Symmetry and Group Theory

Question 1

What are the two types of symmetry operations?

Answer 1

• Proper axis of rotation
• Improper axis of rotation

Question 2

What is the difference between proper and improper axis of rotation?

Answer 2

Proper can be carried out physically while improper cannot be physically performed

Question 3

What the the notation Cn mean?

Answer 3

A rotation of 360/n ° leaves the object unchanged

Question 4

What rotation does every object have?

Answer 4

C1 or E (the identity) - a 360° rotation

Question 5

By convention, in what direction are rotations carried out?

Answer 5

Clockwise

Question 6

What does C3.2 mean?

Answer 6

C3 twice in the clockwise direction

Question 7

What is C4.2 also known as? And therefore how many unique rotations are there about the c4 axis?

Answer 7

C2. There are only two unique rotations; C4.1 and C4.3

Question 8

What is the difference between a symmetry axis and a symmetry element

Answer 8

The symmetry element is the axis or plane that a symmetry operation, like C3.2, is acting on a molecule in

Question 9

What is the principle axis and what does it define?

Answer 9

The axis of the highest rotational symmetry - it defines the z direction

Question 10

What does C2’ and C2’’ mean?

Answer 10

C2’ is through bonds and C2’’ is between bonds

Question 11

What does the notation Sn mean?

Answer 11

A rotation of Cn followed by a reflection in a plane perpendicular to that Cn axis

Question 12

What is an S1 operation also known as?

Answer 12

(A rotation of 360° followed by) a reflection - σ

Question 13

Describe σv, σd and σh

Answer 13

σv - vertical: contains the principle axis
σd - dihedral: contains the principle axis and bisects two C2 axes
σh - horizontal: perpendicular to the principle axis

Question 14

What is an S2 operation also known as?

Answer 14

i = an inversion though an inversion centre

Question 15

For a molecule with an inversion centre, do you label the orbitals gerade and ungerade or does it not matter?

Answer 15

For a molecule with an inversion centre you do label the orbitals gerade and ungerade

Question 16

What does gerade mean?

Answer 16

Even (in German) - sign of the orbital is left unchanged after an inversion

Question 17

What does ungerade mean?

Answer 17

Odd (in German) - sign of the orbital is flipped during inversion

Question 18

For a molecule with 10 proper operations, how many improper operations would it have? (assuming it did have improper operations)

Answer 18

10 - The number of proper operations always matches the number of improper operations given that improper operations are possible

Question 19

Define a point group

Answer 19

A group of symmetry operations which form a closed set

Question 20

Define a closed set

Answer 20

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 21

Do chrial compounds have symmetry?

Answer 21

No - any compound with chirality only has E symmetry; it is asymmetric

Question 22

Chiral objects are dissymmetric and therefore don’t have symmetry. True or False?

Answer 22

Compounds with chiral carbons are asymmetric and therefore don’t have any symmetry, however compounds with only proper symmetry operations are dissymmetric.

Question 23

What is the name of the collection of eigenvalues for a point group known as?

Answer 23

The irreducible representation

Question 24

What does the E label mean in a character table?

Answer 24

Doubly degenerate irreducible representation - two orbitals with the same representation

Question 25

How would you find out the symmetry of the pz-orbital on boron in BF3?

Answer 25

1) Work through the point group flow chart to find the point group
2) Look in the data book for the character table for that point group
3) Find the pz-orbital in the table and read along to the symmetry label
In this case the symmetry is A2’’

Question 26

The s-orbital is not in the character tables, what is it irreducible representation?

Answer 26

It is the irreducible with the highest symmetry - all the eigenvalues are 1 (it is a sphere)

Question 27

What do the A and B labels mean in a character table?

Answer 27

A - singly degenerate and totally symmetric
B - singly degenerate and anti-symmetric

Question 28

What does the T label mean in a character table?

Answer 28

Triply degenerate irreducible representation - three orbitals with the same representation

Question 29

In character tables for point groups, where are the symmetries of the p- and d-orbitals found?

Answer 29

The leftmost column that does not contain eigenvalues are p-orbitals (translations, rotations and IR active bands) and the column to right is the d-orbitals (Raman active bands)

Question 29

What does the Tlabel mean in a character table?

Answer 29

Triply degenerate irreducible representation - three orbitals with the same symmetry

Question 30

What do the subscript labels g and u mean in a character table?

Answer 30

subscript g - symmetric under i (s- and d-orbitals)
subscript u - unsymmetrical under i (p- and f-orbitals)

Question 31

How are the eigenvalues in a character table calculated?

Answer 31

They are the trace from the matrix of the transformation

Question 32

What is the general clockwise rotation matrix?

Answer 32

( cosθ sinθ )
( -sinθ cosθ )
where θ = n

Question 33

In character tables for point groups, where are the symmetries of the p- and d-orbitals found?

Answer 33

The leftmost column that does not contain eigenvalues are p-orbitals (translations, rotations and IR active bands) and the column to right is the d-orbitals (Raman active bands)

Question 34

In character tables for point groups, where are the symmetries of the p- and d-orbitals found?

Answer 34

The leftmost column that does not contain eigenvalues are p-orbitals (translations, rotations and IR active bands) and the column to right is the d-orbitals (Raman active bands)

Question 35

In character tables for point groups, where are the symmetries of the p- and d-orbitals found?

Answer 35

The leftmost column that does not contain eigenvalues are p-orbitals (translations, rotations and IR active bands) and the column to right is the d-orbitals (Raman active bands)

Question 36

In character tables for point groups, where are the symmetries of the p- and d-orbitals found?

Answer 36

The leftmost column that does not contain eigenvalues are p-orbitals (translations, rotations and IR active bands) and the column to right is the d-orbitals (Raman active bands)

Question 37

In character tables for point groups, where are the symmetries of the p- and d-orbitals found?

Answer 37

The leftmost column that does not contain eigenvalues are p-orbitals (translations, rotations and IR active bands) and the column to right is the d-orbitals (Raman active bands)

Question 38

What is Γ and how can it be found?

Answer 38

The reducible representation - by inspection or the sum of the irreducible representations

Question 39

Where can the reduction formula be found?

Answer 39

In the data book - page 81