Chapter 3: Group TheoryChapter 3: Group Theory
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Review of the Previous Lecture
1. Defined symmetry elements and symmetry operations
2. Applied the flow chart to determine the point group of a molecule
3. Determined molecular orientation
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1. Character Tables
Each point group has an associated character table which lists all of the symmetry elements/operations that members of the group possess.
The table is mathematically derived using matrices.
C2V
C2V E C2 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
Symmetry Elements/Operations
IrreducibleRepresentations
BasisFunctions
Characters
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1A. Irreducible Representations
Each point group can be decomposed into basic symmetry properties known as irreducible representations.
An irreducible representation is the simplest way to describe a molecular property.
Let’s deconstruct this soccer ball into irreduciblerepresentations:
1. Shape
2. Color
3. Pattern
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1A. Irreducible Representations
The irreducible representations are defined by a set of characters.
Positive value (i.e. +1): Symmetric behavior
Negative value (i.e. -1): Antisymmetric behavior
C2V E C2 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
IrreducibleRepresentations
Characters
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1A. Irreducible Representations
Total # of irreducible representations = # of classes of symmetry elements/operations in a group
C2V E C2 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
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4
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1A. Irreducible Representations
The Mullikin labels give us information about degeneracies as follows:
A and B labels indicate a non-degenerate state
E label indicates a double degeneracy
T label indicates a triply degeneracy
C2V E C2 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
Mullikin (Symmetry)Labels
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1B. Basis Functions
The basis functions represent certain types of properties and their symmetry will be defined by a specific irreducible representation.
C2V E C2 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
BasisFunctions
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1B. Basis Functions
I. Rx, Ry, Rz : Define rotations around the x, y, or z axes
II. x, y, z : Define translations (movement) along the x, y, or z axes
C2V E C2 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
BasisFunctions
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1B. Basis Functions
III. Orbital Type
Central Atom
Atomic OrbitalBasis Function
sThe first row
(entirely symmetric row)
px x
py y
pz z
dz2 z2 or 2z2 - x2 - y2
dx2-y2 x2 - y2
dxy xy
dxz xz
dyz yz
Binary
Functions
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The d-orbitals
+z
+y
+x
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The d-orbitals
+z
+y
+z
+x
+y
+x
+y
+x
+z
+y
+x
dz2 dyz dxz
dxy dx2-y2
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1B. Basis Functions
C2V E C2 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
BasisFunctions
Central Atom
Atomic OrbitalBasis Function
sThe first row
(entirely symmetric row)
pxx
pyy
pzz
dz2 z2 or 2z2 - x2 -y2
dx2-y2 x2 -y2
dxyxy
dxzxz
dyzyz
Note:
Orbitals of the same type will not necessarily be represented by the same irreducible representation.
i.e. H2O molecule
pz : A1 ; px : B1 ; py : B2
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1C. Application of the character table
C2V E C2 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
Which irreducible representation represents a translation along the y-axis?
+z
+y
+x
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C2V E C2 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
Which irreducible representation represents a translation along the y-axis?
+z
+y
+xLet us formally determine this.
Approach 1- Draw out all operations
I. Draw the molecule in its new location
II. Apply and draw out each symmetry operation from the original position
Determine if the molecule looks unmoved or moved
Label the operation with the characters
Unmoved: +1
Moved: -1
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C2V E C2 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
Which irreducible representation represents a translation along the y-axis?
+z
+y
C2V E C2 σv (xz) σv '(yz)
+z
+y
+z
+y
+z
+y
+z
+y
y translation
(Redrawn 4 times)
E+z
+y
+1
C2
σv (xz)
σv'(yz)
+z
+y
-1
+z
+y
-1
+z
+y
+1
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C2V E C2 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
Which irreducible representation represents a translation along the y-axis?
