Transcript Rotation by 2

CHEM 305:
Group Theory and Spectroscopy
Part 1. Group Theory
© Dr. Jaime Martell
Molecular Symmetry: Group Theory
Symmetry Elements and Symmetry Operations
Point Groups and Multiplication Tables
Reducible and Irreducible Representations
Application to Molecular Vibrations and Infrared and
Raman Spectroscopy
Symmetry Elements and
Symmetry Operations
• A Symmetry Element is a point, line or plane with
respect to which a symmetry operation is
performed.
• A Symmetry Operation is a physical manipulation
of an object which places it in a new orientation
that is indistinguishable from the original.
• NOTE the difference- the symmetry operation is
an action, the symmetry element is a geometric
entity about which an action takes place.
Symmetry operations may be of
two kinds
– Symmetry operations that move the entire
molecule from one location to another
such that it moves to a location previously
occupied by an identical molecule are
required in crystallographic applications of
group theory.
– Symmetry operations that leave the
molecule in the same physical place but
reorient it in space represent the subset of
symmetry operations required for analysis
of bonding.
Elements
Point--------------------
Operations
Inversion through the point.
n-fold Proper Axis------
Rotation by 2/n about the axis.
n-fold Improper Axis---
Rotation by 2/n about the axis
followed by reflection in a plane
perpendicular to the axis.
Plane----------------
None (or All) of the above--
Reflection in the plane.
Do nothing to the object is
also a symmetry operation.
Symmetry Operations and their
Properties
• The Identity Operation:
– Chemists give this the symbol E, while
mathematicians call it I.
– This operation makes no changes in the
object.
• It amounts to leaving the object alone.
• All the other symmetry operations, if done
enough times, will correspond to the
identity. For example, two successive
reflections in a plane correspond to leaving
the object alone.
• Its analog in the real number system is the
number 1 which can be multiplied by
anything leaving the thing it is multiplied by
unchanged.
THE IDENTITY OPERATION – leave alone or
recover to original
0
1
2
3
4
Inversion (i)
• This operation is
inversion through a
center of symmetry.
For each point in the
molecule, move to the
center, and then
move the same
distance to the other
side.
•
Properties of Inversion:
– The operation consists of an exact reversal of
all the unit vectors.
– Doing two successive inversions is equivalent
to the identity operation.
z
y
x
Axes of rotation (Cn)
• Rotation about an
axis of symmetry
through 360/n (or
2/n. The operation
C1 is equivalent to E.
• An H2O molecule has
a two-fold axis, C2 ,
and NH3 has a three
axis, C3 .
•
Properties of n-fold Proper Rotations:
– If the operation is carried out m times, the operation is
designated Cnm.
– Cnn is rotation by 2 radians and is the same as the
identity.
• Many molecules have more than one proper rotation
axis. When this happens,
– The rotation axis with the largest value of n is called
the principal axis or the highest order rotation axis.
– The highest order axis is usually used as the molecular
axis.
– When there is more than one C2 axis, they are
distinguished by primes (‘). The usual convention is
that C2 axes passing through atoms get a single prime
(‘) while those passing between atoms get a double
prime (“)
Proper Rotations
z
y
The behaviour of the
 = C2(y) unit vectors depends on
the angle of rotation.
x
The unit vector lying along the
rotation axis is always left unchanged,
regardless of the value of n.
PROPER ROTATIONS: Principal Axis, Molecular
Axis, and Different C2s
z
C"2
CO
y
x
Ni
OC
CO
C '2
CO
 = C2(y)
C '2
C"2
The Reflection Operation:
– The symbol for this is the Greek letter 
(German Spiegel = mirror).
– This corresponds to reflection in a
plane.
– The unit vectors lying in the plane
remain unchanged, the one
perpendicular to the plane gets
reversed in direction.
– Two successive reflections correspond
to the identity.
Reflection in a Plane
z
y
x
Vertical mirror planes (v)
• If the plane contains
the principal axis, it is
called vertical and
denoted v.
Horizontal mirror planes (h)
• If the plane of
symmetry is
perpendicular to the
principal axis it is
called horizontal and
denoted h . An
example is benzene.
Dihedral mirror planes (d)
• If a vertical plane
bisects the angle
between two C2 axes
it is called a dihedral
plane and denoted d
.
•
Improper Rotations (Sn) :
– The operation is rotation by an angle of 2/n followed by
reflection in a plane perpendicular to the rotation axis.
– The sequential performance of m of these operations is
designated Snm.
– If n is even, Snn corresponds to
• Rotation by 2 radians and
• An even number of reflections
• And therefore is equivalent to the identity.
– If n is odd, Snn contains an odd number of reflections
and is not equivalent to the identity.
– If n is odd, Sn2n corresponds to
• Rotation by 4 radians and
• An even number of reflections
• And therefore is equivalent to the identity.
Improper Rotation
z
y
2/4 = C4(z)
x
x,y
S4