Lecture 4
Lecture 4
Symmetry and group theory
Natural symmetry in plants
Symmetryin animals
Symmetry in the human body
Symmetry in modern artM. C. Escher
Symmetry in arab architectureLa Alhambra, Granada (Spain)
Symmetry in baroque artGianlorenzo BerniniSaint Peter’s ChurchRome
7th grade art projectSilver Star SchoolVernon, Canada
Re2(CO)10
C2F4 C60
Symmetry in chemistry
•Molecular structures•Wave functions•Description of orbitals and bonds•Reaction pathways•Optical activity•Spectral interpretation (electronic, IR, NMR)...
A molecule is said to have symmetry if some parts of it may be interchangedby others without altering the identity or the orientation of the molecule
Molecular structures
Symmetry Operation:
Transformation of an object into an equivalent or indistinguishableorientation
C3, 120º
Symmetry Elements:
A point, line or plane about which a symmetry operation is carried out
5 types of symmetry operations/elements
Identity: this operation does nothing, symbol: E
Operation 1: Identity Operation, do nothing.
Operation 2: Cn, Proper Rotation:Rotation about an axis by an angle of 2/n = 360/n
How about: NFO2?
H2ONH3
C2 C3
180° (2/2)
C2
The Operation: Proper rotation Cn is the movement (2/n)
The Element: Proper rotation axis Cn is the line
Applying C2 twiceReturns molecule to original oreintation
C22 = E
Rotation angle Symmetry operation
60º C6
120º C3 (= C62)
180º C2 (= C63)
240º C32(= C6
4)
300º C65
360º E (= C66)
C2
PtCl4
Proper Rotation:Cn = Rotation about an axis by an angle of 2/n
PtCl4
Proper Rotation:Cn = Rotation about an axis by an angle of 2/n
C4
PtCl4
Proper Rotation:Cn = Rotation about an axis by an angle of 2/n
C2
PtCl4
Proper Rotation:Cn = Rotation about an axis by an angle of 2/n
C2
C2
PtCl4
Proper Rotation:Cn = Rotation about an axis by an angle of 2/n
C2
PtCl4
Proper Rotation:Cn = Rotation about an axis by an angle of 2/n
Operations can be performed sequentially
nnn
nn
CC
EC
1
Can perform operation several times.
mnC
...nnnmn CCCC
Means m successive rotations of 2/n each time. Total rotation is 2m/n
m times
Observe
The highest order rotation axisis the principal axis
and it is chosen as the z axis
Iron pentacarbonyl, Fe(CO)5C3 axis
What other rotational axes do we have here?
Let’s look at the effect of a rotation on an algebraic function
Consider the pz orbital and let’s rotate it CCW by 90 degrees.
px proportional to xe-r where r = sqrt(x2 + y2 + z2) using a coordinate system centered on the nucleus
x
y
x
y
How do we express this mathematically?
The rotation moves the function as shown.
The value of the rotated function, C4 px, at point o is the same as the value of the original function px at the point o .
The value of C4 px at the general point (x,y,z) is the value of px at the point (y,-x,z)
Moving to a general function f(x,y,z) we have C4 f(x,y,z) = f(y,-x,z)
px C4 px
C4
oo
Thus C4 can be expressed as (x,y,z) (y,-x,z). If C4 is a symmetry element for f then f(x,y,z) = f(y,-x,z)
According to the pictures we see that C4 px yields py.
Let’s do it analytically using C4 f(x,y,z) = f(y,-x,z)
We start with px = xe-r where r = sqrt(x2 + y2 + z2) and make the required substitution to perform C4
x
y
x
y
px C4 px
C4
oo
Thus C4 px (x,y,z) = C4 xe-r = ye-r = py
And we can say that C4 around the z axis as shown is not a symmetry element for px
Operation 3: Reflection and reflection planes
(mirrors)
(reflection through a mirror plane)
NH3
Only one ?
H2O, reflection plane, perp to board
What is the exchange of atoms here?
’
H2O another, different reflection plane
What is the exchange of atoms here?
B
F F
F
If the plane containsthe principal axis it is called v
B
F F
F
If the plane is perpendicularto the principal axis
it is called h
n = E (n = even)n = (n = odd)
Classification of reflection planes
Sequential Application:
Operation 4: Inversion: i
Center of inversion or center of symmetry(x,y,z) (-x,-y,-z)
in = E (n is even)in = i (n is odd)
Inversion not the same as C2 rotation !!
Figures with center of inversion
Figures without center of inversion
Operation 5: Improper rotation (and improper rotation axis): Sn
Rotation about an axis by an angle 2/nfollowed by reflection through perpendicular plane
S4 in methane, tetrahedral structure.
Some things to ponder: S42 = C2
Also, S44 = E; S2 = i; S1 =
Summary: Symmetry operations and elements
Operation Element
proper rotation axis (Cn)
improper rotation axis (Sn)
reflection plane (s)
inversion center (i)
Identity (E)
Successive operations, Multiplication of Operators
Already talked about multiplication of rotational Operators
mnC Means m successive rotations of 2/n each
time. Total rotation is 2m/n
But let’s examine some other multiplications of operators
C4
12
3
4
41
2
3
4
12
3
C4 ’
C4
We write x C4 = ’, first done appears to right in this relationship between operators.
Translational symmetry not point symmetry
Symmetry point groups
The set of all possible symmetry operations on a moleculeis called the point group (there are 28 point groups)
The mathematical treatment of the properties of groupsis Group Theory
In chemistry, group theory allows the assignment of structures,the definition of orbitals, analysis of vibrations, ...
See: Chemical Applications of Group Theory by F. A. Cotton
To determinethe point groupof a molecule
Groups of low symmetry