Molecular Symmetry: The Point Group

Molecular shape plays an important role in the spectroscopic properties of complexes. Even if we compare two simple two-coordinate linear complexes one with two identical ligands and one with two different ligands it should be immediately apparent that they do not look the same. For one both sides of the molecule are equivalent, for the other they are different (Figure B.1). Turn one round by rotation 180° around an axis perpendicular to the bond direction and it looks identical to what you had in the first place; do this for the other and it is clearly not the same. Further, consider the situation where the linear molecule with two identical donors is compared with an analogue where the X-M-X unit is bent. Now consider a 180° rotation about an axis containing the X-M-X unit. For the linear molecule the outcome is indistinguishable from the starting situation whereas for the bent molecule this is not the case. In each of these examples, the linear molecule has undergone what is called a symmetry operation. The nonsymmetrical and bent molecules have clearly yielded a different view and thus have behaved differently. What we are seeing is that this process we have undertaken has distinguished between what are clearly different molecules. In effect we can use any molecule's behaviour to a series of operations like those presented below to define the molecular shape (or symmetry). Molecules can be described in terms of a symmetry point group effectively a combination of a limited number of what are termed symmetry elements based on considering a set of operations like those introduced above. There are a limited number of both individual symmetry elements and combinations of these (called the point group). The symmetry elements that can operate are defined below in Table B.1. A legitimate symmetry operation employing these elements occurs when the molecular views before and after the operation are indistinguishable. Operations occur relative to an x. y. z coordinate system arranged around a particular point in the molecule, selected with regard to defining symmetry elements and maximizing operations in such a way that the z axis forms the principal axis, passing through the molecular centre and being the axis around which the highest order operation occurs.

Determination of all possible symmetry elements provides the ability to then assign the point group for the element, which (as the name suggests) is based on the symmetry around a point in the molecule that either coincides with the central atom or the geometric centre of the molecule. A flow chart (Figure B.2) is frequently used to assist in the assignment of all symmetry operations for a molecule. Once all symmetry operations have been identified the point group evolves. Point groups are limited in number, and are identified in Table B.2. To exemplify the concepts, we shall examine two systems: the trigonal bipyramidal MX and the square planar MX4. The shapes axes and operations involving these are shown below in Figures B.3 and B.4.

FigureB.1

Two operations with linear MX2 and MXY molecules that distinguish their differing symmetry.

For MX5, the principal axis passes through the M and the two axial X groups; the M is the point about which the point group is defined. The three X groups are coplanar with the metallic in the xy plane; one axis (designated x) passes through one M–X bond, which then requires the other to pass through M but not include the other X groups. Around the z axis, rotation by 120° leads to an indistinguishable arrangement; this is thus a C3 axis. Rotation by 180° about each of the three M–X bonds in the xy plane leads to an indistinguishable arrangement; thus there are three C2 elements. These operations alone define the molecules as belonging to a D point group. However, there are also three vertical planes of symmetry, each containing one M–X bond in the xy plane and the C3 axis (only one of which is shown as a dotted square in the figure for clarity), as well as one horizontal plane that is the xy plane containing the central equatorial MX3 part of the molecule (shown as a dotted triangle in the figure). Overall then this means the molecule belongs to the D3h point group.

The square planar MX4 has some similarities to the trigonal bipyramidal MX5, as along the z axis there is a C4 operation (rotation by 90° leads to formation of an indistinguishable arrangement), and there are four C2 elements (two about each in-plane X–M–X, and two about imaginary axes bisecting the x and y axes in two directions). There are four vertical planes containing the C4 axis (only one of which is shown as a dotted square in the figure, for clarity) and one horizontal plane of symmetry (shown as a dotted square incorporating the four X groups in the figure). Following the flowchart, this leads to the D4h point group.

Figure B.2

Simplified flow chart for assignment of a symmetry group. Special high symmetry groups (T_d.O_h. I_h) are not included in this chart and need to be identified separately; it applies to other and generally lower symmetry molecules, which are more often met in reality.

It is possible with the flow chart and careful examination of drawings and or three-dimensional models of a complex to assign the point group with reasonable rapidity and success after some practice. To assist further, a list of the point groups of basic higher symmetry structures are collected in Table B.3 below. Those shown assume identical ligands in all sites. You should assume that these shapes with a mixture of different ligands will be of lower symmetry and have a lower symmetry point group. This aspect is illustrated

Figure B.3
The coordinate system and symmetry operations for trigonal bipyramidal MX5 of D3h symmetry point group.

Figure B.4
The coordinate system and symmetry operations for square planar MX4, of D4h symmetry point group. Note that there is a C2 operation colinear with a C4 around the z axis.

In Figure B.5 for moving from square planar MX4 (D4h) to square planar trans-MX2Y2 (D2h) and cis-MX2Y2 (C2v).
Distortions of common geometric shapes lead necessarily to changes in the point group. For example, an octahedron (Oh) can undergo three types of distortion. The first is tetragonal distortion, involving elongation or contraction along a single C4 axis direction, leading to D4h symmetry. The second is rhombic distortion, where changes occur along two C4 axes.

Figure B.5

The coordinate system and symmetry operations for the family of square planar complexes MX4 trans-MX2Y2 and cis-MX2Y2.

so that no two sets of bond distances along each of the three axes are equal, leading to D2h symmetry. The third is trigonal distortion involving contraction or elongation along one of the C3 axes to yield a trigonal antiprismatic shape, reducing the point group to D3d. Similarly, other common stereochemistries may distort, leading to shapes with different point groups. Just as molecules have certain symmetry molecular orbitals likewise have symmetry. Orbital labels such as and relate to the rotational symmetry of the orbital, whereas the labels a.e.eg.t and so on met in complexes arise from considering orbital behaviour in the context of all symmetry operations of the point group of the molecule. A character table, which defines the symmetry types possible in a particular point group provides a way of assigning these labels but this will not be pursued here.

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مواضيع ذات صلة

Organometallic Compounds

Bridging Ligands

Organic Molecules as Ligands

Multipliers

Polydenticity

Crystal Ball Gazing

Taking Stock

Complex Futures

Better Ways to Synthesize Fine Organic Chemicals

Industrial Roles for Ligands and Coordination Complexes

Metal Extraction

Fluoroionophores