C3v point group

Shlang J J ef a/1996 Symmetry of annealed wurtzite CdSe nanocrystals assignment to the C3v point group J. Phys. Chem. 100 13 886... 

For example, the three NH bonding and three NH antibonding orbitals in NH3, when symmetry adapted within the C3V point group, cluster into ai and e mos as shown in the Figure below. The N-atom localized non-bonding lone pair orbital and the N-atom Is core orbital also belong to ai symmetry. 

The ammonia moleeule NH3 belongs, in its ground-state equilibrium geometry, to the C3v point group. Its symmetry operations eonsist of two C3 rotations, C3, 3 ... 

To illustrate sueh symmetry adaptation, eonsider symmetry adapting the 2s orbital of N and the three Is orbitals of H. We begin by determining how these orbitals transform under the symmetry operations of the C3V point group. The aet of eaeh of the six symmetry operations on the four atomie orbitals ean be denoted as follows ... 

We ean likewise write matrix representations for eaeh of the symmetry operations of the C3v point group ... 

We have found three distinet irredueible representations for the C3V symmetry group two different one-dimensional and one two dimensional representations. Are there any more An important theorem of group theory shows that the number of irredueible representations of a group is equal to the number of elasses. Sinee there are three elasses of operation, we have found all the irredueible representations of the C3V point group. There are no more. 

If these rules are applied to the 2px and 2py orbitals of nitrogen within the C3V point group, one obtains... 

This set of eharaeters is the same as Dl2) above and agrees with those of the E representation for the C3V point group. Henee, 2px and 2py belong to or transform as the E representation. This is why (x,y) is to the right of the row of eharaeters for the E representation in the C3V eharaeter table. In similar fashion, the C3V eharaeter table states... 

The symmetry assignment of vibrational states refers to C3V point group. Experimental geometries and wave numbers are taken from [28,29] for CH3 and [30] for CF3. EPR parameters are taken from [31] forCH3 at %K and [32] forCF3..at77K. 

Figure 3 Schematic view of one of the n-hack bonds in an R3PE molecule, from a filled p-orbital of E to one of the empty ct orbital combinations of e-symmetry (C3v point group) on the R3P fragment...

The ammonia molecule is a trigonal pyramid, belonging to the C3v point group. The 2s and 2p orbitals of the nitrogen atom and the Is orbital group combinations of the three hydrogen atoms transform, with respect to the C3v point group, as indicated in Table 6.1. 

Q Look up the transformation properties of the 2s and 2p orbitals of the nitrogen atom in the character tables of the C3v point group in Appendix 1 to confirm the content of Table 6.1. Carry out the procedure for classifying the Is orbitals of the three hydrogen atoms as group orbitals in the pyramidal molecule. 

Formulate the bonding in NH2 in terms of delocalized molecular orbitals. The molecule is trigonal-pyramidal (C3V point group). Compare the general molecular-orbital description with a localized tetrahedral model for NH3. Discuss the values of the following bond angles H—N—H, 107° H— P—H (in PH2), 94° and F—N —F (in NF3 ), 103 °. 

Use of the same rule as that for deriving (E)2 from E x E in the C3v point group gives... 

For a symmetric rotor molecule such as methyl fluoride, a prolate symmetric rotor belonging to the C3v point group, in the zero-point level the vibrational selection rule in Equation (6.56) and the character table (Table A. 12 in Appendix A) show that only... 

As an example, consider IR activity of the six normal vibrations of the NH3 molecule, which are classified into 2A and 2E species of C3V point group. The character table shows that fiz belongs to the A and the pair of (px, fiy) belongs to the E species. Thus, all six normal vibrations are IR-active. 

The multiplication table of the C3v point group is compiled in Table 4-2. Here,... 

Table 4-2. Group Multiplication Table for the C3v Point Group...

To find out what operations belong to the same class within a group, all possible similarity transformations in the group have to be performed. Let us work this out for the C3v point group and begin with the identity operation. Since E commutes with any other elements Z (see under rule 2 above), we have... 

Table 4-4 shows a preliminary character table for the C3v point group. The complete set of symmetry operations is listed in the upper row. Clearly, some of them must belong to the same class since the number of irreducible representations is 3 and the number of symmetry operations is 6. A closer look at this table reveals that the characters of all irreducible representations are equal in C3 and Cf and also in oy, o+ and a", respectively. Thus, according to rule 4 C3 and Cl form one class, and ay, ct and a" together form another class. 

A complete character table is given in Table 4-5 for the C3v point group. The classes of symmetry operations are listed in the upper row, together with the number of operations in each class. Thus, it is clear from looking at this character table that there are two operations in the class of threefold rotations and three in the class of vertical reflections. The identity operation, E, always forms a class by itself, and the same is true for the inversion operation, i (which is, however, not present in the C3v point group). The number of classes in C3v is 3 this is also the number of irreducible representations, satisfying rule 5 as well. 

Thus, the characters of the rotation around the z axis in the C3v point group will be ... 

Wave functions form bases for representations of the point group of the molecule [24], Suppose that f and f are such functions then the new set of functions,/, called the direct product off and f, is also basis for a representation of the group. The characters of the direct product can be determined by the following rule The characters of the representation of a direct product are equal to the products of the characters of the representations of the original functions. The direct product of two irreducible representations will be a new representation which is either an irreducible representation itself or can be reduced into irreducible representations. Tables 4-8 and 4-9 show some examples for direct products with the C2v and C3v point groups, respectively. 

Consider now the construction of the A i symmetry group orbital of the hydrogen s atomic orbitals in ammonia as an example of the application of the projection operator. (The various kinds of orbitals will be discussed in detail in Chapter 6.) The projection operator for the A irreducible representation in the C3v point group is... 

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