Point Group Definition
A classification scheme for finite objects (molecules)
Molecules having the same set of symmetry
elements/operations “belong to” the same point group
Point groups have labels we will learn
We will use the Schoenflies notation (spectroscopy) rather
than the Hermann-Mauguin notation (crystallography)
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Inorganic Chemistry
5.03
Point Group Definition
A classification scheme for finite objects (molecules)
Molecules having the same set of symmetry
elements/operations “belong to” the same point group
Point groups have labels we will learn
We will use the Schoenflies notation (spectroscopy) rather
than the Hermann-Mauguin notation (crystallography)
../../logo
Inorganic Chemistry
5.03
Point Group Definition
A classification scheme for finite objects (molecules)
Molecules having the same set of symmetry
elements/operations “belong to” the same point group
Point groups have labels we will learn
We will use the Schoenflies notation (spectroscopy) rather
than the Hermann-Mauguin notation (crystallography)
../../logo
Inorganic Chemistry
5.03
Point Group Definition
A classification scheme for finite objects (molecules)
Molecules having the same set of symmetry
elements/operations “belong to” the same point group
Point groups have labels we will learn
We will use the Schoenflies notation (spectroscopy) rather
than the Hermann-Mauguin notation (crystallography)
../../logo
Inorganic Chemistry
5.03
Types of Point Groups
Labels are the Schoenflies symbols: here are six limiting symmetry types
High symmetry, multiple higher-order (n > 2) rotation axes.
Examples: Td , Ih , Oh
Low symmetry, only the identity or that plus only a single
mirror plane or an inversion center: C1 , Cs , Ci
Linear molecules: C∞v , D∞h
C groups: Cnv , Cnh , Cn
D groups: Dnd , Dnh , Dn
S groups: S4 , S6 , S8 , etc.; only operations present are based
upon Sn
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Inorganic Chemistry
5.03
Types of Point Groups
Labels are the Schoenflies symbols: here are six limiting symmetry types
High symmetry, multiple higher-order (n > 2) rotation axes.
Examples: Td , Ih , Oh
Low symmetry, only the identity or that plus only a single
mirror plane or an inversion center: C1 , Cs , Ci
Linear molecules: C∞v , D∞h
C groups: Cnv , Cnh , Cn
D groups: Dnd , Dnh , Dn
S groups: S4 , S6 , S8 , etc.; only operations present are based
upon Sn
../../logo
Inorganic Chemistry
5.03
Types of Point Groups
Labels are the Schoenflies symbols: here are six limiting symmetry types
High symmetry, multiple higher-order (n > 2) rotation axes.
Examples: Td , Ih , Oh
Low symmetry, only the identity or that plus only a single
mirror plane or an inversion center: C1 , Cs , Ci
Linear molecules: C∞v , D∞h
C groups: Cnv , Cnh , Cn
D groups: Dnd , Dnh , Dn
S groups: S4 , S6 , S8 , etc.; only operations present are based
upon Sn
../../logo
Inorganic Chemistry
5.03
Types of Point Groups
Labels are the Schoenflies symbols: here are six limiting symmetry types
High symmetry, multiple higher-order (n > 2) rotation axes.
Examples: Td , Ih , Oh
Low symmetry, only the identity or that plus only a single
mirror plane or an inversion center: C1 , Cs , Ci
Linear molecules: C∞v , D∞h
C groups: Cnv , Cnh , Cn
D groups: Dnd , Dnh , Dn
S groups: S4 , S6 , S8 , etc.; only operations present are based
upon Sn
../../logo
Inorganic Chemistry
5.03
Types of Point Groups
Labels are the Schoenflies symbols: here are six limiting symmetry types
High symmetry, multiple higher-order (n > 2) rotation axes.
Examples: Td , Ih , Oh
Low symmetry, only the identity or that plus only a single
mirror plane or an inversion center: C1 , Cs , Ci
Linear molecules: C∞v , D∞h
C groups: Cnv , Cnh , Cn
D groups: Dnd , Dnh , Dn
S groups: S4 , S6 , S8 , etc.; only operations present are based
upon Sn
../../logo
Inorganic Chemistry
5.03
Types of Point Groups
Labels are the Schoenflies symbols: here are six limiting symmetry types
High symmetry, multiple higher-order (n > 2) rotation axes.
