Symmetry and Group Theory Flashcards
What is a proper axis of rotation?
- Rotation around an axis, the rotation is by (360/n)º
- A C2 axis is 360/2= 180º rotation
What is the E axis?
A C1 axis, the object is completely unchanged
What is the trick for determining what coincident axis are contained within a molecule
- If the order of an axis is a factor of the order of an axis contained by the molecule, then the molecule will contain both axes
- For example if a molecule contains a C6 axis, we know it also contains a C3 axis
- We know this because C6 ^ 4 = C3 ^2
What is the symmetry element of a molecule?
The proper and improper axes of rotation and planes of symmetry of a molecule
What is a symmetry operation?
Movement of a molecule such that the molecule appears the same as before eg. rotation by C3 ^2
What is the principal axis of a molecule?
The rotational axis in the molecule of the highest symmetry. Eg. if a molecule has a C4 and a C2 axis, C4 is principle axis as it has a higher order
What is an improper axis of rotation?
A combination of rotation with respect to an axis of rotation (Cn) followed by reflection in a plane perpendicular to that axis of rotation, eg. the rotation can’t be done on a single axis
What is the S1 axis?
Rotation in the C1 axis followed by reflection, C1 = E so S1 is simply a reflection
What is the σv axis?
Plane of reflection that goes through the principle axis, the σv axis passes through the maximum number of atoms
What is the σd axis?
Plane of reflection that goes through the principle axis, the σd axis bisects the angle between two C2 axis, the σd axis passes through less atoms than the σv axis
What is the σh axis?
The plane of reflection perpendicular to the principle axis
What is the S2 axis?
Rotation in the C2 axis followed by reflection in plane perpendicular to rotational axis, this is another special case. It is equal to a centre of inversion denoted i
For improper axes of rotation, do the proper axis of rotation used in the improper operation need to be contained in the molecule?
No, for example if in the improper operation involves a rotation in C2 axis and then reflection in σv plane, the molecule doesn’t have to contain a C2 axis or a σv plane
When are g+u labels used for orbitals?
When the orbital has a centre of inversion
What does gerade mean?
When an orbital is passed through a centre of inversion it remains the same
What does ungerade mean?
When the orbital is passed through a centre of inversion, the negative and positive signs switch
When determining the point group of a molecule, which symmetry operations do you need to take into account?
All the unique symmetry operations, for example if a molecule has one C3 axis, we write 2C3 as that is the number of unique operations on the C3 axis, C3 ^1 and C3 ^2 are unique, while C3 ^3 isn’t as it is identical to E
If we have all the symmetry of a point group, what is the rule linking proper and improper rotations together?
The number of proper operations is equal to the number of improper rotations, with the exception of chiral molecules
What is a point group?
- 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 that is already a property of the molecule
When doing a multiplication table for a point group, which side of the table is the operation that is done first and which is the side that the operation is done second?
The vertical side of the table is the operation that is done first, horizontal operations are done second
When you do a proper operation followed by a proper operation, do you get a proper operation or an improper operation?
Proper
When you do an improper operation followed by a proper operation, do you get a proper operation or an improper operation?
Improper
When you do a proper operation followed by an improper operation, do you get a proper operation or an improper operation?
Improper
When you do an improper operation followed by an improper operation, do you get a proper operation or an improper operation?
Proper
In a multiplication table for a point group, how often does each operation appear in each row/column?
Once in each row and once in each column
What term is used to describe a chiral molecule?
Dissymmetric
What term is used to describe a chiral carbon?
Asymmetric
What is axial chirality?
Where rotating in the C2 axis produces a non-superimposable image of the original molecule
What does the eigen value ‘a’ tell us about the transformation of an orbital?
The transformation that happens on application of a symmetry operation. For example, if you apply C2 to a 2px orbital assuming the rotation is in the z axis, the value of a = -1 as the positive and negative signs of the orbital are switched
What is an irreducible representation?
A collection of eigen values that can be added to another irreducible representation to form a reducible representation
What does the number of operations of a point group tell us about the order of the point group?
The sum of the number of operations tells us the order of the point group
What does the label A mean for an irreducible representation?
