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5 The Vibrations of Polyatomic Molecules
5.4.4 Molecular Point Groups
We may group together molecules that possess the same symmetry elements. These
groups of symmetry elements are called point groups.
Some important point groups are given in Table 5.1 using the Schoenflies notation.
As an example, we will discuss the case of the planar molecule XeF 4 that belongs
to the symmetry group D 4h , see Fig. 5.1. It has a C 4 axis perpendicular to the plane
called principal axis because it has the largest n. It has four different C 2 axes. It has
two planes of symmetry parallel to the principal axis: σ v , two planes of symmetry
parallel to the principal axis and bisecting the angle between two C 2 axes: σ d , and
one plane of symmetry perpendicular to the principal axis: σ h . It also has an inversion
center at the Xe atom. Finally, it has a C 2 and a S 4 axis (both coincident with C 4 ).
Table 5.1 Some important point groups
Point group Symmetry operations
Examples
C 1
E (no symmetry)
CHBrClF
C s
E, σ h (mirror plane)
CH 2 BrCl
C i
E, i (inversion center)
CHBrCl-CHBrCl, anti
C 2
E, C 2 (twofold axis)
H 2 O 2
C 2v
E, C 2 , 2σ v
H 2 O, CH 2 F 2
C 3v
E, C 3 , 3σ v
NH 3 , CH 3 Cl
C ∞v
E, C ∞, ∞σ v (linear)
HCN, OCS
C 2h
E, C 2 , σ h , i (planar with inversion center)
CHF=CHF, trans
D 2d
E, 3C 2 (mutually perp.), S 4 (coincident with one C 2 ),
2σ d (through the S 4 axis)
H 2 C=C=CH 2
D 3d
E, C 3 , 3C 2 (perp. to C 3 axis), S 6 (coincident with C 3
axis), i, 3σ d
C 6 H 12 (cyclohexane)
D 2h
E, 3C 2 (mutually perp.), 3σ (mutually perp.), i (planar,
with inversion center)
CH 2 =CH 2
D 3h
E, C 3 , 3C 2 (perp. to C 3 axis), 3σ v , σ h (trigonal)
BF 3 (planar molecule)
D 6h
E, C 6 , 6C 2 (perp. to C 6 axis), 6σ v , σ h , C 2 and C 3 and S 6
(all coincident with the C 6 axis), i (hexagonal)
C 6 H 6 (benzene)
D ∞h
E, C ∞ , ∞C 2 (perp. to the C ∞ axes), ∞σ v , σ h , i (linear
with inversion center)
CO 2 , HC≡CH
T d
E, 3C 2 (mutually perp.), 4C 3 , 6σ, 3S 4 (coincident with
the C 2 axes) (tetrahedral)
CH 4
O h
E, 3C 4 (mutually perp.), 4C 3 , 3S 4 and 3C 2 (coincident
with the C 4 axes), 6C 2 , 9σ, 4S 6 (coincident with the C 3
axes), i (octahedral or cubic)
SF 6
5 The Vibrations of Polyatomic Molecules
5.4.4 Molecular Point Groups
We may group together molecules that possess the same symmetry elements. These
groups of symmetry elements are called point groups.
Some important point groups are given in Table 5.1 using the Schoenflies notation.
As an example, we will discuss the case of the planar molecule XeF 4 that belongs
to the symmetry group D 4h , see Fig. 5.1. It has a C 4 axis perpendicular to the plane
called principal axis because it has the largest n. It has four different C 2 axes. It has
two planes of symmetry parallel to the principal axis: σ v , two planes of symmetry
parallel to the principal axis and bisecting the angle between two C 2 axes: σ d , and
one plane of symmetry perpendicular to the principal axis: σ h . It also has an inversion
center at the Xe atom. Finally, it has a C 2 and a S 4 axis (both coincident with C 4 ).
Table 5.1 Some important point groups
Point group Symmetry operations
Examples
C 1
E (no symmetry)
CHBrClF
C s
E, σ h (mirror plane)
CH 2 BrCl
C i
E, i (inversion center)
CHBrCl-CHBrCl, anti
C 2
E, C 2 (twofold axis)
H 2 O 2
C 2v
E, C 2 , 2σ v
H 2 O, CH 2 F 2
C 3v
E, C 3 , 3σ v
NH 3 , CH 3 Cl
C ∞v
E, C ∞, ∞σ v (linear)
HCN, OCS
C 2h
E, C 2 , σ h , i (planar with inversion center)
CHF=CHF, trans
D 2d
E, 3C 2 (mutually perp.), S 4 (coincident with one C 2 ),
2σ d (through the S 4 axis)
H 2 C=C=CH 2
D 3d
E, C 3 , 3C 2 (perp. to C 3 axis), S 6 (coincident with C 3
axis), i, 3σ d
C 6 H 12 (cyclohexane)
D 2h
E, 3C 2 (mutually perp.), 3σ (mutually perp.), i (planar,
with inversion center)
CH 2 =CH 2
D 3h
E, C 3 , 3C 2 (perp. to C 3 axis), 3σ v , σ h (trigonal)
BF 3 (planar molecule)
D 6h
E, C 6 , 6C 2 (perp. to C 6 axis), 6σ v , σ h , C 2 and C 3 and S 6
(all coincident with the C 6 axis), i (hexagonal)
C 6 H 6 (benzene)
D ∞h
E, C ∞ , ∞C 2 (perp. to the C ∞ axes), ∞σ v , σ h , i (linear
with inversion center)
CO 2 , HC≡CH
T d
E, 3C 2 (mutually perp.), 4C 3 , 6σ, 3S 4 (coincident with
the C 2 axes) (tetrahedral)
CH 4
O h
E, 3C 4 (mutually perp.), 4C 3 , 3S 4 and 3C 2 (coincident
with the C 4 axes), 6C 2 , 9σ, 4S 6 (coincident with the C 3
axes), i (octahedral or cubic)
SF 6