Chimica Inorganica 3
The symmetry properties of molecules (i.e. the atoms of a molecule form a basis set) are described by point groups, since all the symmetry elements in a molecule will intersect at a common point, which is not shifted by any of the symmetry operations. There are also symmetry groups, called space groups, which contain operators involving translational motion.
The point groups are listed below along with their distinguishing symmetry elements
C1 : E (h = 1) no symmetry
Chimica Inorganica 3
Cs : σ (h = 2) only a mirror plane
Ci : i (h = 2) only an inversion center (rare point group)
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Cn : Cn and all powers up to Cnn = E (h = 2) a cyclic point group
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Cnv : Cn and nσv (h = 2n) … by convention a σv contains Cn (as opposed to σh which is ⊥ to Cn). For n even, there are n/2σv and n/2 σv’ with σv containing the most atoms and σv’ containing the least atoms
σv
σv’
a dihedral mirror plane … the σds bisect σvs (and σv’); σd designation is more common in D point groups
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two σvs: the σ (xz) and σ (yz) mirror planes; two σds bisect σ (xz) and σ (yz) planes
Cnh : Cn and σh (normal to Cn) are generators of Sn operations as well (h = 2n)
Note the inversion center; generated from C2 ⋅σh = S2 = i
Chimica Inorganica 3
Chimica Inorganica 3
The D point groups are distiguished from C point groups by the presence of rotation axes that are perpindicular to the principal axis of rotation.
Dn : Cn and n⊥C2 (h = 2n)
Example: Co(en)33+ is in the D3 point group
3+
Reorient the molecule along the (1,1,1) axis, i.e the C3 axis
In identifying molecules belonging to this point group, if a Cn is present and one ⊥C2 axis is identified, then there must necessarily be (n–1)⊥C2s generated by rotation about Cn.
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Dnd : Cn, n⊥C2, nσd (dihedral mirror planes bisect the ⊥C2s)
Example: allene is in the D2d point group
Reorient the molecule along the Cn axis
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Note: Dnd point groups will contain i, when n is odd
Chimica Inorganica 3
Chimica Inorganica 3
Dnh : Cn, n⊥C2, nσv, σh (h = 4n)
When n is even, n/2 σv and n/2 σv’
Chimica Inorganica 3
Chimica Inorganica 3
C∞v : C∞ and ∞ σv (h = ∞) linear molecules without an inversion center
a σv is easily identified as the plane of the paper, by virtue of the C∞, ∞ σvs are generated
D∞h : C∞, ∞⊥C2, ∞ σv, σh, i (h = ∞) linear molecules with an inversion center
Chimica Inorganica 3
Chimica Inorganica 3
Td : E, 8C3, 3C2, 6S4, 6σd (h = 24) a cubic point group; the cubic nature of the point group is easiest to visualize by inscribing the tetrahedron within a cube
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Oh : E, 8C3, 6C2, 6C4, 3C2 (=C42), i, 6S4, 8S6, 3σh, 6σd (h = 48)
a cubic point group; an octahedron inscribed within a cube
Chimica Inorganica 3
Chimica Inorganica 3