Transcript Lecture 2

Lecture 4
Symmetry and group theory
Natural symmetry in plants
Symmetry
in animals
Symmetry in the human body
Symmetry in modern art
M. C. Escher
Symmetry in arab architecture
La Alhambra, Granada (Spain)
Symmetry in baroque art
Gianlorenzo Bernini
Saint Peter’s Church
Rome
7th grade art project
Silver Star School
Vernon, 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)
...
Molecular structures
A molecule is said to have symmetry if some parts of it may be interchanged
by others without altering the identity or the orientation of the molecule
Symmetry Operation:
Transformation of an object into an equivalent or indistinguishable
orientation
C3, 120º
Symmetry Elements:
A point, line or plane about which a symmetry operation is
carried out
5 types of symmetry operations/elements
Operation 1: Identity Operation, do nothing.
Identity: this operation does nothing, symbol: E
Operation 2: Cn, Proper Rotation:
Rotation about an axis by an angle of 2/n = 360/n
C2
H2O
How about:
C3
NH3
NFO2?
The Operation: Proper rotation Cn is the movement (2/n)
The Element: Proper rotation axis Cn is the line
180° (2/2)
Applying C2 twice
Returns molecule to original oreintation
C 22 = E
C2
Rotation angle Symmetry
operation
60º
C6
120º
C3 (= C62)
180º
C2 (= C63)
240º
C32(= C64)
300º
C65
360º
E (= C66)
Proper Rotation:
Cn = Rotation about an axis by an angle of 2/n
PtCl4
C2
Proper Rotation:
Cn = Rotation about an axis by an angle of 2/n
PtCl4
C4
Proper Rotation:
Cn = Rotation about an axis by an angle of 2/n
PtCl4
C2
Proper Rotation:
Cn = Rotation about an axis by an angle of 2/n
PtCl4
C2
Proper Rotation:
Cn = Rotation about an axis by an angle of 2/n
PtCl4
C2
Proper Rotation:
Cn = Rotation about an axis by an angle of 2/n
PtCl4
C2
Operations can be performed sequentially
Can perform operation several times.
C
m
n
Means m successive rotations of 2/n each
time. Total rotation is 2m/n
Cnm  Cn  Cn  Cn ...
Observe
C E
n
n
C
n 1
n
 Cn
m times
C3 axis
Iron pentacarbonyl, Fe(CO)5
The highest order rotation axis
is the principal axis
and it is chosen as the z 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-ar where r = sqrt(x2 + y2 + z2) using a coordinate
system centered on the nucleus
y
y
C4
x
o
o
px
x
C4 px
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)
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-ar where r = sqrt(x2 + y2 + z2) and make the
required substitution to perform C4
y
y
C4
x
o
o
px
x
C4 px
Thus C4 px (x,y,z) = C4 xe-ar = ye-ar = 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)
s
s
s (reflection through a mirror plane)
s
NH3
Only one
s?
H2O, reflection plane, perp to board
s
What is the exchange of
atoms here?
H2O another, different reflection plane
s’
What is the exchange of
atoms here?
Classification of reflection planes
F
F
B
If the plane contains
the principal axis it is called sv
F
F
If the plane is perpendicular
to the principal axis
it is called sh
Sequential Application:
sn = E (n = even)
sn = s (n = odd)
F
B
F
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/n
followed by reflection through perpendicular plane
S4 in methane, tetrahedral structure.
Some things to ponder: S42 = C2
Also, S44 = E; S2 = i; S1 = s
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
C
m
n
Means m successive rotations of 2/n each
time. Total rotation is 2m/n
But let’s examine some other multiplications of operators
C4
1
2
1
C4
s
3
C4
We write s x C4
operators.
2
2
3
4
1
4
4
3
s’
s
= s’, first done appears to right in this relationship between
Translational symmetry
not point symmetry
Symmetry point groups
The set of all possible symmetry operations on a molecule
is called the point group (there are 28 point groups)
The mathematical treatment of the properties of groups
is 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 determine
the point group
of a molecule
Groups of low symmetry