1 CHEM2210 Structure Elucidation in Chemistry Prof Peter Gill Part 1. Molecular Symmetry Symmetry Operations Point Groups Character Tables & Symmetry Species Orbital Symmetry Part 2. Molecular Orbital Theory General Rules Diatomic Molecules Linear Polyatomic Molecules Non-Linear Polyatomic Molecules Part 3. Molecular Vibrations and Electronic Transitions Reducible and Irreducible Representations Reduction Formula Application to Molecular Vibrations Application to Electronic Transitions
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1 CHEM2210
Structure Elucidation in Chemistry Prof Peter Gill
Part 1. Molecular Symmetry Symmetry Operations Point Groups Character Tables & Symmetry Species Orbital Symmetry
Part 2. Molecular Orbital Theory General Rules Diatomic Molecules Linear Polyatomic Molecules Non-Linear Polyatomic Molecules
Part 3. Molecular Vibrations and Electronic Transitions Reducible and Irreducible Representations Reduction Formula Application to Molecular Vibrations Application to Electronic Transitions
2
A molecule with symmetry has two or more orientations in space that are indistinguishable from one another.
Molecular symmetry is measured by the number and type of symmetry operations. Five different types of symmetry operations are important to us:
Symmetry Operations
3
1. Identity (E). Trivial operation that does nothing.
2. Proper Rotation (Cn). Rotation by 360º/n.
Symmetry Operations
4 Symmetry Operations
Two-fold C2 (180º), three-fold C3 (120º), four-fold C4 (90º), five-fold C5 (72º), six-fold C6 (60º), etc. proper rotations are possible.
The highest-fold rotation axis is known as the principal rotation axis of the molecule and is generally designated as the molecular z axis.
3. Reflection (σ). A reflection in a mirror plane that maps one half of the molecule onto the other half.
5 Symmetry Operations
There are three types of mirror planes:
Horizontal mirror plane (σh) is a mirror plane which is perpendicular to the principal rotation axis.
6 Symmetry Operations
Vertical Mirror Plane (σv) is a mirror plane which contains the principal rotation axis.
7 Symmetry Operations
Dihedral Mirror Plane (σd) is a mirror plane which contains the principal rotation axis and also bisects the angle between two C2 axes or two σv mirror planes.
8 Symmetry Operations
4. Inversion ( i ). Turning a molecule “inside out” by moving each atom through a point (the centre of inversion) in a straight line to an identical atom on the other side at the same distance.
9 Symmetry Operations
5. Improper Rotation (Sn). A proper rotation by 360o/n, followed by reflection through a plane perpendicular to the rotation axis.
Examples: A2′ a non-degenerate symmetry species, which is • symmetric with respect to the principal axis • antisymmetric with respect to σv • symmetric with respect to σh
B1u a non-degenerate symmetry species, which is
• antisymmetric with respect to principal axis
• symmetric with respect to σv
• antisymmetric with respect to inversion
33 Character Tables
Linear molecules with no inversion centre (e.g. CO)
• All can be found in the “Symmetry Tables” file on Wattle
• In an exam, any required character tables will be provided.
36 Orbital Symmetry
The rules for determining the character χ associated with a symmetry operation on an orbital are: 1. If the operation moves the centre of the orbital,
then χ = 0.
37 Orbital Symmetry
2. If the operation swaps the orbital with another orbital on the same atom, then χ = 0.
38 Orbital Symmetry
3. If the operation leaves the orbital unchanged, then χ = +1.
39 Orbital Symmetry
4. If the operation changes the sign of the orbital, then χ is -1.
40 Orbital Symmetry
What are the characters of the oxygen 2p orbitals in the C2v water molecule?
41 Orbital Symmetry
χ = +1
χ = -1
χ = +1
χ = -1
42 Orbital Symmetry
χ = +1
χ = -1
χ = -1
χ = +1
43 Orbital Symmetry
χ = +1
χ = +1
χ = +1
χ = +1
44 Orbital Symmetry
C2v E C2 σv(xz) σv'(yz) Species
Γ(2px) +1 -1 +1 -1 B1
Γ(2py) +1 -1 -1 +1 B2
Γ(2pz) +1 +1 +1 +1 A1
C2v E C2(z) σv(xz) σv'(yz)
A1 +1 +1 +1 +1 z x2, y2, z2 A2 +1 +1 -1 -1 Rz xy B1 +1 -1 +1 -1 x, Ry xz B2 +1 -1 -1 +1 y, Rx yz
45 Orbital Symmetry
An orbital belonging to a degenerate symmetry species (E or T) will not transform correctly in isolation. Degenerate orbitals must be considered collectively when generating the characters under a symmetry operation. Example: the 4p orbitals on bromine in BrF5 which has C4v symmetry:
Individually, the px and py orbitals on the Br atom do not transform as a symmetry species of the C4v point group. However, together, they transform as the doubly degenerate E species.
In generating the above characters, the same σv and σd reflection planes are used for all three p orbitals.
49 Orbital Symmetry
The transformation of s, p and d orbitals on the central atom in a molecule can be determined directly from the point group character tables.
An s orbital always transforms as the totally symmetric representation (the first listed symmetry species with all χ = +1). The p and d orbitals transform the same as the following basis functions:
Orbital: px py pz dxy dxz dyz dx²-y² dz² Basis fn: x y z xy xz yz x2-y2 z2
In some point groups e.g. C2v, the dx²-y² orbital transforms as two separate functions x2 and y2.