CHAPTER SEVENTEEN MOLECULAR SYMMETRY AND BONDING 17-1 Introduction The chemical bond presents quite a problem to the writer of a physical chemistry text. The numerous rather empirical rules that exist and are useful do not have enough solid theoretical content to be transferred out of their organic or inorganic chemical context and treated as true theory. Yet the exact wave mechanical approach becomes hopelessly bogged in mathematical complexities if one departs from diatomic molecules of the lightest elements. What, then, should be done ? It turns out that if we know the geometry of a molecule, it is much easier to set up the wave mechanical treatment; also, a number of qualitative results can be demonstrated without calculation. The emphasis of this chapter is therefore on that discipline of mathematics which allows a concise and penetrating descrip­ tion of the essential elements of molecular geometry, that is, group theory. As a branch of mathematics, group theory requires serious and extensive study. There are numerous theorems and lemmas to be proved and many special cases to be examined in detail. We will take a very pragmatic approach here, however. Theorems will be stated rather than rigorously proved; their validity will lie in the success of their practical as opposed to their abstract demonstration. By allowing ourselves this latitude, it will be possible to present the group-theoretical approach to the symmetry properties of molecules in a rather efficient, and, this writer hopes, a stimulating way. We proceed therefore to describe symmetry operations and to develop the concept of a set of such operations as defining a symmetry group. Some of the basic attributes of groups necessarily are introduced. The result will be a compact description of those aspects of molecular geometry that determine the qualitative application of wave mechanics to a set of chemical bonds. The group-theoretical approach also provides a means for simplifying quantitative wave mechanical calculations. This is illustrated in the Commentary and Notes section with a dis­ cussion on the crystal field treatment of transition metal complexes. The wave mechanics of chemical bonding, treated qualitatively and from the symmetry point of view in this chapter, is taken up more quantitatively in Chapter 18. 717 718 CHAPTER 17: MOLECULAR SYMMETRY AND BONDING 17-2 Symmetry and Symmetry Operations A. Symmetry Elements and Stereographic Projection Diagrams The objects that we shall be dealing with, namely simple molecules, will generally have a considerable degree of symmetry. The ammonia molecule, for example, can be viewed as a triangular pyramid whose base is formed by the hydrogen atoms and whose apex is a nitrogen atom, as illustrated in Fig. 17-l(a). There is an axis of symmetry, labeled C3, in recognition of the fact that if the molecule were rotated 120° about this axis, the result would be an equivalent configuration. That is, if the rotation were performed while the observer's eyes were closed, on looking again he could not tell that anything had occurred. The rotation is a symmetry operation—it gives an equivalent configuration. Were each hydrogen atom labeled, however, one could then tell that the rotation had been made. Only a rotation of 0°, 360°, and so on, would lead to a totally unobservable change; such an operation is called the identity operation E, since it results in a configuration identical to the starting one. The ammonia molecule has other symmetry elements. A plane passing through the nitrogen and H3 atoms and midway between Hx and H2 is a plane of symmetry. As illustrated in Fig. 17-1(b), if some molecular feature, such as , occurs at a perpendicular distance χ in front of the plane, then an equivalent feature is found by extending the perpendicular to a distance — χ to the other side of the plane, which in this case locates H2. A plane of symmetry is designated by the symbol σ, and if it contains the principal axis of symmetry, C3 in this case, it is called a vertical plane and written σν. There are evidently three equivalent such planes in the case of NH3. Three-dimensional perspective drawings are difficult to work with, and it has become conventional to use a projection down the principal axis of symmetry c3 (d) FIG. 17-1. (a-c) Symmetry elements for the molecule NH3 . (d) Other objects having the same symmetry. 