Ethane symmetry elements

D2d A molecule is D2d if it has a C2 axis and two perpendicular C2 axes (as for D2 above), plus two dihedral mirror planes these are mirror planes that bisect two C2 axes (in general, that bisect the C2 axes perpendicular to the principal axis). Example allene (propadiene). Staggered ethane is D3d (it has D3 symmetry elements plus three dihedral mirror planes. Dnd symmetry can be hard to spot. 

With a total of fourteen valence electrons to accommodate in molecular orbitals, ethane presents a more complicated picture, and we now meet a C—C bond. We will not go into the full picture—finding the symmetry elements and identifying which atomic orbitals mix to set up the molecular orbitals. It is easy enough to see the various combinations of the Is orbitals on the hydrogen atoms and the 2s, 2px, 2py and 2pz orbitals on the two carbon atoms giving the set of seven bonding molecular orbitals in Fig. 1.19. 

Bis(tridentate) complexes (i.e. octahedral complexes containing two identical linear tridentate ligands) may exist in three stereoisomeric forms, and there will be more if the tridentate ligands do not themselves contain some symmetry elements. The three isomers of the simplest case are represented below (Examples 1, 2 and 3), along with their polyhedral symbols (Section IR-9.3.2.1) and configuration indexes (Section IR-9.3.3.4). Complexes of A-(2-aminoethyl)ethane-1,2-diamine and iminodiacetate can be described by these diagrams. 

Fig. 2.4 Improper rotation. Axis connecting points C-C is a rotatirai—reflection axis S5. An ethane molecule has a symmetry element including two subsequent opt tirais, the rotation of the whole structure through an angle of 30° with a subsequent reflection by plane o. After this the left and right sketch became identical...

The symmetry elements of the point groups (a) >3, (b) D2d and (c) Dgh- (a) A twisted form of ethane, neither perfectly staggered nor eclipsed, viewed along the C3 axis, (b) Allene, where the planes xz and yz are planes and the C 2 axes lie at 45° to the x and y axes in the xy plane, (c) Benzene. Note that the Cg axis includes C3, C2, Sg and S3 axes. Also shown are the three axes (one of which lies along x) and the three C axes (one of which lies along y). The xy plane is CTh and there are vertical mirror planes including each C 2 and C axis the center of the molecule is an inversion center, i. 

The following two examples should help clarify the nature of these symmetry elements. Consider first the ethane molecule in its eclipsed conformation (I). 

Now let us consider staggered ethane (II) (i.e., one methyl group rotated 60° from its eclipsed position). Its symmetry elements are... 

Improper axes can also be associated with several symmetry operations. We noted earlier that e, applied twice in succession, results in a simple 2tt/3 rotation about the S6 axis. In other words, we can write the set 5e, 5g, Sl, S, as Se, C3, S2, C. We stop at s because S = E due to the combination of C and an even number of reflections. 5g is equivalent to S2 because it contains three rotations by 2 r/6 and an odd number of reflections, and S2 means one rotation by n and one reflection. The operation S2, however, is easily shown to be equivalent to an inversion, and so we have, using a compressed notation, 25e, 2C3, i associated with the Se axis. Since we have already explicitly listed the elements C3 and i in our set of elements for staggered ethane (or any other system containing an Se axis) only the ISe operations are unique to the Se axis. The generalization of this case is that an S2n axis with odd n implies that elements C and i are also present. Of the 2 — 1 operations associated with S2 , — 1 are preempted by the C axis and 1 by the element i leaving — 1 operations to be attributed to the S2a axis. [If = 1, we have S2, C, and i as elements. But Ci = E, so we ignore it. There is only one operation here (21 — 1 = 1) and it is preempted by i. Therefore, S2 has no unique operations and it is not listed as a symmetry element.] For S2n with n even, i is not implied and there are n unique operations. 

The set of symmetry operations is contrasted to the set of symmetry elements for eclipsed and staggered ethane in Table 13-6. Note that there are 12 symmetry operations in each case. By setting up the multipUcation table for either of these sets of 12 operations, we can show that the mathematical requirements for a group are satisfied. Thus, each of these sets of symmetry operations constitutes a separate group of order 12. 

Which one of the following is not a symmetry element for eclipsed ethane ... 

Figure 2.5 Symmetry elements and example operations for ethane in the staggered conformation each example is shown both in flying wedge representation and in Newman projection. (a) The C3 principal axis, defining the vertical direction (b) C2 axis for H,—C—C—He plane there are three equivalent C2 axes, (c) one of the three equivalent

Center of Symmetry (C,). A center of symmetry exists in a molecule if one half of the molecule is obtained from the other by inversion through the center of symmetry there may or may not be an atom at the center of symmetry. There are a few molecules with a center of symmetry as the sole symmetry element. One such molecule would be a substituted ethane HXYC-CYXH, where each pair of similar atoms is in the trans position. If a line is drawn from one atom to the center of symmetry and continued on, it would intercept a similar atom at the same distance from the center of symmetry as the original atom. The molecule HXYC-CYXH would be described as belonging to the point group C -, since it has only the symmetry element C -. 

The types of vibrations (species) of molecules with related symmetry can be correlated and these correlations used to assign vibrations. For example, ethane and its various derivatives have somewhat similar symmetry elements. Ethane belongs to point group while CH3CCI3 has symmetry. The D d groups have... 

Chemically equivalent nuclei are those that have exactly identical environments in the molecule. Another way of saying this is that if we list all the distances between every pair of atomic nuclei in the molecule, two equivalent nuclei will have the same set of distances to nuclei of the same elements. So, for example, all six H atoms in ethane (CH3CH3) are equivalent, but propane (CH3CH2CH3) has two groups the six H atoms at the ends are all equivalent, and the two in the middle are equivalent. Equivalent nuclei experience the same chemical shifts because the symmetry of the molecule requires the electron density at each nucleus to be the same. 

Usually it is not difficult to see most elements of symmetry in molecules, the exception being improper axes, which tend to be a little tricky. The six-fold improper rotation in staggered ethane is not too hard to envision when it is applied once. If we rotate clockwise by 60° and reflect. Hi replaces H5, H5 replaces H2, etc. It is a little... 

---