C3v character table

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

From the information on the right side of the C3V character table, translations of all four atoms in the z, x and y directions transform as Ai(z) and E(x,y), respectively, whereas rotations about the z(Rz), x(Rx), and y(Ry) axes transform as A2 and E. Hence, of the twelve motions, three translations have Ai and E symmetry and three rotations have A2 and E symmetry. This leaves six vibrations, of which two have Ai symmetry, none have A2 symmetry, and two (pairs) have E symmetry. We could obtain symmetry-adapted vibrational and rotational bases by allowing symmetry projection operators of the irreducible representation symmetries to operate on various elementary cartesian (x,y,z) atomic displacement vectors. Both Cotton and Wilson, Decius and Cross show in detail how this is accomplished. 

The coefficients of the symmetry elements along the top of the above classification (the same as those across the top of the C3v character table), Le. 1,2 and 3, give a total of six which is the order of the point group, denoted by h. The relationship used to test the hypothesis that the reducible representation contains a particular irreducible representation is ... 

Indeed, Rz belongs to the irreducible representation A2 in the C3v character table. In other words, Rz transforms as A2, or, it forms a basis for A2. 

Table 6-4. The C3v Character Table and the Reducible Representation of the Hydrogen Group Orbitals of Ammonia...

The representatives of , ov, and Cy are therefore all one-dimensional matrices with characters 1,-1. 1 respectively. It follows that I- is a basis for A2 (see the C3v character table). 

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. 

While we have chosen to proceed here by reducing representations for the full group D3h, it would have been simpler to take advantage of the fact that D3h is the direct product of C3u and C where the plane in the latter is perpendicular to the principal axis of the former. The behaviour of any atomic basis functions with respect to the C3 subgroup is trivial to determine, and there are only two classes of non-trivial operations in C3v. In more general cases, it is often worthwhile to look for such simplifications. It is seldom useful, for instance, to employ the full character table for a group that contains the inversion, or a unique horizontal plane, since the symmetry with respect to these operations can be determined by inspection. With these observations and the transformation properties of spherical harmonics given in the Supplementary Notes, it should be possible to determine the symmetries spanned by sets of atomic basis functions for any molecular system. Finally, with access to the appropriate literature the labour can be eliminated entirely for some cases, since... 

We list here full matrix representations for several groups. Abelian groups are omitted, as their irreps are one-dimensional and hence all the necessary information is contained in the character table. We give C3v (isomorphic with D3) and C4u (isomorphic with D4 and D2d). By employing higher 1 value spherical harmonics as basis functions it is straightforward to extend these to Cnv for any n, even or odd. We note that the even n Cnv case has four nondegenerate irreps while the odd n Cnv case has only two. 

Example 4.4-3 Using the partial character table for C3v in Table 4.3, show that the character systems ixi and xf satisfy the orthonormality condition for the rows. 

Table 4.3. Partial character table for C3V obtained from the matrices of the IRs and T3 in Table 4.1.

Exercise 4.4-1 Check the orthogonality of the columns in the character table for C3v which was completed in Example 4.4-4. 

Determine correlation relations between the IRs of (a) Td and C3v, and (b) Oh and D3d. [Hints. Use character tables from Appendix A3. For (a), choose the C3 axis along [111] and select the three dihedral planes in Td that are vertical planes in C3v. For (b), choose one of the C3 axes (for example, that along [11 1]) and identify the three C2 axes normal to the C3 axis.]... 

Example 12.8-1 Determine the IRs and the character table for the point group D3. Hence find the IRs of C3v. 

Table 12.8. Character tables for the isomorphous point groups D3 = E 2 C3 3C2 and C3v = 2C3 3uv. ...

Exercise 17.4-5 AtX, P(k) is C2v the character table for which is shown in Table 17.7. The basis functions shown are those for the IRs of C2v when the principal axis is along a. Table 17.8 contains the character table for C3V with basis functions for a choice of principal axis along [1 1 1], The easiest way to transform functions is to perform the substitutions shown by the Jones symbols in these two tables. The states X2 and S4 are antisymmetric with respect to Jones symbol for ah in Table 17.7). Note that vertical planes at A and, as Table 17.8 shows, A2 is antisymmetric with respect to dihedral planes in C4v and from Table 17.2 we see that the bases for T j and T2 are antisymmetric with respect to fold axis at T is parallel to kz, and carrying out the permutation y —> z, z —> x, x y on the bases for A[ and A2 gives xy(x2 y2) and x2 y2 for the bases of Tj and T2, which are antisymmetric with respect to ab.)... 

Table 17.8. Character table for C3V with principal axis along [1 1 1],...

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... 

Prior to interpreting the character table, it is necessary to explain the terms reducible and irreducible representations. We can illustrate these concepts using the NH3 molecule as an example. Ammonia belongs to the point group C3V and has six elements of symmetry. These are E (identity), two C3 axes (threefold axes of rotation) and three crv planes (vertical planes of symmetry) as shown in Fig. 1-22. If one performs operations corresponding to these symmetry elements on the three equivalent NH bonds, the results can be expressed mathematically by using 3x3 matrices. ... 

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. 

Symmetry selection rules for Raman spectrum can be derived by using a procedure similar to that for the IR spectrum. One should note, however, that the symmetry property of symmetry species of six components of polarizability are readily found in character tables. In point group C3V, for example, normal vibrations of the NH3 molecule (2A1 and 2E) are Raman-active. More generally, the vibration is Raman-active if the component(s) of the polarizability belong(s) to the same symmetry species as that of the vibration. 

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. 

Table 4-4. A Preliminary Character Table for the C3v Point Group...

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