CHEM 515 Spectroscopy

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CHEM 515Spectroscopy

• Vibrational Spectroscopy III

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Normal Modes in Water Molecule
x
Q3
z
Q2
Q1
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Internal Coordinates

• Mathematically, the mass-weighted Cartesian
  coordinates (Cartesian force constant model) are
  very convenient. Ab initio calculation utilizes
  such a model in molecule optimization.
• A more recognizable way to define the atomic
  displacement in molecular modeling program is by
  utilizing internal coordinates.

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Internal Coordinate Definitions

• It is much convenient to transform from Cartesian
  coordinates to internal coordinates.
• These are the main four internal coordinate out
  of which other internal coordinated can be
  defined.

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Internal Coordinate for Water Molecule

• The internal displacement of atoms can be defined
  as change in the three coordinates.
• ?r1
• ?r2
• ??

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Symmetry Coordinates

• More preferably than internal coordinates is the
  use of symmetry coordinates.
• The concept of molecule symmetry group theory
  is applied with help of projection operators to
  generate the required 3N 6 symmetry coordinates.

Symmetry specie
Normalization factor
Projection operator
Operation
New coordinate obtained from Inti upon operation R
Character for symmetry specie ?
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Symmetry Coordinates

• This totally symmetric projection operator is
  used to get a set of symmetric coordinates by
  linear combinations of internal coordinates.
• This method is also known as Symmetry-Adapted
  Linear Combinations SALCs) as proposed by A. F.
  Cotton.

Symmetry specie
Normalization factor
Operation
New coordinate obtained from Inti upon operation R
Character for symmetry specie ?
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Symmetry Coordinates of Ethylene Using SALC Method
D2d E C2 (z) C2 (y) C2 (x) i s (xy) s (xz) s (yz)
Ag 1 1 1 1 1 1 1 1
B1g 1 1 -1 -1 1 1 -1 -1
B2g 1 -1 1 -1 1 -1 1 -1
B3g 1 -1 -1 1 1 -1 -1 1
Au 1 1 1 1 -1 -1 -1 -1
B1u 1 1 -1 -1 -1 -1 1 1
B2u 1 -1 1 -1 -1 1 -1 1
B3u 1 -1 -1 1 -1 1 1 -1
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Symmetry Coordinates of Ethylene Using SALC Method
D2d E C2 (z) C2 (y) C2 (x) i s (xy) s (xz) s (yz)
OR (r1) r1 r4 r2 r3 r4 r1 r3 r2
Ag 1 1 1 1 1 1 1 1
B1g 1 1 -1 -1 1 1 -1 -1
B2g 1 -1 1 -1 1 -1 1 -1
B3g 1 -1 -1 1 1 -1 -1 1
Au 1 1 1 1 -1 -1 -1 -1
B1u 1 1 -1 -1 -1 -1 1 1
B2u 1 -1 1 -1 -1 1 -1 1
B3u 1 -1 -1 1 -1 1 1 -1
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Determining the Symmetry Species for the
Vibrations in a Molecule

• We are very concerned with the symmetry of each
  normal mode of vibration in a molecule.
• Each normal mode of vibration will form a basis
  for an irreducible representation (G) of the
  point group of the molecule.
• The objective is to determine what the character
  (trace) is for the transformation matrix
  corresponding to a particular operation in a
  specific molecule.

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Symmetry of Normal Modes of Vibrations in H2O

• H2O has C2v symmetry.
• Operation E results in the following
  transformations

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Symmetry of Normal Modes of Vibrations in H2O

• The transformations in the x, y and z modes can
  be represented with the following matrix
  transformation
• Trace of E matrix is equal to 9.

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Symmetry of Normal Modes of Vibrations in H2O

• The operation C2 is more interesting!
• Operation C2 results in the following
  transformations

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Symmetry of Normal Modes of Vibrations in H2O

• The transformations in the x, y and z modes can
  be represented with the following matrix
  transformation
• Trace of C2 matrix is equal to 1.

