Group Theory II

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Group Theory II
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Group Theory II

• Today
• Repetition
• Block matrices
• Character tables
• The great and little orthogonality theorems
• Irreducible representations
• Basis functions and Mulliken symbols
• How to find the symmetry species
• Projection operator
• Applications

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Repetition

• We already know
• Symmetry operations obey the laws of group
  theory.

• Great, we can use the mathematics of group
  theory.

• A symmetry operation can be represented by a
  matrix operating on a base set describing the
  molecule.

• Different basis sets can be choosen, they are
  connected by similarity transformations.

• For different basis sets the matrices describing
  the symmetry operations look different. However,
  their character (trace) is the same!

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Repetition

• We already know
• Matrix representations of symmetry operations
  can often be reduced into block matrices.
  Similarity transformations may help to reduce
  representations further. The goal is to find the
  irreducible representation, the only
  representation that can not be reduced further.

• The same type of operations (rotations,
  reflections etc) belong to the same class.
  Formally R and R belong to the same class if
  there is a symmetry operation S such that
  RS-1RS. Symmetry operations of the same class
  will always have the same character.

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Block Matrices
Block matrices are good
C
C
C
AAA BBB CCC
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Block Matrices
If a matrix representing a symmetry operation is
transformed into block diagonal form then each
little block is also a representation of the
operation since they obey the same multiplication
laws.
When a matrix can not be reduced further we have
reached the irreducible representation. The
number of reducible representations of symmetry
operations is infinite but there is a small
finite number of irreducible representations.
The number of irreducible representations is
always equal to the number of classes of the
symmetry point group.
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Group Theory II
Reducing big matrices to block diagonal form is
always possible but not easy. Fortunately we do
not have to do this ourselves.
As stated before all representations of a certain
symmetry operation have the same character and we
will work with them rather than the matrices
themselves. The characters of different
irreducible representations of point groups are
found in character tables. Character tables can
easily be found in textbooks.
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Character Tables
The C3v character table
Symmetry operations
The order h is 6 There are 3 classes
Irreducible representations
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Character Tables
Operations belonging to the same class will have
the same character so we can write
Classes
Irreducible representations (symmetry species)
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The Great Orthogonality Theorem
Consider a group of order h, and let D(l)(R) be
the representative of the operation R in a
dl-dimensional irreducible representation of
symmetry species G(l) of the group. Then

Read more about it
in section 5.10.
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The Little Orthogonality Theorem


Heres a smaller one, where c(l)(R) is the
character of the operation (R). Or even more
simple if the number of symmetry operations in a
class c is g(c). Then since all operations
belonging to the same class have the same
character.
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character Tables
There is a number of useful properties of
character tables

• The sum of the squares of the dimensionality of
  all the irreducible representations is equal to
  the order of the group

• The sum of the squares of the absolute values of
  characters of any irreducible representation is
  equal to the order of the group.

• The sum of the products of the corresponding
  characters of any two different irreducible
  representations of the same group is zero.

• The characters of all matrices belonging to the
  operations in the same class are identical in a
  given irreducible representation.

• The number of irreducible representations in a
  group is equal to the number of classes of that
  group.

