Symmetry elements, operations and point groups (‘in the molecular
world’)
Symmetry concept is extremely useful in chemistry in that it can help to predict infra-red
spectra (vibrational spectroscopy) and optical activity. It can also help in describing
orbitals involved in bonding, and in interpreting electronic spectra of molecules.
The five fundamental symmetry elements (operators) or operations:
The following symbols (Schoenflies) are used to describe both symmetry elements as
well as the symmetry operations (application of the symmetry element)
E (Identity) – all molecules have E (or C1); causes no change in the molecule
Cn (Rotation axis, clockwise by definition) – 360° (or 2pi)/n rotation about a rotation axis
‘Highest order’ or ‘principal’ rotation axis: the axis with the highest n value
e.g. C3 (in CHCl3), C6 (snowflake, benzene); C∞ (linear molecules)
σ (Reflection plane) – mirror plane (exchange of ‘left and right’)
four different types of reflections: σv, σh, σd, σ
σv: vertical mirror plane (contains principal rotation axis)
example: BF3 (planar)
F
σv
B
F
F
B
F
σv
F
F
F
σv
B
F
F
σh: horizontal mirror plane (plane perpendicular to principal axis)
example: BF3 (trivial because its planar)
σd: dihedral mirror plane (planes that cut the angle between 2 C2 axes in half)
example: allene
C2'
C2'
H
C2
C
H
σd
H
C
C
H
σd
σ: Mirror plane, but no rotational axis
ex: aniline (because the 2 hydrogen atoms on NH2 are out of plane)
N
H
H
σ
i (Inversion centre) – position of all atoms remains unchanged after their reflection
through the inversion centre
ex: [PtCl4]2-
Cl
Cl
Pt
Cl
Cl
Sn (Mirror-rotation or rotation-reflection axis or improper axis) – 360° /n rotation
followed by a reflection through a plane perpendicular to the rotation axis
Ex: staggered ethane – S6
1. rotation by 60 deg
2. reflection through plane
H
H
Note: S1 ≡ σ; S2 ≡ i
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Multiplication of symmetry operations
One of the important features of symmetry operations is that they can be performed
multiple times and they can also be combined:
1. Repetition of symmetry operations:
Example 1: C22 ≡ E (the superscript indicates how many times C2 is performed)
Obviously C22 does not lead to a new symmetry operation
Another example of this: σh2 ≡ E
Example 2: C32 is a new symmetry operation (whereas C33 ≡ E)
Î the symmetry element C3 creates the symmetry operations C3 and C32
H
Cl1
H
C3
Cl3
Cl2
Cl2
Cl1
Cl3
H
C3
Cl2
Cl1
Cl3
C32
Example 3: C6 (benzene): only C6 and C65 are symmetry operations of C6
C62 = C3, C63 = C2, C64 = C32, C65, C66 = E
2. Combination of different sym. operations:
B x A (A first, B second)
Example 1: CH2Cl2
σv'
H
Cl
Cl
H
σv
Example 2: Sn = σh x Cn
3
3. Inverse operation:
For each operation there is one that brings the molecule back into its original position.
Example 1: C2-1 (counterclockwise rotation by 180 deg), C2-1 = C2 and C2 x C2-1 = E
Generally: Cn-1 = Cnn-1
Example 2: C3-1 = C32
Example 3: σ-1 = σ; i-1 = i
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Point Groups
Each molecule has at least one point which is unique and which remains unchanged, no
matter how many or what type of symmetry operations are performed. Such point is
termed singular point. (e.g. centre of the benzene molecule)
•
Point group = The complete set of symmetry operations that characterize a
molecule’s overall symmetry.
•
Space groups = Symmetry classes characterizing entities with translational
symmetry. [Space groups don’t have singular points!]
