Physical Chemistry
15. Molecular Symmetries

Ruifeng ZHOU

Faculty of Science &Institute for the Advancement of Higher Education
Hokkaido University

Table of Content

1. Symmetry operations and elements
2. Molecular point groups
3. Characters of typical groups

12

1. Symmetry operations and elements• Every molecule has some symmetry elements,
� wrt which, theafter the symmetry operations (𝑅𝑅)
positions of nuclei stay unchanged (nuclei ofisotopes are the same or different).
• Symmetry operations and elements are:
• 𝐸𝐸� is no change. 𝐸𝐸 is identity.
2𝜋𝜋̂• 𝐶𝐶𝑛𝑛 (n-fold rotation) is rotation wrt an axis by .
𝑛𝑛• The axis, 𝐶𝐶𝑛𝑛 , is called n-fold axis of symmetry.
• The 𝐶𝐶𝑛𝑛 with the highest 𝑛𝑛 is defined to be theprincipal axis of the molecule.

1. Symmetry operations and elements• Water has one 𝐶𝐶2 . Methane has 4 𝐶𝐶3 s.

Pictures from https://symotter.org/

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1. Symmetry operations and elements• 𝚤𝚤̂ (centre reflection) is inversion wrt the centre.
• The centre is 𝑖𝑖.
• Difluorodiazene and benzene have 𝑖𝑖.

41. Symmetry operations and elements• 𝜎𝜎� (plane reflection) is reflection wrt a plane. If theplane is perpendicular to the principal axis, it is 𝜎𝜎�ℎ(horizontal).
• PF5 and benzene have 𝜎𝜎ℎ .

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1. Symmetry operations and elements• If the plane contains the principal axis, it is 𝜎𝜎�𝑣𝑣(vertical).
• Water has 2 𝜎𝜎𝑣𝑣 s. Ammonia has 3 𝜎𝜎𝑣𝑣 s.

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1. Symmetry operations and elements

7• If the reflection plane bisects the angle between 2fold axes that are perpendicular to the principal axis,
it is 𝜎𝜎�𝑑𝑑 (dihedral).
• Benzene and ethane have 3 𝜎𝜎𝑑𝑑 s.

1. Symmetry operations and elements
2𝜋𝜋̂• 𝑆𝑆𝑛𝑛 (improper rotation) is rotation wrt an axis by𝑛𝑛and reflection wrt a plane perpendicular to this axis.
𝑆𝑆𝑛𝑛 is called 𝑛𝑛-fold rotation-reflection axis ofsymmetry.
• Sulphur has 𝑆𝑆8 . Ethane has 𝑆𝑆6 .

8

1. Symmetry operations and elements

9 All the symmetry operations are unitarytransformations, which preserve the inner productsof vectors (e.g., functions in Hilbert space).
• The symmetry operations of a molecule and theirconsecutive operations (right first) form a 3D finite
point group (math) or molecular point group (chem).
• There are 30 molecular point groups of chemicalsignificance. They are named after their mainsymmetry elements.
• Molecular point groups are classified according totheir symmetry elements.

2. Molecular point groups

•
•
•
•

Nonaxial groups.
𝐶𝐶1 ： 𝐸𝐸.
𝐶𝐶𝑠𝑠 ： 𝐸𝐸, 𝜎𝜎ℎ .
𝐶𝐶𝑖𝑖 ： 𝐸𝐸, 𝑖𝑖.

C2H2Cl2Br210Bromochlorofluoromethane(CHFClBr)

cis-1-bromo-2-chloroethene (C2H2BrCl)

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2. Molecular point groups

•
•
•
•

𝐶𝐶𝑛𝑛 (cyclic groups): 𝐸𝐸, 𝐶𝐶𝑛𝑛 .
𝐶𝐶𝑛𝑛𝑛 (reflection groups): 𝐶𝐶𝑛𝑛 and 𝜎𝜎ℎ .
Examples: 𝐶𝐶2ℎ : 𝐸𝐸, 𝐶𝐶2 , 𝑖𝑖, 𝜎𝜎ℎ .
𝐶𝐶3ℎ : 𝐸𝐸, 𝐶𝐶3 , 𝜎𝜎ℎ , 𝑆𝑆3 .

