Reducible representation: How the basis objects transform. By adding and subtracting the basis objects the reducible representation can be reduced to a combination irreducible representations.

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Lecture 7
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Basis of a Representation A set of objects
capable of demonstrating the effects of the all
the symmetry operations in a Point Group. A set
of functions, atomic orbitals on a central atom
or ligands, or common objects.
Representation How the operations affect the
basis objects by transforming them into
themselves or each other by the operations. The
representation is an array showing how the basis
objects transform. Ususally we do not need the
full matrices but only the trace of the matrices.
These values are called the characters. The
collected set of character represntations is
called the character table.

• Irreducible representations A representation
  using a minimal set of basis objects. A single
  atomic orbital (2s, 2pz), a linear combination of
  atomic orbitals or hybrids (h1 h2), etc. An
  irreducible representation may be minimally based
  on
• A single object, one dimensional, (A or B)
• Two objects, two dimensional, (E)
• Three objects, three dimensional, (T)

Reducible representation How the basis objects
transform. By adding and subtracting the basis
objects the reducible representation can be
reduced to a combination irreducible
representations. of irreducible
representations of classes of symmetry
operations
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Effect of the 4 operations in the point group
C2v on a translation in the x direction. The
translation is simply multiplied by 1 or -1. It
forms a 1 dimensional basis to show what the
operators do to an object.
Operation E C2 sv sv
Transformation 1 -1 1 -1
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Character Table
Verify this character. It represents how a
function that behaves as x, Ry, or xz behaves for
C2.
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Another point group, C3v.
x, y, z Symmetry of translations (p orbitals)
Classes of operations
Rx, Ry, Rz rotations
dxy, dxz, dyz, as xy, xz, yz dx2- y2 behaves as
x2 y2 dz2 behaves as 2z2 - (x2 y2) px, py, pz
behave as x, y, z s behaves as x2 y2 z2
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Symmetry of Atomic Orbitals
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Naming of Irreducible representations

• One dimensional (non degenerate) representations
  are designated A or B. (A basis object is only
  changed into itself or the negative of itself by
  the symmetry operations)
• Two-dimensional (doubly degenerate) are
  designated E. (Two basis object are required to
  repesent the effect of the operations for an E
  representation. In planar PtCl42- the px and py
  orbitals of the Pt, an E representation, are
  transformed into each other by the C4 rotation,
  for instance.)
• Three-dimensional (triply degenerate) are
  designated T. (Three objects are interconverted
  by the symmetry operations for the T
  representations. In tetrahedral methane, Td, all
  three p orbitals are symmetry equivalent and
  interchanged by symmetry operations)
• Any 1-D representation symmetric with respect to
  Cn is designated A antisymmétric ones are
  designated B
• Subscripts 1 or 2 (applied to A or B refer) to
  symmetric and antisymmetric representations with
  respect to C2 ? Cn or (if no C2) to ? sv
  respectively
• Superscripts and indicate symmetric and
  antisymmetric behavior respectively with respect
  to sh.
• In groups having a center of inversion,
  subscripts g (gerade) and u (ungerade) indicate
  symmetric and antisymmetric representations with
  respect to i

But note that while this rationalizes the naming,
the behavior with respect to each operation is
provided in the character table.
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Character Tables

• You have been exposed to symmetry considerations
  for diatomic molecules s or p bonding.
• Characters indicate the behavior of an orbital or
  group of orbitals under the corresponding
  operations (1 orbital does not change -1
  orbital changes sign anything else more
  complex change)
• Characters in the E column indicate the dimension
  of the irreducible representation (of degenerate
  orbitals having same energy)
• Irrecible representations are represented by
  CAPITAL LETTERS (A, B, E, T,...) whereas orbitals
  of that symmetry behavior are represented in
  lowercase (a, b, e, t,...)
• The identity of orbitals which a row represents
  is found at the extreme right of the row
• Pairs in brackets refer to groups of degenerate
  orbitals and, in those cases, the characters
  refer to the properties of the set

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Definition of a Group

• A group is a set, G, together with a binary
  operation, , such that the product of any two
  members of the group is a member of the group,
  usually denoted by ab, such that the following
  properties are satisfied
• (Associativity) (ab)c a(bc) for all a, b, c
  belonging to G.
• (Identity) There exists e belonging to G, such
  that eg g ge for all g belonging to G.
• (Inverse) For each g belonging to G, there exists
  the inverse of g, g-1, such that g-1g gg-1
  e.
• If commutativity ( ab ba) for all a, b
  belonging to G, then G is called an Abelian
  group.