Examples: Td , Ih , Oh
Low symmetry, only the identity or that plus only a single
mirror plane or an inversion center: C1 , Cs , Ci
Linear molecules: C∞v , D∞h
C groups: Cnv , Cnh , Cn
D groups: Dnd , Dnh , Dn
S groups: S4 , S6 , S8 , etc.; only operations present are based
upon Sn
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Inorganic Chemistry
5.03
The Platonic Solids: Polyhedra with Regular Polygon Faces
Symmetry lowered from the sphere, but still present are multiple higher-order axes
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Inorganic Chemistry
5.03
High Symmetry Groups
These have multiple higher order (n > 2) rotation axes. Example: C60 , icosahedral
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Inorganic Chemistry
5.03
High Symmetry Groups
These have multiple higher order (n > 2) rotation axes. Example: [B12 H12 ]2− , icosahedral
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Inorganic Chemistry
5.03
High Symmetry Groups
The tetrahedron has four C3 axes but lacks inversion center
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Inorganic Chemistry
5.03
High Symmetry Groups
The group Th has four C3 axes (through octahedral faces) and adds the inversion center
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Inorganic Chemistry
5.03
High Symmetry Groups
The group T is a pure rotation group with no mirror planes or inversion centers
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Inorganic Chemistry
5.03
High Symmetry Groups
The group Oh has three C4 axes and an inversion center: [Mo6 Cl14 ]2−
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Inorganic Chemistry
5.03
High Symmetry Groups
The group Oh has three C4 axes and an inversion center: SF6
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Inorganic Chemistry
5.03
Groups of Low Symmetry
The identity alone, or together with one mirror or an inversion center
The identity only, C1
The identity plus one mirror plane: Cs
The identity plus an inversion center: Ci
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Inorganic Chemistry
5.03
Groups of Low Symmetry
The identity alone, or together with one mirror or an inversion center
The identity only, C1
The identity plus one mirror plane: Cs
The identity plus an inversion center: Ci
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Inorganic Chemistry
5.03
Groups of Low Symmetry
The identity alone, or together with one mirror or an inversion center
The identity only, C1
The identity plus one mirror plane: Cs
The identity plus an inversion center: Ci
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Inorganic Chemistry
5.03
Groups of Low Symmetry
The identity alone, or together with one mirror or an inversion center
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Inorganic Chemistry
5.03
Groups of Low Symmetry
The identity alone, or together with one mirror or an inversion center
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Inorganic Chemistry
5.03
Linear Molecules
Distinguish based upon presence or absence of ⊥ C2 axes
A linear molecule has a C∞ axis of rotation
Nitrous oxide, N2 O, N=N=O, the two ends are different so no
C2 ⊥ to the C∞ , the point group assignment is C∞v
Carbon dioxide, CO2 , O=C=O, the two ends are “symmetry
related” and exchangeable by ⊥ C2 or by σh so the point
group assignment is D∞h
In general, D groups have nC2 axes ⊥ to the Cn (single
principal rotation axis)
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Inorganic Chemistry
5.03
Linear Molecules
Distinguish based upon presence or absence of ⊥ C2 axes
A linear molecule has a C∞ axis of rotation
Nitrous oxide, N2 O, N=N=O, the two ends are different so no
C2 ⊥ to the C∞ , the point group assignment is C∞v
Carbon dioxide, CO2 , O=C=O, the two ends are “symmetry
related” and exchangeable by ⊥ C2 or by σh so the point
group assignment is D∞h
In general, D groups have nC2 axes ⊥ to the Cn (single
principal rotation axis)
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Inorganic Chemistry
5.03
Linear Molecules
Distinguish based upon presence or absence of ⊥ C2 axes
A linear molecule has a C∞ axis of rotation
Nitrous oxide, N2 O, N=N=O, the two ends are different so no
C2 ⊥ to the C∞ , the point group assignment is C∞v
Carbon dioxide, CO2 , O=C=O, the two ends are “symmetry
related” and exchangeable by ⊥ C2 or by σh so the point
group assignment is D∞h
In general, D groups have nC2 axes ⊥ to the Cn (single
principal rotation axis)
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Inorganic Chemistry
5.03
Linear Molecules
Distinguish based upon presence or absence of ⊥ C2 axes
A linear molecule has a C∞ axis of rotation
Nitrous oxide, N2 O, N=N=O, the two ends are different so no
C2 ⊥ to the C∞ , the point group assignment is C∞v
Carbon dioxide, CO2 , O=C=O, the two ends are “symmetry
related” and exchangeable by ⊥ C2 or by σh so the point
group assignment is D∞h
In general, D groups have nC2 axes ⊥ to the Cn (single
principal rotation axis)
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Inorganic Chemistry
5.03
Examples of C groups
A single Cn plus n vertical mirror planes
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Inorganic Chemistry
5.03
Examples of C groups
A single Cn plus n vertical mirror planes
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Inorganic Chemistry
5.03
Examples of C groups
A single Cn plus a horizontal mirror plane
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Inorganic Chemistry
5.03
Examples of C groups
A single C2 with no mirror planes: ansa metallocene example of point group C2
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Inorganic Chemistry
5.03
Examples of D groups
Dnh has n C2 ⊥ to the Cn , plus a σh
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Inorganic Chemistry
5.03
Examples of D groups
Dnh has n C2 ⊥ to the Cn , plus a σh
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Inorganic Chemistry
5.03
Examples of D groups
Dnd has n C2 ⊥ to the Cn , plus nσd but no σh ; example is S4 N4
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Inorganic Chemistry
5.03
Examples of D groups
Dnd has n C2 ⊥ to the Cn , plus nσd but no σh ; example is S8
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Inorganic Chemistry
5.03
Examples of D groups
Dn has n C2 ⊥ to the Cn , but no mirror planes; example is [Fe(C2 O4 )3 ]3−
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Inorganic Chemistry
5.03
Example of Sn groups
Sn only for n = 2, 4, 6, ... but note, S2 = inversion so that is Ci
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Inorganic Chemistry
5.03
Types of Point Groups
Labels are the Schoenflies symbols: here are six limiting symmetry types
High symmetry, multiple higher-order (n > 2) rotation axes.