𝜎𝜎ℎTrans-1,2-difluoroethylene(C2H2F2)

𝜎𝜎ℎBoric acid (H3BO3)

Hydrogen peroxide (H2O2)

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2. Molecular point groups• 𝐶𝐶𝑛𝑛𝑛𝑛 (pyramidal groups): 𝐶𝐶𝑛𝑛 and 𝑛𝑛 𝜎𝜎𝑣𝑣 s.
• Examples: 𝐶𝐶2𝑣𝑣 : 𝐸𝐸, 𝐶𝐶2 , 𝜎𝜎𝑣𝑣 , 𝜎𝜎𝑣𝑣 ′.
• 𝐶𝐶3𝑣𝑣 : 𝐸𝐸, 𝐶𝐶3 , 3𝜎𝜎𝑣𝑣 .
𝜎𝜎𝑣𝑣 ′
𝜎𝜎𝑣𝑣
Water (H2O)

Ammonia (NH3)

𝜎𝜎𝑣𝑣

𝜎𝜎𝑣𝑣 ′𝜎𝜎𝑣𝑣 ′′2. Molecular point groups• 𝑆𝑆𝑛𝑛 (improper rotation group): 𝑆𝑆𝑛𝑛 .
• Examples: 𝑆𝑆4 : 𝐸𝐸, 𝑆𝑆4 , 𝐶𝐶2 .

Tetrabromoneopentane (C5H8Br4)

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2. Molecular point groups• 𝐷𝐷𝑛𝑛 (dihedral groups): 𝐶𝐶𝑛𝑛 and 𝑛𝑛 𝐶𝐶2 s normal to 𝐶𝐶𝑛𝑛 .
• Examples: 𝐷𝐷2 : 𝐸𝐸, 𝐶𝐶2 , 𝐶𝐶2 ′, 𝐶𝐶2 ′′.
• 𝐷𝐷3 : 𝐸𝐸, 𝐶𝐶3 , 𝐶𝐶2 , 𝐶𝐶2 ′, 𝐶𝐶2 ′′.

Tris(oxalato)iron(III) (Fe(C2O4)23-)

14

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2. Molecular point groups• 𝐷𝐷𝑛𝑛𝑛 (prismatic groups): 𝐶𝐶𝑛𝑛 , 𝑛𝑛 𝐶𝐶2 s normal to 𝐶𝐶𝑛𝑛 and𝜎𝜎ℎ . Examples: 𝐷𝐷2ℎ : 𝐸𝐸, 𝐶𝐶2 , 𝐶𝐶2 ′, 𝐶𝐶2 ′′, 𝑖𝑖, 𝜎𝜎ℎ , 𝜎𝜎𝑣𝑣 , 𝜎𝜎𝑣𝑣 ′.
• 𝐷𝐷3ℎ : 𝐸𝐸, 𝐶𝐶3 , 3𝐶𝐶2 , 𝜎𝜎ℎ , 𝑆𝑆3 , 3𝜎𝜎𝑣𝑣 .
𝜎𝜎𝑣𝑣

𝜎𝜎𝑣𝑣 ′

𝜎𝜎ℎ

1,4-dibromobenzene (C6H4Br2)

𝜎𝜎𝑣𝑣 𝜎𝜎𝑣𝑣 ′𝜎𝜎𝑣𝑣 ′′𝜎𝜎ℎBorane (BH3)16

2. Molecular point groups• 𝐷𝐷4ℎ : 𝐸𝐸, 𝐶𝐶4 , 𝐶𝐶2 , 2𝐶𝐶2 ′, 2𝐶𝐶2 ′′, 𝑖𝑖, 𝑆𝑆4 , 𝜎𝜎ℎ , 2𝜎𝜎𝑣𝑣 , 2𝜎𝜎𝑑𝑑 .
• 𝐷𝐷6ℎ : 𝐸𝐸, 𝐶𝐶6 , 𝐶𝐶3 , 𝐶𝐶2 , 3𝐶𝐶2 ′, 3𝐶𝐶2 ′′, 𝑖𝑖, 𝑆𝑆6 , 𝑆𝑆3 , 𝜎𝜎ℎ , 3𝜎𝜎𝑣𝑣 , 3𝜎𝜎𝑑𝑑 .
𝜎𝜎𝑑𝑑 ′𝜎𝜎𝑣𝑣𝜎𝜎𝑑𝑑𝜎𝜎𝑣𝑣 ′𝜎𝜎ℎTetrachloroplatinate (PtCl42-)