The symmetry operations of a Point Group comprise
a group.
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Example
Consider the set of all integers and the
operation of addition ( )
Is this set of objects (all integers) associative
under the operation? (ab)c a(bc)
Yes, (3 4) 5 3 (45)

• Is there an identity element, e? ae a

Yes, 0
For each element is there an inverse element,
a-1? a-1 a e
Yes, 4 (-4) 0
We have a group.
Abelian? Is commutativity satisfied for each
element? a b b a
Yes. 3 (-5) (-5) 3
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As applied to our symmetry operators.

• For the C3v point group

What is the inverse of each operator? A A-1
E
E C3(120) C3(240)
sv (1) sv (2) sv (3)
E C3(240) C3(120)
sv (1) sv (2) sv (3)
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Examine the matrix representation of the elements
of the C2v point group
E
C2
sv(yz)
sv(xz)
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Multiplying two matrices (a reminder)
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C2
sv(xz)
sv(yz)
E
What is the inverse of C2?
C2

What is the inverse of sv?
sv

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What of the products of operations?
C2
sv(xz)
sv(yz)
E
C2
E C2 ?

sv C2 ?
sv

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Classes
Two members, c1 and c2, of a group belong to the
same class if there is a member, g, of the group
such that gc1g-1 c2
Consider PtCl4

C2(x)
C2(y)
So these operations belong to the same class?
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C2(y)
C2(x)

C4
Since C4 moves C2(x) on top of C2(y) it is an
obvious choice for g
C43
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C2(x)
C2(y)
C4
C43

Belong to same class! How about the other two C2
elements?

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Properties of Characters of Irreducible
Representations in Point Groups

• Total number of symmetry operations in the group
  is called the order of the group (h). For C3v,
  for example, it is 6.

1 2 3 6

• Symmetry operations are arranged in classes.
  Operations in a class are grouped together as
  they have identical characters. Elements in a
  class are related.

This column represents three symmetry operations
having identical characters.
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Properties of Characters of Irreducible
Representations in Point Groups - 2

• The number of irreducible reps equals the number
  of classes. The character table is square.

1 2 3 6 h
3 by 3
1 1 22 6 h
The sum of the squares of the dimensions of the
each irreducible rep equals the order of the
group, h.
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Properties of Characters of Irreducible
Representations in Point Groups - 3
For any irreducible rep the squares of the
characters summed over the symmetry operations
equals the order of the group, h.
A1 12 2 (12) 3 (12) 6 of sym
operations 123
A2 12 2 (12) 3((-1)2) 6
E 22 2 (-1)2 3 (0)2 6
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Properties of Characters of Irreducible
Representations in Point Groups - 4
Irreducible reps are orthogonal. The sum over the
symmetry operations of the products of the
characters for two different irreducible reps is
zero.
For A1 and E 1 2 2 (1 (-1)) 3 (1 0)
0
Note that for any single irreducible rep the sum
is h, the order of the group.
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Properties of Characters of Irreducible
Representations in Point Groups - 5
Each group has a totally symmetric irreducible
rep having all characters equal to 1
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Reduction of a Reducible Representation. Given a
Reducible Rep how do we find what Irreducible
reps it contains?
Irreducible reps may be regarded as orthogonal
vectors. The magnitude of the vector is h-1/2 Any
representation may be regarded as a vector which
is a linear combination of the irreducible
representations.
Reducible Rep S (ai IrreducibleRepi) The
Irreducible reps are orthogonal. Hence for the
reducible rep and a particular irreducible rep
i S(character of Reducible Rep)(character of
Irreducible Repi) ai h Or ai S(character
of Reducible Rep)(character of Irreducible Repi)
/ h
Sym ops
Sym ops
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Reducible Representations in Cs E and sh
Use the two sp hybrids as the basis of a
representation
h1
h2
sh operation.
E operation.
h1 becomes h1 h2 becomes h2.
h1 becomes h2 h2 becomes h1.


