Symmetry and Group Theory Flashcards
What are the two types of symmetry operations?
- Proper axis of rotation
- Improper axis of rotation
What is the difference between proper and improper axis of rotation?
Proper can be carried out physically while improper cannot be physically performed
What the the notation Cn mean?
A rotation of 360/n ° leaves the object unchanged
What rotation does every object have?
C1 or E (the identity) - a 360° rotation
By convention, in what direction are rotations carried out?
Clockwise
What does C3.2 mean?
C3 twice in the clockwise direction
What is C4.2 also known as? And therefore how many unique rotations are there about the c4 axis?
C2. There are only two unique rotations; C4.1 and C4.3
What is the difference between a symmetry axis and a symmetry element
The symmetry element is the axis or plane that a symmetry operation, like C3.2, is acting on a molecule in
What is the principle axis and what does it define?
The axis of the highest rotational symmetry - it defines the z direction
What does C2’ and C2’’ mean?
C2’ is through bonds and C2’’ is between bonds
What does the notation Sn mean?
A rotation of Cn followed by a reflection in a plane perpendicular to that Cn axis
What is an S1 operation also known as?
(A rotation of 360° followed by) a reflection - σ
Describe σv, σd and σh
σv - vertical: contains the principle axis
σd - dihedral: contains the principle axis and bisects two C2 axes
σh - horizontal: perpendicular to the principle axis
What is an S2 operation also known as?
i = an inversion though an inversion centre
For a molecule with an inversion centre, do you label the orbitals gerade and ungerade or does it not matter?
For a molecule with an inversion centre you do label the orbitals gerade and ungerade
What does gerade mean?
Even (in German) - sign of the orbital is left unchanged after an inversion
What does ungerade mean?
Odd (in German) - sign of the orbital is flipped during inversion
For a molecule with 10 proper operations, how many improper operations would it have? (assuming it did have improper operations)
10 - The number of proper operations always matches the number of improper operations given that improper operations are possible
Define a point group
A group of symmetry operations which form a closed set
Define a closed set
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
Do chrial compounds have symmetry?
No - any compound with chirality only has E symmetry; it is asymmetric
Chiral objects are dissymmetric and therefore don’t have symmetry. True or False?
Compounds with chiral carbons are asymmetric and therefore don’t have any symmetry, however compounds with only proper symmetry operations are dissymmetric.
What is the name of the collection of eigenvalues for a point group known as?
The irreducible representation
What does the E label mean in a character table?
Doubly degenerate irreducible representation - two orbitals with the same representation
How would you find out the symmetry of the pz-orbital on boron in BF3?
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’’
The s-orbital is not in the character tables, what is it irreducible representation?
It is the irreducible with the highest symmetry - all the eigenvalues are 1 (it is a sphere)
What do the A and B labels mean in a character table?
A - singly degenerate and totally symmetric
B - singly degenerate and anti-symmetric
What does the T label mean in a character table?
Triply degenerate irreducible representation - three orbitals with the same representation
In character tables for point groups, where are the symmetries of the p- and d-orbitals found?
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)
What does the Tlabel mean in a character table?
Triply degenerate irreducible representation - three orbitals with the same symmetry
What do the subscript labels g and u mean in a character table?
subscript g - symmetric under i (s- and d-orbitals)
subscript u - unsymmetrical under i (p- and f-orbitals)
How are the eigenvalues in a character table calculated?
They are the trace from the matrix of the transformation
What is the general clockwise rotation matrix?
( cosθ sinθ )
( -sinθ cosθ )
where θ = n
In character tables for point groups, where are the symmetries of the p- and d-orbitals found?
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)
In character tables for point groups, where are the symmetries of the p- and d-orbitals found?
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)
In character tables for point groups, where are the symmetries of the p- and d-orbitals found?
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)
In character tables for point groups, where are the symmetries of the p- and d-orbitals found?
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)
In character tables for point groups, where are the symmetries of the p- and d-orbitals found?
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)
What is Γ and how can it be found?
The reducible representation - by inspection or the sum of the irreducible representations
Where can the reduction formula be found?
In the data book - page 81