𝜎𝜎𝑣𝑣 ′′

𝜎𝜎𝑑𝑑 ′

𝜎𝜎𝑣𝑣

𝜎𝜎𝑑𝑑𝜎𝜎𝑣𝑣 ′𝜎𝜎𝑑𝑑 ′′

𝜎𝜎ℎBenzene (C6H6)

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2. Molecular point groups• 𝐷𝐷𝑛𝑛𝑑𝑑 (antiprismatic group): 𝐶𝐶𝑛𝑛 , 𝑛𝑛 𝐶𝐶2 s normal to 𝐶𝐶𝑛𝑛
and 𝑛𝑛 𝜎𝜎𝑑𝑑 s. Examples: 𝐷𝐷2𝑑𝑑 : 𝐸𝐸, 𝐶𝐶2 , 𝑆𝑆4 , 2𝐶𝐶2 ′, 2𝜎𝜎𝑑𝑑 .
• 𝐷𝐷3𝑑𝑑 : 𝐸𝐸, 𝐶𝐶3 , 3𝐶𝐶2 , 𝑖𝑖, 𝑆𝑆6 , 3𝜎𝜎𝑑𝑑 .
𝜎𝜎𝑑𝑑 ′

𝜎𝜎𝑑𝑑

Staggered ethane (C2H6)

𝜎𝜎𝑑𝑑 ′𝜎𝜎𝑑𝑑𝜎𝜎𝑑𝑑 ′′Allene (C3H4)

2. Molecular point groups

18• Polyhedral groups: more than one 𝐶𝐶𝑛𝑛 s, 𝑛𝑛 > 2.
• 𝑇𝑇𝑑𝑑 (tetrahedral group): 𝐸𝐸, 4𝐶𝐶3 , 3𝐶𝐶2 , 6𝑆𝑆4 , 6𝜎𝜎𝑑𝑑 .
• 𝑂𝑂ℎ (octahedral group): 𝐸𝐸, 4𝐶𝐶3 , 3𝐶𝐶2 , 3𝐶𝐶4 , 3𝐶𝐶2 ′, 𝑖𝑖, 3𝑆𝑆4 ,4𝑆𝑆6 , 3𝜎𝜎ℎ , 3𝜎𝜎𝑑𝑑 .

Methane (CH4)

Sulphur hexafluoride (SF6)2. Molecular point groups• Icosahedral groups: with 𝐶𝐶5 .
• 𝐼𝐼ℎ : 𝐸𝐸, 12𝐶𝐶5 , 20𝐶𝐶3 , 15𝐶𝐶2 , 𝑖𝑖, 12𝑆𝑆10 , 20𝑆𝑆6 , 15𝜎𝜎.

Buckminsterfullerene (C60)