The reflection operation interchanges the two
hybrids.
The hybrids are unaffected by the E operation.
Proceed using the trace of the matrix
representation.
0 0 0
1 1 2
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Lets observe one helpful thing here. Only the
objects (hybrids) that remain themselves, appear
on the diagonal of the transformation of the
symmetry operation, contribute to the trace.
They commonly contribute 1 or -1 to the trace
depending whether or not they are multiplied by
-1.
h1 h2 do not become themselves, interchange
h1 , h1 become themselves


The reflection operation interchanges the two
hybrids.
The hybrids are unaffected by the E operation.
Proceed using the trace of the matrix
representation.
0 0 0
1 1 2
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The Irreducible Representations for Cs.
Cs E sh
A A 1 1 -1 x, y,Rz z, Rx,Ry x2,y2,z2,xy yz, xz
The reducible representation derived from the two
hybrids can be attached to the table.
G 2 0 (h1, h2)
Note that G A A
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The Irreducible Representations for Cs.
Cs E sh
A A 1 1 -1 x, y,Rz z, Rx,Ry x2,y2,z2,xy yz, xz
The reducible representation derived from the two
hybrids can be attached to the table.
G 2 0 (h1, h2)
Lets verify some things. Order of the group
sym operations 2 A and A are orthogonal
11 1(-1) 0 Sum of the squares over sym
operations order of group h The magnitude of
the A and A vectors are each (2) 1/2
magnitude2 ( 12 (/- 1)2)
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Now lets do the reduction.
We assume that the reducible rep G can be
expressed as a linear combination of A and A G
aA A aA A our task is to find out the
coefficients aA and aA
Cs E sh
A A 1 1 -1 x, y,Rz z, Rx,Ry x2,y2,z2,xy yz, xz
G 2 0 (h1, h2)
aA (1 2 1 0)/2 1
aA (1 2 1 0)/2 1
Or again G 1A 1A. Note that this holds
for any reducible rep G as above and not limited
to the hybrids in any way.
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These are block-diagonalized matrices (x, y, z
coordinates are independent of each other)
Reducible Rep
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Water is C2v. Lets use the Character Table
Symmetry operations
Point group
Characters 1 symmetric behavior -1 antisymmetric
Mülliken symbols
Each row is an irreducible representation
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Lets determine how many independent vibrations a
molecule can have. It depends on how many
atoms, N, and whether the molecule is linear or
non-linear.
of atoms degrees of freedom Translational modes Rotational modes Vibrational modes
N (linear) 3 x N 3 2 3N-5
Example 3 (HCN) 9 3 2 4
N (non- linear) 3N 3 3 3N-6
Example 3 (H2O) 9 3 3 3
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Symmetry and molecular vibrations
A molecular vibration is IR active only if it
results in a change in the dipole moment of the
molecule A molecular vibration is Raman
active only if it results in a change in the
polarizability of the molecule
In group theory terms A vibrational mode is IR
active if it corresponds to an irreducible
representation with the same symmetry of a x, y,
z coordinate (or function) and it is Raman
active if the symmetry is the same as A quadratic
function x2, y2, z2, xy, xz, yz, x2-y2 If the
molecule has a center of inversion, no vibration
can be both IR Raman active
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How many vibrational modes belong to each
irreducible representation?
You need the molecular geometry (point group) and
the character table
Use the translation vectors of the atoms as the
basis of a reducible representation. Since you
only need the trace recognize that only the
vectors that are either unchanged or have become
the negatives of themselves by a symmetry
operation contribute to the character.
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A shorter method can be devised. Recognize that
a vector is unchanged or becomes the negative of
itself if the atom does not move. A reflection
will leave two vectors unchanged and multiply the
other by -1 contributing 1. For a rotation
leaving the position of an atom unchanged will
invert the direction of two vectors, leaving the
third unchanged. Etc.
Apply each symmetry operation in that point
group to the molecule and determine how many
atoms are not moved by the symmetry
operation. Multiply that number by the character
contribution of that operation E 3 s 1 C2
-1 i -3 C3 0 That will give you the
reducible representation
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Finding the reducible representation
E 3 s 1 C2 -1 i -3 C3 0
3x3 9
1x-1 -1
3x1 3
1x1 1
( atoms not moving x char. contrib.)
G
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Now separate the reducible representation into
irreducible ones to see how many there are of
