A is combined with B or whether B is combined with A. This is taken care of by saying that “B is left multiplied by A” or “B is right multiplied by A” – Note: When we begin combining symmetry operations from character tables the element on the right is taken first.

•

(2) One element in the group MUST commute with all other elements in the group AND leave them unchanged.

•

Convention designates this element with the letter E and is known as the identity element – So, if we take E and some arbitrary element in the group we have… » EX = XE = X

• (3) The Associative Law of multiplication must hold •

In other words, if we take three arbitrary elements in the group we will have the following… » A(BC) = (AB)C or… » (AB) (CD) (EF) = A (BC) (DE) F » Etc…

•

(4) Every element must have a reciprocal, which is ALSO an element of the group.

•

An element…say R…is the reciprocal of some element…say A… if the following is true… » RA = AR = E • Note that E is its own reciprocal

•

Allow us to revisit an important theorem involving reciprocals…namely… • The reciprocal of a product of two or more elements is equal to the product of the reciprocals, in reverse order. • In other words… » (ABC…YZ)-1 = Z-1 Y-1…C-1 B-1 A-1

Subgroups • As the name implies…a subgroup is simply a group within a group. ¾It must still posses ALL of the properties of a group ¾The order of any subgroup MUST be an integral divisor of the order of the main group ¾The identity operator forms the trivial subgroup of order 1 ¾While a certain number of subgroups may be allowed to exist (3 for our example) they do not HAVE to exist!! • Highlighted below is the subgroup of order 2 B

C

T

A

G

J

B

B

C

T

A

G

J

C

C

B

J

G

A

T

T

T

G

B

J

C

A

A

A

J

G

B

T

C

G

G

T

A

C

J

B

J

J

A

C

T

B

G

Subgroup of order 1 B

The Subsets of our group… B

B

Subgroup of order 2 B

C

B

B

C

C

C

B

Subgroup of order 3 B

G

J

B

B

G

J

G

G

J

B

J

J

B

G

Is this group of order 3 also a subgroup of our main group? B

C

T

B

B

C

T

C

C

B

J

T

T

G

B

Homework Assignment…Demonstrate that this is or is not a subgroup of our main group by demonstrating that it has the four properties needed to form a group.

Similarity Transformations • We will define a similarity transformation as the consecutive application of three operations to yield a third. ¾These three operations are the following ¾Z-1 ¾Z So… ¾X Z-1·X·Z = Y ¾In order to get ¾Y

• X and Y are related by a similarity transformation and are said to be conjugate to one another. • In our example abstract group…G and J are conjugate to one another C-1·G·C = J

C-1·G·C = J B

C

T

A

G

J

B

B

C

T

A

G

J

C

C

B

J

G

A

T

T

T

G

B

J

C

A

A

A

J

G

B

T

C

G

G

T

A

C

J

B

J

J

A

C

T

B

G

Is this true?

• Let’s work through this… ¾Note that C is its own inverse…C-1 = C ¾From this we then get the following from the character table C-1·(G·C) = C·(G·C) = C·(T) =J B

C

T

A

G

J

B

B

C

T

A

G

J

C

C

B

J

G

A

T

T

T

G

B

J

C

A

A

A

J

G

B

T

C

G

G

T

A

C

J

B

J

J

A

C

T

B

G

Cyclic Groups • A cyclic group of order n is defined as an element X and all of its powers up to

X =E n

• A group of order 3 is the simplest, nontrivial member (group of order 1always there)

G3

E

A

B

E

E

A

B

A

A

B

E

B

B

E

A

We can consider the entire group to be generated by taking the element A and its powers An important property of cyclic groups is that they are Abelian

Classes • First…let’s go back to the idea of similarity transformations… Z-1·X·Z = Y recall that X and Y are related by a similarity transformation and are therefore said to be conjugate to one another. The following properties of conjugate elements may come in handy… ¾Every element is conjugate with itself ¾If A is conjugate with B, then B is conjugate with A ¾If A is conjugate with B and C, then B and C are conjugate with each other

Classes A complete set of elements that are conjugate to one another is called a CLASS of the group • The classes that may be present within a group are easy to find… 1. Begin with one element and work out all of its transforms using all of the elements in the group, including itself 2. Take the second element which is NOT found to be a conjugate of the first element looked at and determine all of its transforms 3. Repeat step 2 until done!

