What are proper axes of rotation?

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 C_{1} axis which is the identity (E). By convention, rotations are clockwise.

What is a C_{2} rotation?

For a C_{2} axis, rotating by 180° leaves the object unchanged. There is only 1 unique C_{2} operation (C_{2}) as C_{2}^{2} = E.

What is a C_{3} rotation?

For a C_{3} axis, rotating by 120° leaves the object unchanged. There are 2 unique C_{3} rotations (C_{3} and C_{3}^{2}) as C_{3}^{3} = E.

What is a C_{4} rotation?

For a C_{4} axis, rotating by 90° leaves the object unchanged. There are 2 unique C_{4} rotations (C_{4} and C_{4}^{3}) as C_{4}^{4} = E and C_{4}^{2} = C_{2}.

What is a C_{5} rotation?

For a C_{5} axis, rotating by 72° leaves the object unchanged. All of the rotations are unique.

What is a C_{6} rotation?

For a C_{6} axis, rotating by 60° leaves the object unchanged. There are 2 unique C_{6} rotations (C_{6} and C_{6}^{5}) as C_{6}^{2}^{ }= C_{3} and C_{6}^{3} = C_{2} and C_{6}^{4} = C_{3}^{2} and C_{6}^{6} = E.

What is a symmetry element?

It is the axis of rotation.

What is a symmetry operation?

It is what is done to the object.

What is a principal axis?

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

What are improper axes of rotation?

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.

What is a reflection?

A reflection (S_{1}) is a rotation of C_{1} 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.

What is an inversion?

An inversion (S_{2}) is a rotation of C_{2} followed by a reflection. Travel from 1 atom to the equivalent position on the other side of the centre of inversion.

What are the S_{3} operations?

There are 2 unique S_{3} operations (S_{3} and S_{3}^{5}) as S_{3}^{2} = C_{3}^{2} and S_{3}^{3} = σ_{h} and S_{3}^{4} = C_{3}.

What are the S_{4} operations?

There are 2 unique S_{4} operations (S_{4} and S_{4}^{3}) as S_{4}^{2} = C_{2}.

What are inversion centres?

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.

What is a point group?

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.

What are chiral molecules?

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.

What are the symbols used for irreducibles?

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 C_{2}' or σ_{v}

Subscript 2 = anti-symmetric under C_{2}' 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

How are matrices used in group theory?

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.

What are reducible representations?

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

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

a_{1} is the symmetric representation:

1_{s} (H_{A}) + 1_{s} (H_{B}) so the AOs are in-phase

b_{1} is the lower symmetry representation:

1_{s} (H_{A}) - 1_{s} (H_{B}) so the AOs are out-of-phase

How does mixing in H_{2}O affect the MO diagram?

E (2a_{1}) does not equal E (1b_{2}).

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

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- Write down all symmetry operations of the group
- Label the orbitals with σ or π
- Apply the symmetry operations to σ
_{1} or π_{1} and record the result to give row 1
- Multiply row 1 by the irreducible to give row 2
- Add up row 2
- Normalise (ignore the number outside the bracket)
- aσ
_{1} + bσ_{2} + cσ_{3} = (1/(a^{2}+b^{2}+c^{2})^{1/2})(σ_{1}+σ_{2}+σ_{3})
- If they are positive, they are in-phase
- If they are negative, they are out-of-phase

_{1}or π_{1}and record the result to give row 1_{1}+ bσ_{2}+ cσ_{3}= (1/(a^{2}+b^{2}+c^{2})^{1/2})(σ_{1}+σ_{2}+σ_{3})How do you find a second explicit SALC?

Start with σ_{2} then σ_{3}, then subtract the second result from the first.