1920

2. Molecular point groups
𝐷𝐷∞ℎMolecule𝐶𝐶𝑠𝑠yesno𝑖𝑖

yes

𝐶𝐶∞𝑣𝑣

𝐶𝐶∞𝑇𝑇𝑑𝑑𝐼𝐼ℎNonpolar
Polar

no> 1𝐶𝐶𝑛𝑛>2

yes

no

yes𝑖𝑖

yes𝐶𝐶5𝑂𝑂ℎnono𝐶𝐶𝑛𝑛>1noyes𝜎𝜎ℎyesno𝐶𝐶𝑖𝑖
𝑖𝑖

yes
nono𝑛𝑛𝐶𝐶2 ⊥ 𝐶𝐶𝑛𝑛𝜎𝜎ℎ

yes

𝜎𝜎ℎno
yes

𝐷𝐷𝑛𝑛𝑛

𝑛𝑛𝜎𝜎𝑣𝑣no

yes𝐷𝐷𝑛𝑛𝑛𝑛

𝐶𝐶1

𝐷𝐷𝑛𝑛
𝑆𝑆2𝑛𝑛𝑛𝑛𝜎𝜎𝑣𝑣
yes

𝑆𝑆2𝑛𝑛

yes
no
yes
nono

𝐶𝐶𝑛𝑛𝑛
𝐶𝐶𝑛𝑛𝑛𝑛𝐶𝐶2

2. Molecular point groups• The group theory can beused to analyse molecule
properties.
• For example, the dipole⃑
moment of molecule, 𝓅𝓅.
• 𝓅𝓅⃑ is a vector in space, so𝚤𝚤𝓅𝓅̂ ⃑ = −𝓅𝓅⃑𝜎𝜎�ℎ 𝓅𝓅𝑧𝑧 = −𝓅𝓅𝑧𝑧
𝑛𝑛−1𝑘𝑘̂� 𝐶𝐶𝑛𝑛 𝓅𝓅𝑥𝑥𝑦𝑦 = 0

𝑘𝑘=0212. Molecular point groups• If a molecule has 𝑖𝑖,𝚤𝚤𝓅𝓅̂ ⃑ = −𝓅𝓅⃑ = 𝓅𝓅⃑• Then, 𝓅𝓅⃑ = 0.
• If a molecule has 𝜎𝜎ℎ ,𝜎𝜎�ℎ 𝓅𝓅𝑧𝑧 = −𝓅𝓅𝑧𝑧 = 𝓅𝓅𝑧𝑧
𝓅𝓅𝑧𝑧 = 0• Meanwhile,𝑛𝑛−1𝑘𝑘̂� 𝐶𝐶𝑛𝑛 𝓅𝓅𝑥𝑥𝑦𝑦 = 𝑛𝑛𝓅𝓅𝑥𝑥𝑦𝑦 = 0𝑘𝑘=0• Molecules with 𝑖𝑖 or 𝜎𝜎ℎ are nonpolar.

222. Molecular point groups• If a molecule has more than one 𝐶𝐶𝑛𝑛 (at least 3),
𝑛𝑛𝑛−1𝑘𝑘̂� 𝐶𝐶𝑛𝑛 ′𝓅𝓅′ = 𝑛𝑛′ 𝓅𝓅′ = 0𝑘𝑘=0𝑛𝑛𝑛𝑛−1𝑘𝑘̂� 𝐶𝐶𝑛𝑛 ′′𝓅𝓅′′ = 𝑛𝑛′′ 𝓅𝓅′′ = 0

𝑘𝑘=0
𝑛𝑛𝑛𝑛𝑛−1𝑘𝑘̂� 𝐶𝐶𝑛𝑛 ′′′𝓅𝓅′ = 𝑛𝑛′′′ 𝓅𝓅′′′ = 0

𝑘𝑘=0• 𝓅𝓅⃑ is 0 in 3 directions. 𝓅𝓅⃑ = 0.

23

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3. Characters of typical groups• Multiplication table and character table of 𝐶𝐶2𝑣𝑣 .

