Symmetry II

We shall often utilize spectroscopic data in order to elucidate the detailed nature of structure and bonding in molecules as continuous lattices. In many cases the spectroscopic transitions can be mathematically formulated with a knowledge of the ground state (WG), the appropriate excited state (WE) and the form of the interaction (HI). This means that the transition probability will depend on the excitation quanta having the correct energy (for conventional spectrometers) and that the integral of the products of WG x HI x WE is not equal to zero. Usually we will not know the form of the wavefunction but we often will know something about their symmetry properties. For this section let us limit the discussion of the interaction to electric dipole transitions. These are proportioned to the vector r and therefore to its components x, y or z. Now in a given point group we will know how x, y and z transform - that is to what irreducible representation they belong. So what we really are going to do is to factor the transition probability integral into two parts: a part dependent on symmetry and everything else; we shall assume that the second part is non-zero which is usually a good approximation. Then the problem reduces to a determination of whether or not the symmetry part is zero or not. The details are given below in another subsection. Here we seek mainly to illustrate the kind of reasoning involved, not the detailed mathematical method, which, in any event, can be learned in a routine mechanical way.

Lets us begin by recalling that the integral of an odd function over an interval that is symmetrical about the origin is always zero. For example: integral of SM x dx from - a to +a is always zero for any value of a.

On the other hand for an even function the integral over a symmetrical interval generally is not zero. For example integral of cos x is generally nonzero.

In a way what we do with the transition probability integral is to break it up and examine the symmetry part for each operation in the group. If the integral is "even" for each operation then the product of the integral is taken as non-zero. If the integral is "odd" for even one operation then the total is zero.

The final statement is that the product of the symmetry parts of the ground state wavefunctions, the interaction Hamiltonian, and the excited state wavefunction must contain the total symmetric irreducible representation. That is, it must contain Al (or Alg) and this representation is found in all groups. The details are given below but it is worth emphasizing that the analogy of the even/odd integral with the other symmetry operation is not perfect but a good way to start thinking about what is the physical significance of the mathematical forms employed.

Special Cases

1. Most of the molecular point groups we are concerned with contain a finite number of symmetry operations and therefore a finite number of irreducible representations. However, there are a reasonable number of linear molecules and these belong to one of two special point groups. In linear molecules there is always a C,. axis which is the long axis of the molecule. If in addition to this axis there is a plane perpendicular to it (and hence an infinite number, of C2 axis perpendicular to it) then the molecule belongs to the group D-h. If there is no such horizontal plane, then the group is C(v. Both of these groups have an infinite number of irreducible representations. Examples of molecules in D (h are C02 and N2 while for C(v examples include HCN and C0.

2. Another case where there are an infinite number of irreducible representations is in the full rotational group. Of course, no molecules belong to this group, but it is the appropriate atomic point group. The spherical harmonics are the irreducible representations for this group and they are of fundamental importance in the study of angular momentum and in the specification of the atomic orbital contributions to chemical bonding.

3. A common experimental "rule of thumb" is that there are no bands of the Raman and IR spectrum in common for a given molecule then the molecule probably possesses a center of symmetry. The correct statement would be that if i is present then no Raman bonds and IR bonds can coincide (except for accidental degeneracies). This follows from the observation that groups which contain a center of: symmetry always have their irreducible representations classified as either g (German: gerade-even) or u (German: ungerade-odd). Inspection of the character tables reveals that x, y and z are always of type a while the products of coordinates (e.g., X2 or Xz) are always of type g. It turns out that the appropriate description of the interaction of electromagnetic radiation with molecules is proportional to either the coordinates (in IR) or their products (in Raman spectroscopy). As a result the old rule of thumb is solidly based in a group theory formulation. We will see examples of this later in this course.

4. Another experimentalist rule is that optical activity can be found in molecules that have no plane of symmetry (6) or no center of inversion (i). This rule can be expanded but first we should examine the symmetry element Sn in greater detail. Inspection of the character tables reveals that an S1 or S2 axis is never explicitly listed. This is perhaps due to historical circumstances. An Sl operation corresponds to rotation by 360° (or 0°) followed by a reflection and there is just a 6. The S2 operation is a rotation by 180° followed by a reflection in a plane perpendicular to the axis of rotation. This is just equivalent to an inversion i. Thus there are not really four nontrivial chemically important symmetry operations (cs i Sn Cn) but only two (Sn and Cn).

It turns out that the rule of thumb should be expanded to say that optical isomerism can occur when molecules are not superimposible on their mirror images and this implies the existence of no Sn axis.

Classification Scheme

With a little experience, most chemists classify molecules into their point groups by inspection.

It is often convenient however, to have a set of rules to check on the result. Of course, with a set of character tables it is usually necessary to verify the existence of all the symmetry elements once the point group has been tentatively established. Even then it may be that the full symmetry of the molecule has not been found and the molecule can be classified into a lower point group (that is a "subgroup"), the one to which it properly belongs. Sometimes it is advantageous to utilize subgroup rather than the full group. Some examples will be given below.

In this section we set out a systematic scheme for classifying molecules. It is usually convenient to look for the highest rotational axis first.

1. If the highest rotational axis is infinite then the molecule is linear and belongs to D (h or C (v (vida supra)

2a. If the highest Cn axis is a five-fold axis and there are several of these then check the groups I or Ih. These are the icosahedral groups. An example of an ion in Ih is dodecaborane B12H122

b. If the highest axis is C4 and there are three of these then it is one of the octahedral group. SF6 is an example.

c. If there are four 3-fold axes, then it is probably one of the tetrahedral groups. CH4 is an Td.

In steps (1) and (2) the emphasis was on molecules with very high symmetry. These are often easy to identify as are molecules with very little symmetry, which we consider next.

3a. If a molecule has no symmetry then it is in group C1 (e.g., CFBrClH). If it has only Sl = a then the group is Cs. If it has only S2 = i then the group is Ci and an example is staggered CFHC1-uCFHCl.

b. If there are only even ordered Sn axes then the group is also called Sn. Staggered C2H6 is an example of S6.

Now we come to the two broad categories of point groups that contain many molecules.

4a. If the highest rotational axis, Cn, has n 2-fold axes perpendicular to it then it is a D group. If there is a horizontal plane the group is Dnh, a very common group. Benzene C6H6 is D6b, PC15 is D3h as is BF3, and ferrocene is D5h in the eclipsed configuration.

b. If there is no 6h but a set of n vertical planes which bisect the two-fold axes then the group is Dnd. The staggered form of ferrocene is D5d.

c. Finally in the D groups if there are only the Cn and nC2 axes the group is Dn.

5a. If only a Cn axis is found in a molecule then the group is also called Cn. It is a cyclic group with all members committing. While few molecules belong to this group they are useful subgroup for several applications. The molecules in this point group are optically active.

b. If, in addition to Cn, the molecules possess a 6h then the group is Cnh. B(OH)3 is an example of C3h.

c. Finally, if the molecule possesses a Cn and n planes containing the Cn axis then the point group is Cnv which is quite common. SF5 Cl, H20 and NH3 are in C5v, C 2v and C3v respectively.

This completes the classification scheme based on molecular symmetry. It is important to recognize that in the gas phase or liquid phase, molecular symmetry is often sufficient to determine spectroscopic symmetry. However, in the solid state and in some viscous media, intermolecular interactions may remove or augment molecular symmetry considerations. The classification of crystallized solids into their space groups is an important subject, but one that is reserved for a further section in this course.