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Determining the Symmetry Species for the
Vibrations in a Molecule a Shorter Way

• The matrix transformation method is very
  cumbersome. However, it can be streamlined
  tremendously another procedure.
• Alternative Method
• Count unshifted atoms per each operation.
• Multiply by contribution per unshifted atom to
  get the reducible representation (G).
• Determine (G) for each symmetry operation.
• Subtract Gtrans and Grot from Gtot.
• Gvib Gtot Gtrans Grot .

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Determining the Irreducible Representation for
the H2O Molecule

• 1. Count unshifted atoms per each operation.

C2v E C2 s (xz) s (yz)
Unshifted atoms 3 1 1 3
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Determining the Irreducible Representation for
the H2O Molecule

• 2. Multiply by contribution per unshifted atom to
  get the reducible representation (G).

C2v E C2 s (xz) s (yz)
Unshifted atoms 3 1 1 3
Contribution per atom (Gxyz) 3 1 1 1
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Determining the Irreducible Representation for
the H2O Molecule

• 2. Multiply by contribution per unshifted atom to
  get the reducible representation (G).

C2v E C2 s (xz) s (yz)
Unshifted atoms 3 1 1 3
Contribution per atom (Gxyz) 3 1 1 1
G 9 1 1 3
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Determining the Irreducible Representation for
the H2O Molecule

• 3. Determine (G) for each symmetry operation.
• ?i number of times the irreducible
  representation (G) appears for the symmetry
  operation i.
• h order of the point group.
• R an operation of the group.
• ?R character of the operation R in the
  reducible represent.
• ?iR character of the operation R in the
  irreducible represent.
• CR number of members of class to which R
  belongs.

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Determining the Irreducible Representation for
the H2O Molecule
C2v E C2 s (xz) s (yz)
G 9 1 1 3

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Determining the Irreducible Representation for
the H2O Molecule
C2v E C2 s (xz) s (yz)
G 9 1 1 3

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Determining the Irreducible Representation for
the H2O Molecule

• 3. Determine (G) for each symmetry operation.

• Gtot 3A1 A2 2B1 3B2
• Number of irreducible representations Gtot must
  equal to 3N for the molecule.

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Determining the Irreducible Representation for
the H2O Molecule

• Subtract Gtrans and Grot from Gtot.

Gtot 3A1 A2 2B1 3B2
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Determining the Irreducible Representation for
the H2O Molecule

• Gvib 2A1 B2
• The difference between A and B species is that
  the character under the principal rotational
  operation, which is in this case C2, is always 1
  for A and 1 for B representations. The
  subscripts 1 and 2 are considered arbitrary
  labels.

A1
A1
B2
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Determining the Irreducible Representation for
the H2O Molecule

• Gvib 2A1 B2
• None of these motions are degenerate. One can
  spot the degeneracy associated with a special
  normal mode of vibration when the irreducible
  representation has a value of 2 at least, such as
  E operation in C3v and C4v point groups.

A1
A1
B2
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Determining the Irreducible Representation for
Ethene
D2d E C2 (z) C2 (y) C2 (x) i s (xy) s (xz) s (yz)
Ag 1 1 1 1 1 1 1 1
B1g 1 1 -1 -1 1 1 -1 -1 Rz
B2g 1 -1 1 -1 1 -1 1 -1 Ry
B3g 1 -1 -1 1 1 -1 -1 1 Rx
Au 1 1 1 1 -1 -1 -1 -1
B1u 1 1 -1 -1 -1 -1 1 1 z
B2u 1 -1 1 -1 -1 1 -1 1 y
B3u 1 -1 -1 1 -1 1 1 -1 x
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Determining the Irreducible Representation for
Ethene
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Determining the Irreducible Representation for
Ethene
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Normal Modes in Ethene
Physical Chemistry  By Robert G. Mortimer
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Mutual Exclusion Principle

• For molecules having a center of symmetry (i),
  the vibration that is symmetric w.r.t the center
  of symmetry is Raman active but not IR active,
  whereas those that are antisymmetric w.r.t the
  center of symmetry are IR active but not Raman
  active.

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Vibrations in Methyl and Methylene Groups

• Ranges in cm-1
• C-H stretch 2980 2850
• CH2 wag 1470 1450
• CH2 rock 740 720
• CH2 wag 1390 1370
• CH2 twist 1470 - 1440