Symmetry and group theory

Question 1

What are proper axes of rotation?

Answer 1

Proper rotations can be done physically. They are given the symbol C_n, where n is the order of the axes. Rotating the molecule by (360/n)° will leave it unchanged. All objects have a C₁ axis which is the identity (E). By convention, rotations are clockwise.

Question 2

What is a C₂ rotation?

Answer 2

For a C₂ axis, rotating by 180° leaves the object unchanged. There is only 1 unique C₂ operation (C₂) as C₂² = E.

Question 3

What is a C₃ rotation?

Answer 3

For a C₃ axis, rotating by 120° leaves the object unchanged. There are 2 unique C₃ rotations (C₃ and C₃²) as C₃³ = E.

Question 4

What is a C₄ rotation?

Answer 4

For a C₄ axis, rotating by 90° leaves the object unchanged. There are 2 unique C₄ rotations (C₄ and C₄³) as C₄⁴ = E and C₄² = C₂.

Question 5

What is a C₅ rotation?

Answer 5

For a C₅ axis, rotating by 72° leaves the object unchanged. All of the rotations are unique.

Question 6

What is a C₆ rotation?

Answer 6

For a C₆ axis, rotating by 60° leaves the object unchanged. There are 2 unique C₆ rotations (C₆ and C₆⁵) as C₆² = C₃ and C₆³ = C₂ and C₆⁴ = C₃² and C₆⁶ = E.

Question 7

What is a symmetry element?

Answer 7

It is the axis of rotation.

Question 8

What is a symmetry operation?

Answer 8

It is what is done to the object.

Question 9

What is a principal axis?

Answer 9

It is the axis of highest symmetry and it defines the z-direction.

Question 10

What are improper axes of rotation?

Answer 10

Improper roatations cannot be done physically. They are given the symbol S_n. They are a rotation of C_n followed by a reflection in the plane perpendicular to C_n. An S_n symmetry axis can be associated with a C_n or C_n/2 rotational axis.

Question 11

What is a reflection?

Answer 11

A reflection (S₁) is a rotation of C₁ followed by a reflection. It is given the symbol σ. σ_v contains the principal axis. σ_d contains the principal axis but it bisects a pair of bonds. σ_h is perpendicular to the principal axis.

Question 12

What is an inversion?

Answer 12

An inversion (S₂) is a rotation of C₂ followed by a reflection. Travel from 1 atom to the equivalent position on the other side of the centre of inversion.

Question 13

What are the S₃ operations?

Answer 13

There are 2 unique S₃ operations (S₃ and S₃⁵) as S₃² = C₃² and S₃³ = σ_h and S₃⁴ = C₃.

Question 14

What are the S₄ operations?

Answer 14

There are 2 unique S₄ operations (S₄ and S₄³) as S₄² = C₂.

Question 15

What are inversion centres?

Answer 15

Octahedral complexes have a centre of inversion so the orbitals are labelled gerade (the wavefunction remains the same under inversion) or ungerade (the wavefunction changes under inversion). Tetrahedral complexes do not have a centre of inversion.

Question 16

What is a point group?

Answer 16

A point group is a group of symmetry operations which form a closed set. A closed set is a set of symmetry operations such that successive applications of the operations is equivalent to another operation, which is also a property of the object.

Question 17

What are chiral molecules?

Answer 17

The number of proper operations is equal to the number of improper operations, if the object has improper operations. If there are no improper oberations, the object is chiral. They belong to the C_n and D_n point groups.

Question 18

What are the symbols used for irreducibles?

Answer 18

A = singly degenerate and totally symmetric about the principal axis

B = singly degenerate and anti-symmetric about the principal axis

E = doubly degenerate

T = triply degenerate

Subscript 1 = symmetric under C₂’ or σ_v

Subscript 2 = anti-symmetric under C₂’ or σ_v

Superscript ‘ = symmetric under σ_h

Superscript “ = anti-symmetric under σ_h

Subscript g = symmetric under i

Subscript u = anti-symmetric under i

Question 19

How are matrices used in group theory?

Answer 19

If 2 properties of an object can be interconvereted then they must be treated together and are degenerate. The character of the irreducible comes from the trace of the transformational matrix.

Question 20

What are reducible representations?

Answer 20

A reducible representation can be reduced to irreducibles. If a property moves when an operation is applied, a=0. If a property changes sign when an operation is applied, ‘a’ is negative. In the equation to find the irreducibles:

n(r) = number of times an irreducible occurs

h = order of group (total number of operations)

X_R = character in reducilbe

X_i = character in irreducible

N = number of operations in a given class

Question 21

What are the linear combinations of hydrogen’s atomic orbitals?

Answer 21

a₁ is the symmetric representation:

1_s (H_A) + 1_s (H_B) so the AOs are in-phase

b₁ is the lower symmetry representation:

1_s (H_A) - 1_s (H_B) so the AOs are out-of-phase

Question 22

How does mixing in H₂O affect the MO diagram?

Answer 22

E (2a₁) does not equal E (1b₂).

Question 23

How do you use the projection operator method to calculate the SALCs?

Answer 23

1. Write down all symmetry operations of the group
2. Label the orbitals with σ or π
3. Apply the symmetry operations to σ₁ or π₁ and record the result to give row 1
4. Multiply row 1 by the irreducible to give row 2
5. Add up row 2
6. Normalise (ignore the number outside the bracket)
7. aσ₁ + bσ₂ + cσ₃ = (1/(a²+b²+c²)^1/2)(σ₁+σ₂+σ₃)
8. If they are positive, they are in-phase
9. If they are negative, they are out-of-phase

Question 24

How do you find a second explicit SALC?

Answer 24

Start with σ₂ then σ₃, then subtract the second result from the first.