each type
S
A1 1/4 (1x9x1 1x(-1)x1 1x3x1 1x1x1) 3
A2
1/4 (1x9x1 1x(-1)x1 1x3x(-1) 1x1x(-1)) 1
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Symmetry of molecular movements of water
Vibrational modes
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Which of these vibrations having A1 and B1
symmetry are IR or Raman active?
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Often you analyze selected vibrational modes
Example C-O stretch in C2v complex.
n(CO)
2 x 1 2
0 x 1 0
2 x 1 2
0 x 1 0
G
Find vectors remaining unchanged after
operation.
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Both A1 and B1 are IR and Raman active
A1 B1
A1 1/4 (1x2x1 1x0x1 1x2x1 1x0x1) 1
A2 1/4 (1x2x1 1x0x1 1x2x-1 1x0x-1) 0
B1 1/4 (1x2x1 1x0x1 1x2x1 1x0x1) 1
B2 1/4 (1x2x1 1x0x1 1x2x-1 1x0x1) 0
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What about the trans isomer?
Only one IR active band and no Raman active bands
Remember cis isomer had two IR active bands and
one Raman active
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Symmetry and NMR spectroscopy
The of signals in the spectrum corresponds to
the of types of nuclei not related by
symmetry The symmetry of a molecule may be
determined From the of signals, or vice-versa
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Molecular Orbitals
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Atomic orbitals interact to form molecular
orbitals Electrons are placed in molecular
orbitals following the same rules as for atomic
orbitals
In terms of approximate solutions to the
Scrödinger equation Molecular Orbitals are linear
combinations of atomic orbitals (LCAO) Y caya
cbyb (for diatomic molecules)
Interactions depend on the symmetry
properties and the relative energies of the
atomic orbitals
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As the distance between atoms decreases
Atomic orbitals overlap
Bonding takes place if the orbital symmetry
must be such that regions of the same sign
overlap the energy of the orbitals must be
similar the interatomic distance must be short
enough but not too short
If the total energy of the electrons in the
molecular orbitals is less than in the atomic
orbitals, the molecule is stable compared with
the atoms
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Combinations of two s orbitals (e.g. H2)
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Both s (and s) notation means symmetric/antisymme
tric with respect to rotation
s
s
s
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Combinations of two p orbitals (e.g. H2)
s (and s) notation means no change of sign upon
rotation
p (and p) notation means change of sign upon C2
rotation
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Combinations of two p orbitals
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Combinations of two sets of p orbitals
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Combinations of s and p orbitals
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Combinations of d orbitals
No interaction different symmetry
d means change of sign upon C4
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Is there a net interaction?
NO
NO
YES
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Relative energies of interacting orbitals must be
similar
Weak interaction
Strong interaction
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Molecular orbitals for diatomic molecules From H2
to Ne2
Electrons are placed in molecular
orbitals following the same rules as for atomic
orbitals Fill from lowest to highest Maximum
spin multiplicity Electrons have different
quantum numbers including spin ( ½, - ½)
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O2 (2 x 8e)
1/2 (10 - 6) 2 A double bond
Or counting only valence electrons 1/2 (8 - 4)
2
Note subscripts g and u symmetric/antisymmetric up
on i
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Place labels g or u in this diagram
su
pg
pu
sg
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su
sg
g or u?
pg
pu
du
dg
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Orbital mixing
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s orbital mixing
When two MOs of the same symmetry mix the one
with higher energy moves higher and the one with
lower energy moves lower
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Molecular orbitals for diatomic molecules From H2
to Ne2
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Bond lengths in diatomic molecules
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Photoelectron Spectroscopy
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O2
N2
sg (2p)
pu (2p)
pu (2p)
sg (2p)
pu (2p)
Very involved in bonding (vibrational fine
structure)
su (2s)
su (2s)
(Energy required to remove electron, lower energy
for higher orbitals)