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radian 2πrad =360° 1πrad =180°
1°=π/180 rad(about 57.29578°)
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(群論) H2O of point group

 

※ character table
App_C_Pg731_UnTable_5.jpg
04_15_Figure.jpg









※ H2O molecular vibrations mode
04_21_Figure.jpgvibration1.png
※ The transformation matrix for C2v
vibr2.png
reducible representation : 
blockd.gif
irred.gif
Set 1 gives the effects of the symmetry operations on the x axis.
Set 2 gives the effects of the symmetry operations on the y axis.
Set 3 gives the effects of the symmetry operations on the z axis.

※ The characters for each symmetry element
 
 
 
 
 irreducible repesentations :    
 

   E    

   C2      

 σv(xz)

σv(yz)

1

-1

1

-1

1

-1

-1

1

1

1

1

1

reducible representation : 

-1 

1

 

E

C2v

σv(xz)

σv′(yz)

3

-1

1

1


Γ:  a generic notation for any reducible representation
 

※ the number of each type of irreducible representation contained in any reducible representation
form1.gif
※ All Motions:
 formula to the reducible representation above
reduced1.gif
  Γ = 3 A1 + A2 + 3 B1 + 2 B2
→   9 modes of motion for the water molecule.
→  a molecule with N atoms, each atom can move in 3 directions, so 3N total motions which are called degrees of freedom
→ For a non-linear molecule, there will also be 3 rotational modes which will leave 3N – 6 vibration modes
→ water molecule (N = 3), there will be 3 vibrational modes
※ H2O vibration and IR
04_11_Table.jpg
→  vibrational modes of water are A1 + B1.
→  IR (active or not ?)                               § hint: transforms as xy, or z will be IR active
 A1 irreducible representation transforms as z
    B1 irreducible representation transforms as x s
 IR  active 
04_12_Table.jpg
資料來源: 
and 群論課
 

 


 

 
 
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