Point groups example molecules

Point group Dpi. Molecules with a / -fold axis and /7-vertical planes of symmetry lying in the rotation axis plus a horizontal plane of symmetry perpendicular to the Cp axis fall in this point group. The group is identical with C2V and therefore not listed as such. Boron trichloride [Fig. 3.1(e)] is an example of the 1)3, point group. 

Property 3 ean be confirmed by inspecting any of the character tables in Appendix 12. For example, the Ad point group for molecules such as ethane in the staggered conformation has the headings... 

As an example, we again consider the PH molecule. In its pyramidal equilibrium configuration PH has all tlnee P-H distances equal and all tlnee bond angles Z(HPH) equal. This object has the point group synnnetry where the operations of the group are... 

We now turn to electronic selection rules for syimnetrical nonlinear molecules. The procedure here is to examme the structure of a molecule to detennine what synnnetry operations exist which will leave the molecular framework in an equivalent configuration. Then one looks at the various possible point groups to see what group would consist of those particular operations. The character table for that group will then pennit one to classify electronic states by symmetry and to work out the selection rules. Character tables for all relevant groups can be found in many books on spectroscopy or group theory. Ftere we will only pick one very sunple point group called 2 and look at some simple examples to illustrate the method. 

The functions put into the determinant do not need to be individual GTO functions, called Gaussian primitives. They can be a weighted sum of basis functions on the same atom or different atoms. Sums of functions on the same atom are often used to make the calculation run faster, as discussed in Chapter 10. Sums of basis functions on different atoms are used to give the orbital a particular symmetry. For example, a water molecule with symmetry will have orbitals that transform as A, A2, B, B2, which are the irreducible representations of the C2t point group. The resulting orbitals that use functions from multiple atoms are called molecular orbitals. This is done to make the calculation run much faster. Any overlap integral over orbitals of different symmetry does not need to be computed because it is zero by symmetry. 

Figure 4.11 Examples of molecules belonging to various point groups...

Examples are rare except for the S2 point group. This point group has only an S2 axis but, since S2 = i, it has only a centre of inversion, and the symbol generally used for this point group is C,. The isomer of the molecule ClFHC-CHFCl in which all pairs of identical FI, F or Cl atoms are trans to each other, shown in Figure 4.11(b), belongs to the C, point group. 

Apart from these two important examples molecules belonging to point groups are rare. 

Figure 4.11j), for example, and all homonuclear diatomic molecules belong to this point group. 

The dipole moment vector /i must be totally symmetric, and therefore symmetric to all operations of the point group to which the molecule belongs otherwise the direction of the dipole moment could be reversed by carrying out a symmetry operation, and this clearly cannot happen. The vector /i has components fiy and along the cartesian axes of the molecule. In the examples of NH3 and NF3, shown in Figures 4.18(b) and 4.18(e), respectively, if the C3 axis is the z-axis, 7 0 but = 0. Similarly in H2O and cis-... 

In Section 4.3.f it was shown that there are 3N — 5 normal vibrations in a linear molecule and 3N — 6 in a non-linear molecule, where N is the number of atoms in the molecule. There is a set of fairly simple rules for determining the number of vibrations belonging to each of the symmetry species of the point group to which the molecule belongs. These rules involve the concept of sets of equivalent nuclei. Nuclei form a set if they can be transformed into one another by any of the symmetry operations of the point group. For example, in the C2 point group there can be, as illustrated in Figure 6.18, four kinds of set ... 

In a molecule belonging to a degenerate point group, for example C, , the non-degenerate vibrations of the various sets of equivalent nuclei can be treated as in Section 6.2.2.1. 

The vibrations of acetylene provide an example of the so-called mutual exclusion mle. The mle states that, for a molecule with a centre of inversion, the fundamentals which are active in the Raman spectmm (g vibrations) are inactive in the infrared spectmm whereas those active in the infrared spectmm u vibrations) are inactive in the Raman spectmm that is, the two spectra are mutually exclusive. Flowever, there are some vibrations which are forbidden in both spectra, such as the torsional vibration of ethylene shown in Figure 6.23 in the >2 point group (Table A.32 in Appendix A) is the species of neither a translation nor a component of the polarizability. 

The size of the group attached to the main chain carbon atom can influence the glass transition point. For example, in polytetrafluoroethylene, which differs from polyethylene in having fluorine instead of hydrogen atoms attached to the backbone, the size of the fluorine atoms requires the molecule to take up a twisted zigzag configuration with the fluorine atoms packed tightly around the chain. In this case steric factors affect the inherent flexibility of the chain. 

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