Chapter 4 Symmetry and Group
Theory
4.1 Symmetry Elements and
Operations
• Symmetry element – a point, line, or plane
about which a symmetry operation is carried
out.
• Symmetry operation – A geometry
transformation that leaves object in an
indistinguishable configuration
• This chapter covers point groups symmetry –
at least one point is not moved by the
symmetry operation.
Examples of Symmetry
Five Point Group Symmetry Operations
• E – Identity – no movement – all objects have this
symmetry
• Cn – Proper rotation – rotation by 360°/n about an axis.
Always counter clockwise when viewed down the axis.
• s – a plane of symmetry. Sometimes called a mirror
plane
• i – inversion center. There are equivalent points in any
direction in opposite directions equidistant from the
center.
• Sn – Improper rotation – Rotation by 360°/n about an
axis, followed by a reflection perpendicular to the axis.
Rotations
Six-Fold Rotation combinations
Bilateral Symmetry – a Plane of
Symmetry in a Human
Multiple Planes of Symmetry in a
Pencil
Examples of a Inversion Center,
Staggered Ethane, and Lack of an
Inversion Center in Methane
Figures with an Inversion Center
Figures Lacking an Inversion Center
Improper Rotation in Methane
An Excellent Interactive Symmetry Site
• Symmetry at Otterbien
4.2 Point Groups
First Step
• Assign a molecule to a point group
• A point group has a set of symmetry
operations for the molecule.
• A flow chart (next page) is used to assign the
point group.
Molecule whose Point Groups are
Assigned in the Text
4.2.1 Point Groups of Low and High
Symmetry
We will go through the flow chart step
by step
• MAW tips
• If there is a unique atom (only one atom of an
element or one atom connected a specific way),
all the symmetry elements must run through
that atom
• If there are more than one atom of an element
look for operations relating that set of atoms
• Sometime atoms of the same element are not
related by symmetry operations
First look for low symmetry or high
symmetry point groups
Next Look
for
Groups of
High
Symmetry
Including
Linear
Molecules
4.2.2 Other Groups
• For Other Groups There Must be Some
Detailed Examination of the Molecules
• First Find the Highest Order RotationAxis
• Examples on Next Slide
The Order of the Rotation Axis will be
a Subscript in the Group Name
• Dn Dnh Dnv Cn Cnh Cnv Sn
• D groups – dihedral groups contain C2 axes
perpendicular to the highest order axes.
• Note for D2 groups the highest order axis is
arbitrary. There are 3 mutually perpendicular
axes.
Perpendicular C2 axes?
Yes  a D group, No a C or S group
Look for a Horizontal Mirror Plane that
is Perpendiciular to the Highest Order
RotationAxes  Dnh or Cnh
Next Look for A Mirror Plane That
Runs along the Highest Order Rotation
Axis  Dnd or Cnv
• Any molecules not assigned will be in pure
rotation groups Dn or Cn or the very rare Sn if
there is additional Sn axes.
• These assignments are on the next slide
Additional Point Group Assignments
1 of 3 slides for Table 4.4
A Nice Summary of Non-High or NonLow Symmetry Groups
Additional High Symmetry Groups
4.3 Properties and Representations of
Groups
Symmetry Operations for NH3
Group
• Needs set of objects (a,b,c,d,e,…) and a
combining operation x
• Has the following properties
• Identity a X E = E X a = a
• Inverse a X a-1 = E
• Closure a X b = a member of the group
• Associative property (a X b) X c = a X (b X c)
• Note Commutative Property is not required a X b = b X a is not
necessarily true
4.3.1 Matrices
Symmetry operations in water may be repersented by matrices
4.3.2 Representations of Point Groups
Review of matrix multiplication
•
•
•
•
Matrix i rows and j columns
Left multiply a i x k by a k x j = i x j resultant
Example on board
Symmetry operations as matrices
Character sum of diagonal
elements of a matrix
Example water symmetry matrices
Irreducible Reresentations in water
from block diagonalization
4.3.3 Character Tables
Character table for water
Matrices for NH3 in C3v
C3v character table
Additional Features of Character
table on page 107
•Why are there two C3 axes?
• What do the primes mean? C2’C2”
• What is the difference between σh σv and σd
• What do Rx Ry Rz mean?
• How do the orbitals transform?
• Match symmetry operations with molecule
• What does symmetric ( = 1) vs
antisymmetric ( = -1) mean?
•What do the labels A1 B2 mean?
