P.A. Cox - Inorganic chemistry (793955), страница 16
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Identification of the point group and its designation in theSchönflies nomenclature involves finding the major symmetryelements and their orientations with respect to each other.Specifying the symmetry of a molecule can be a useful way ofsummarizing important aspects of its structure, and enables theprediction of some properties such as chirality and polarity.However, the information is limited and symmetry alone does notspecify the full structure, or even the stoichiometry uniquely.DescribinginorganicLigand field theory (H2)compounds (B5)Complexes, structure andMolecular shapes: VSEPRisomerism (H6)(C2)Symmetry operations and symmetry elementsSymmetry is a property of molecules containing more than one atom of the same kind, with equal bond lengths and/orbond angles.
For example the high symmetry of the SF6 molecule (Fig. 1. arises from the six equal S-F bonds disposed atangles of 90° to each other. In order to make the notion more precise we use the idea of a symmetry operation. Forexample rotating SF6 by 90° about an appropriate axis, it appears indistinguishable after the rotation. The axisconcerned is known as the symmetry element. Rotations which do not leave the molecule looking the same are notsymmetry operations.The different types of symmetry operation and their corresponding symmetry elements are listed in Table 1. Theidentity operation E is included by convention for mathematical completeness. It involves no change at all and is asymmetry operation for any system.
The remaining, non-trivial types of operation are illustrated for the case of SF6 inFig. 1. The C4 rotation is the same 90° rotation referred to above. SF6 has two other C4 axes and also several C2 (180°)and C3 (120°) axes not shown. The effects of reflection (σ) and inversion (i) should be clearly distinguished.
In theformer case the symmetry element is a plane that reflects like a mirror, and does not affect atoms lying in the plane. The66SECTION C—STRUCTURE AND BONDING IN MOLECULESTable 1. Symmetry operations and symmetry elementsoperation of inversion through a center however takes everything through the center and out to the same distance onthe opposite side.
A rotation-reflection (Sn) is an operation that combines a Cn rotation with a reflection in a planeperpendicular to the rotation axis. Sometimes the individual components are themselves symmetry operations: forexample 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 combininga 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 areimproper. Proper symmetry operations may be performed physically using molecular models, whereas improperoperations can only be visualized.
A molecule possessing no improper symmetry elements is distinguishable from itsmirror image and is known as chiral. Chiral molecules have the property of optical activity which means that whenpolarized light is passed through a solution, the plane of polarization is rotated. In organic molecules, chirality ariseswhen four different groups are tetrahedrally bonded to a carbon atom.
Inorganic example of chiral species include sixcoordinate complexes with bidentate ligands (see Topic H6, structures 10 and 11). Molecules with improper symmetryelements cannot be chiral as the operations concerned convert the molecule into its mirror image, which is thereforeindistinguishable from the molecule itself. Most often such achiral molecules have a reflection plane or an inversioncenter, but more rarely they have an Sn rotation-reflection axis without reflection or inversion alone.Point groupsPerforming two symmetry operations sequentially generates another symmetry operation. For example, two sequentialC4 operations about the same axis make a C2 rotation; reflecting twice in the same plane gives the identity E. Everysymmetry operation also has an inverse operation which reverses its effect.
For example, the inverse of a reflection isthe same reflection; the inverse of an anticlockwise C4 operation is a clockwise rotation about the same axis. Theseproperties mean that the complete set of symmetry operations on a given object form a mathematical system known as agroup. Groups of symmetry operations of molecules are called point groups, in distinction to space groups whichare involved in crystal symmetry and include operations of translation, shifting one unit cell into the position ofanother.
(Unit cells are discussed in Topic D1, but space groups are required only for advanced applications incrystallography and are not treated in this book.)Chemists use the Schönflies notation for molecular point groups, the labels used being listed in the ‘flow chart’shown in Fig. 2 and explained below.
For a non-linear molecule with at least one rotation axis, the first importantquestion is whether there is a principal axis, a unique Cn axis with highest n. For example SF6 has no principal axis, asthere are several C4 axes. The molecules shown in Fig. 3 however all have a principal C3 axis as there is no other of thesame 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 (σh) isoneperpendicularto the principal axis, thus being horizontalif the molecule is oriented so that the axis is vertical. The molecules B(OH)3 and BF3 in Fig. 3 do have a σh plane andC3—MOLECULAR SYMMETRY AND POINT GROUPS67Fig. 1.
Symmetry operations and elements illustrated for SF6. The effect of each operation is shown by the numbering of the F atoms.have the point groups C3h and D3h, respectively. In a Cn group, planes which contain the principal axis are known asvertical 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 pointgroup Dnd (e.g. D3d for the staggered conformation of ethane as shown in Fig. 3). Dnd groups can be difficult to identifybecause the dihedral axes are hard to see.Linear molecules fit into the above classification by using the designation C∞ for the molecular axis, implying that arotation of any angle whatever is a symmetry operation. Thus, we have C∞v for a molecule with no centre of inversion(examples being CO and N2O) and D∞h if there is an inversion center (examples being N2 and CO2), the presence ofsuch a center implying also that there are dihedral axes.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 havetetrahedral groups (T), the commonest example being Td, the point group of a regular tetrahedral molecule such asCH4 with reflection planes but no inversion center. Octahedral groups (O) arise with n=4, most often having an68SECTION C—STRUCTURE AND BONDING IN MOLECULESFig. 2. Flow chart for identification of point groups.
See text for explanation.inversion center giving Oh (e.g. SF6, Fig. 1). The highest Cn allowed without a principal axis is n=5 giving icosahedralgroups I.Uses and limitationsSpecifying the point group is a useful way of summarizing certain aspects of the structure of a molecule: for example theC3v symmetry of NH3 implies a pyramidal structure as distinct from the planar D3h molecule BF3. However, it must beC3—MOLECULAR SYMMETRY AND POINT GROUPS69Fig. 3. Illustrating the important symmetry elements of four molecules each having a 3-fold principal axis, but with different points groups.recognized that certain important features are not implied by symmetry alone. Even molecules with differentstoichiometry may have the same symmetry elements, for example BF3 and trigonal bipyramidal PF5 share the D3h pointgroup.
The C3v point group tells us that the three N-H bonds in ammonia are equal, but says nothing about their actuallength.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 iszero if the symmetry is too high. A molecule with a net dipole moment can have no inversion center and at most onerotation axis, and any reflection planes present must contain that axis. The only point groups compatible with theserequirements are C1, Cs, Cn, Cnv and C∞v.
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 wavefunctionsunder symmetry operations. For example in a molecule with a centre of inversion (such as a homonuclear diatomic, seeTopic C4), molecular orbitals are classified as u or g (from the German, ungerade and gerade) according to whether ornot they change sign under inversion. In ligand field theory (Topic H2) the t2g and eg classification of d orbitals in anoctahedral complex relates to their behavior under the symmetry operations of the Oh point group. Molecular vibrationsmay be classified in similar ways and such analysis can be valuable in using vibrational spectroscopy (infrared andRaman, see Topic B7) to determine the point group of a molecule.Section C—Structure and bonding in moleculesC4MOLECULAR ORBITALS: HOMONUCLEAR DIATOMICSKey NotesBonding andantibonding orbitalsMO diagramsSecond period diatomicsRelated topicsMolecular orbitals (MOs) are wavefunctions for electrons inmolecules, often formed by the linear combination of atomic orbitals(LCAO) approximation.
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