Understanding Character Tables

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Understanding Character Tables !

• The top row consists of the symmetry operations.
• The first column consists of irreducible
  representations for the group.
• The table elements are the characters.
• The final two columns show the first and second
  order combinations of Cartesian coordinates.
• Infinitesimal rotations are listed as Rx, Ry, and
  Rz in the second to last column.

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Reducing By Decomposition Formula
ai the number of contributions for the
irreducible representation (i.e.
the number of A1 in the total representation) ,
R refers to one particular operation (e.g. C2)
h the order of the group (the number of
symmetry operations in the point group,
i.e. E, C2,C3, etcbe sure to 3sv's as three)
g is the number of elements in the class (e.g.
how many sv or C2 axes we have) Xi (R) is the
character of the irreducible representation
(that operations value in the character
table) and XG (R) is the character for the
analogous operation R in the total
representation. (?r value for that
operation),
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 ?r 9 -1 1 3
aA1 1/4 g XA1 (E) Xt (E) g XA1 (C2) Xt
(C2) g XA1 (sv) Xt (sv) g XA1 (sv') Xt
(sv') 1/4 1 19 11 (-1) 1 1
1 113 1/4 9 1 -1 3
12/4 3A1 aA2 1/4 g XA2 (E) Xt (E) g
XA2 (C2) Xt (C2) g XA2 (sv) Xt (sv) g XA2
(sv') Xt (sv') 1/4 1 19 11
(-1) 1 1 (-1) 1(1-)3 1/4
9 - 1 -1 - 3 A2 aB1 1/4 1 19
1 (-1) (-1) 1 1 1 1(-1)3
1/4 9 1 1 - 3 2B1 aB2 1/4 1
19 1 (-1) (-1) 1(-1)1 113
1/4 9 1 -1 3 3B2  
The total representation consists of the sum of
the irreducible representations 3A1, 1A2, 2B1 and
3 B2 which describes all the degrees of freedom
of the molecule.
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• Selection Rules for Fundamental Vibrations
• A fundamental will be infra-red active if the
  normal mode which is excited belongs to the same
  representation as any one or several of the
  Cartesian coordinates
• A fundamental will be Raman active if the normal
  mode involved belongs to the same representation
  as any one or more components of the
  polarizability tensor of the molecule
• Mutual Exclusion Principle
• In a centrosymmetric molecule,
• no Raman-active vibration is also infrared
  active, and no infrared-active vibration is also
  Raman active

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Raman vs IR

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