Lecture 12 February 11, 2019
Symmetry in Nature and in Molecules
Symmetry Operations
Symmetry Elements
Point Groups and Assignments
, Symmetry
Intuitively, we know symmetry when we see it.
But how do we put in quantitative terms that allows us to compare, assign, classify?
, Symmetry in Nature and in Molecules
The symmetry of a molecule is determined by the existence of symmetry operations
performed with respect to symmetry elements. A symmetry element is a line, a
plane or a point within or through an object, intersecting at a specific point (hence
point groups) about which a rotation or reflection leaves the object in an
orientation indistinguishable from the original. A plane of symmetry is designated by
the symbol σ (or sometimes s), and the reflection operation is the coincidence of
atoms on one side of the plane with corresponding atoms on the other side, as
though reflected in a mirror. A center or point of symmetry is labeled i, and the
inversion operation demonstrates coincidence of each atom with an identical one on
a line passing through and an equal distance from the inversion point. Finally, a
rotational axis is designated Cn, where the degrees of rotation that restore the
object is 360/n (C2= 180º rotation, C3= 120º rotation, C4= 90º rotation, C5= 72º
rotation). C1 is called the identity operation E because it returns the original
orientation.
An object having no symmetry elements other than E is called asymmetric. Such an
object is necessarily chiral. Since a plane or point of symmetry involves a reflection
operation, the presence of such an element makes an object achiral. One or more
rotational axes of symmetry may exist in both chiral, dissymmetric, and achiral
objects.
, Symmetry Operations and Symmetry Elements
Definitions:
A symmetry operation is an operation on a body such that, after the operation
has been carried out, the result is indistinguishable from the original body (every
point of the body is coincident with an equivalent point or the same point of the
body in its original orientation).
A symmetry element is a geometrical entity such as a line, a plane, or a point,
with respect to which one or more symmetry operations may be carried out
Symmetry Operation Symmetry Element Notation
Identity ‐ E
Reflection in a plane Plane of symmetry σvσ, σd, σh
Proper rotation Rotation axis (line) Cn ; whereC=n 360/angle
Rotation followed by reflection in Improper rotation axis Sn
the plane perpendicular to the (line)
rotation axis
Inversion Center of inversion I
Symmetry in Nature and in Molecules
Symmetry Operations
Symmetry Elements
Point Groups and Assignments
, Symmetry
Intuitively, we know symmetry when we see it.
But how do we put in quantitative terms that allows us to compare, assign, classify?
, Symmetry in Nature and in Molecules
The symmetry of a molecule is determined by the existence of symmetry operations
performed with respect to symmetry elements. A symmetry element is a line, a
plane or a point within or through an object, intersecting at a specific point (hence
point groups) about which a rotation or reflection leaves the object in an
orientation indistinguishable from the original. A plane of symmetry is designated by
the symbol σ (or sometimes s), and the reflection operation is the coincidence of
atoms on one side of the plane with corresponding atoms on the other side, as
though reflected in a mirror. A center or point of symmetry is labeled i, and the
inversion operation demonstrates coincidence of each atom with an identical one on
a line passing through and an equal distance from the inversion point. Finally, a
rotational axis is designated Cn, where the degrees of rotation that restore the
object is 360/n (C2= 180º rotation, C3= 120º rotation, C4= 90º rotation, C5= 72º
rotation). C1 is called the identity operation E because it returns the original
orientation.
An object having no symmetry elements other than E is called asymmetric. Such an
object is necessarily chiral. Since a plane or point of symmetry involves a reflection
operation, the presence of such an element makes an object achiral. One or more
rotational axes of symmetry may exist in both chiral, dissymmetric, and achiral
objects.
, Symmetry Operations and Symmetry Elements
Definitions:
A symmetry operation is an operation on a body such that, after the operation
has been carried out, the result is indistinguishable from the original body (every
point of the body is coincident with an equivalent point or the same point of the
body in its original orientation).
A symmetry element is a geometrical entity such as a line, a plane, or a point,
with respect to which one or more symmetry operations may be carried out
Symmetry Operation Symmetry Element Notation
Identity ‐ E
Reflection in a plane Plane of symmetry σvσ, σd, σh
Proper rotation Rotation axis (line) Cn ; whereC=n 360/angle
Rotation followed by reflection in Improper rotation axis Sn
the plane perpendicular to the (line)
rotation axis
Inversion Center of inversion I