z

𝐸𝐸�
𝐸𝐸�

𝐸𝐸�
𝐶𝐶̂2

𝐶𝐶̂2𝜎𝜎�𝑣𝑣

𝜎𝜎�𝑣𝑣 ′

𝜎𝜎�𝑣𝑣

𝐶𝐶2𝑣𝑣

𝐸𝐸�

𝐵𝐵1

1𝐴𝐴1𝐴𝐴2𝐵𝐵2

1
11𝜎𝜎�𝑣𝑣 ′𝐶𝐶̂2
𝐶𝐶̂2𝐸𝐸�𝜎𝜎�𝑣𝑣 ′𝜎𝜎�𝑣𝑣

𝐶𝐶̂211−1−1𝜎𝜎�𝑣𝑣

𝜎𝜎�𝑣𝑣 ′

𝐶𝐶̂2𝐸𝐸�𝜎𝜎�𝑣𝑣

𝜎𝜎�𝑣𝑣 ′𝐸𝐸�𝜎𝜎�𝑣𝑣
1−11−1O𝜎𝜎�𝑣𝑣 ′𝜎𝜎�𝑣𝑣
𝐶𝐶̂2𝜎𝜎�𝑣𝑣 ′1−1−11𝜎𝜎𝑣𝑣 ′H*x𝐶𝐶2𝜎𝜎𝑣𝑣

H**𝑧𝑧

𝑥𝑥 2 , 𝑦𝑦 2 , 𝑧𝑧 2𝑦𝑦, 𝑅𝑅𝑥𝑥

𝑦𝑦𝑧𝑧

𝑅𝑅𝑧𝑧

𝑥𝑥, 𝑅𝑅𝑦𝑦𝑥𝑥𝑦𝑦𝑧𝑧𝑥𝑥y3. Characters of typical groups• *The vectors and rotations which belong to thesymmetry of the irreps.
• **The quadratic functions which belong to thesymmetry of the irreps.
• 𝐴𝐴 and 𝐵𝐵: 1 dimensional irreps.
• 𝐴𝐴: when the vector belonging to the symmetry ofthe irrep is symmetric wrt rotation wrt 𝐶𝐶𝑛𝑛 (𝑛𝑛 > 2).
That is,𝜒𝜒A 𝐶𝐶̂𝑛𝑛 = 1• 𝐵𝐵: when 𝜒𝜒𝐵𝐵 𝐶𝐶̂𝑛𝑛 = −1.

25

3. Characters of typical groups

26

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3. Characters of typical groups•�𝑣𝑣 = 1.
1 : when 𝜒𝜒1 𝜎𝜎•
�𝑣𝑣 = −1.
2 : when 𝜒𝜒2 𝜎𝜎• 𝐸𝐸: 2 dimensional irreps.
• 𝑇𝑇: 3 dimensional irreps.
• Character table of 𝐶𝐶3𝑣𝑣 .
𝐶𝐶3𝑣𝑣

𝐸𝐸�

2𝐶𝐶̂33𝜎𝜎�𝑣𝑣

𝐸𝐸

2−10𝐴𝐴1

𝐴𝐴2

1
1

1
1

1−1𝑧𝑧

𝑅𝑅𝑧𝑧

𝑥𝑥, 𝑦𝑦 , 𝑅𝑅𝑥𝑥 , 𝑅𝑅𝑦𝑦𝑥𝑥 2 + 𝑦𝑦 2 , 𝑧𝑧 2𝑥𝑥 2 − 𝑦𝑦 2 , 𝑥𝑥𝑥𝑥 , 𝑥𝑥𝑥𝑥, 𝑦𝑦𝑦𝑦

28

3. Characters of typical groups• Character table of 𝐶𝐶2ℎ .
𝐶𝐶2ℎ𝐸𝐸�𝐴𝐴𝑢𝑢

1𝐴𝐴𝑔𝑔
𝐵𝐵𝑔𝑔

𝐵𝐵𝑢𝑢

111𝐶𝐶̂21−11−1

𝚤𝚤̂

1

1−1−1

• 𝑔𝑔: when 𝜒𝜒𝑔𝑔 𝚤𝚤̂ > 0.

• 𝑢𝑢: when 𝜒𝜒𝑢𝑢 𝚤𝚤̂ < 0.

𝜎𝜎�ℎ1−1−11𝑅𝑅𝑧𝑧

𝑅𝑅𝑥𝑥 , 𝑅𝑅𝑦𝑦𝑧𝑧

𝑥𝑥, 𝑦𝑦𝑥𝑥 2 , 𝑦𝑦 2 , 𝑧𝑧 2 , 𝑥𝑥𝑦𝑦𝑥𝑥𝑧𝑧, 𝑦𝑦𝑧𝑧

29

3. Characters of typical groups• Character table of 𝐷𝐷3ℎ .
𝐷𝐷3ℎ𝐸𝐸�2𝐶𝐶̂33𝐶𝐶̂2𝜎𝜎�ℎ2𝑆𝑆̂33𝜎𝜎�𝑣𝑣

𝐸𝐸𝐸

2−102−10𝑥𝑥, 𝑦𝑦𝑥𝑥 2 − 𝑦𝑦 2 , 𝑥𝑥𝑥𝑥

2−1−210𝑅𝑅𝑥𝑥 , 𝑅𝑅𝑦𝑦𝑥𝑥𝑧𝑧, 𝑦𝑦𝑦𝑦

𝐴𝐴1 ′𝐴𝐴2 ′𝐴𝐴1 ′′𝐴𝐴2 ′′𝐸𝐸𝐸𝐸111111111−11−101

1−1−1• ′: when 𝜒𝜒′ 𝜎𝜎�ℎ > 0.
• ′′: when 𝜒𝜒′′ 𝜎𝜎�ℎ < 0.
̂•
1 : when 𝜒𝜒1 𝐶𝐶2 = 1.
•̂

2 : when 𝜒𝜒2 𝐶𝐶21

1−1−1= −1.

