38: Selection Rules rial in this lecture covers the following in Atkins ecular Symmetry 15.5 Vanishing integrals and orbital ov (a) The criteria for vanishing integral (b) Orbitals with nonzero overlaps (c) Symmetry-adapted linear combination 15.6 Vanishing integrals and selection on-line election Rules (PowerPoint) Selection Rules (PDF) outs for this lecture
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Lecture 38: Selection Rules The material in this lecture covers the following in Atkins.
Lecture 38: Selection Rules The material in this lecture covers the following in Atkins. 15 Molecular Symmetry 15.5 Vanishing integrals and orbital overlaps (a) The criteria for vanishing integrals - PowerPoint PPT Presentation
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Lecture 38: Selection Rules The material in this lecture covers the following in Atkins.15 Molecular Symmetry 15.5 Vanishing integrals and orbital overlaps (a) The criteria for vanishing integrals (b) Orbitals with nonzero overlaps (c) Symmetry-adapted linear combinations 15.6 Vanishing integrals and selection rules
Lecture on-line Selection Rules (PowerPoint) Selection Rules (PDF) Handouts for this lecture
Thus, the three mirror planesshown here are related by threefoldrotations : C3v=v’
Character Table
Symmetry operations in the same class are related to one another by the symmetry operations of the group.
and the two rotations shown here are related by reflection in v.vC3=C3
-1
C3v
The dimension is 6 since we have 6 elements.
We have three different symmetryrepresentations as we have threedifferent classes of symmetry elements
Character Table C3v
The pz orbitaldoes not change
with E, C3, C3-1
σv, σ'v ,σ"vThe symmetryrep. is A1
px pydoes change
with E, C3, C3-1
σv, σ'v ,σ"v
X
Y
X
Y
Character Table C3v
X
Y
X
Y
X
Y
px p'x p''x
Epx =px ; C3px =p'x; C3-1 px =p"x
X
Y
X
Y Y
Character Table C3v
Epy =py ; C3py =p'y; C3-1 py =p"y
py p'y p''y
px py( )D(C3)= px py( )−
12
32
3
2−
1
2
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
px py( )D(C3−1 )= px py( )
−12
−32
−3
2−
1
2
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
The trace is -1 forboth matrices
px py( )D(E)= px py( )1 0
0 1
⎛
⎝ ⎜
⎞
⎠ ⎟
The trace is 2which is also thedimension ofthe representation
Character Table C3v
px py( )D(σv)= px py( )−1 0
0 1
⎛
⎝ ⎜
⎞
⎠ ⎟
px py( )D(σv' )= px py( )
12
−32
−3
2−
1
2
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
px py( )D(σv" )= px py( )
12
32
3
2−
1
2
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
The trace is -1 forboth matrices
Character Table C3v
Typical symmetry-adapted linear combinations of orbitals in aC 3v molecule.
Character Table Constructing Linear combinationsHow are they constructed
a1
a2
ex
ey
C3v Original basis SA SB SC E SA SB SCC3
+ SB SC SA
C3− SC SA SB
σv SA SC SBσv' SB SA SCσv" SC SB SA
Character Table
Construct a table showing the effect of each operation on each orbitalof the original basis
To generate the combination of aSpecific symmetry species, takeEach column in turn and
(I) Multiply each member of theColumn by the character of the Corresponding operator
SA
SB
SC
Constructing Linear combinations
C3v
Original basis SA SB SC E 1SA 1SB 1SCC3
+ 1SB 1SC 1SA
C3− 1SC 1SA 1SB
σv 1SA 1SC 1SBσv' 1SB 1SA 1SCσv" 1SC 1SB 1SA
Character Table
(I) Multiply each member of theColumn by the character of the Corresponding operator
The polarizations of the allowed transitions in a C2v molecule.The shading indicates the structure of the orbitals of the specified symmetry species. The perspective view of the molecule makes it look rather like a door-stop; however, from the side, each `door-stop' is in fact an isosceles triangle.
What you should learn from this lecture
1.You should be able with the help of a charactertavle to decide which components are differentfom zero in :<