Because there is a limited number of symmetry elements (at least in the molecular world,
because n is usually <10) and their combinations, there must be a limited number of point
groups! Therefore, so many different molecules belong to the same point group:
Example: H2O and CH2Cl2 both C2v (E, C2, σv, σv’)
Three point groups of low symmetry:
Point group Description
Example
C1
asymmetry
CHFClBr
Cs
only 1 symmetry (mirror) plane
C2H2ClBr
H
Cl
Br
H
Ci
only 1 inversion centre (very rare)
1,2-dibromo-1,2dichloroethane
(staggered)
H
Cl
Br
x
Br
Cl
H
Five point groups of high symmetry:
Point group Description
Example
C∞v
linear
H-F
infinite number of rotations
infinite number of reflection planes containing the principal axis
(easy criterion: linear but no i)
D∞h
linear (like C∞v but with additional σh, C2, and i)
(easy criterion: linear + i)
CO2
Td
tetrahedral symmetry
Contains 24 symmetry operations
(E, 4C3, 4C32, 3S4, 3S43, 3C2, 6σd)
CH4
Oh
octahedral symmetry (48 symmetry operations)
SF6
Ih
icosahedral symmetry (120 symmetry operations)
B12H122-
dodecahydridododecaborane
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Groups of high symmetry
Symmetry operations for high symmetry point groups and their rotational subgroups
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Diagram of the Point Group Assignment Method
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Cn and Sn point groups
Point groups without any mirror planes:
Cn
the only symmetry element is Cn;
symmetry operations: Cn, Cn2, Cn3, Cnn-1
C1 complete asymmetry
C2, C3, … are (extremely) rare and difficult to visualize
Example of C2: H2O2 (C2 cuts angle between the two OOH planes, 94 deg,
in half)
94 deg
H
O
O
C2
H
Sn
(only possible with n = 2, 4, 6, …)
symmetry operations: Sn, Sn2, Sn3, Snn-1
S2 ≡ Ci
example: staggered 1,2-dibromo-1,2-dichloroethane
S4: Rare! Contains symmetry operations S4, S42 (≡ C2), S43
Example: 1,3,5,7-tetrafluorocyclooctatetraene
S4
F
F
F
F
Point groups with mirror planes:
Cnv
symmetry operations: Cn, ..., Cnn-1, n x σv
C1v ≡ Cs
C2v (very common): H2O, 1,2-dichlorobenzene
σv'
C2
Cl
O
H
Cl
σv
H
C3v (very common): trigonal-pyramidal molecules (NH3, CH3Cl)
Cl
N
H
H
H
H
H
H
C4v: square-pyramidal molecules (BrF5)
F
F
F
Br
F
F
C5,6,..v: very rare
C∞v: all linear molecules without inversion centre (HF, HCl)
Cnh
symmetry elements: Cn, σh, i if n even, Sn
C2h: has C2, σh and i
example: difluorodiazene (planar)
F
N
N
F
2
C3h: B(OH)3 boric acid (planar)
H
O
B
H
O
O
H
C4,5...h: extremely rare
D point groups
Criterion: in contrast to Cn groups, D groups have n C2 axes perpendicular to Cn!
Dn:
symmetry operations: Cn, ..., Cnn-1, n x C2
(rare point group!)
D2: one of the excited states of ethylene (slight deviation from planarity!)
C2v fragment
C2v fragment
H
H
H
H
fragmentation method: D2 can be regarded as consisting of 2 identical
C2v fragments joined back to back, so that one half is rotated (with respect
to other) by any degree (except m x π/n with m = 2,4,6,..; n = order of C)
D3: [Co(en)3]3+
en = ethylenediamine
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Dnd:
symmetry elements: Cn, n x C2, S2n, n x σd
D2d: allene
C 2'
C 2'
H
C2 S4
σd
H
C
C
C
H
H
σd
D3d: symmetry operations: C3, C32, 3 C2, S6, S63 = i, S65, 3 σd
example: staggered ethane (2 C3v fragments: H3C, rotation by 60 deg)
C2
σd
C2
σd
σd
C2
D4d: S8
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
D5d: Ferrocene
Fe
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Dnh:
symmetry elements: Cn, n x C2, σh, Sn, n x σ, i if n is even
Note: the horizontal mirror plane is usually easily identifiable!
D2h: e.g. C2H4 (has 3 equivalent C2 axes; therefore: assigning a v or h
subscript to σ is redundant)
D3h: PF5 (trigonal-bipyramidal), BF3 (planar)
F
F
F
P
F
F
B
F
F
F
D4h: molecules with the geometry of a square (XeF4)
F
F
Xe
F
F
D5h: planar with 5-fold rotational symmetry (e.g. cyclopentadienyl anion)
D6h: planar with 6-fold rotational symmetry (benzene)
D∞h: linear symmetrical molecules (N2, C2H2)
Three tips for assigning point groups
1. Most of the low and high symmetry point groups are relatively easy to assign.
2. When it comes to D vs. C or S: look for n C2 axes perpendicular to Cn.
3. Then look for σh.
Chirality
Group-theoretical criterion for chirality: Absence of Sn!
Td, Oh, Cnh, Dnd, Dnh possess Sn. Molecules belonging to these point groups can therefore
not be chiral!
Note: Molecules are usually chiral when neither i (S2) nor σh is present (σh ≡ S1)
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