1−1−1
1𝑅𝑅𝑧𝑧𝑧𝑧𝑥𝑥 2 + 𝑦𝑦 2 , 𝑧𝑧 2

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3. Characters of typical groups• Character table of 𝐷𝐷4ℎ .
𝐷𝐷4ℎ𝐸𝐸�2𝐶𝐶̂4𝐶𝐶̂22𝐶𝐶̂2 ′ 2𝐶𝐶̂2 ′′𝐵𝐵1𝑔𝑔1−111𝐴𝐴1𝑔𝑔

𝐴𝐴2𝑔𝑔
𝐵𝐵2𝑔𝑔
𝐸𝐸𝑔𝑔

𝐴𝐴1𝑢𝑢

𝐴𝐴2𝑢𝑢
𝐵𝐵1𝑢𝑢

𝐵𝐵2𝑢𝑢
𝐸𝐸𝑢𝑢

11121111211−11110−2−1111−10110−1111211−1−11−11−200−1−1−211−1−1

1−2−1−110

1−1−111−1112𝜎𝜎�𝑣𝑣 2𝜎𝜎�𝑑𝑑

101𝜎𝜎�ℎ1−112𝑆𝑆̂4

1−1−1𝚤𝚤̂−101

1−1−1𝑅𝑅𝑧𝑧

00𝑅𝑅𝑥𝑥 , 𝑅𝑅𝑦𝑦

−1−1
1−1−1−1−11−1−121−10110𝑧𝑧

𝑥𝑥, 𝑦𝑦𝑥𝑥 2 + 𝑦𝑦 2 , 𝑧𝑧 2𝑥𝑥 2 − 𝑦𝑦 2𝑥𝑥𝑦𝑦𝑥𝑥𝑧𝑧, 𝑦𝑦𝑦𝑦

313. Characters of typical groups
𝐷𝐷6ℎ𝐸𝐸�2𝐶𝐶̂6 2𝐶𝐶̂3 𝐶𝐶̂23𝐶𝐶̂2 ′ 3𝐶𝐶̂2 ′′

𝐵𝐵1𝑔𝑔

1−1

1𝐴𝐴1𝑔𝑔

𝐴𝐴2𝑔𝑔

𝐵𝐵2𝑔𝑔
𝐸𝐸1𝑔𝑔

𝐸𝐸2𝑔𝑔

𝐴𝐴1𝑢𝑢

𝐴𝐴2𝑢𝑢

𝐵𝐵1𝑢𝑢

𝐵𝐵2𝑢𝑢

𝐸𝐸1𝑢𝑢

𝐸𝐸2𝑢𝑢

11
12

1

1−1

1

111

1
1−1

1−1 −21

1−1

1−1 −11−1

1−12−1 −1211211

12112𝑆𝑆̂6 2𝑆𝑆̂3 𝜎𝜎�ℎ 3𝜎𝜎�𝑑𝑑 3𝜎𝜎�𝑣𝑣

1

1

1−1−11002−1 −1211𝚤𝚤̂01−110112111−1111−1 −11−20−1 −1−1 −1

2−1 −1 −1 −1−1 −200−20

100−1100−1 −1 −1 −1 −1 −1−1−1 −1

1−1 −1 −1−1−111111−1 −1−1 −1−211

11−11

11−1−201210𝑅𝑅𝑧𝑧
𝑅𝑅𝑥𝑥 , 𝑅𝑅𝑦𝑦

1𝑧𝑧

0𝑥𝑥, 𝑦𝑦1−10𝑥𝑥 2 + 𝑦𝑦 2 , 𝑧𝑧 2𝑥𝑥𝑥𝑥, 𝑦𝑦𝑦𝑦𝑥𝑥 2 − 𝑦𝑦 2 , 𝑥𝑥𝑦𝑦