symmetry and molecular structuresocw.nctu.edu.tw/course/ichemistry/ichemistry_lecture...symmetry...
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Symmetry and MolecularStructures
Some Readings
Chemical Application of Group TheoryF. A. Cotton
Symmetry through the Eyes of a ChemistI. Hargittai and M. Hargittai
The Most Beautiful Molecule - an Adventure in ChemistryH. Aldersey-Williams
Perfect SymmetryJ. Baggott
Some LinksThe Point Group Tutorialhttp://www.emory.edu/CHEMISTRY/pointgrp/index.htmlTables for Group Theoryhttp://www.oup.com/uk/orc/bin/9780199264636/01student/tables/
Symmetry Groupshttp://www.chemistry.nmsu.edu/studntres/chem639/symmetry/group.html
For FunWallpaper Groups (Plane Symmetry Groups)http://www.clarku.edu/~djoyce/wallpaper/The Geometry Junk Yardhttp://www.ics.uci.edu/~eppstein/junkyard/
Shapes, Geometry and Symmetry
Shapes, Geometry and Symmetry
Shapes, Geometry and Symmetry
Shapes, Geometry and Symmetry
Shapes, Geometry and Symmetry
Shapes, Geometry and Symmetry
Shapes, Geometry and Symmetry
Shapes, Geometry and Symmetry
Shapes, Geometry and Symmetry
Shapes, Geometry and Symmetry
Shapes, Geometry and Symmetry
Shapes, Geometry and Symmetry
Types of Symmetry and Group
Point, Plane, SpaceMolecular Symmetry –Point Symmetry
–Symmetry elements–Symmetry operations
Symmetry Operations•Defined: A well-defined, non-translational movement
of an object that produces a new orientation that isindistinguishable from the original object.
•Orientation A is indistinguishable from Orientation B,but not necessarily identical.
•These are Equivalent Configurations.
Symmetry Element•Defined: A point, line or plane about which
the symmetry operation is performed.
SymmetryElement
n-fold symmetry axisMirror planeCenter of inversionn-fold axis ofimproper rotation
SymmetryOperation
IdentityRotation by 2/nReflectionInversionRotation by 2/nFollowed by reflectionperpendicular to rotation axis
Symbol
ECn
iSn
Identity (E)All the points in a molecule can be described withCartesian coordinates x, y and z.These points can be transformed by the followingmatrix equation:
E operation
C2Operation –Rotation by 180o
Symmetry Element:
2 fold rotation axis C2
*
*
C3 Operation –Rotation by 120o
Symmetry Element:3 fold rotation axis C3
*
*
*
Proper Rotation (Cn)
The proper rotation (clockwise direction) aboutthe z axis can be described by the followingmatrix equation.
is the angle of rotation
Cn operation
A Cn axis generates n operations
The four operations generated are:C4
1, C2 (C42), C4
3 and E (C44).
v Operation –Reflection
Symmetry Element:Mirror plane v
Parallel to rotation axis
*
*
* *
Mirror Planes ( σ )
If the mirror plane coincides with the xy, xz, oryz Cartesian planes, they can be described bythe following matrix equations:
operation
σ x σ = E
Some Mirror Planes of Benzene
Mirror plane vParallel to rotation axis
Mirror plane dParallel to rotation axisBisects the angles of C-C
Mirror plane hPerpendicular to rotation axis
Inversion Operation
Symmetry Element:Center of inversion ii × i = E
Center of Inversion (i)
The inversion operation changes the sign of allthe x, y and z coordinates:
i operation
i Operation
C2 Operation
S4 OperationRotation by 90o
- followed by reflectionperpendicular to rotationaxisSymmetry Element: S4
Improper Rotation (Sn)
The improper rotation about the z axis can bedescribed as a proper rotation followed bychanging the sign of the z coordinate.
is the angle of rotationSn operation
When n is even, Sn generates n operations.When n is odd,
Sn generates 2n operations.Sn
n is equivalent to σ , and Sn2n is equivalent to E.
S1 = h S2 = i
When n is even, Sn generates n operations.