Examples: Td , Ih , Oh
Low symmetry, only the identity or that plus only a single
mirror plane or an inversion center: C1 , Cs , Ci
Linear molecules: C∞v , D∞h
C groups: Cnv , Cnh , Cn
D groups: Dnd , Dnh , Dn
S groups: S4 , S6 , S8 , etc.; only operations present are based
upon Sn
../../logo
Inorganic Chemistry
5.03
Types of Point Groups
Labels are the Schoenflies symbols: here are six limiting symmetry types
High symmetry, multiple higher-order (n > 2) rotation axes.
Examples: Td , Ih , Oh
Low symmetry, only the identity or that plus only a single
mirror plane or an inversion center: C1 , Cs , Ci
Linear molecules: C∞v , D∞h
C groups: Cnv , Cnh , Cn
D groups: Dnd , Dnh , Dn
S groups: S4 , S6 , S8 , etc.; only operations present are based
upon Sn
../../logo
Inorganic Chemistry
5.03
Types of Point Groups
Labels are the Schoenflies symbols: here are six limiting symmetry types
High symmetry, multiple higher-order (n > 2) rotation axes.
Examples: Td , Ih , Oh
Low symmetry, only the identity or that plus only a single
mirror plane or an inversion center: C1 , Cs , Ci
Linear molecules: C∞v , D∞h
C groups: Cnv , Cnh , Cn
D groups: Dnd , Dnh , Dn
S groups: S4 , S6 , S8 , etc.; only operations present are based
upon Sn
../../logo
Inorganic Chemistry
5.03
Types of Point Groups
Labels are the Schoenflies symbols: here are six limiting symmetry types
High symmetry, multiple higher-order (n > 2) rotation axes.
Examples: Td , Ih , Oh
Low symmetry, only the identity or that plus only a single
mirror plane or an inversion center: C1 , Cs , Ci
Linear molecules: C∞v , D∞h
C groups: Cnv , Cnh , Cn
D groups: Dnd , Dnh , Dn
S groups: S4 , S6 , S8 , etc.; only operations present are based
upon Sn
../../logo
Inorganic Chemistry
5.03
Types of Point Groups
Labels are the Schoenflies symbols: here are six limiting symmetry types
High symmetry, multiple higher-order (n > 2) rotation axes.
Examples: Td , Ih , Oh
Low symmetry, only the identity or that plus only a single
mirror plane or an inversion center: C1 , Cs , Ci
Linear molecules: C∞v , D∞h
C groups: Cnv , Cnh , Cn
D groups: Dnd , Dnh , Dn
S groups: S4 , S6 , S8 , etc.; only operations present are based
upon Sn
../../logo
Inorganic Chemistry
5.03
Types of Point Groups
Labels are the Schoenflies symbols: here are six limiting symmetry types
High symmetry, multiple higher-order (n > 2) rotation axes.
Examples: Td , Ih , Oh
Low symmetry, only the identity or that plus only a single
mirror plane or an inversion center: C1 , Cs , Ci
Linear molecules: C∞v , D∞h
C groups: Cnv , Cnh , Cn
D groups: Dnd , Dnh , Dn
S groups: S4 , S6 , S8 , etc.; only operations present are based
upon Sn
../../logo
Inorganic Chemistry
5.03
Molecular
Structure
Is the molecule
linear?
Does the
molecule contain an
inversion center?
Yes
No
Yes
D!h
No
C!v
Does the
molecule contain
two or more unique
C3 axes?
Does the
molecule contain
two or more unique
C5 axes?
Yes
No
Does the
molecule contain an
inversion center?
Yes
No
Yes
Ih
Yes
Oh
Yes
Th
No
I
Does the
molecule contain
two or more unique
C4 axes?
Does the
molecule contain an
inversion center?
Yes
No
No
O
Does the molecule
contain one or more
reflection planes?
Does the
molecule contain an
inversion center?
Yes
No
No
Td
T
Does the
molecule contain a
proper rotation axis
(Cn)?
Identify the
highest order Cn.
Yes
No
Does the
molecule contain a
reflection plane?
Does the
molecule contain a
horizontal reflection
plane ("h)?
Yes
Are there n perpendicular C2 axes?
No
Yes
Cs
No
Does the
molecule contain a
horizontal reflection
plane ("h)?
Yes
Cnh
No
Does the
molecule contain n
dihedral reflection
planes ("d)?
No
Dn
Does the
molecule contain an
inversion center?
No
Yes
Ci
Does the
molecule contain n
vertical reflection
planes ("v)?
Yes
Cnv
Yes
S2n
No
C1
Does the
molecule contain a
2n-fold improper
rotation axis?
No
Cn
Yes
Dnh
Yes
Dnd
No