Lecture 34: Symmetry Elements
The material in this lecture covers the following in Atkins.
15 Molecular Symmetry
The symmetry elements of objects
15.1 Operations and symmetry elements 15.2 Symmetry classification of molecules (a) The groups C1,Ci, and Cs
(b) The groups Cn,Cnv, and Cnh (c) The groups Dn,Dnh, and Dnd Lecture on-line
Symmetry Elements (PowerPoint) Symmetry Elements (PDF)
Some of the symmetry elements of a cube. The twofold, threefold, and fourfold axes are labelled with the conventional symbols.
Symmetry elements The systematic discussion of
symmetry is called
Some objects are more
symmetrical than others
Any rotation of sphere around axis through
center brings sphere
over into itself Only some rotations of sphere brings cube into itself
(b) an H2O
molecule has a twofold (C2) axis. Both have other symmetry
elements too.
Symmetry elements
(a) An NH3 molecule has a threefold (C3) axis
NH
O
3 has more rotation
Symmetry elements
An action that leaves an object the same after it has been carried out is called a :
Symmetry operations are :
For each symmetry operation there is a : symmetry element Which is
Operators and Symmetry elements The classification of objects according to
symmetry elements corrsponding to
operations that leave at least one common point unchanged gives rise to the :
There are five kinds of symmetry operations and corresponding symmetry elements
I F Cl
Br
1. The identity E Consists of doing nothing
Operators and Symmetry elements
(b) an H2O
molecule has a twofold (C2) axis.
C and C3 3-1 C6 C ; C ; C and C ; C 6 3 2 6 -1 3 -1 C5
The principle rotation axis
Operators and Symmetry elements
The H O molecule has two mirror
planes. They are both vertical
(that is contain the vertical axis) and as so are denoted and 2 v v σ σ '
Benzene has one mirror plane
perpendicular to the principle C - axis (6 σh)
Operators and Symmetry elements
C2
C2
C2 C6
Dihedral mirror planes ( bisect the C axis perpendicular to the
principle axis
d 2
A regular octahedron has a centre of inversion (i).
(a). A CH4 molecule has a fourfold improper rotation axis (S4):
the molecule is indistinguishable after a 90° rotation followed by a reflection across the horizontal plane, but neither operation
alone is a symmetry operation.
Operators and Symmetry elements
C4
(b) The staggered form of ethane has an S6 axis
composed of a 60° rotation followed by a reflection.
The Symmetry Classification of Molecules The Schoenflies system
The Symmetry Classification of Molecules The groups C , C , C1 i s
A molecule belongs to C if it has only the identitity E
1 I
F Cl
Br
A molecule belongs to C if it has only the identitity E and i
i HO H COOH OH HOOC H Meso-tartaric acid N Quinoline A molecule belongs to Cs if it
has only the identity and a mirror plane
The Symmetry Classification of Molecules The groups Cn, Cnv, and C nh
A molecule belongs to C if it
has a C axis and the identitity E only
n n O O H H C2 H2O2 C2 C3 C4 C5 C6 Cn
The Symmetry Classification of Molecules The groups Cn, Cnv, and C nh
A molecule belongs to C if it has a C axis and n vertical mirror planes nv n C2v C3v C4v C5v C6v C∞v cone CO Heteronuclear diatomics are C∞v Cnv H O2 C v 2 C3v NH3
The Symmetry Classification of Molecules The groups Cn, Cnv, and C nh
A molecule belongs to C if it has a C axis and a horizontal mirror plane nh n C2h C3h C4h C5h C6h C C Cl H H Cl Trans CHCl=CHCl B O O O H H H B(OH)3 Cnh
The presence of a twofold axis and a horizontal mirror plane jointly imply the presence of a centre of inversion in the
molecule.
The Symmetry Classification of Molecules The groups Cn, Cnv, and C nh
Cnh
C2
σ
The Symmetry Classification of Molecules The groups Dn, Dnv, and D nh
A molecule belongs to D if it
has a C axis and n two fold axes (C ) perpendicular to C .
n n
2 n
A molecule with n twofold rotation axes perpendicular to an n-fold rotation axis belongs to the group Dn.
Dn Pt N N N N N N D3
The Symmetry Classification of Molecules The groups Dn, Dnv, and D nh
D2 D
3 D4 D5 D6
The Symmetry Classification of Molecules The groups Dn, Dnv, and D nh
A molecule belongs to D if it
has a C axis and n two fold axes (C ) perpendicular to C as well as a horizontal mirror plane.
nh n 2 n A molecule with a Mirror plane perpendicular to a Cn axis, and with n twofold axes in the plane, belongs to the group
Dnh.
The Symmetry Classification of Molecules The groups Dn, Dnv, and D nh
A molecule belongs to D if it has a C axis and n two fold axes (C ) perpendicular to C as well as a horizontal mirror plane.
nh n 2 n Dnh D2h D3h D4h D5h D6h C C H H H H C4 C2 C2 σh D2h Cl Cl Cl Au Cl C4 C2 C2 C'2 C2 σh D4h
The Symmetry Classification of Molecules The groups Dn, Dnv, and D nh
A molecule belongs to D if it
has a C axis and n two fold axes (C ) perpendicular to C as well as a horizontal mirror plane.
nh n 2 n E , C , 3C , 3C'6 2 2 , σh B F F F E , C , 3C 3 2 , σh Dnh
The Symmetry Classification of Molecules Dnh F P F F F F C3 C2 C2 C2 σh
The groups Dn, Dnv, and D nh
The Symmetry Classification of Molecules The groups Dn, Dnv, and D nh
A molecule belongs to D if in addition to the elements of
D it posesses n dihedral mirror planes nd n Dnd D2d D3d D4d D5d D6d C C C C2 C2 C2 D2d
What you must learn from this lecture
1. Given a molecule, you should be able to identify the different symmetry elements (Cn,σv,σh,σd,i,Sn, etc. )
2. Having identified the symmetry elements of a molecule, you should be able to establish the point group to which the molecule belong