S6 has generated six operations:S6
1, C31 (S6
2), i (S63), C3
2 (S64), S6
5 and E (S66).
When n is odd, Sn generates 2n operations.
The 10 operations generated by S5:S5
1, C52 (S5
2), S53, C5
4 (S54), σ,
C51 (S5
6), S57, C5
3 (S58), S5
9 and E (S510).
Symmetry Elements ofC2v and C3v
Groups
What is Group Theory?•A fairly “recent”branch of mathematics. Fedorov
pioneered application of group theory incrystallography.
•Group Theory is the closest many chemists get totruly “modern”mathematics.
•Can be used to simplify complicated geometricsystem (structures) and derive physical properties
•Used in spectroscopy and molecular orbital theory
Properties of Group•The product of any two elements in the group and the
square of each element must be an element in thegroup.
•One element in the group must commute with allothers and leave them unchanged.
•The associative law of multiplication must hold.•Every element must have a reciprocal, which is also
an element of the group.•The reciprocal of a product of two or more elements
is equal to the product of the reciprocals, in reverseorder.
(AB)-1 = B-1A-1 (ABC)-1 = C-1B-1A-1
Types of Point Groups IGroups with a single symmetry element:
C1 (E only) Cs (E and σ) Ci (E and i)
Cn (E and Cn) Sn (n is even and generates n operations)
Types of Point Groups IIGroups with more than one symmetry element:Dn Cn axis, and n C2 axes
perpendicular to Cn
Dnd Cn axis, and n C2 axesperpendicular to Cn, and n dihedralσ parallel to Cn and bisect theangles between the n perpendicularC2 axes
Dnh Cn axis, and n C2 axes perpendicularto Cn, and a σ perpendicular to Cn
Types of Point Groups IIGroups with more than one symmetry element:
Cnh Cn axis, and one σ perpendicular to Cn
Cnv Cn axis, and two or more σ that contain Cn
Types of Point Groups IIISpecial groups:C∞v linear molecules lacking a center of symmetry
D∞h linear molecules with a center of symmetry
Types of Point Groups IIISpecial groups:
Td tetrahedral groups –also includes Th and TOh octahedral groups –includes O
Ih icosahedral groups –includes I
Decision Tree for Molecular Point Group
Polar Molecules
A molecule cannot be polar if it has1. a center of inversion
–any group with i
2. an electric dipole moment perpendicular toany mirror planes–any of the groups D and their derivatives
3. an electric dipole moment perpendicular toany axis of rotation–the cubic groups T, O, the icosahedral I, and
their modifications
Chiral Molecules
A molecule is not chiral if1. it posses an improper rotation axis Sn
2. it belongs to the group Dnh or Dnd
3. it belongs to Td or Oh
Some Molecules
Some Molecules
Total Representation for C2vIndividually block diagonalized matrices
Reduced to 1D matricesx [ 1] [-1] [ 1] [-1]y [ 1] [-1] [-1] [ 1]
z [ 1] [ 1] [ 1] [ 1]
irreducible representationΓx = 1 -1 1 -1Γy = 1 -1 -1 1Γz = 1 1 1 1ΓRz = 1 1 -1 -1z
Character Tables
I: Mulliken symbol.A, B: 1D E: 2D T: 3DA: 1D symmetric about the principal axis (1)B: 1D unsymmetric about the principal axis (1)
II: Irreducible representations for the groupIII: Transformation properties of vectors and
rotations along the x, y and z axisIV: Transformation properties of squares and binary
products of the coordinates
(yz)
WEB Pages
Point Group Theoryhttp://newton.ex.ac.uk/people/goss/symmetry/Symmetry.html
The Point Group Tutorialhttp://www.emory.edu/CHEMISTRY/pointgrp/index.html
Exercises in Point Group Symmetryhttp://origin.ch.ic.ac.uk/vchemlab/symmetry/index.htm
Tables for Group Theoryhttp://www.oup.com/uk/orc/bin/9780199264636/01student/tables/
Symmetry Groupshttp://www.chemistry.nmsu.edu/studntres/chem639/symmetry/group.html