+z
+y
C2V E C2 σv (xz) σv '(yz)
B2 1 -1 -1 1
+z
+y
+z
+y
+z
+y
+z
+y
y translation
(Redrawn 4 times)
E+z
+y
+1
C2
σv (xz)
σv'(yz)
+z
+y
-1
+z
+y
-1
+z
+y
+1
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C2V E C2 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
Which irreducible representation represents a translation along the y-axis?
+z
+y
+xLet us formally determine this.
Approach 2- Determine an appropriate basis set
I. The basis set will represent the y-translation
II. Observe how the vectors transform according to the symmetry operations
Determine if the vectors look unmoved or moved
Label the operation with the characters
Unmoved: +1
Moved: -1
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C2V E C2 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
Which irreducible representation represents a translation along the y-axis?
+z
+y
C2V E C2 σv (xz) σv '(yz)
B2 1 -1 -1 1
+z
+y
+z
+y
+z
+y
+z
+y
Basis Set fory-translation
(Redrawn 4 times)
E
+1
C2
σv (xz)
σv'(yz)
-1
-1
+1
+z
+y
+z
+y
+z
+y
+z
+y
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2. Reducible Representations
Molecular structure and dynamics can be defined by a net sum of irreducible representations.
Reducible Representations = Σ Irreducible representations
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3. Molecular motion
The motion of a molecule containing N atoms can be described in terms of the 3 Cartesian axes (x,y,z).
The molecule has 3N degrees of freedom, which describe the translational, rotational, and vibrational motions of the molecule.
3N- 3 due to translations in the x, y, z directions- 3 due to rotations around the x, y, z axes (Rx, Ry, Rz)
3N-6 : # of vibrational modes for nonlinear molecules
3N-5 : # of vibrational modes for linear moleculesNormal vibrational modes
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3A. Molecular vibration and spectroscopy
Infrared (IR): Molecular vibrations due to excitation
Raman: Molecular vibrations that result in differences in scattered light due to relaxation to different lower energy vibrational levels.
E
ν = 0
ν = 1
ν = 2
ν = Very high
states
Rayleigh
Raman
Scattering
Stokes
Raman
Scattering
Anti-stokes
Raman
Scattering
IR
Absorption
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E
ν = 0
ν = 1
ν = 2
ν = Very high
states
IR
Absorption
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E
ν = 0
ν = 1
ν = 2
ν = Very high
states
Rayleigh
Raman
Scattering
Stokes
Raman
Scattering
Anti-stokes
Raman
ScatteringIn
ten
sity
ννs < νl νs = νl νs > νl
Taken from Edinburgh Instruments
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3B. Vibrational spectroscopy
3N- 3 due to translations in the x, y, z directions- 3 due to rotations around the x, y, z axes (Rx, Ry, Rz)
3N-6 : # of vibrational modes for nonlinear molecules
3N-5 : # of vibrational modes for linear moleculesNormal vibrational modes
Is concerned with the observation of the degrees of vibrational freedom.
Each normal mode of vibration will transform as an irreducible representation of the molecule’s point group.
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3C. Determining if a mode is IR and/or Raman active
I. IR : Results in a change in dipole moment. According to group theory, an IR active mode is symmetric with the x, y, z basis functions.
No Δdm Yes Δdm; IR Active
D∞h C∞v
Dipole Moment
Irred. rep. = Σg+
for the vibrationIrred. rep. = A1 ; transforms like zfor the vibration
Symmetric Stretch
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3C. Determining if a mode is IR and/or Raman active
II. Raman : Results in a change in polarizability or distortion of the electron cloud. According to group theory, a Raman active mode is symmetric with the binary basis functions (i.e. x2, xy, etc.)
Both are Raman active
Symmetric Stretch
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3D. Rule of mutual exclusion
For centrosymmetric molecules, the rule of mutual exclusion applies. IR active vibrations are Raman inactive and vice versa.
Identify a centrosymmetric molecule by identifying the presence of an inversion center (i)
The Raman active vibrationis IR inactive
i No i