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Chimica Inorganica 3 - unipd.it. 2017... · Chimica Inorganica 3 The symmetry properties of...

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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 C 1 : E (h = 1) no symmetry
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Page 1: Chimica Inorganica 3 - unipd.it. 2017... · Chimica Inorganica 3 The symmetry properties of molecules (i.e. the atoms of a molecule form a basis set) are described by point groups,

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

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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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Chimica Inorganica 3

Cn : Cn and all powers up to Cnn = E (h = 2) a cyclic point group

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Chimica Inorganica 3

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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Chimica Inorganica 3

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

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Chimica Inorganica 3

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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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Chimica Inorganica 3

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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Chimica Inorganica 3

Note: Dnd point groups will contain i, when n is odd

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Chimica Inorganica 3

Page 11: Chimica Inorganica 3 - unipd.it. 2017... · Chimica Inorganica 3 The symmetry properties of molecules (i.e. the atoms of a molecule form a basis set) are described by point groups,

Chimica Inorganica 3

Dnh : Cn, n⊥C2, nσv, σh (h = 4n)

When n is even, n/2 σv and n/2 σv’

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Chimica Inorganica 3

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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

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Chimica Inorganica 3

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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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Chimica Inorganica 3

Oh : E, 8C3, 6C2, 6C4, 3C2 (=C42), i, 6S4, 8S6, 3σh, 6σd (h = 48)

a cubic point group; an octahedron inscribed within a cube

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Chimica Inorganica 3

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Chimica Inorganica 3


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