1. Begin with one element and work out all of its transforms using all of the elements in the group, including itself 2. Take the second element which is NOT found to be a conjugate of the first element looked at and determine all of its transforms 3. Repeat step 2 until done!

Lets find the classes of this 8th order abstract group… M8

G

K

H

J

L

M

O

P

G

G

K

H

J

L

M

O

P

K

K

M

J

P

G

L

H

O

H

H

O

G

L

J

P

K

M

J

J

H

K

G

P

O

M

L

L

L

G

O

H

M

K

P

J

M

M

L

P

O

K

G

J

H

O

O

P

L

M

H

J

G

K

P

P

J

M

K

O

H

L

G

1. Begin with one element and work out all of its transforms using all of the elements in the group, including itself 2. Take the second element which is NOT found to be a conjugate of the first element looked at and determine all of its transforms 3. Repeat step 2 until done!

Find the classes of our example abstract group for Tuesday… B

C

T

A

G

J

B

B

C

T

A

G

J

C

C

B

J

G

A

T

T

T

G

B

J

C

A

A

A

J

G

B

T

C

G

G

T

A

C

J

B

J

J

A

C

T

B

G

Molecular Symmetry and The Symmetry of Groups

Molecular Symmetry and The Symmetry of Groups We have been discussing groups and have been using abstract elements which constitute a group > We will be using symmetry elements whose operations constitute a group • Symmetry Operation An operation applied to an object such that it leaves it unchanged. In other words…A symmetry operation carries…or MAPS…an entity onto an equivalent configuration • Symmetry Element A symmetry element is a geometrical entity…a line, point, plane, etc…with respect to which one or more symmetry operations may be carried out Symmetry Operations and Symmetry Elements are inextricably connected and so are easily confused…THEY ARE DIFFERENT THINGS…and it is important to understand the difference between them

• Symmetry Group Classification of a molecule according to its symmetry elements • Point Group Symmetry operations which leave one point of the molecule unchanged There are four types (five if you include the identity operator) of symmetry elements and operations that need to be considered in dealing with molecular symmetry ¾Plane Reflection in the plane ¾Center of symmetry or center of inversion Inversion of all atoms through the center ¾Proper axis One or more rotations about the axis ¾Improper axis One or more repetitions of the sequence: rotation followed by reflection in a plane perpendicular to the rotation axis

Element

Operation

Example

E, the identity

Does nothing

All objects posses E

Cn, n-fold rotation

•Rotate object by 2π/n about the n-fold axis. •The axis with the highest n is called the principle axis. •For n>2 the sense is important…Cn+ - clockwise rotation…Cn--counterclockwise rotation

•H2O posses a C2 axis •NH3 posses a C3 axis •Benzene posses a C2, C3, C6, and two sets of three C2 axes perpendicular to the C6 axis

σ, reflection plane

Reflects through the plane

Three types

σv, vertical plane

Plane containing the principle axis

σh, horizontal plane

Plane perpendicular to the principle axis

•Benzene

σd, dihedral plane

Vertical plane which bisects two C2 axes perpendicular to the principal axis

i, inversion center

Sends each atom through the origin along a straight line so that x→-x, y→-y, and z→-z

•H2 •CO2 •Benzene •SF6

Rotate object by 2π/n, then reflects through plane perpendicular to the axis of rotation

•CH4

Sn, n-fold improper rotation axis

•H2O •Benzene

Let’s take the ECLIPSED conformation of ethane

• One three-fold axis coincident with the C-C bond •Three two-fold axes perpendicular to the C-C bond and intersecting its midpoint •Three reflection planes, each containing the C-C bond and a pair of C-H bonds •One reflection plane perpendicular to the C-C bond and bisecting it •NO CENTER OF INVERSION IS PRESENT •One three-fold improper axis coincident with the C-C bond

Let’s take the STAGGERED conformation of ethane

• One three fold axis coincident with the C-C bond •Three two-fold axes perpendicular to the C-C bond and intersecting its midpoint •Three reflection planes, each containing the C-C bond and a pair of C-H bond •No reflection plane perpendicular to the C-C bond…we lost it! •One point of inversion at the midpoint of the C-C bond…we gained it! •One six-fold improper axis coincident with the C-C bond…changed its order!