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Dr. S. M. Condren
Symmetry Elements and Symmetry Operations
• Identity
• Proper axis of rotation
• Mirror planes
• Center of symmetry
• Improper axis of rotation
Dr. S. M. Condren
Symmetry Elements and Symmetry Operations
• Proper axis of rotation => Cn
– where n = 2, 180o rotation– n = 3, 120o rotation– n = 4, 90o rotation– n = 6, 60o rotation– n = , (1/)o rotation
• principal axis of rotation, Cn
Dr. S. M. Condren
Symmetry Elements and Symmetry Operations
Mirror planes =>h => mirror plane perpendicular to a
principal axis of rotationv => mirror plane containing principal
axis of rotationd => mirror plane bisects dihedral angle made
by the principal axis of rotation and two adjacent C2 axes perpendicular to principal rotation axis
Dr. S. M. Condren
Symmetry Elements and Symmetry Operations
• Improper axis of rotation => Sn
– rotation about n axis followed by inversion through center of symmetry
Dr. S. M. Condren
Selection ofPoint Group from Shape
• first determine shape using Lewis Structure and VSEPR Theory
• next use models to determine which symmetry operations are present
• then use the flow chart Figure 3.9, Pg. 81 text to determine the point group
Dr. S. M. Condren
Selection ofPoint Group from Shape
1.determine the highest axis of rotation
2.check for other non-coincident axis of rotation
3.check for mirror planes
Dr. S. M. Condren
Orbital Symmetry, pz C2v
z E + X(E) = +1
- +
+ C2(z)
x
- + - X(C2(z)) = +1
y v(xz)
- X(v(xz)) = +1
v(yz) +
- X(v(xz)) = +1
Dr. S. M. Condren
Orbital Symmetry, py C2v
+
-
+-
-+
-
+
+-
zE
x
y
X(E) = +1
C2(z)X(C2(z)) = -1
v(xz)
X(v(xz)) = -1
X(v(xz)) = +1
v(yz)
Dr. S. M. Condren
Orbital Symmetry, px C2v
- +
- +
+ -
- +
+ -
z
x
y
EX(E) = +1
C2(z)
X(C2(z)) = -1v(xz)
v(yz)X((xz)) = +1
X(v(xz)) = -1
Dr. S. M. Condren
Water, C2v Point GroupTranslational motion in y
z
y o o
H H H H
x v(xz)
“asymmetric” => -1
Dr. S. M. Condren
Water, C2v Point GroupTranslational motion in y
z
o
y H H
x o
H H v(yz)
“symmetric” => +1
Dr. S. M. Condren
Water, C2v Point GroupTranslational motion in y
z
y C2(z)
x
O
H H
“asymmetric” = - 1
Dr. S. M. Condren
Water, C2v Point GroupTranslational motion in y
Representation:
E C2(z) v(xz) v(yz)
3 +1 -1 -1 +1
Dr. S. M. Condren
Water, C2v Point GroupRotation about z axis
z
O
Ha Hb
- movement out of plane towards observer
- movement out of plane away from observer
a,b - labeling to distinguish hydrogens before and after symmetry operations
Dr. S. M. Condren
Water, C2v Point GroupRotation about z axis
Representation
E C2(z) v(xz) v(yz)
4 +1 +1 -1 -1
Dr. S. M. Condren
Water, C2v Point Group
Representations:
Rotation
E C2(z) v(xz) v(yz)
4 +1 +1 -1 -1
Dr. S. M. Condren
Water, C2v Point Group
Representation:
Translation
E C2(z) v(xz) v(yz)
1 +1 +1 +1 +1 Tz
2 +1 -1 +1 -1 Tx
3 +1 -1 -1 +1 Ty
Dr. S. M. Condren
Water, C2v Point Group
Representation:
Rotation
E C2(z) v(xz) v(yz)
4 +1 +1 -1 -1 Rz
5 +1 -1 +1 -1 Ry
6 +1 -1 -1 +1 Rx
Dr. S. M. Condren
Water, C2v Point Group
Character Table
E C2(z) v(xz) v(yz)
A1 +1 +1 +1 +1 Tz 1
A2 +1 +1 -1 -1 Rz 4
B1 +1 -1 +1 -1 Ry, Tx 2 , 5
B2 +1 -1 -1 +1 Rx,Ty 3, 6