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Symmetry in modern art
M. C. Escher
Symmetry in Arabic architecture
La Alhambra, Granada (Spain)
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Symmetry in baroque art
Gianlorenzo Bernini
Saint Peter’s Church
Rome
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Platonic solids (5 of the 8 shown)
Archimedean solids (3 of the 8 shown)
cuboctahedron, icosidodecahedron, truncated octahedron
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Molecular Symmetry
• Group theory is an important aspect for
spectroscopy. It is used to explain in details
the symmetry of molecules.
• Group theory is used to:
– label and classify molecule’s energy levels /
molecular orbitals (electronic, vibrational and
rotational)
– look up the possibility of molecular and electronic
transitions between energy levels / molecular
orbitals.
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Point Groups
Molecules are classified and grouped
based on their symmetry. Molecules with
similar symmetry are but into the same
point group. A point group contains all
objects that have the same symmetry
elements.
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Symmetry Operations
• A symmetry operation is geometrical action
that leaves the nuclei in a molecule in
equivalent positions. (leaves them
indistinguishable).
• Five main classes of symmetry operations:
– Reflections (σ).
– Rotation (Cn).
– Rotation-reflection “Improper rotation” (Sn).
– Inversion (i).
– Identity (E). “do nothing”
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Symmetry Operations and Symmetry
Elements
Symmetry Operation Symmetry Element
Reflection (σ) Plane of reflection (σh, σv, σd)
Rotation (Cn) Axis of rotation (principal and
non-principle)
Improper rotation (Sn) Rotation followed by reflection
Inversion (i) Center of inversion
Identity (E) E itself “does nothing”
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Operator Algebra
• Operator algebra is similar in many aspects
to ordinary algebra.
• For: Af1 f2 ,
operator A is said to transform functions f1 to
f2 by a sort of operation.
• Addition of operators:
Cf = (A + B)f = Af + Bf
or
C = (A + B) = A + B
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Operator Algebra
• Multiplication of operators:
Cf = (AB)f = A(Bf)
or
C = (AB) = AB
However, it is important to note that:
A(Bf) is not necessarily equivalent to B(Af).
We say operators A and B do not necessarily
commute.
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Identity Operator (E)
• The identity operator leaves a molecule
unchanged. It is applied for all molecule with
any degree of symmetry or asymmetry.
• The identity operator does nothing. Why are
we still dealing with it?
• It is important not by itself but for specific
operator algebra as going to be discussed
later.
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Identity Operator (E)
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Rotation Operator (Cn)
• Cn rotates a molecule by an angle of 2π/n
radians in a clockwise direction about a Cn axis.
• If a rotation of 2π/n leaves out the molecule
indistinguishable, the molecule is said to have
an n-fold axis of rotation.
1 2
C2
Rotation by
2π/2 radians
2 1
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Rotation Operator (Cn)
• When a molecule has several rotational axes
of symmetry, the one with the largest value
of n is called the principle axis.
Example: Trifluoroborane
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Rotation Operator (Cn)
• Successive Rotations (Cnk).
Cnk = Cn Cn … Cn (k times)
Also:
Cnn = E Cn
n+1 = Cn
• Example: BF3
Rotation by
2π/3 radians
C3
Rotation by
4π/3 radians
C32
Rotation by - 2π/3 radians C3-1
1 2
3
3 1
2
2 3
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Rotation angle Symmetry operation
60º C6
90º C4
120º C3 (= C62)
180º C2 (= C63 = C4
2)
240º C32 (= C6
4)
270º C43
300º C65
360º E
Rotation Operator (Cn)
Cn is shorthand for (n1) rotation operators.
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Reflection Operator (σ)
• σ reflects a molecule through a plane passing
through the center of the molecule. The
molecule is said to have a plane of symmetry.
1 2
C2
Reflection
through σv
plane
2 1
σv
Some simple algebra: σ2 = E
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Reflection Operator (σ)
• There are three types of mirror
planes:
– σv vertical mirror plane which
contains the principle axis.
– σh horizontal mirror plane
which is perpendicular to the
principle axis.
– σd dihedral mirror plane which
is vertical and bisects the
angle between two adjacent
C2 axes that are perpendicular
to the principle axis.
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Inversion Operator (i)
• This operator inverts all atoms through a
point called “center of inversion” or “center
of symmetry”.
i (x,y,z) (-x,-y,-z)
Inversion not the same as C2 rotation !!
Inversion Operator (i)
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Inversion Operator (i)
Figures with center of inversion
Figures without center of inversion
Inversion Operator (i)
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Improper Rotation Operator (Sn)
• This operator applies a clockwise rotation on
the molecule followed by a reflection in a
plane perpendicular to that axis of rotation.
Sn = σhCn
• Example: Methane
C4σh
S4
The staggered
conformation of
ethane has an S6
axis that goes
through both carbon
atoms.
Improper Rotation Operator (Sn)
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Note that an S1
axis does not exist;
it is same as a
mirror plane.
Improper Rotation Operator (Sn)
Likewise, an S2
axis is a center of
inversion.
Improper Rotation Operator (Sn)
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An S3 is equivalent
to simultaneous C3
and s.
S32 = C3
2
S33 = s
S34 = C3
Improper Rotation Operator (Sn)
S42 = C2 S4
4 = E S2 = i S1 = s
Improper Rotation Operator (Sn)
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Symmetry Operator Algebra
• Symmetry operators can be applied
successively to a molecule to produce new
operators.
σv’’’ = σv’’ C3 σv’ = C3 σv’’
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A group multiplication must satisfy the following
conditions in regard with the group’s elements:
1 Closure: If P and Q are elements of a group and
PQ = R , then R must be also an element of that
group.
2 Associative Law: The order of multiplication is
not important. (PQ)R = P(QR).
3 Identity Element: There must be an identity
element (E) in the group so that: RE = ER = R.
4 Inverse: Every element has an inverse in the
group so that: RR-1 = R-1R = E
More Operator Algebra
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If the group elements commute, i.e. PQ = QP,
then the group is said to be “Abelian group”.
For point symmetry groups, we have non-
Abelian groups.
Point groups retain the center of mass of the
molecules unchanged under all symmetry
operations and all of the symmetry elements
meet at this point
More Operator Algebra
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Point Group for Ammonia
• The ammonia molecule has six symmetry
operators.
E , C3, C3-1 (or C3
2), σv’ , σv’’ and σv’’’
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Multiplication Table for NH3
Each operator appears just once in a given row
or column in the table but in a different position.
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Classes
• The members of a group can be divided into
classes.
The members of a class within a group have
a certain type of a geometrical relationship.
For ammonia with the C3v symmetry, the
three classes are:
E , C3 and σv
• The point group C3v will contain E , 2C3 and
3σv elements.