The Symmetry Point Groups

We first describe the proper point groups, P, that is the point groups that contain the identity and proper rotations only. 

Schonflies symbol Full International symbol Abbreviated symbol 

This completes the list of proper point groups, P. A summary is given in the first column of Table 2.6. All the remaining axial point groups may be generated from the proper point groups P by one or other of two methods. 

The subscript d denotes the presence of dihedral planes which bisect the angles between C 2 axes that are normal to the principal axis. If n is even, 

Exercise 2.3-1 Confirm the DP D3 0 C, in eq. (9) by constmcting the (labeled) projection diagram for D3d. Identify the dihedral planes. 

A short description of the 32 crystallographic point groups is given below. 

Suppose that we have, by inspection, compiled a list of all of the symmetry elements possessed by a given molecule. We can then list all of the symmetry operations generated by each of these elements. Our first objective in this section is to demonstrate that such a complete list of symmetry operations satisfies the four criteria for a mathematical group. When this has been established, we shall then be free to use the theorems concerning the behavior of groups to assist in dealing with problems of molecular symmetry. 

Let us first specify what we mean by a complete set of symmetry operations for a particular molecule. A complete set is one in which every possible product of two operations in the set is also an operation in the set. Let us consider as an example the set of operations which may be performed on a planar AB3 molecule. These are E, C3, Cjj, C2, C2, CJ, r,., g v, ah, S3, and Sj. It should be clear that no other symmetry operations are possible. If we number the B atoms as indicated, we can systematically work through all binary products for example  

Hence we see that avC3 = a v. Proceeding in this way, we can check all the combinations and we will find that the set given is indeed complete. This is suggested as a useful exercise. 

we can see that, because our set of operations is complete in the sense defined above, it satisfies the first requirement for mathematical groups, if we take as our law of combination of two symmetry operations the successive application of these operations. 

The second requirement—that there must exist a group element E such that for every other element in the group, say X, EX = XE = X—is also seen to be satisfied. The operation of performing no operation at all, or that which results from a sequence of operations which sends the molecule into a configuration identical with the original (e.g., cr, C ), is our identity, , and we have been calling it that all along. 

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CO, CO, co, and o, respectively. The integrals in Eqs. (E.9) and (E.IO) will then be different from zero only if the integrands are invariant under all symmetry operations allowed by the symmetry point group, in particular under C3. It is readily seen that the linear terms in Q+ and Q- vanish in and H In turn. 

Exercise 2.4-1 Identify the symmetry point groups to which the following molecules belong. [Hint. For the two staggered configurations, imagine the view presented on looking... 

The electronic charge density in an MO extends over the whole molecule, or at least over a volume containing two or more atoms, and therefore the MOs must form bases for the symmetry point group of the molecule. Useful deductions about bonding can often be made without doing any quantum chemical calculations at all by finding these symmetry-adapted MOs expressed as linear combinations of AOs (the LCAO approximation). So we seek the LCAO MOs... 

When the wave functions and 02) belong to different IRs (A r2) of the symmetry point group G(Qo), then the matrix element... 

The wavefunctions V /, may be classified according to their symmetry properties. If we take the symmetry point group of this system to be C3V, there are three symmetry species in this group A (symmetric with respect to all operations of this group), A2 (symmetric with respect to the threefold rotations but antisymmetric with respect to the vertical symmetry planes), and E (a two-dimensional representation). 

To help students to determine the symmetry point group of a molecule, various flow charts have been devised. One such flow chart is shown in Table 6.2.3. However, experience indicates that, once we are familiar with the various operations and with visualizing objects from different orientations, we will dispense with this kind of device. 

Determine the symmetry point groups of the following compounds. In each case show the symmetry elements in the structural formula. Use the flow chart in the appendix to assist you. 

Deduce the symmetry point groups of all the isomers of [CrCl2(NH3)4]+ and assign a precise stereodescriptor for each isomer. 

Deduce the symmetry point group of (S,S)-tartaric acid in the +synclinal conformation. 

Show that the ethylenediaminetetraacetate complex of Ca2+ belongs to the symmetry point group C2 and assign a suitable stereodescriptor to the compound. 

Show that [CoCl2(en)2] (en = ethane-1,2-diamine) can be racemic. Determine the symmetry point groups of each isomer and assign suitable stereodescriptors. 

Deduce the symmetry point group of the following idealised representation of the structure of copper(I) benzoate. (Assume that the plane of the phenyl rings lies parallel to the carboxylate groups.)... 

The isomers of butene are but-l-ene (with only a plane of symmetry, symmetry point group Cs), ( )-but-2-ene (with a horizontal plane of symmetry, a twofold axis of symmetry perpendicular to it, and a centre of symmetry, symmetry point group C2h), (Z)-but-2-ene (with a twofold proper axis of symmetry and two planes of symmetry containing this axis, symmetry point group C2V) and isobutene (2-methylpropene) with two mutually perpendicular planes of symmetry and on the line of intersection of these two planes there is a twofold axis of symmetry. The symmetry point group is therefore C2v. These results can be verified using the flow chart in the appendix. 

Both diastereomers of [CrCl2(NH3)4]+ are shown below, these are usually differentiated by the stereodescriptors cis and trans. The trans isomer belongs to the symmetry point group D4h. The symmetry elements are the main fourfold axis of symmetry C4, a horizontal plane of symmetry ah (perpendicular to the C4 axis), four C2 axes also perpendicular to the C4 axis and four planes of symmetry av the intersection of which is the main axis of symmetry. The cis isomer belongs to the symmetry point group C2v. The associated symmetry elements are a C2 axis and two vertical planes of symmetry av intersecting at the C2 axis. Verify this using the flow chart in the appendix. 

S,S)-Tartaric acid has a single symmetry element, a C2 axis, and therefore belongs to the symmetry point group C2. On rotation through 180° the pairs of carbon centres 1 and 4, and 2 and 3 are transformed into each other. This is also the case if the molecule adopts another conformation as illustrated by the two examples shown below. 

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