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Irreducible representations
Each irreducible representation of a group has a
label called a symmetry species, generally noted
G. When the type of irreducible representation is
determined it is assigned a Mulliken
symbol One-dimensional irreducible
representations are called A or
B. Two-dimensional irreducible representations
are called E. Three-dimensional irreducible
representations are called T (F). The basis for
an irreducible representation is said to span the
irreducible representation. Dont mistake the
operation E for the Mulliken symbol E!
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Irreducible representations
The difference between A and B is that the
character for a rotation Cn is always 1 for A and
-1 for B.
The subscripts 1, 2, 3 etc. are arbitrary labels.
Subscripts g and u stands for gerade and
ungerade, meaning symmetric or antisymmetric with
respect to inversion.
Superscripts and denotes symmetry or
antisymmetry with respect to reflection through a
horizontal mirror plane.
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character Tables
Example The complete C4v character table
These are basis functions for the irreducible
representations. They have the same symmetry
properties as the atomic orbitals with the same
names.
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character Tables
Example The complete C4v character table
A1 transforms like z. E does nothing, C4 rotates
90o about the z-axis, C2 rotates 180o about the
z-axis, sv reflects in vertical plane and sd in a
diagonal plane.
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character Tables
A2 transforms like a rotation around z.
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Reducible to Irreducible representation
Given a general set of basis functions describing
a molecule, how do we find the symmetry species
of the irreducible representations they span?
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Reducible to Irreducible representation
If we have an interesting molecule there is often
a natural choice of basis. It could be cartesian
coordinates or something more clever.
From the basis we can construct the matrix
representations of the symmetry operations of the
point group of the molecule and calculate the
characters of the representations.
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Reducible to Irreducible representation
How do we find the irreducible representation? Let
s use an old example from two weeks ago
C3v in the basis (Sn, S1, S2, S3)
To find the characters of the symmetry operations
we look at how many basis elements fall onto
themselves (or their negative self) after the
symmetry operation.
C3 c1
sv c2
E c4
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Reducible to Irreducible representation
So C3v in the basis (Sn, S1, S2, S3) will have
the following characters for the different
symmetry operations.
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Reducible to Irreducible representation
So C3v in the basis (Sn, S1, S2, S3) will have
the following characters for the different
symmetry operations.
Lets add the character table of the irreducible
representation
By inspection we find Gred2A1E
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Reducible to Irreducible representation
The decomposition of any reducible representation
into irreducible ones is uniqe, so if you find
combination that works it is right.
If decomposition by inspection does not work we
have to use results from the great and little
orthogonality theorems (unless we have an
infinite group).
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Reducible to Irreducible representation
From LOT we can derive the expression (see
section 5.10) where ai is the number of times
the irreducible representation Gi appears in
Gred, h the order of the group, l an operation of
the group, g(c) the number of symmetry operations
in the class of l, cred the character of the
operation l in the reducible representation and
ci the character of l in the irreducible
representation.
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Reducible to Irreducible representation
Lets go back to our example again.
So once again we find Gred2A1E
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Projection Operator
Symmetry-adapted bases The projection operator
takes non-symmetry-adapted basis of a
representation and and projects it along new
directions so that it belongs to a specific
irreducible representation of the group.

where Pl is the projection operator of the
irreducible representation l, c(l) is the
character of the operation R for the
representation l and R means application of R to
our original basis component.

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Applications?
Can all of this actually be useful? Yes, in many
areas for example when studying electronic
structure of atoms and molecules, chemical
reactions, crystallography, string theory
(Lie-algebra) etc
Lets look at one simple example concering
molecular vibrations. Martin Jönsson will tell
you a lot more in a couple of weeks.
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Molecular Vibrations
Water Molecular vibrations can always be
decomposed into quite simple components called
normal modes.
Water has 9 normal modes of which 3 are
translational, 3 are rotational and 3 are the
actual vibrations.
Each normal mode forms a basis for an irreducible
representation of the molecule.
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Molecular Vibrations
First find a basis for the molecule. Lets take
the cartesian coordinates for each atom.
Water belongs to the C2v group which contains
the operations E, C2, sv(xz) and sv(yz).
The representation becomes E C2
sv(xz) sv(yz) Gred
9
-1
1
3
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Molecular Vibrations
Character table for C2v.
Now reduce Gred to a sum of irreducible
representations. Use inspection or the formula.
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Molecular Vibrations
The representation reduces to Gred3A1A22B13B2
Gtrans A1B1B2
GrotA2B1B2
Modes left for vibrations
Gvib2A1B2
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Molecular Vibrations
Modes with translational symmetry will be
infrared active while modes with x2, y2 or z2
symmetry are Raman active.
Thus water which has the vibrational modes
Gvib2A1B2 will be both IR and Raman active.
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Integrals
A last example Integrals of product functions
often appear in for example quantum mechanics and
symmetry analysis can be helpful with them to.
An integral will be non-zero only if the
integrand belongs to the totally symmetric
irreducible representation of the molecular point
group.
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Summary

• Molecules (and their electronic orbitals,
  vibrations etc) are invariant under certain
  symmetry operations.

• The symmetry operations can be described by a
  representation determined by the basis we choose
  to describe the molecule.

• The representation can be broken up into its
  symmetry species (irreducible representations).

• In character tables we find information about
  the symmetry properties of the irreducible
  representations.

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More (and better) reading
The group theory chapter in Atkins is not very
good (in my opinion). More understandable
descriptions can be found in Harris and
Bertolucci, Symmetry and spectroscopy Hargittai
and Hargittai, Symmetry through the eyes of a
chemist
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