Inorganic Chemistry
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Inorganic Chemistry

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however takes everything through the center and out to the 
same distance on the opposite side. A rotation-reflection (Sn) is an operation that combines a Cn 
rotation with a reflection in a plane perpendicular to the rotation axis. Sometimes the individual 
components are themselves symmetry operations: for example the C4 axes of SF6 are also S4 axes as 
the molecule has reflection planes perpendicular to each C4 axis. However, in the case illustrated in 
Fig. 1 that is not so. The axis illustrated is a C3 axis but not a C6. However combining a 60° rotation 
with a reflection creates the S6 symmetry operation shown. 
Rotations are known as proper symmetry operations whereas the operations involving reflection 
and inversion are improper. Proper symmetry operations may be performed physically using 
molecular models, whereas improper opera- 
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Fig. 1. Symmetry operations and elements illustrated for SF6
. The effect of each 
operation is shown by the numbering of the F atoms.
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tions can only be visualized. A molecule possessing no improper symmetry elements is 
distinguishable from its mirror image and is known as chiral. Chiral molecules have the property of 
optical activity which means that when polarized light is passed through a solution, the plane of 
polarization is rotated. In organic molecules, chirality arises when four different groups are 
tetrahedrally bonded to a carbon atom. Inorganic example of chiral species include six-coordinate 
complexes with bidentate ligands (see Topic H6, structures 10 and 11). Molecules with improper 
symmetry elements cannot be chiral as the operations concerned convert the molecule into its mirror 
image, which is therefore indistinguishable from the molecule itself. Most often such achiral 
molecules have a reflection plane or an inversion center, but more rarely they have an Sn rotation-
reflection axis without reflection or inversion alone. 
Point groups 
Performing two symmetry operations sequentially generates another symmetry operation. For 
example, two sequential C4 operations about the same axis make a C2 rotation; reflecting twice in 
the same plane gives the identity E. Every symmetry operation also has an inverse operation which 
reverses its effect. For example, the inverse of a reflection is the same reflection; the inverse of an 
anticlockwise C4 operation is a clockwise rotation about the same axis. These properties mean that 
the complete set of symmetry operations on a given object form a mathematical system known as a 
group. Groups of symmetry operations of molecules are called point groups, in distinction to space 
groups which are involved in crystal symmetry and include operations of translation, shifting one 
unit cell into the position of another. (Unit cells are discussed in Topic D1, but space groups are 
required only for advanced applications in crystallography and are not treated in this book.) 
Chemists use the Schönflies notation for molecular point groups, the labels used being listed in 
the \u2018flow chart\u2019 shown in Fig. 2 and explained below. For a non-linear molecule with at least one 
rotation axis, the first important question is whether there is a principal axis, a unique Cn axis with 
highest n. For example SF6 has no principal axis, as there are several C4 axes. The molecules shown 
in Fig. 3 however all have a principal C3 axis as there is no other of the same kind. Given a principal 
axis, the only other axes allowed are C2 axes perpendicular to it, called dihedral axes. Point 
groups with and without such axes fall into the general classes Dn and Cn, respectively. If there are 
reflection planes, these are additionally specified. A horizontal plane (\u3c3h) isoneperpendicularto the 
principal axis, thus being horizontal if the molecule is oriented so that the axis is vertical. The 
molecules B(OH)3 and BF3 in Fig. 3 do have a \u3c3h plane and have the point groups C3h and D3h, 
respectively. In a Cn group, planes which contain the principal axis are known as vertical and give 
the point group Cnv, for example C3v for NH3. However, in a Dn group without a horizontal plane, 
any planes containing the principal axis lie between the dihedral axes and are called diagonal thus 
giving the point group Dnd (e.g. D3d for the staggered conformation of ethane as shown in Fig. 3). 
Dnd groups can be difficult to identify because the dihedral axes are hard to see. 
Linear molecules fit into the above classification by using the designation C\u221e for the molecular 
axis, implying that a rotation of any angle whatever is a symmetry operation. Thus, we have C\u221ev for 
a molecule with no centre of inversion (examples being CO and N2O) and D\u221eh if there is an 
inversion center (examples being N2 and CO2), the presence of such a center implying also that there 
are dihedral axes. 
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Fig. 2. Flow chart for identification of point groups. See text for explanation. 
If there are several equivalent Cn axes of highest n, the designation depends on n. Groups with n=2 
are of type D2; commonly there are also reflection planes and an inversion center giving D2h (for 
example, C2H4). With n=3 we have tetrahedral groups (T), the commonest example being Td, the 
point group of a regular tetrahedral molecule such as CH4 with reflection planes but no inversion 
center. Octahedral groups (O) arise with n=4, most often having an inversion center giving Oh (e.g. 
SF6, Fig. 1). The highest Cn allowed without a principal axis is n=5 giving icosahedral groups I. 
Uses and limitations
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Specifying the point group is a useful way of summarizing certain aspects of the structure of a 
molecule: for example the C3v symmetry of NH3 implies a pyrami- 
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Fig. 3. Illustrating the important symmetry elements of four molecules each having a 3-
fold principal axis, but with different points groups. 
dal structure as distinct from the planar D3h molecule BF3. However, it must be recognized that 
certain important features are not implied by symmetry alone. Even molecules with different 
stoichiometry may have the same symmetry elements, for example BF3 and trigonal bipyramidal PF5
share the D3h point group. The C3v point group tells us that the three N-H bonds in ammonia are 
equal, but says nothing about their actual length. 
Symmetry may be useful for predicting molecular properties. The example of chirality has been 
discussed above. Another example is polarity resulting from the unequal electron distribution in 
polar bonds (see Topics B1 and C10). The overall polarity of a polyatomic molecule arises from the 
vectorial sum of the contributions from each bond, and is zero if the symmetry is too high. A 
molecule with a net dipole moment can have no inversion center and at most one rotation axis, and 
any reflection planes present must contain that axis. The only point groups compatible with these 
requirements are C1, Cs, Cn, Cnv and C\u221ev. Thus, of the molecules shown in Fig. 3 only NH3 can 
have a dipole moment. 
More advanced applications of symmetry (not discussed here) involve the behaviour of molecular 
wavefunctions under symmetry operations. For example in a molecule with a centre of inversion 
(such as a homonuclear diatomic, see Topic C4), molecular orbitals are classified as u or g (from the 
German, ungerade and gerade) according