• What do gerade (g) vs ungerade (u) mean
• Dimensions of the irreducible representations
Transformations of orbitals
4.4 Examples of Applications of
Symmetry
4.4.1 Chirality
• Chiral objects are not superimposable on their
mirror images
Each Enantiomer Rotates Polarized
light in Opposite Directions
Criteria for chirality
•The point group the molecule is in must have
only proper rotations
•C1 = E, C2, C3, C4, etc.
•If the point group has improper rotations, the
molecule can not be chiral
•S1 = σ, S2 = i, S3, S4, S5, S6, etc.
•In organic you used a partial definition,
looking for σ or i. Which works in
99.9999999% of cases.
• Point group S4 has no σ or i but is not chiral.
A common chiral structure in
transition metal chemistry
• Three rings are like a propeller. The point
group is D3 . Molecule has only E, 2C3 a 3C2
• Look for a D3 molecule on the Otterbein site
4.4.2 Molecular Vibrations
• Molecules have 3N degrees of freedom. N =
number of atoms. N x 3-Dimensions of space.
• 6 of these degrees are translations (x,y, and z)
and Rotations (Rx, Ry, and Rz). Linear
molecules have one less rotation.
• So there are 3N-6 vibration modes possible.
(3N-5 for a linear molecule)
Coordinate system for water
Vibrational analysis
• Create a coordinate system
• Carry out the symmetry operations for the molecule to create
a reducible representation, Γ .
• If a vector is the same : it contributes +1 to the character for
that operation
• If a vector goes to the negative of itself: it contributes -1 to
the chaacter for that operation.
• If the atom moves, It contribute zero to the character for that
operation.
• Reduce the reducible representation (Γ) to a sum of
irreducible representations of the point group. This is the
symmetry of all degrees of freedom.
Water generation of Γ on the board
Math interlude: Formula to reduce
reducible representations
• A reducible representation may be expresses as a sum on
irreducible representation of the point group in only one way.
• # an irred. rep. occurs in the sum =
• 1/order[Σ (# symm. ops.)(irred)(red) ]
• Symm. Ops. = 3 of symmetry operation in the class
• irred = Character of the irreducible representation from the
chartacter table
• red = Character of the reducible representation ,Γ
• The sum is over all the classes of the point group and must be done
for each irreducible representation.
Examples
Water
•
•
•
•
•
Reduction
#A1 = 1/4[(9)(1) +(-1)(1)+(3)(1)+(1)(1)] = 12/4 = 3
#A2 = 1/4[(9)(1) +(-1)(1)+(3)(-1)+(1)(-1)] = 4/4 = 1
#B1 = 1/4[(9)(1) +(-1)(-1)+(3)(1)+(1)(-1)] = 12/4 = 3
#B2 = 1/4[(9)(1) +(-1)(-1)+(3)(-1)+(1)(1)] = 8/4 = 2
Proof
•
•
•
•
•
•
4 A1
1A2
3B1
2B2
Γ
E
3
1
3
2
9
C2(z)
3
1
-3
-2
-1
σ(xz)
3
-1
3
-2
3
σ(xz)
3
-1
-3
2
1
Subtraction of Translations and
Vibrations
Irreducible representations for x,y,z and Rx, Ry, Rz are found in the
character tables
Vibrational modes of water
Demonstration of symmetry on the board and illustration of
vibrations on Models 360
A more complicated molecule XeF4
3(5) – 6 = 9 vibrations
Reducible Representation for XeF4
Reduction is cumbersome, skip to
the Results
Translations of XeF4
Rotations in XeF4
Selected Vibrations in XeF4
Vibrational analysis of NH3 in C3v
-
Reduces to
3A1 + A2
A1(x)
A2(Rz)
2A1
A1 symmetric stretch,
(z) (x2+y2, z2)
E asymmetric stretch,
(x,y) ((x2-y2, xy) (xz, yz))
+
symmetric bend
asymmetric bend
4E
E(x,y)
E(Rx, Ry)
2E
IR and Raman active
IR and Raman active
Selected Vibration Modes
D2h
C2v
Reducible Representation cis
isomer
CO stretching modes cis isomer
Note both bands are IR and Raman active.
Reducible Representation for the
trans isomer
Note: B3u is IR active only, and Ag is Raman active only.
Octahedral M(CO)3 facial complex
C3v
Reduces to A1 + E
A1  z
one band IR active and Raman
E  (x,y) one band IR active and Raman
XeO4 tetrahedral
Raman only
IR and Raman