D,h character table

We may illustrate these rules, using the carbonate ion. We see in the D h character table that (x, y) form a basis for the E representation and z for the Ay representation. For the polarizability tensor components we see that one or more of these belong to the A, , and E" representations. Thus, for any molecule of Dv, symmetry, we have the following selection rules ... 

The D- h character table (Table A.37 in Appendix A) shows that, of the 1-0 bands of acetylene (Figure 6.20), only lg, 2g and 4g are allowed in the Raman spectrum. 

The set of Cartesian displacement vectors as basis for a representation is shown in Figure 5-9. The symmetry operations of the point group are also shown. The D h character table is given in Table 5-3. Recall (Chapter 4) that the matrix of rotation by an angle is... 

Multiplying each character in the row by the corresponding character in the Ag or 3 representations in the D h character table gives the unnormalized wavefunctions for the LGOs. The normalized wavefunctions are represented by... 

Linear molecules belong to either the D h (with an inversion centre) or the (without an inversion centre) point group. Using the vibrational selection rule in Equation (6.56) and the D h (Table A.37 in Appendix A) or (Table A. 16 in Appendix A) character table we can see that the vibrational selection rules for transitions from the zero-point level (LjJ in Di0o/l, A1 in ) allow transitions of the type... 

The reduction formula can only be applied to finite point groups. For the infinite point groups, D h and C h, the usual practice is to reduce the representations by inspection of the character table. 

The next step is to determine how these group orbitals transform in the D(,h point group. The D6h character table is given in Table 6-7. Since most of the AOs in the suggested group orbitals are transformed into another AO by most of the symmetry operations, the representations will be quite simple, though still reducible ... 

C atom may form molecular orbitals with the four His orbitals. In Ty the d orbitals of the central atom span E + T2 (character table, final column), and so only the Ti set (dxy,dy-,d ) may contribute to molecular orbital formation with the H orbitals. 

The reduction formula (6.5), in combination with the characters of the irreducible representations of the D h point group found in Table 6.21, leads to ... 

We shall limit ourselves here to the determination of the symmetry-adapted A" and E" orbitals in the Fj,x representation. Table 6.25 shows the action of all the symmetry operations of the D h point group on the generating functions pxi and (px2 — Pxs), as well as the characters of the A" and E" irreducible representations. 

In a molecule with D h symmetry, the choice of C 2 and C 2 directions is arbitrary, but, once the twofold directions are chosen, the and reflection planes are also fixed, as was indicated in Fig. 3.10. Show that these spatial relationships between the axes and the planes is in line with the character table for Deh-... 

The character of natural oscillations in a bay or harbor is strongly controlled by the aperture ratio d = h/l, which can vary from = 1.0 to t = 0.0. These two asymptotic cases represent a fully open harbor and a closed basin, respectively. It is evident that the smaller is d (i.e., the smaller the width of the entrance), the slower water from the external basin (open sea) penetrates into the harbor. Thus, as decreases, the periods of all harbor modes for n > 1 in Table 9.1 increase, tending to the periods of the corresponding eigen modes for a closed basin, while the period of the fundamental (Helmholtz) harbor mode tends to infinity. This is one of the important properties of harbor oscillations. 

We are now able to determine the total number of modes of each symmetry species, and to say which of these are infrared- and which are Raman-active. Often we want to go further than this, and assign each band in the spectra first to specific symmetry species, and then to particular modes. We illustrate the first of these stages with the example of PF5 (Table 8.3), which has a trigonal bipyramidal structure of D h symmetry (see character table in the on-line supplement to chapter 2). 

Show that the five M-L a bonding orbitals in a MLj trigonal bipyramidal complex have 2A l + A 2 + E symmetries. For this problem the character table of the D h point group is... 

Figure 3.23 shows the two point groups that arise for linear molecules. If the two ends of the molecule are different, then the only symmetry elements are and the vertical mirror planes, so the point group is Coov by analogy with C2v, Csv, etc. If the molecule has equivalent points at either end of the axis then it will also have a horizontal mirror plane

Here is the square root of — I. As in equation 12.9, j runs from 0, I, 2. . . . We shall see below, and, very importantly, in Chapter 13, that this complex form of the wavefunction is very useful. It is interesting to see where this expression comes from. Group theory provides the answer. The molecular point group of, for example, benzene is D h- However, the group is the simplest one we can use to generate the tt orbitals of the molecule. Table 12.1 shows its character table. The reducible representation for the basis set of six tt orbitals is... 

This set of valence orbitals has the symmetry D h] the character table for this group is reproduced in Table 12 1. The table also includes the transformation properties of the coordinates and the pertinent combinations of the coordinates. The atomic orbitals are functions of r multiplied by the function written as subscript. Hence they transform under the operations of the group in the same way as their subscripts. The atomic orbitals therefore form bases for representations of the group as given in the table. It is now necessary... 

These differences have been attributed to various factors caused by the introduction of new structural features. Thus isopentane has a tertiary carbon whose C—H bond does not have exactly the same amount of s character as the C—H bond in pentane, which for that matter contains secondary carbons not possessed by methane. It is known that D values, which can be measured, are not the same for primary, secondary, and tertiary C—H bonds (see Table 5.3). There is also the steric factor. Hence, it is certainly not correct to use the value of 99.5 kcal mol (416 kJ mol ) from methane as the E value for all C—H bonds. Several empirical equations have been devised that account for these factors the total energy can be computed if the proper set of parameters (one for each structural feature) is inserted. Of course these parameters are originally calculated from the known total energies of some molecules that contain the structural feature. 

Table 4.3. Natural bond angles (aacute and a0htUseA percentage d character, and hybrid concentration ( h max) of equivalent sd/l hybrids...

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