17-2 SYMMETRY AND SYMMETRY OPERATIONS 719 of the object. The ammonia molecule then looks as in Fig. 17-l(c), where the triangle imposed on the nitrogen atom stands for a C3 rotation axis and the lines labeled σν locate the vertical planes of symmetry. Even Fig. 17-1(c) is too specific; quite a variety of objects would have C3 and three av's as the only symmetry elements, as illustrated in Fig. 17-l(d), and therefore belong to the same sym­ metry class as NH3. A yet more abstract representation is desirable. The next step is then to devise a geometric representation or pattern from which all superfluous features have been removed. It is customary to show the projection of this abstract figure as viewed down the principal axis—this is called a stereo- graphic projection diagram. Figure 17-2(a) shows such a diagram for the symmetry elements C3 and 3σν . The triangle represents the C3 axis and the three full lines represent the three σν planes. The symmetry pattern produced by these symmetry elements is generated by supposing the circle to lie in the plane of the paper and that there is initially a mathematical point lying above the plane, denoted by the plus sign. The full pattern of points is then generated by carrying out the symmetry operations. As shown in Fig. 17-2(b), the operation σν generates point 2 from 1 point 1 ; application of Q (meaning one unit of rotation on the C3 axis) generates 2 point 3 from point 1 and point 4 from point 2 and application of C3 similarly produces points 5 and 6. We now have obtained Fig. 17-2(a). Disregarding the labeling, no new points are produced by σν' or . A second example of a symmetric molecule is H20, or Η—Ο—H. There is a twofold axis of symmetry passing through the oxygen atom, and a 180° rotation about this axis gives an equivalent configuration. In addition to the C2 axis, there are two σν planes. Both contain the C2 axis, and one is just the Η—Ο—Η plane, while the other is perpendicular to it, as illustrated in Fig. 17-3(a). The stereographic projection diagram for an object having just these symmetry ele­ ments is shown in Fig. 17-3(b), the lens denoting a twofold rotation axis. The four plus points are generated from a single initial point by application of σν and then C2 (or vice versa). A square planar molecule, such as PtCl^", possesses a number of symmetry features. There is a fourfold rotation axis C4 passing through the platinum atom and perpendicular to the molecular plane—successive rotations of 90° give equiva­ lent configurations. As illustrated in Fig. 17-4(a), there are also two pairs of C2 FIG. 17-2. Stereographic projection showing the symmetry features o/NH3. 720 CHAPTER 17: MOLECULAR SYMMETRY AND BONDING (a) (b) FIG. 17-3. Stereographic projection showing the symmetry features ofU20. C4 FIG. 17-4. (a) Symmetry elements ofPtCl!" and (b) corresponding stereographic projection. (c) Stereographic projection if no oh and no av's are present. 17-2 SYMMETRY AND SYMMETRY OPERATIONS 721 axes, labeled C2 and C2'. The fourfold axis, being the highest-order one, is called the principal axis. The two pairs of σν planes contain the C4 axis and either the C2 or the C2' ones; in addition, the molecular plane itself is a plane of symmetry. This last is a plane perpendicular to the principal axis, and is denoted ah . The stereographic projection diagram generated by the symmetry features C4, C2, C2', 2σν , 2σν' (or 2ad), and ah is shown in Fig. 17-4(b). The set of plus points is generated by carrying out σν on the initial one, followed by the successive C4 operations. The circles denote mathematical points lying below the plane of the paper and are produced by the ah operation. The set of points implies all of the symmetry features—none of the other operations generate any further points. Figure 17-4(b) also illustrates some additional conventions. The presence of ah is signified by drawing the projection as a full rather than a dashed circle. The secondary axes C2 and C2' are indicated by the lenses located radially around the circle; the fact that the lines to the lenses are full means that corresponding av's are also present; were they not, the lines would be dashed. Thus Fig. 17-4(c) shows the presence of the symmetry elements C4, C2, and C2' only (no ah and no av's). A few remaining conventions will be illustrated later as they come up. β. Symmetry Operations A given symmetry element may generate more than one symmetry operation, and these will now be spelled out. E: The identity operation has already been described; it is that which leaves the figure in an identical configuration to the initial one. Cn : A rotation axis is one about which the molecule may be rotated to give equivalent positions, η denoting the number of positions. Thus C4 denotes a 1 2 fourfold rotation axis; it generates the individual symmetry operations Q , C4, 3 4 C4, and C4, the superscript denoting the number of rotational increments of 2 4 2π/η degrees each.