Singly degenerate and totally symmetric about the principle axis
What does the label B mean for an irreducible representation?
Singly degenerate and anti-symmetric (sign changes when rotated) about principle axis
What does the label E mean for an irreducible representation?
Doubly degenerate
What does the label T mean for an irreducible representation?
Triply degenerate
What does the label subscript 1 mean for an irreducible representation?
Symmetric under C2 (perpendicular to the principle axis, if C2 is the principle axis then effectively there is no C2 axis), or σv if there is no C2 axis
What does the label subscript 2 mean for an irreducible representation?
Anti-symmetric under C2 (perpendicular to the principle axis, if C2 is the principle axis then effectively there is no C2 axis), or σv if there is no C2 axis
What does the label superscript ‘ mean for an irreducible representation?
Means symmetric under σh
What does the label superscript ‘’ mean for an irreducible representation?
Means anti-symmetric under σh
What does the label subscript g mean for an irreducible representation?
Symmetric under inversion
What does the label subscript u mean for an irreducible representation?
Anti-symmetric under inversion
When two properties of a molecule can be interconverted by a symmetry operation, what must we treat them as?
Degenerate, for example: if a px orbital can be converted into a py orbital by a C4 rotation then we consider them degenerate
If two orbitals are degenerate, what can we use to describe the transformation?
Matrices
What does the character in the irreducible representation come from?
The trace of the transformation matrix, for example if the trace of the transformation of an orbital when a C4 operation is applied is 0 then the character of the C4 part of the irreducible representation will be 0
What is the general clockwise rotation matrix?
(X1) = (Cosθ Sinθ) (X1) = (X1') (Y1) = (-Sinθ Cosθ) (Y1) = (Y1')
What is the trace of a matrix?
The sum of the leading diagonal, top left and bottom left in a 2x2 matrix
When there is an xyz axis with z pointing up, y pointing away from you and x pointing to the right, which way does a pz orbital point?
Upwards in the plane of the paper
When there is an xyz axis with z pointing up, y pointing away from you and x pointing to the right, which way does a px orbital point?
Sideways in the plane of the paper
When there is an xyz axis with z pointing up, y pointing away from you and x pointing to the right, which way does a py orbital point?
In and out of the plane of the paper
If you rotate a py orbital 90º so that it is now a px orbital, what is the eigen value to describe this rotation?
0 as the orbital is no longer a py orbital
How do you determine the reducible representation of a point group?
Look at the 1s orbitals of the molecule, when you apply an operation to it, how many 1s orbitals are left unchanged. That gives you the character of the reducible representation for that operation
What is the equation for determining what the reducible representation reduces to?
- n(Γ) = 1/h x Sum of all classes (χR • χi • N)
- You repeat this equation for each of the irreducible representations, if you get a value that isn’t 0 then you know the reducible representation can be reduced to that particular irreducible representation
When forming an orbital overlap diagram, which orbitals can overlap?
Orbitals with the same symmetry, for example if one element has a1, b1 and b2 symmetry but one element only has a1 and b1 symmetry, the b2 orbital will be a non-bonding orbital
What is the equation for calculating bond order?
(Electrons in bonding orbitals - electrons in anti-bonding orbitals) / 2
How do you calculate a SALC?
- Work out which position the atom at σ1 moves to for each operation in a point group
- Multiply this by each irreducible representation and create a σ1(irreducible representation column)
- Total up the number of each σ and factorise to leave (for example): 4(σ1 + σ2 + σ3)
- Ignore the number in front to normalise and use the equation 1/√(a^2 +b^2 +c^3) where a, b and c are the coefficients in front of σ1, σ2 and σ3 respectively
- The equation gives us the normalisation constant, you write this where the 4 was in front of (σ1 + σ2 + σ3)
Why do you need to normalise the number of (σ1 + σ2 + σ3)?
It allows for contribution of an orbital to more than one SALC, the total contribution of an orbital across all SALCs must add up to 1
How is the second SALC obtained for doubly degenerate irreducibles?
The same process as obtaining the first SALC, however instead of using σ1, you use σ2 and then you repeat with σ3 and then take your values for σ3 away from σ2 and then you can normalise your answer
